Will update as I go.

shore

ive been so tired lately
like the tide just keeps pulling me under
and I can't do anything about
the sand and debris it drags over me

but I know, I know
that one day the moon will
come, pull the tide back,
and maybe for a moment
I'll be free of it all,
face up to the sky,
eyes glistening in moonlight
before it all comes rushing back — 12th of Feb, '26

irony

an Iranian student,
studying at an American institution,
as Iranian missiles carve streaks of fire into the sky above me.

i am told i am being ‘liberated’
by an American leader
via the assassination of Iran’s leaders.

i receive updates from the American embassy,
and also messages of safety
from the Iranian-American president of my university.

ahhhh that hyphen, that damn hyphen. Iranian. American.

god bless cosmopolitanism,
god bless that part of me that insists on belonging,
belonging to places that do not belong to each other,
god bless having to carry two names in one body,
and having no where to call home. — 3rd of March, '26

rise

when I die,
bury me under the great cypress
don’t erect a headstone for my grave
and when I die,
do not visit my grave
under the great cypress.

for I am not dead.

when the great cypress’ roots wrap my body
when they meet my eyes,
the cypress will see through them all I have seen.
it will see the love, the rage, the contempt yes,
but it will also see itself, standing tall.

for when its roots caress my fingers,
the cypress will feel through them
the pain of gripping a knife, of carving something of your own flesh and bones—
but also the rough certainty of its own bark.

and when its roots coil around my heart,
they will drink from it—
the same love,
the same rage,
the same old contempt for man.

I will rise with it,
taller than I ever stood in life,
my eyes fixed forever on the skyline.

so do not visit me when I die.

if you wish to see me,
look to the cypress—
the one that will not stop staring
into the skyline. — 26th of March, '26

geranium

in my cold, bitter world
i thought of you.
so that your scent could come and
permeate through my garden
i thought of you,
so that i could hear your laughs,
so that i could perch them like the geraniums,
on the windowsill.

little did I know,
that your laughter was the sky.

the whole sky. — 11th of April, '26

postscript to self

To you, 7 semesters in the future;

Spread your wings, yes,
but don't forget that certainty fractures without warning.

Even Icarus laughed as he fell, as death whispered in his ear.

Did he laugh at the symmetry of ambition,
that the moment of ascent is always one breath
away from the fall,

Or did he laugh at the way the world keeps turning,
even as a fallen body cuts through the air? — 12th of December, '25

to start empty

in 12 hours, I become a university student.

I will, once again, have something to chase.
something to carve my nights and days around.

and finally, rid myself of the weight of weightlessness,

and feel the edge of purpose cut into me.

in 12 hours, I will, once more,
work for the knife,
let it claim its due,
cut by cut,
hour by hour,

until I am shaped into something it can use. — 19th of August, '25

and to end empty

slowly

i thought i was done
and i am,
but not in the way you think.

yes,
i don’t wake up early on sundays anymore.
nor do i sleep as soon as i get home.
instead,
i’ve forgotten how to sleep at all.

slowly

you ask me why i’m still stressed.
yes,
there’s nothing left to work toward,
nothing left to carve out of myself

but im still sitting here
staring at what has become of me,
at the edges of what could have been.

slowly

every time i think i am finished,
my body reminds me
consequences don’t arrive on time.
decisions surface only later,
after the storm has already passed.

the truth is
i am being immobilized.
paralyzed by nothing,
which is also everything.

i thought i was done,
and i am
just not in the way you think.

i am spent.
exhausted.
learning how to exist
as nothing
after once being everything.

slowly

i am realizing
i am still working for the knife—
that i still start the day lying
and end with the truth

that my fingers are still clenched.
only now
clenched around nothing

so i drag nothing
along the edges of nothing,
hoping to catch something

slowly,

trying to conjure
what was lost. — 1st of May, '26

hayes

you don’t understand
you give me chills,
the ones that climb up my spine,
the ones that leave my mouth dry,
like I’ve swallowed the sea.

you gave me chills last week.
i was bent over an exam, but then
i remembered us,
on that beach, that night,
you told me to pounce on anything, everything
that arrived.

but I didn’t.
I watched the waves instead,
how they pounced at each other,
endlessly
with nothing to lose.

eventually, I told you
your advice sounded familiar
to something I was learning:
the gale-shapley algorithm,
two groups, reaching, choosing.

i said

the ones who move first
who pounce
they get the best they’re allowed to have.

the ones who wait
end up with what’s left

i continued staring at the waves

you gave me chills last week
in that exam room,
when the question found me.

salt in my throat,
graphite blurring,
the tide rising

and suddenly
the waves again,
curving over themselves,
and your voice through the breeze,
insisting,
again and again
pounce.

you gave me chills yesterday.
when i got my exam back,

i noticed it was as if something in me
had moved
before I even knew how to.

like, somewhere between fear
and memory,
i had leapt.

you don’t understand
you give me chills — 5th of May, '26

speed

as a child I never understood why one would speed

why place your life
in the hands of something built to leave
why gamble on momentum
why

but i dropped you off at the airport yesterday
and i would be lying if i said i didn’t gamble.

as a child i never understood what it felt like

to have the wind kiss my face
to have its fingers run through my hair
the way yours once did.

to watch streetlights disappear
before they come into sight
much like you did.

to feel nothing as you gamble everything
on a pile of metal.
to bet on a losing dog,
much like i did.

i still don’t understand why people speed

but it felt great knowing my body was moving at
one hundred eighty
while yours moved at twice that

did you see me from up there?
the car threading between all the others

did you see how i was
blurring the lines in my periphery
blurring the lines between life and death
blurring

blurring until the only thing i felt
was how badly i wanted you back

i still don’t understand speeding
but maybe i loved you like that
and now i understand why it’s hard to stop — 9th of May, '26

a relationship with the sun

to heal is to have once loved,
i tell the sun and maybe myself.

i close my eyes and see yours
through the sunlight kissing my eyelids.
i open them and look down at the waves:
dark, serene.

i remember looking into your eyes and realizing that
when you shut them,
the sky loses two of its brightest stars.

i don't even know what i'm healing from.

this place feels like a person,
warm, much like you—
the sun is rising on this port,
where crickets synchronize with my heartbeat, where the lingering night breeze brushes my hair,
where the palm's crown shuffles and shushes my grief,
and the birds periodically pass by to check on me,

i know that i am healing.
the shore approaches to tell me so,
the sun's warmth on my skin tells me so.

to heal is to have once loved, i tell the sun,
and it understands.

i can manage with two fewer stars in the sky.

the sun rises higher now,
and still every time i close my eyes,
there they are:
those two lustrous stars.

ahhhh forough said it best- her name translates to luster;

say something to me, please.
i am at the mercy of this window.
i have a relationship with the sun.

to heal is to have once loved,
the shore would tell you that.
the sun would tell you that. — 17th of June, '26

afloat

throw me to the wind
I yearn to feel its touch
have it kiss my brow and whisper tales
of devotion, sacrifice, and love.

throw me to the wind
I beg to feel the morning dew
gather on my skin,
mix with the salt of my eyes
which will finally know freedom.

throw me to the wind
I ache to rise and to fall
To burn and to suffer
To disappear and be forgotten.

throw me to the wind
I will carry a smile into its vastness
I will tear out my own roots
And let it take me to its garden.

And when they find me
face in the grass, smiling
They will find that not even Death
could smother the Love in me.

what a ruinous thing. — 10th of July, '26

sand, salt, and the memory of you.

the water keeps washing over
your footsteps in the sand

one
two
three
four
five fingers

filling up with seawater
i stare at the ebbs and flows
how the sea comes in and washes over
the memory of you

one
two
three
four

do you remember when you thought you would die?
scribbling your will on the back of a Cornflakes box
do you remember how afraid you were?

i do, i think i do

one
two
three

when i held you in my arms, kissed your neck
your long hair got stuck in my mouth
we talked about that later
and laughed at how absurd everything had become
do you remember?

i do, i think i do

one
two

we set fire to our lives
let everything burn, until only we remained
do you remember when we also
set fire to your physics textbook
we both hated mechanics
do you remember?

i do, i think i do

one

before you left
i took your left with my right,
took you to the shore
you complained that the sand got in your eyes
I blew into them as hard as i could
you started tearing up
and so did i

do you remember?

the tide has risen.
the sea has washed over the memory of you
and brought with it enough sand
to patch your footsteps.

do i remember?
i don’t. do you? — 15th of July, '26

love

I met Love, after the show,
pressed against the barricade,
it looked like a person still leaning on,
long after the encore
love was the shape of their eyes, still staring,
silently still.

I met Love, while thrifting
I just caught it in my periphery:
a vintage postcard from indochina
depicting two ladies, shoulder over shoulder
smile over smile.
Love was not shaped like a heart.
it was a 3:2 piece of glossy paper

I met Love, while driving
both hands on the wheel as it sang
from the passenger seat,
“I got a pocket, got a pocketful of sunshine”
Love looked like singing this song, on a cloudy day.
a windy one, too, where the wind sang along.

I met Love waiting to get my tetanus shot.
it was a young man holding a phone down to his friend’s ear.
the friend, in a wheelchair, motorcycle burns and right arm in a sling:
“it’s just a small burn mom, im fine, i promise”
Love looked like a bruised hand,
squeezing a shoulder wrapped in gauze

Oh, I also met Love on Discord,
A deliberate, private friend of three years whom I’ve never really seen,
Never really heard,
telling me there are temperatures to silence.
Warm and cold silences.
I met love realizing theirs was the warm kind.

I met Love watching my friend shake salt onto his Pad Thai,
looking away from his hovering hands,
I saw the people I had met only a few days ago.
I met Love watching them talk about their lives,
I met Love watching one of them wrap her arm around his, as the salt still fell,
I met Love saying goodbye to them after that meal.
I met Love knowing that was the last I’d see them.

What is the shape of Love?
Is it a heart?
Oh, definitely not.

Love hops and hides in concert stages,
in thrift stores, in hospital waiting rooms- waiting for you to notice
that Love is you.
Love is the shape of your outline.
You, Lover, with or without a Lover. — 1st of August, '26

In June, I was forcefully made aware of the illusion of peace. you showed me what it truly looked like.

That month, an all out war broke out in my home country, Iran. My 18th summer— the only summer I had before my first year in university— took on the bleakest of colors and lost all its vibrance.

I had been there for 2 weeks, excited, making plans for the rest of summer. But soon enough, I was looking for guides on preventing window glass from shattering. Thoughts of enjoyment and adventure turned into ones of survival and safety.

On more than one occasion I wondered to myself if I would live to see tomorrow or not.

And when a missile struck 2 kilometers away from home, my family decided I was not to stay in Iran any longer. With the airspace shut down, they arranged for me to travel to Bandar Abbas alone to search for any vessel that could take me out.

And that’s when I heard about your ships, Omani ships, carrying civilians back to Oman. After the 8 hour bus ride to the port, I visited your makeshift embassy at Ghods hotel. I pleaded with your delegation. I wasn’t an Omani resident, but I promised I would go straight to Qatar. I begged. Hours of back and forth passed, until I was finally offered a spot on your ship.

It was all over. I had found a way out. I was safe.

Until again, life reminded us all of the illusion of peace.

The port your boat was docked in was struck. I remember the entire boat shaking. I pulled the window open and saw the scene outside.

The water rippled and reflected the bright hue of chaos. I took a picture before being told to turn my phone to flight mode.

Screams, questions, and cries burst throughout the ship.

You instructed your ship to go into blackout mode. For a moment, everyone went silent. The lights went out. The only light entering the ship was that of the blazing scene outside— peering through the blinds. The ship remained docked.

And yet again, life reminded us all of the illusion of peace.

Another blast, this one closer. Off to the right. This time the windows rattled. People fled from the windows.

Your ship started moving this time. We floated through the water, away from the port. The window blinds glowed an eerie red, illuminated by the destruction outside.

Your officers asked me how I felt. 4 weeks later, I still don’t know what I was feeling, really; at first I thought I was numb to it. That if it was in the stars for me to perish, I would’ve done so that night. One moment alive, the next under water. But I realize I was not numb to it. In moments like these, your body takes control of your conscience and you enter a fugue state. Autopilot.

Only afterward do you grasp the damage done to your conscience. For that reason, I am grateful I found myself in your country in the aftermath.

The rest of the ride there was oddly serene. Some slept; others stared at the horizon, confronting how fragile and helpless we truly are. We were adrift in a boat, in the middle of the Strait of Hormuz. The Strait of Hormuz. The same place I had constantly read about on the news. “Volatile body of water”. “Threatens closure and disruption”.

When we finally got to Khasab, your captain informed us that because of the situation at the port, none of our luggage could make it. You had kept it safe back in Iran, away from the port.

For the residents of Oman, this was fine— they had homes and visas to stay. I was not a resident. The fact I was even on your boat was because of the sympathy of your officers. And now I was in your country, with my passport, phone, and practically nothing else. I was stunned and alone, too confused to fathom what was happening and what I would even begin to do.

And then, you reminded me I was in safe hands.

At customs, your officers assured me it would all be fine. They gave me numbers to communicate with and people to reach out to. You issued a temporary visa for me. Most of all, they gave me what I desperately needed—assurance.

I believe this protected my conscience, or at the very least mitigated the impact of the sights I had just seen.

The sun had started rising at the port. I remember recalling how beautiful the landscape was. On my right were rugged brown cliffs and shrubbery, and on my left was the wide ocean, a pale blue.

You directed us towards buses, and we started heading away from the airport. Conversation soon filled the aisles as us passengers— bound by shared terror and relief— began to process what had happened. Everyone had a different perspective, be were all grateful; grateful that we were alive, and we had you to thank for that.

We made it to Atana Khasab— the hotel you had allocated for us. We were rushed to the buffet while the receptionists prepared rooms for us.

It was grim, really. But I remember finding it so funny at the time. Perhaps it’s in poor taste to find humor in war and destruction, but I was genuinely amused by the absurdity of it: just hours ago, I had been trapped in a warring Iran, running on nothing but willpower, begging for a place on your boat. And now, here I was in peaceful Oman, eating braised salmon at a luxury hotel.

What I suppose I mean is— I felt so untethered from my life. I felt like an observer, watching forces shove and nudge my body across the world. I had quite literally lost control of the steering wheel. I didn’t know what the next hour would bring, let alone tomorrow. And the stark difference between that feeling— the lack of control, the helplessness— and the life I once knew struck me as, in some macabre way, funny.

Artist Bo Burnham once called it “that funny feeling”— total disassociation, fully out your mind. googling derealization, hating what you find.

I didn’t really know what I was, only that I was fully disassociated. A violent quake had come and split my life into pre-war and post-war Faisal— and I wasn’t in control of what was happening to post-war Faisal.

But I was, yet again, assured by your delegation that everything would be alright. I suppose I had lost control of the steering wheel, but I’m grateful you had reached out and taken hold.

In that way, you reminded me yet again, that I was in safe hands.

In my room, I rinsed the chaos from my skin, hand-washed my only set of clothes, and fell onto the bed, weightless. Through the window, I watched the sea breathe, the waves catching the first hints of dawn. Birds began to sing and pierced the stillness of my mind. The sky, bruised with the memory of night, began to brighten.

And somewhere between the ocean’s breath and the chorus of mo(u)rning, I drifted to sleep.

I woke up to the ring of the hotel telephone.

“Breakfast is served! Make sure to come grab a bite before leaving for Muscat.”

I slipped into my barely dry clothes, gathered the few belongings I had, headed downstairs. Despite all that had happened, I had an insatiable appetite. I piled two plates up with food and sat in the smoker’s lounge. I don’t smoke; I just wanted to be around other people. I didn’t want to be alone with my thoughts.

I started feeling alive again. Breakfast in front of the turquoise sea— the horizon splitting the view into two different shades of blue— one noticeably darker, more mysterious than the other. pre-war and post-war me, I thought. The panorama of blue was ocassionally joined by the earthy wafts of steam rising from my cup of coffee. Now and then, a thread of cigarette smoke pierced through the lounge, sooty and deep, keeping me grounded to reality. The sounds of people conversing in Farsi, talking about their relatives or homes. And maybe, just maybe,

I saw someone’s eyes crease as they let out a hearty laugh.

And for that instant, I let myself believe in safety again,

because of you.

Like all good things, my moment of peace eventually came to an end. But it filled me with warmth and security— feelings I craved far more intensely than my gnawing hunger.

We boarded your buses yet again, this time bound for the airport. As we pulled into the departures entrance, I quickly realized that this was no regular terminal. You were escorting us aboard your own Royal Airforce.

Again, that funny feeling bubbled up in me. How surreal this journey had been, and how much stranger it was about to become, being carried to safety by a royal aircraft. Who would’ve even thought? Just 2 weeks ago, I was enjoying Iran, surrounded by family. Now, I was somehow in Oman, alone, slowly losing control. But perhaps losing control of the trajectory of my life wasn’t so bad, as long as you were steering the course.

We were guided across the tarmac towards the silhouette of your Lockheed C-130 Hercules. I remember thinking of how well the sand-green camouflauge worked. The military plane almost melted into the mountains behind it, only your flag interrupting the illusion. That funny feeling possessed my body. I sank into the seat, weightless, resigned to forces far greater than myself.

As the engine started and rumbled, our bodies were pressed into the seat, subjected to force of weightlessness. Oh, the mysterious mechanics through which life operates. How did I end up on the Omani Royal Air Force’s warplane?

Answering myself, I whispered part of an old quatrain I had memorized.

با چرخ مکن حواله کاندر ره عقل
چرخ از تو هزار بار بیچاره‌تر است

Blame not the heavens through the reasoning mind,
For the heavens are a thousand times more wretched than you.

— Khayyam, Quatrain 48

Perhaps that illusion of peace— the one life had shattered without hesitation— came from thinking the heavens had our best interests in mind. That they were gentler than our world. Our world, enraptured by war, riddled with destruction, steeped in inequality. But they were not. Their decrees were, if anything, a thousand times more chaotic and indifferent than we could bear to imagine.

I repeated this couplet to myself, until I felt the stillness of sleep take over my body once more. Through clenched teeth, I whispered it again and again, a chorus of mourning against the engine’s chorus of war.

And I drifted off to this duet of chaos and resignation.

I instantly knew that we had landed when the ramp of plane opened and the humid, sticky air filled the cargo hold. I jolted awake and looked around, half-expecting it all to be a dream.

We were led to the arrivals terminal. For many, the journey was over— they were residents. But what home did I have in a city I had never set foot in before?

I spoke with your officers about what would come next. They told me that you would be in touch and inform us as soon as our luggage arrived. I got the feeling that I was on my own now, finally responsible for what would come next.

I installed your country’s analog to Uber, Yango, and ordered a cab to the cheapest hotel I could find.

I assumed the pleasantries were over. That your plan for me had run its course. I’d been handed back the steering wheel, sent off in a direction unknown.

But then Ahmad— the Omani Yango driver picking me up from the airport— noticed.

Ahmad realized something was off immediately and asked what was wrong. I wasn’t exactly in the mood to describe everything; I just said I was going through a rough spot. He asked how I’d learned Arabic despite not being Arab, and I briefly explained my upbringing. He listened with so much sincerity and patience, then said:

“I live here and know people. You are our guest. If you ever need anything, we are here to serve you.”

And perhaps he was just offering. Being polite. But it stuck with me.

Ahmad was the first person I met outside of the chaos of war, and he reminded me life could still be normal, peaceful even. But beyond that, he also set the tone for my unplanned stay in your country.

I didn’t know it yet, but I would feel that same generosity again in Oman. And again. And again. In fact, every single local I met was unbelievably welcoming. It didn’t feel like mere formality— these people, your people were genuinely concerned, eager to help in any way they could.

And it wasn’t even out of pity— I didn’t tell any of them how I ended up in Oman. These people rushed to help someone they didn’t even know. All they knew was I was visiting, and that was enough for them to immediately offer whatever they could.

I wasn’t expecting pity— let alone kindness. I wasn’t expecting anything at all, really. After everything that had unfolded, I had almost forgotten what kindness really felt like. What surprised me was that the kindness I encountered wasn’t out of pity or obligation— it was genuine kindness, a sincere eagerness to help someone out, just because you can. A choice to be good. Your people reminded me of that.

I could write a whole book about the sheer kindness I experienced during my stay in Oman, but even that would fail to describe what it felt like being surrounded by your people.

If I were to try to describe it, however, the word I’d use is peace. I felt at peace. Despite everything, despite the displacement, despite the bombing, despite it all, I felt at peace in your land.

Ahmad, like almost every other cab driver, did not accept payment for the ride. They insisted. Gave me their phone numbers.

At the hotel, the manager upgraded my room for free. The receptionist gave me his charger until I could buy one.

Other cab drivers offered coffee, tea, dinner. Very few of them accepted payment.

Sales associates at Amouage gave me extra samples, never got bored of seeing me. When I paid with cash and they didn’t have change, they ran to an exchange spot to make change for me.

When I was alone at the beach, people came up and talked, bought me drinks.

One elderly gentleman even insisted I pray with him at your Sultan Qaboos Mosque.

These interactions made me feel at peace, but I was also dumbfounded. The contrast between what my life was a few days ago and what it had started to become, what had formed around me, was abrupt. War had shattered my perception of peace. But you reframed it. Rebuilt it.

And with the real war quieting down, I started to feel in control over myself again. Over where I was, where I’d go, what’d happen next. I began tracking my luggage using Find My, since my MacBook was inside one of the bags.

And what was initially a feeling of dread— of being alone in a foreign country— turned into one of odd serenity. I was still alone, sure. But you and your people helped me take back the wheel, and I was so deeply grateful for that. At one point, I even found myself hoping the luggage would take longer to arrive, just so I could keep wandering Muscat and meet more people.

After 8 days of wandering Oman in awe, my screen lit up: my MacBook was in Muscat. A notification came through. My luggage had arrived.

It was a bittersweet moment, but I booked my flight back to Doha for later that afternoon.

8 days ago, I was on board a military transport plane. Now I was sitting on the window seat of a SalamAir commercial airplane. I reflected on what I had felt last time. That funny feeling. I had come full circle. Oh, the quiet way in which life— almost mechanically— becomes symmetric. How funny, to be airborne again. I recalled the Khayyam quatrain once more, but now the full quatrain, complete with the first couplet.

نیکی و بدی که در نهاد بشر است
شادی و غمی که در قضا و قدر است

با چرخ مکن حواله کاندر ره عقل
چرخ از تو هزار بار بیچاره‌تر است

The good and evil that dwell in human nature,
The joy and sorrow decreed by fate and destiny—

Blame not the heavens through the reasoning mind,
For the heavens are a thousand times more wretched than you.

— Khayyam, Quatrain 48

I had previously resigned myself to the wretched decrees of heaven in that warplane. Blamed them for the mercilessness of the universe. But now, suspended over a calmer sky, I realized that the quatrain began inwards, not outwards. It doesn’t open with the stars or heaven or fate, but with the good and evil in human nature. Before destiny ever intervenes, we are already made of choice. Even if sorrow is written in the stars, goodness is not. Goodness is a choice. It lives in the same human nature that holds cruelty, and it cannot be imposed by the heavens or excused by them either.

Your people chose goodness. Against the cosmos’ decree of sorrow, they choose it. Again and again. And that taught me that even if fate may define the conditions, it is human nature that determines the response.

And now I realize that the quatrain never blamed the heavens— they are a thousand times more wretched than us. It pitied them. They are a thousand times more wretched than us. That line used to feel bitter to me. Resigned. But I think on this SalamAir commercial plane– of all places— I finally understood it. The line wasn’t about surrender, it was about letting go of blame, and finding meaning in the only thing we ever truly hold: the self.

I repeated the full couplet to myself, softly this time. I felt the peacefulness of sleep take over my body, slowly. Through relaxed teeth, I sang it again and again, letting its completeness— my newfound understanding of it— carry me. The engines on this plane hummed too, but they no longer drowned me out. I let the couplet join their hum, not as a chorus of mourning, but one of meaning.

And I drifted off to this duet, knowing that being carried was enough.

And so, dearest Oman, I am eternally grateful for what you did to me.

In June, I was forcefully made aware of the illusion of peace. you showed me what it truly looked like.

That peace was not silence or distance.

Not numbness or indifference either.

That peace was the quiet strength of people who choose goodness against the odds.

That peace was in the warmth of the tea offered to strangers.

You taught me that peace is not the absence of sorrow—

Sorrow was an inextricable part of peace,

peace is what we do with it

Yours truly,

Faisal Toosan

Perfume was one of them.

I never thought I would spend such a significant chunk of my money on perfume. Or that I’d spend hours reading about the linearity of a perfume, its sillage, or its projection. Past me would cackle and label present me pretentious. Sillage? Really?

However this obsession with perfumery and fragrances snuck up on me silently, and before I knew it I was buying stands for my perfumes.

Since then, I’d like to think my addiction has slowed down. But there is still one perfume house I always circle back to. One whose sillage and projection I’d gladly be pretentious over.

That perfume house is Amouage.

Amouage & Omani Heritage

The House of Amouage was born in Oman in 1983, under the rule of Sultan Qaboos bin Said Al Said. Commisioned with the aim of preserving Oman’s rich history in global frankincense trade, Amouage has grown into one of the most luxurious and sought after perfume brands.

Oman’s history as a hub in the trade of frankincense and myrrh is central to Amouage’s identity— many of its perfumes feature a smoky frankincense note as a nod to that heritage.

Interestingly, while the house’s roots are deeply Omani, the noses behind its creations are often Western— perfumers like Guy Robert, Pierre Negrin, and Quentin Bisch craft Amouage’s scents using rare and traditional Eastern ingredients: oud, amber, myrrh, and most iconically, frankincense.

Branded as “The Gift of Kings”, Amouage offers fragrances that are bold, nuanced, and unapologetically luxurious. In reviving the traditions of Arabian perfumery and marrying them with modern expertise, the house has built a legacy on the meeting point between East and West.

Frankincense is part of Oman’s identity. You smell it at the airport, in people’s homes— it’s everywhere. It’s also the signature that unifies our work at Amouage.

— Renaud Salmon, Creative Director at Amouage

Discovering Amouage

I had always known of Amouage— their small stand at Doha’s airport always seemed full of people— but I had never really experienced any of their fragrances.

That was until earlier this year when my perfume hobby started. I started researching Amouage and their fragrance collections.

Amouage— waves in Arabic— had, by far, the most overwhelmingly positive reviews. People would gush over it in the reviews, and I thought they were simply exaggerating, as many perfume enthusiasts do.

I decided to pick up a bottle, just to see what it was all about. I checked and found a retail store selling a full bottle of Interlude Man for a decent price.

Interlude

I interpret what I see and feel into scents. All the social and natural chaos and disorder surrounding us today can be translated to a much more intimate level. The interlude moment is a reflection of all the trials and tribulations one overcomes to attain personal satisfaction and achievement.

— Christopher Chong, former Creative Director at Amouage

When I recieved Interlude, I was immediately blown away by the packaging.

With niche perfumery, you expect purposeful, intentful presentation, especially when you dish out over 380 USD for a perfume. And of all niche perfumes I’ve purchased, Amouage easily takes the cake for the most gorgeous packaging.

The bottle itself blew me away too.

Styled like a traditional Omani dagger— a Khanjar— the flacon is adorned with the Amouage emblem, a 12 pointed star. The cap is weighty and crowned with a gorgeous sapphire blue Swarovski crystal, also reflecting Khanjar patterns.

Even before spraying, Interlude made its presence known. As I removed the cap, a wave of incense hit me; resinous, deep incense.

I sprayed it once on my wrist and took a whiff.

I still don’t know what Interlude smells like, exactly. But I had read about how frankincense was harvested, and I think it serves as a great analogy to the fragrance’s profile.

Interlude & harvesting frankincense

a lone boswellia sacra tree stands in the sun-parched Omani landscape. the frankincense tree stands, alone. a testament to resilience.

a harvester approaches.

he is immediately hit by notes of sharp, medicinal greenery. not lush, ripe greenery, but rather sharpened, angular, dry, spiced greenery which draws him closer— not gently though.

the harvester draws his mengaf, a curved blade. he strikes the tree, cutting through the bark, exposing its pale flesh. the tree bleeds.

frankincense, a resin meant to protect the tree against foreign predators, bleeds from the tree. it also serves to prevent the tree from losing its most valuable resource: water.

as the exposed resin dries, it crystallizes into frankincense.

over time, the tree heals and regenerates a layer of bark where it was struck.

then, the harvester returns. he strikes the tree again, harvesting the frankincense and creating new cuts. this process is repeated several times. each time, the frankincense becomes more concentrated, more fragrant.

The harvester is gone.

And the tree, again, begins to heal.

Interlude performance

As with all Amouage perfumes, the performance on Interlude is stellar. In fact, Interlude might be the best performing Amouage.

On skin, I get 6-8 hours of consistent projection before it settles into a bitter-herbal skin scent.

On clothes, it is eternal: it literally will not dissipate until you wash the article of clothing.

I use 2-3 sprays max on pulse points during the colder seasons. This would probably suffocate in closed, hot spaces.

Very lingering sillage too, and the perfume exudes presence as you move. You don’t wear Interlude as much as you carry it with you.

Overall, it is very easy to let Interlude wear you, not the other way around. It is genuinely a very challenging perfume to wear. It is, definitionally, a statement perfume.

Unapologetically dark and smokey, Interlude does not demand you understand it at all. But in my opinion Interlude tells a tale of pain, rupture, chaos, repair, beauty. And if you want to witness that, then Interlude is a masterpiece worth enduring.

Decision

Uncompromising, distinctive and incandescent, Decision captures the paradox that lies at the core of all of life’s most powerful decisions: it is focused on its goal while remaining open to countless possibilities.

— Amouage

After falling in love with Interlude, all other perfumes smelled dull to me. I have not bought any perfumes since buying Interlude.

Except Decision, also by Amouage.

Against all odds and decisions, I found myself in Oman recently. Oman, the birthplace of Amouage.

There, I got to wander the old souq of Mutrah. I was bombarded by a thousand different scents. Piles of real frankincense, sold by the kilogram. Vendors begging you to try their perfume oils. Potent spices piled up in neat dunes.

I also visited several Amouage boutiques. I think I visited so many times that the representatives genuinely got bored of seeing me there so often.

And 2 days before I left Oman, I smelled Decision.

I had walked in to buy a dark, brooding perfume like Interlude, maybe Memoir or Epic. But the representative, after hearing I already owned Interlude, recommended Decision.

I decided to try Decision.

Apex of Oman, Jabal Shams

Renaud Salmon, Creative Director at Amouage, shares the scene that inspired this perfume.

He describes his journey to Jabal Shams, the highest peak in Oman, rising over 3000 meters.

“When I arrived at the base of the mountain and started walking,” says Salmon, “something caught my eye. Across the terrain were what looked like conifers at first, but when I got closer to them, I saw they were juniper trees – their branches twisted into unusual shapes.” Many junipers are survivors of electrical storms, struck by lightning that turns their sap into steam, causing the bark to split and fossilise. Legend says that these trees carry a mystical quality, and their wood is sometimes also used to burn at home like frankincense.

— Renaud Salmon, to Vogue Arabia

After this striking opening, the light spreads further, aided by the gleaming sheen of Atlas Cedarwood.

— Amouage

Finally, a warm Vanilla in the base conveys the profoundly liberating effect of acceptance.

— Amouage

Most trees die when struck by lightning.

But the Juniper tree decides to live.

Decision performance

Decision is no Interlude in projection and sillage. Its more calm, less theatrical and doesn’t really seek to prove anything loudly as with Interlude. If Interlude was the scent of tragedy and healing, then Decision is the breath of calm after deciding to heal.

On skin, Decision performs moderately well, projecting loudly for 3-4 hours before becoming a present yet tamed scent. After 9 hours, it becomes a creamy-spicy skin scent.

On clothes, this might perform even better than Interlude. It sticks around forever and refuses to let go. I sprayed Decision once on my watch’s strap when I bought it, and almost 2 weeks later, I can still smell it.

I can see Decision worn in every season, just maybe not during peak heat, where the sweetness of the resins could become overbearingly cloying. Otherwise, go heavier on the trigger with 4-6 sprays.

The sillage is moderate on Decision, and I help it linger a bit more by doing a spray on my nape.

However, this perfume is still not an easy wear. The metallic-sweet note can be off-putting or clinical to some, but if you found Interlude wearable, then Decision comes as gentler, cooler, and more restrained.

Some believe lightning is punishment, divine wrath.

But in Decision, Amouage reinterprets lightning as revelation.

The fragrance strikes first with unripe citrus, split suddenly by metal. And yet, it never feels angry. The frankincense and resins ground it— composed, restrained.

If Interlude was a wound torn open, a chaotic bleeding of heat and smoke, Decision is the scar that remains after healing.

It doesn’t shout. It doesn’t demand.

It simply is— clear, measured, whole.

Wearing Interlude feels like being possessed by fire.

Wearing Decision feels like standing in the aftermath of lightning, unburnt.

Decision is not about destruction. It is about clarity,

about choosing to live after the storm— and finding beauty in what remains.

tangent on the ‘pretentious’ nature of perfume hobbies

I am well aware how out of touch and ostentatious a perfume hobby may seem. Spending hours reading about fragrances, spending unimaginable sums of money on what, to most people, is scented alcohol; all this talk of projection and sillage. It all seems so bourgeois and insane.

But I encourage you to look at it in a different lens: Our sense of smell is one of our main 5 senses that structure how we experience the world.

We spend hundreds— even thousands— on sight (fashion, apparel), sound (headphones, concert tickets), touch (fabrics, furniture), and taste (fine dining).

All of these are demonstrations of consumerism, I understand. But we appease all these senses with little social consequence. No one bats an eye when you spend 300 USD+ on a pair of Sony Headphones or 100 USD on a night out.

But when it comes to smell, one of the most intimate and emotional senses, indulgence becomes ‘pretentious’.

A common counterargument is that perfumes are rapid consumables. However, a well-made fragrance offers years of use. At 100ml, you’re looking at around 1000–1200 sprays— that’s three years of scent if you apply it three times a week.

Perfume, in that sense, is actually one of the more enduring luxuries.

AMOUAGE

Amouage is “only expensive once, after that it’s priceless”.

Beyond marketing flourish, I think this line captures the ethos of the house perfectly.

From the beginning, Amouage set out to revive the forgotten grandeur of Arab perfumery and introduce it to the West— not as mimicry, but as a dialogue. And in doing so, it forged its own identity.

An identity linked with fragrances that are challenging, transcendental, and ritualistic.

An identity rooted in Omani identity, built upon the backs of local artisans, sourcing ingredients like frankincense from protected Omani regions.

An identity which brings massive pride to the Omani people and challenges Western perfumery giants like Dior.

It has formed this identity all while still costing less compared to niche brands like Creed (which costs far more per milliliter). All while maintaining the quality of ingredients and artistry of the brand.

I truly believe that Amouage is a perfume house people should try at least once. If price is a barrier, be on the lookout for samples. You could even split a bottle with someone. Or wait for a discounter selling on clearance: I got Interlude for 180 USD. I wear it once a week or so, and it changes the color of my day every time.

Perfume may not be essential. But it can be transformative.

And Amouage, the Gift of Kings, reminds you why scent matters.

Sometime in early April, a former teacher of mine let me know about Stanford’s Code In Place Program, an online 6-week course offered by Stanford aimed at teaching introductory CS content from their flagship course— CS106A— for free.

I first considered joining as a student— I mean, It looked great! Attend a live zoom with a teacher every week, get certification, and work on a final project (I LOVE final projects).

But then looking at the content, it seemed pretty rudimentary.. This was meant to be an introductory course after all. I had known Python pretty well, so I decided to instead volunteer to become a Section Leader (SL)— the person hosting each of the “sections”: Zoom meetings where 10-15 students join and work on a problem together and clarify any of their concerns.

But then again, I’d have to teach for 6 weeks, spanning my school exams, APs, and even graduation. I was worried I wouldn’t be able to keep up with the responsibilities.

Still, I applied anyway. And now, two months later, I’m sitting here writing this reflection—having just finished leading a section of 16 students through CS106A. Fittingly, at the time of writing this, I would usually receive a reminder telling me section is in 24 hours :(

Applying

The application process to become a section leader was pretty simple:

• I needed to provide a few short responses about myself and why I’d like to teach, what prior experience I have, etc.

• Solve a few Python problems using their Karel tool (more on this later!)

• Provide some sample help with debugging where I’d be given some erroneous code and I’d have to help debug it without saying what is wrong explicitly.

• And most significantly, I’d have to film a short, 5-6 minute teaching demo where I go through a problem and explain how to solve it with Python. This video me the longest to complete, and I had to retake it a handful of times. After completing the application, I received an email and waited for the decision to come out. I got in!



Onboarding

After getting accepted, I had to pledge that I would make and prepare for section over 6 weeks. I then had the option to choose what time and date I wanted to hold section, and there were offerings from Wednesday to Saturday if I recall correctly. I personally picked Friday 4 PM my time. I also notably had an option to refer two students to the program and they would be automatically accepted.

I now had access to the Code In Place platform. A small on screen tutorial showed me around the website. I will attempt to summarize each tab.

The Website

• Home:

  Here, you are able to either pick the Teacher Home or Student Home.

The Teacher Home:

would allow you to view all your students, the number of assignments they’ve completed, the number of lectures (online videos) they’ve watched, and which sections they have attended. Additionally, it would allow you to go to their actual solutions for each assignment and run it for yourself on the Stanford provided IDE.

The Student home:

emulates what a student would see, as expected. It is neatly organized into week by week lectures, assignments, and any additional tests or events.

• Code:

In this tab, you would be able to create your own projects and run them with the Stanford IDE or access a list of all assignments and work on them. There are also optional questions which are much more algorithmically challenging to solve.

• Lessons:

  Here, you could access all the pre-recorded lectures in case you want to brush up on concepts before section. You also had access to “readers” for Karel and Python, which are basically self-contained textbooks outlining everything (and more!) you needed to know about them. I used the lessons here to see what the students knew exactly and adjust section accordingly.

• SL Training:

  In this tab, you had access to a SL “teacher’s handbook” with a bunch of advice and content. You also had an option to open a testing Zoom room, just to get used to Zoom and troubleshoot. Additionally, there were also 3 pre-recorded training videos you had to watch in order to actually lead section. They involved topics such as: orienting your students, interacting with them, and also organizing section.

• Section:

  This is the main directory you’d be operating in. This tab has all of the data you need to lead and operate section. On it, you would have to RSVP for section each week, and if you’d be able to make it, you would get an option to open a Zoom meeting 15 minutes before your section time. We also had an option here to make an announcement which would appear on all of your students’ section tabs as well. There was also a table indicating each week’s topics, resources, code, and solutions. The resources included the main lesson plan, a slides deck, objectives, code, etc.

It also had bonus optional extensions for each of the problems, which my students found particularly stimulating.

• Forum:

  The forum was a particularly useful way of communicating with the entire platform. Similar to Stack Exchange, you would make a post on the appropriate forum and people would be able to like and reply to it.

Main Forum:

Everyone in the program can access this forum

Section Forum:

Only you and students in your own section had access to this forum. On the Section forum, you also had an option to email all of your students and make announcements which would appear at the top of the forum.

Teacher’s Forum:

A forum only with Code In Place staff and teachers.

Main X Forum:

A forum with experiences students: students not assigned to a section who would navigate the course alone.

The forum/platform’s WYSIWYG text editor was very convenient, as you could format text in several different ways and also embed links, code, quotes, images, and a lot more. You also had the option to make your posts private or anonymous.

• TeachNow:

  This was one of the most exciting features for me. You could, at any time, use TeachNow to connect to a student who has been struggling on a problem (they would be able to request for help) via Zoom and help them out. You would be able to share your screens and go through it together. This was an amazing opportunity and it was open to other students to volunteer as well. In fact, if a section leader missed their section for any reason, one of the ways they could compensate was by completing 4 TeachNow sessions for each missed section.

• Connections:

  A lesser used feature for me. In this tab, you would be able to see the profiles of 4 other Code In Placers (Section Leaders, Students, Experienced Students, etc.) and ask to connect with them on platforms of your choosing (most did LinkedIn). These profiles would refresh every 4 hours. After making connections, they would appear in the bottom of this tab.

• About:

  This tab is similar to an FAQ: it has a bunch of information about Code in Place, section, responsibilities, and everything in between. More on responsibilities later.

• Events:

  This tab has a collection of Zoom events Stanford highlights regarding Code In Place and CS in general. I saw a few interesting ones about pedagogy, a hackathon, and a final project showcase. There is also (at the time of writing) an upcoming event on training AI auto-graders, which I am planning to participate in.

Training for Section

Before leading my first section, all section leaders had to join a 1.5 hour live training session where we went through the course expectations and standards. It was led by a head Teaching Assistant (TA) who was a former SL herself. She was our main point of contact regarding any issues or problems we had with section, including if we couldn’t make section. We were also put into breakout rooms and practiced teaching with other section leaders and shared feedback together. We had to do this twice throughout the course— once before our first section and once after— for a total of 3 hours of live training. The second session was particularly productive as we all shared resources we were planning to use for section. Further practice sessions were also carried out at the same time weekly, but they were not mandatory.

First Section

Finally, it was time for my first section. I remember being insanely nervous for it, so much so I thought I’d develop a case of restless leg syndrome because of how much I was fidgeting. I also remembered the founders of Code in Place— Mehran Sahami and Chris Piech— sharing tips they had for what to do before teaching and tried to go through them. I played some of my favorite music, and when I could finally open the meeting, I did.

At first, I only had 2-3 students join despite having 15 students in my section. I told them (and myself!) people would start joining and that we were quite early. I was luckily right, and soon I had to constantly accept people from the Zoom waiting room. This was actually one of the technical difficulties I had in the first section— I would have to constantly accept people manually. Luckily it was resolved by my second section.

What struck me the most was the great diversity of people wanting to code— medical students, entrepreneurs, chemistry majors, flooring professionals, and even other CS students looking to learn Python. This diversity was motivating, but beyond that it also became a great way to show how in-demand basic coding skills have become. I tailored a few analogies to the students’ backgrounds; I showed the flooring professional a Karel assignment we had where we had to tile the grid in a specific pattern, similar to actual tile layouts.

I really emphasized setting the culture for my section in the first session we had, and it percolated over to the next sessions very well. When we had everyone in the meeting (I think 13-15) people, we all introduced ourselves for a good 10 minutes before moving on to the actual problem, which was a vestige of Code in Place’s COVID origins: We had to use Karel (a small robot guided by Python) to “build hospitals” on a grid of cells given a certain condition was met.

Karel only had four commands, which made it very easy to learn. It could turn_left(), turn_right(), move(), put_beeper(), pick_beeper(), and also had a few methods which returned booleans like front_is_clear(). This made it really useful for practicing functional decomposition and teaching core programming paradigms. Furthermore, you weren’t allowed to use variables or return statements with Karel, forcing experienced students to come up with algorithms they wouldn’t have thought of before (see Section 2 problem!).

Before starting to code, I made it clear that I would prefer if everyone had their cameras on, just so I could gauge their understanding by looking at their faces every now and then. I think this was super important and it set the standard for the rest of section, as I would always have a majority with cameras on.

The problem itself went by pretty well, we went through different routes and explained why they wouldn’t work or how we would adjust them. I also tried as much as possible to let the students themselves write the code, so in the end we could have a chimera of code and functions working together to solve the problem.

Another aim was to handle off-by-one errors. The Stanford IDE made this very convenient: it allowed us to see code execute line by line. I think a great advantage of Code in Place was the ability to code in front of the students. It allowed us to actually run the code and see what happens in real time and how simple changes, even in code order, can lead to very different results. We also had built-in Karel & basic Python documentation which allowed us to quickly check for what method names or outputs. Overall the coding experience was great and I did not run into any technical difficulties on the IDE.

Teaching the problem and seeing how curious and willing to learn the students were quickly chipped away at my angst and moved me to the “zone” I’m sure you’ve felt if you’ve taught before. Everything just comes out smoothly and if it does not, it becomes a learning opportunity.

We finished teaching the problem and I clarified any concerns the students had. We finished just in time, and I had a bit of free time afterwards so I decided to stay in the Zoom meeting just to see if anyone would warm up and feel more comfortable to ask more questions or just have a little chat. Thus, the after-section-yap-session was born, a huge part of our section’s culture. After every section, I would let the students know that section was over and they were free to leave, but I would remain in the meeting if anyone would like to ask anything or chat. This worked super well, as people started asking me more about coding and also myself. Maybe 8-9 students stayed every time, and we would start by clearing up some more important concerns about Code in Place or CS, then moved on to just chatting about our lives. We got to know each other very well as a result of these “yap sessions”, and a student later said they always looked forward to them after section.

Missing Section Protocol

Before I move onto the remaining sections, I’d also like to highlight what would happen if you missed a section. Thankfully I made all of my sections (there was no WAY I was missing my yap sessions), but Stanford had outlined what would happen pretty clearly:

• If you missed your very first section, then it would automatically be “disbanded”— I’m assuming this means you would be dismissed as section leader.
• For the remainder of sections, if you miss section by RSVP’ing that you can’t show up and also letting the head TA know, then you would be given a choice of covering for another SL who couldn’t make their section or completing 4 TeachNow sessions. In both cases, you would still receive your certificate at the end.
• If you do not RSVP and don’t inform your head TA, then Stanford reserved the right to disband your section.

Students evidently had similar conditions, I think it went something like:

• Attend a make-up session
• Complete final project
• Do the optional diagnostic test at the end of the course

Subsequent sections and some section culture

After the first section, I realized how special section really was to me. I first thought that the responsibility of managing section would be too great for me to manage— that I’d be overburdening myself in peak exam season. But section was a place where I could go to give back to the community and also “zoom out” from all the stress of exams. It genuinely had such a positive impact on me that I’ve vowed to volunteer to lead section again next year. However, I will always look back at this first group of students with beaming pride and gratitude. They truly made section so lively and enjoyable.

In my next sections, we had 2 new students who were joining from other sections— one to join a temporary make-up session and another looking for a new section. I wear it as a token of great pride that the second student liked us so much (I hope!) that he joined us for the rest of the course! So I now had 16 students every section.

Section 2

The second section had a more algorithmically challenging problem– Karel had to distribute a pile of $n$ beepers across the row it stood on. Again, Karel cannot count (no variables or returns), so this problem was a bit harder for the students.

We went through different algorithmic ideas and wrote faulty code before reaching the canonical solution. Writing the faulty code was very productive– both the students and I got to brush up our debugging skills!

I also started trying different things for upcoming sections.

bCode

I needed a platform allowing several people to collaborate on one IDE while also allowing me to monitor everyone’s work. bCode fit the bill perfectly.

For my third section, I decided to use bCode to allow people to all work on the problems in either breakout rooms or individually. The UI is pretty intuitive, and you’re able to set starter code up and allow students to run their code. We organized everyone into groups based on their breakout room number, and everyone got to work. I think this was one of the most productive tools I used. I would check in on all the breakout rooms periodically to check for issues and ensure everyone was collaborating. It worked especially well with some of the people who used chat to communicate, because they began to open up and interact via mic in these breakout rooms.

Beyond highlighting the obvious wide variety of solutions that existed for a single problem, it also allowed for some pretty neat (and funny) learning moments afterwards when we reviewed everyone’s code.

GPT antics

One of the funniest moments (in my opinion) in section occurred when we were working on building a program to convert weight on Earth to any of the 7 other planets (sorry Pluto..). For context, the conversion factors were given in the problem. But we had also recently learnt how to call GPT within the problem. So when we were reviewing everyone’s code at the end of the bCode session, I moved to someone’s code and noticed the first line was from ai import call_gpt.

Oh no. Had they just asked GPT to solve the entire problem?

No. They had taken the planetary input and asked GPT to generate the conversion factor for them. This avoided checking against the user input, something I’m assuming they struggled with. After realizing what he had done, the entire section burst into laughter and the incident became a running joke in section.

Menti Polls

Another tool I used were polls, via Menti. They allowed me to give the students some freedom, and I allowed them to vote on if they wanted to work individually or in breakout rooms. It was also completely anonymous. We also used Menti to simulate the behavior of a short game we were tasked with building. Overall, Menti was easy to use and I would definitely use it again for polling.

Extensions to problems

One of the most enjoyable and productive things we also did, in my opinion, was build our own extensions to the problems. It allowed us to really get into the programming mindset and think of how the different parts of our program worked together, beyond what was given in the problem. A notable example of this was in a problem where we were tasked with using graphics to draw a bunch of random circles. The function generating the random color was already given to the students, but our extension involved building it ourselves with random.randint() instead. And later, we extended that beyond the predefined colors and into any hex color.

I also tried encouraging this mindset by telling them to think of how some of the predefined methods (eg: for the random package: shuffle(), etc.) worked under the hood before telling them to try building their own implementations of them.

Discord Server

In our 5th section, I decided to make a small discord server for the students, just so we could stay connected in the future and also if any of them needed help with their final projects, they could communicate together there. This worked pretty well, and people are sharing their projects and also asking for help decently often. We also made a resources channel with some resources for if they want to further their knowledge.

Moving away from Karel

Another fun moment in section was when we finally moved away from Karel and into actual Python. In the last Karel section, I had expressed my disdain for Karel’s unintuitive orientation.

See; at first glance, you’d assume Karel’s “left” refers to the cell to our left. But no—Karel is viewed from above, and her left is actually the cell above her in our perspective. Still, the little feet at the bottom suggest a side view. It’s confusing, and methods like left_is_clear() don’t help.

By our final Karel session, I changed the Karel icon in the IDE settings to a turtle—which made it clearer that we were viewing the robot from the top. This led to a cathartic rant during our yap session… and a celebratory meme in the next week’s Python section:



Yes, I’m very proud of my meme-making skills :P

Final Section

And finally, one week ago, I hosted my final section. It was bittersweet, but the problem we worked on was quite fitting for the final section— it involved making a terminal game of Heads Up, and everyone would get a turn to describe a word that I had to guess. After finishing section, I had planned out a Menti game where we would all vote on a few superlatives for each other (“most likely to forget an indent..”, etc.), which gave us a bit more closure. We also had a longer than usual yap session and also took a few pictures together.

And that was it— 6 weeks of Stanford Code In Place, ending with a click on “End Meeting for All” :(

What I would do differently

Honestly, this was one of the best experiences I think I could’ve had. My students were ideal, the technical issues were minimal, it was easy to navigate and explain the content. I would genuinely recommend volunteering as SL to anyone reading this, regardless if you have prior teaching experience or not.

I’m having to think really hard in order to point to something specific to change, but here’s a (non-)comprehensive list:

• Deviate from the Stanford problems a bit more. I think my students learnt best when we strayed a bit from the course and tried applying things on our own. Not to say I won’t use the problems, but I definitely will dedicate more time next time to expanding on them and creating more elaborate extensions.
• Make it more interactive for people who only used chat. I think I only had 3-4 people who didn’t use mic at first, but in the final sections they had warmed up. But I will definitely try addressing it next time by coaxing them into turning it on if possible. The Zoom chat as is isn’t very productive for the style of programming we were doing. Maybe I could somehow give them access to directly comment on the IDE.

I suppose I will update this as list as things come to mind :)

Closure

Leading section was an incredible opportunity that I am infinitely grateful for discovering. It genuinely surprised me how close we grew in such little time— how rewarding it felt to be able to give back to a group of people looking to learn how to code. Despite starting off quite anxious, I found myself energized each week by the diversity and also growth of my students. Teaching really challenged me to think more clearly, communicate effectively, and build a fun environment worth coming back to every week. It was also a lot of dedication for these students: show up every week, do 30+ assignments, watch lectures, build a final project, and do a final test.

I’m leaving this experience with a full heart, a screenshot of a zoom grid, and the certainty that these students— whether they keep coding or not— will do incredible things. What started as a spontaneous application between my exam prep ended up becoming one of the highlights of my year. I’m incredibly thankful to my students (I know I’ve said this ad nauseam), but also to the entire Code In Place team who made all of this possible behind the scenes.

I walked in hoping to give back— what I didn’t expect was how much I’d take away.

I can’t wait to do it all over again next year.

Thank you, CIP 2025 <3

$\newcommand{\surs}[2]{\langle\, #1 \, \mid \, #2 \, \rangle}$ $\newcommand{\sur}[2]{\langle\, \{#1\} \, \mid \, \{#2\} \, \rangle}$ $\newcommand{\surr}[1]{\langle\, \{#1\} \, \mid \, \null \, \rangle}$ $\newcommand{\surl}[1]{\langle\, \null \, \mid \, \{#2\} \, \rangle}$ A few days ago, I was chatting with a friend about some math. We were both bored and had a severe itch to discover something new. They sent me a meme ¹ which challenged the reader to try proving $1+1=2$. That can’t be hard can it? We both knew of the infamous Whitehead & Russell proof ² which took 162 pages to prove this seemingly self-explanatory proposition, but surely there was an easier way to do it… right?

Well, I set out to find this way, if it even existed. And honestly, I’m grateful I did, because I discovered something so much more interesting. The Surreal Numbers.

Conway, Knuth, and Affairs in Hotels

I found out that John Conway, an English mathematician who also created the Conway Game of Life ³had come up with 2 simple rules which give birth to all the numbers we know of and more.

I decided to look into it and found out about Donald Knuth’s ⁴ book Surreal Numbers: How Two Ex-students Turned on to Pure Mathematics and Found Total Happiness : a Mathematical Novelette.

He claims to have wrote it on a Hotel getaway with his wife ⁵ after meeting Conway, who described to him the rules and nuances of The Surreals. Conway says he wrote the vital parts on a tissue and then reconstructed them, over the course of 7 days, in his hotel. He published his work as a novella afterwards.

The book is beautifully written as a dialogue between two characters who discover a tablet containing inscriptions about Conway’s rules for the Surreals. They then reconstruct the surreals– just as Knuth himself does– and discover their properties in a very natural, authentic way. I could not recommend it more! ⁶

This blog post will follow my recent exploration of this book, as well as the Surreals and any notes I have made on them. I will most likely create a second post on $\omega$, so look out for that!

Constructing the Surreals

The entire Surreal number system can be defined by two rules. We will be revisiting these rules extremely often, so get familiar with their every nuance.

For now, these rules may seem nebulous, but they will make more sense as we make use of them.

Day 0: but where are all the numbers?

As it stands, we have 2 rules and… well, nothing to apply them to. Nothing. nothing. Here’s an idea: let’s let the first number be composed of nothing!

As in, let our first number, call it $X_0$ for now, have:

And so,

To verify that $X_0$ is actually a number by the surreal rules, let’s check that $X_0$ does indeed satisfy rule 1:

Okay well.. we need to prove that for all $x_R \in X_R$, there does not exist any $x_L \in X_L$ such that $x_L>=x_R$..

But there’s a problem: there doesn’t even exist any $x_L$ and $x_R$ in $\null$ for us to compare. Hence, Rule 1 holds vacuously! $X_0$ is indeed a surreal number.

This is great! Because we can use the null set for comparisons, it makes our work so much easier. We can create so many forms now.

As our conclusion for Day 0, let’s call this progenitor number $0$. We will see that the surreal zero also behaves like the real zero soon.

Day 1: comparisons with nothing

Time to use combinations of $0$ and $\null$ to create new numbers! Let’s call ⁷

Since we have a total of 3 numbers ⁸ now, we can begin comparing them with the binary relation described in Rule 2!

Nice! We’ve ordered some of the surreals. But notice that this $X_1, X_2$ behave just like our usual $1,-1$ in the reals…

As in, in the reals, we have $0\leq 1$ and here too, $0\leq X_1$, and a similar reasoning follows for $X_2$.

One can also intuit that $X_1 \nleq X_2$, as well as $X_1 \nleq 0$ and $0\nleq X_2$, and it then intuitively holds that $0<X_1$ and $X_2<0$, but we still haven’t really defined what it means for two numbers to be “equal”. We’ll get to this later.

So, by Day 1, we conclude:

interlude; infinite birth and tosets

Before we begin comparing our newly generated numbers with each other and make new surreals on Day 2, it is useful for us to start reflecting on the nature of the rules.

First off, you may’ve noticed that when we compare two numbers $x\leq y$, we are quantifying every number in the first’s leftmost set. That is, we say $\forall x_L \in X_L, x_L \ngeq y$. But intuitively, if we pick $\max(X_L)=x_L$ instead, we can do just one comparison instead of as many are in $X_L$! Because we are guaranteed to be comparing the biggest element of this left set and ensuring it is not bigger than $y$, then no others in the left set are bigger than $y$.

A similar reasoning follows where we pick $\min(Y_R)$ to compare with $x$.

2/8/25: guess i need to complete this huh

int randInt = (int) ((b-a+1)*Math.Random()+a) 

Pretty neat, right?

So I went on and decided to generate a few random numbers to test out what I had:

int a = 1;
int b = 10;
int numSamples = 20;
ArrayList<Integer> l = new ArrayList<>();
for (int i=0; i<numSamples; i++){
    int rand = (int) ((b - a + 1) * Math.random() + a);
    l.add(rand);
}
for (int num: l){
    System.out.print(num+", ");
}

3, 2, 8, 4, 1, 5, 5, 3, 10, 1, 9, 1, 1, 2, 9, 1, 2, 3, 2, 7, 
Process finished with exit code 0

To my surprise, this ‘random’ number generator had generated $1$ 5 times, making $\frac 1 4$ of all numbers generated a $1$…

Perhaps it was just the scarce sample size, I thought, and I decided to better visualize my data:

int a = 1;
int b = 10;
int numSamples = 30;
ArrayList<Integer> l = new ArrayList<>();
HashMap<Integer, Integer> hashMap = new HashMap<>();

// initialize map 
for (int i = a; i <= b; i++) {
    hashMap.put(i, 0);
}

// insert random ints into list
for (int i = 0; i < numSamples; i++) {
    int rand = (int) ((b - a + 1) * Math.random() + a);
    l.add(rand);
}

// move list to map
for (int i : l) {
    hashMap.put(i, hashMap.get(i) + 1);
}

// print
System.out.println("Results:");
for (int key : hashMap.keySet()) {
    System.out.println(key + ": " + hashMap.get(key));
}

Results:
1: 1
2: 1
3: 6
4: 5
5: 3
6: 0
7: 4
8: 6
9: 1
10: 3

Process finished with exit code 0

$3,8$ 6 times?! That doesn’t seem random at all! Repeating with 100 samples:

Results:
1: 4
2: 15
3: 10
4: 15
5: 10
6: 11
7: 6
8: 13
9: 6
10: 10

Process finished with exit code 0

Hmm… Still very out of proportion..

I eventually tried a million samples and the results, although more uniform, still seemed unpredictable. Why was this? I went and asked ChatGPT for answers.

Introduction to LCGs

GPT postulated that the inaccuracy comes from my several floating arithmetic² operations perhaps resulting in errors— exacerbated by my casting of the double to an int.

However, it also mentioned that these results may just be coincidental, as Math.Random() uses a Linear Congruential Generator (LCG) to generate numbers, which is statistically reliable but not cryptgraphically so.

This concession piqued my interest. Why wasn’t something random seeming not reliable? And why was the distinction between statistical reliability and cryptographical reliablity important? So I set out to find more about these LCGs, coming across their definition.

The state in this definition is derived from some source of entropy, like the system’s time.

So it then takes the state, and using some absurdly large modulus and weird addends and multipliers, determines the next state. For example, for $X_0=23, a=13, b=5, m=48$, we have:

In fact, this sequence has been proven to only repeat after $m$ terms by the Hull-Dobell Theorem. That is, this will generate a sequence of random numbers for 48 times, after which it returns to $X_0$ and repeats. You can test this out yourself by implementing an lcg() method yourself and running it in a loop. ³

And, as it turns out the sequences generated by such an algorithm are indeed statistically random⁴. So what about cryptographic reliability? We know the method is deterministic, that is, we can find the next numbers of the sequence given some— but only if we also know the parameters $a, b, m$. But we don’t!

Well, it turns out that a little bit of number theory magic and wizardry can be performed to find $a, b, m$ by manipulating terms of the sequence.

Breaking LCGs

This section will be dedicated to reviewing a method on how to break LCGs, motivated (but heavily adjusted and standardized⁵) by James Reeds’ initial demonstration in 1977.

To break an LCG, it suffices to be able to predict the next term in a sequence of given consecutive terms by finding $a, b, m$.

Finding the modulus

We begin by getting any 3 numbers generated by a LCG, call them $X_n, X_{n+1}, X_{n+2}$. Then, by the definition of an LCG:

Recall that the definition of modular arithmetic is as follows:

So we may subtract $\text{(I)}$ from $\text{(II)}$ and $\text{(II)}$ from $\text{(III)}$ to obtain the following, eliminating $b$:

Then, multiply $\text{(IV)}$ by $(X_{n+1}-X_n)$ and $\text{(V)}$ by $(X_n-X_{n-1})$ and subtract the resulting equations to eliminate $a$:

So many $X$s! But now we have gotten rid of both $a, b$, albeit at the cost of our eyes having to see some of the most disgusting notation on Earth. So let’s try to clean this up;

You may notice that $(X_{n+2}-X_{n+1})$ in (VI) can be rewritten as $(X_{(n+1)+1}-X_{(n+1)})$…

Also, $(X_{n}-X_{n-1})=(X_{(n-1)+1}-X_{(n-1)})$

This is great because now we can set:

Okay, now take a deep breath. The hardest part is almost over.

Now, this indicates that $Q_n$ is a multiple of $m$. That is, $Q_n =k_n m$ for some $0<=k<Q_n$. Furthermore, we notice that $Q_n/m=k_n$ implies that $m$ is a divisor of $Q_n$.

Consider $L_n$, a set of all divisors of $Q_n$. Then:

Where $m$ must be in this list. Now, if $m$ is in the interior of the set, then it isn’t really mathematically striking. However, if $m$ is the largest element of this set, then it is called the Greatest Divisor of $Q_n$, which can be forcefully computed by several algorithms⁶.

These algorithms can be extended to find the Greatest Common Divisor of two (or more) numbers, denoted by $\gcd(a,b)$.

With this in mind, consider $\gcd(Q_{n},Q_{n+1})$. Observe⁷:

But we wish to seek $\gcd(Q_{n},Q_{n+1})=m$…

To those of you who are familiar with number theory, this just lit the brightest bulb in your head.

To those of you who aren’t, here’s why:

In number theory, it is a well-known fact that the probability that 2 integers are co-prime⁸ is $\frac{6}{\pi^2}$, $\pi$ sneaking its way into the probability through the Reimann-Zeta function, which can be used to express the probability that $n$ integers are co-prime with $\frac{1}{\zeta(n)}$⁹

Euler, in one of his many strokes of genius, proved that $\zeta(2)=\frac{\pi^2}{6}$¹⁰. Thus the probability of our two integers being co-prime is $\frac{1}{\zeta(2)}=\frac{6}{\pi^2}\approx 60\%$. And this probability increases exponentially for greater numbers of integers given, allowing us to find $m$ with fantastic certainty— if we simply generate $\gcd(Q_1, Q_2, Q_3, \ldots, Q_{n-3})$.

Luckily for us, any good computer is able to do this with high efficiency. Thus, $m$ is found.¹¹

class modulusFinder{
    private int[] outputs; // the lcg outputs, len>=3
    private int[] Sn; // Xn+1-Xn
    private int[] Qn; // Sn+1^2 -Sn+2*Sn
    private int mod;

    public modulusFinder(int[] outputs){
        this.outputs=outputs;
        Sn= new int[outputs.length-1];
        for (int i=0; i<outputs.length-1; i++){
            Sn[i]=outputs[i+1]-outputs[i];
        }
        Qn=new int[Sn.length];
        for (int i=0; i< Sn.length-2; i++){
            Qn[i]=Sn[i+1]*Sn[i+1]-Sn[i+2]*Sn[i];
        }
        mod=gcdArr(Qn); // call gcd of all Qn
    }
    /* finds the gcd of two numbers using a modified euclidean algo */
    public static int gcd(int x, int y){
        if (y == 0) return x;
        return gcd(y, x%y);
    }
    /* calls gcd on all elements of an array */
    public static int gcdArr(int[] arr) {
        int result = arr[0];
        for (int i : arr) {
            result = gcd(result, i);
            if (result == 1) return 1;
        }
        return result;
    }

    /* what we built up to. */
    public void getModulus(){
        System.out.println("The modulus m is: "+mod);
    }
}

Finding the multiplier

From here on out, everything becomes quite trivial. Recall $\text{(V)}$:

where $(X_{n+1}-X_n)^{-1}$ is the modular inverse of the argument¹² with respect to $m$. Thus, $a$ is found.

/* returns coefficients of bezouts identity & the gcd. */
public static int[] extendedgcd(int a, int b){
    if (b==0){
        int[] base = {a, 1, 0}; // the base case has gcd(a,0)=a(1)+0(0)
        return base;
    }
    int[] subdiv = extendedgcd(b, a%b); // break up the problem 
    int gcd =subdiv[0]; 
    int subdivX = subdiv[1];
    int subdivY = subdiv[2];
    int divX = subdivY;
    int divY= subdivX-(a/b)*subdivY;
    int[] div = {gcd, divX, divY};
    return div;
}
/* takes the inverse, with precondition of coprimality of parameters */
public static int modInv(int a, int m){
    int[] result = extendedgcd(a,m);
    int inv = result[1];
    return (inv%m +m) % m;
}

Finding the addend

Finally, recall $\text{(II)}$.

Plain and simple, we are done!

Final Code ¹³

The following code breaks any LCG, implementing the math we outlined in this blog, provided you have over 5 outputs of the LCG and there is no processing done to the outputs.

class lcgCracker{
    private int[] outputs; // the lcg outputs, len>=5, preferably more
    private int[] Sn; // Xn+1-Xn
    private int[] Qn; // Sn+1^2 -Sn+2*Sn
    private int mod;
    private int mult;
    private int add;
    private int next;

    public lcgCracker(int[] outputs){
        this.outputs=outputs;
        Sn= new int[outputs.length-1];
        for (int i=0; i<outputs.length-1; i++){
            Sn[i]=outputs[i+1]-outputs[i];
        }
        Qn=new int[Sn.length];
        for (int i=0; i< Sn.length-2; i++){
            Qn[i]=Sn[i+1]*Sn[i+1]-Sn[i+2]*Sn[i];
        }
        mod=gcdArr(Qn); // call gcd of all Qn
        // compute multiplier (a)
        mult = ((outputs[1] - outputs[2]) * modInv((outputs[0] - outputs[1]), mod)) % mod;
        if (mult < 0) mult += mod; // ensure positive

        // compute addend (b)
        add = (outputs[1] - mult * outputs[0]) % mod;
        if (add < 0) add += mod; // ensure positive

        // Predict the next term
        next = (mult * outputs[outputs.length - 1] + add) % mod;
        if (next < 0) next += mod; // ensure positive
    }

    /* returns coefficients of bezouts identity & the gcd. */
    public static int[] extendedgcd(int a, int b){
        if (b==0){
            int[] base = {a, 1, 0}; // the base case has gcd(a,0)=a(1)+0(0)
            return base;
        }
        int[] subdiv = extendedgcd(b, a%b); // break up the problem
        int gcd =subdiv[0];
        int subdivX = subdiv[1];
        int subdivY = subdiv[2];
        int divX = subdivY;
        int divY= subdivX-(a/b)*subdivY; // see footnote 13
        int[] div = {gcd, divX, divY};
        return div;
    }

    /* calls gcd on all elements of an array */
    public static int gcdArr(int[] arr) {
        int result = arr[0];
        for (int i : arr) {
            result = Math.abs(extendedgcd(result, i)[0]);
            if (result == 1) return 1;
        }
        return result;
    }

    /* takes the inverse, with precondition of coprimality of parameters */
    public static int modInv(int a, int m){
        a = (a % m + m) % m; // this code is because of java's modulo operator shenanigans.
        int[] result = extendedgcd(a,m);
        int inv = result[1];
        return (inv%m +m) % m;
    }
    public void runCracker(){
        System.out.println("The multiplier is: "+mult);
        System.out.println("The addend is: "+add);
        System.out.println("The modulus is: "+mod);
        System.out.println("And the next term in the lcg is: "+ next);
    }
}

Java’s LCG: Attempts at cryptographic reliability

I encourage you to try running this code with your own LCG implementation and test it out! See if our math checks out!

However, if you try to call this class on outputs generated from, say Java’s LCG, it won’t return the right parameters at all. Why is this?

After some digging, I found the code responsible for Java’s Math.Random() method:

/**
     * The internal state associated with this pseudorandom number generator.
     * (The specs for the methods in this class describe the ongoing
     * computation of this value.)
     */
    private final AtomicLong seed;

    private static final long multiplier = 0x5DEECE66DL; // 25214903917
    private static final long addend = 0xBL; // 11
    private static final long mask = (1L << 48) - 1; // 1L is 1, shift to left 48 units for 10000000.... then -1 to get 01111... (48 times)

    protected int next(int bits) {
        long oldseed, nextseed;
        AtomicLong seed = this.seed;
        do {
            oldseed = seed.get();
            nextseed = (oldseed * multiplier + addend) & mask;
        } while (!seed.compareAndSet(oldseed, nextseed));
        return (int)(nextseed >>> (48 - bits)); // shift nextseed by 48-bits right ; fill the left with 48-bits zeroes
    }

    @Override
    public long nextLong() {
        // it's okay that the bottom word remains signed.
        return ((long)(next(32)) << 32) + next(32); //gens a 64 bit number, upper half is next(32), lower half is next(32) (second call)
    }

    default double nextDouble() {
        return (nextLong() >>> (Double.SIZE - Double.PRECISION)) * 0x1.0p-53; //extract top 53 bits only, and multiply by 2exp-53 to map to a decimal in 0,1
    }
// the math class then calls this nextdouble method for .Random()

So much processing of next(bits)! The makers of Java’s LCG probably realized that LCGs… well, aren’t the most secure, and decided to add several layers of processing to make it a bit more secure;

• protected int next(int bits): this is the actual call to the LCG, and the code is self explanatory, except perhaps for the return statement. What the return statement does is it takes the new generated state, and actually shift it 48 minus the input to the right. This means whatever our binary representation for the state was, would be preceded by $48-\text{bits}$ $0$s.
• public long nextLong(): this method takes a LCG state with 16 leading 0s, shift it 32 bits to the left (remove those leading 0s), then add that to another, newly generated state with 32 leading zeroes. This creates a new number whose binary representation has an upper half consisting of the first call's binary representation and lower second call, generating a 64-bit number.
• default double nextDouble(): Finally, this is the method called by Math.Random(). It takes the output of nextLong(), an already processed state, and take only the top 53 bits. Then, it maps that to the interval $[0,1)$ by division as needed.

Despite all this processing, however, the output is still not random.

Frieze, Kannan, Lagarias outlined an algorithm in a 1984 IEEE presentation which mathematically proved that all LCGs are predictable in polynomial time, given that two fifths of the original sequence’s leading bits were provided. This built on prior solutions to breaking LCGs with all the bits of the outputs, and demonstrated that—contrary to prior belief— manipulation of LCG outputs does NOT make them secure.

Frieze, Kannan, Lagarias, et al. later went on to strengthen their result to any linear congruences manipulating the outputs in 1988.

Conclusion

Well wasn’t that one hell of a trip? We’ve just built a Java class to break LCGs!

What I hope to convey through this post is the immense power that a solid grasp of mathematics— in this case, number theory— can have in Computer Science.

If you’ve taken a look at the timeline on my home page, you’ll know that my passion for math predates my interest in CS. But once I saw how seamlessly math and CS complement¹⁴ each other, I found myself yearning to dive deeper into the intersection¹⁵ between these two fields. It’s a space that continues to fuel my curiosity, and I’m excited to explore it further.

All images of Sepideh Zarrat's artwork are her intellectual property. All rights are reserved, and no one is permitted to use, reproduce, or distribute these images for any purpose without her explicit permission. This includes, but is not limited to, their use in NFT productions or any other commercial or non-commercial projects. Any unauthorized copying or misuse is illegal and will result in legal repercussions.

تمامی تصاویر آثار سپیده ذرات متعلق به او بوده و تمامی حقوق محفوظ است.
هیچ‌کس بدون اجازه‌ی صریح او حق استفاده، تکثیر، یا انتشار این تصاویر را برای هیچ منظوری ندارد.
این شامل، اما محدود به، استفاده در تولید NFT یا هر پروژه‌ی تجاری و غیرتجاری دیگر نیز می‌شود.
هرگونه کپی‌برداری یا سوءاستفاده غیرمجاز غیرقانونی بوده و با پیگرد قانونی مواجه خواهد شد.

featured artist

هنرمند برجسته

@sepideh.zarrat

Your work has a very organic, almost untamed quality. Do you start with a structured vision, or do the forms emerge spontaneously as you create? کارهاتون یه کیفیت ارگانیک و رها دارن. آیا با یه ایده مشخص شروع می‌کنین، یا فرم‌ها حین خلق اثر به‌صورت خودجوش شکل می‌گیرن؟

Sepideh Zarrat:

Many of your pieces have an ethereal, almost surreal quality. Are there specific influences—artistic, philosophical, or literary—that shape your aesthetic? خیلی از آثارتون یه حس اثیری و سوررئال دارن. چه چیزهایی روی سبک و زیبایی‌شناسی شما تأثیر گذاشتن؟ (هنر، فلسفه، ادبیات…)

Sepideh Zarrat:

Your use of line and movement is incredibly dynamic. How do you balance fluidity and precision in your technique? استفاده شما از خط و حرکت خیلی پویاست. چطور بین رهایی و دقت تعادل برقرار می‌کنین؟

Sepideh Zarrat:

How does your cultural background influence your art? Do Persian poetry and symbolism often inspire your work? پیشینه فرهنگی‌تون چقدر روی هنرتون تأثیر گذاشته؟ آیا شعر و نمادگرایی ایرانی نقش بزرگی توی کارهاتون دارن؟

Sepideh Zarrat:

What message or experience do you hope viewers take away from your paintings? دوست دارین بیننده‌ها از آثار شما چه تجربه‌ای بگیرن یا چه حسی رو با خودشون ببرن؟

Sepideh Zarrat:

What inspired you to depict the saffron crocus in such an intricate and abstract way? چی شد که گل زعفران رو به این شکل به تصویر کشیدین؟

Sepideh Zarrat:

In one of your interviews, you mentioned that the fact that saffron farmers sing to the saffron plant before it blooms impacted your collection. How so? تو یکی از مصاحبه‌هاتون گفتین که قبل از اینکه زعفرون سر از خاک دربیاره براش ساز می‌زنن. این موضوع چه تأثیری روی مجموعه‌تون گذاشت؟

Sepideh Zarrat:

Saffron has deep cultural and historical significance in many places—does your painting connect to any personal or symbolic meaning? زعفران توی فرهنگ‌ها و تاریخ‌های مختلف جایگاه خاصی داره—آیا این نقاشی برای شما یه معنای شخصی یا نمادین داره؟

Sepideh Zarrat:

The delicate balance between chaos and structure in your work is fascinating. How do you decide how much control to impose on the lines? توی کارتون یه تعادل خاص بین آشوب و نظم وجود داره که خیلی جذابه. چطور تصمیم می‌گیرین که چقدر کنترل روی خطوط داشته باشین؟

Sepideh Zarrat:

How did this particular Mowlana poem influence your artistic choices? این شعر خاص از مولانا چطور روی انتخاب‌های هنری شما تأثیر گذاشت؟

Sepideh Zarrat:

Your use of color and flowing, tangled forms evokes both motion and emotion. How do you translate poetry into visual language? ترکیب رنگ‌ها و فرم‌های درهم‌تنیده توی کارهاتون حس حرکت و احساس رو منتقل می‌کنه. چطور شعر رو به زبان بصری ترجمه می‌کنین؟

Sepideh Zarrat:

The swirling dark lines forming the faces create an almost dreamlike or fragmented identity. Is there a particular story behind this piece? خطوط درهم و تیره‌ای که چهره‌ها رو شکل می‌دن، یه حس رؤیایی یا هویت تکه‌تکه‌شده ایجاد می‌کنن. آیا داستان خاصی پشت این اثر هست؟

Sepideh Zarrat:

The contrast between the bold yellow background and the dark forms is striking. What role does color play in shaping the mood of your work? تضاد بین پس‌زمینه زرد و فرم‌های تیره خیلی چشمگیره. رنگ چه نقشی توی شکل دادن حال و هوای این کار داره؟

Sepideh Zarrat:

In this post, I intend to demonstrate how the binary system is the smallest such system which can be used to represent every single integer uniquely in one way, providing motivation for why binary is often called the “language of computers” (other than beep boop).

The word binary comes from the latin word bini, which means “two together”. It hence makes sense that the binary number system has only two numerals, represented by $0$ and $1$. For comparison, the number system we are all used to—the base-10 system— has 10 different numerals which we can count on our fingers ¹.

This means numbers in binary only contain these two numerals, for example $110101$ is a binary number. But how does one count with this system?

Counting in Binary

Counting using the binary number system is quite simple. Like our base-10 system, the leftmost symbol has the highest value in binary, and as you proceed rightward, the values become smaller. In base-10, however, you only ‘increment the right’ after the left symbol has surpassed 9, resetting that symbol. This is contrasted to binary, where you only have 0s and 1s and ‘increment the right’ immediately if the left symbol is a 1, resetting it back to a zero.

For example, in base-10, when counting, we proceed as:

However, in base-2:

But what does each symbol, called a bit, represent?

You may correctly guess that it is something related to the digit 2— after all our entire counting system is based on 2 symbols. Its helpful to think of a binary number as a board of switches— either on ($1$) or off ($0$).

So, the example we gave in the introduction is:

and in this system, what is on or off is a power of 2! That is;

So the value of $110101$ in base-10 is:

where we only sum the powers which were $\text{on}$.

Now we are ready to give a formal definition to any binary number $k$:

But are we sure this is unique?

That is, how can we be sure that every $k$ has exactly one and only one binary representation which no other $k$ has? We seek to prove this in the following section.

Proof of Uniqueness

We proceed by induction.

Our base case is $1$, which can be expressed as $2^0$ in base-10 and also $1$ in binary.

This is a unique representation as $1$ is the smallest natural, positive number. Hence all other numbers are greater than 1, and hence have more bits ‘to the left’.

Now assume that the conjecture holds for all naturals $1,2,3,\ldots,b$. The number $b+1$ is our focus. One idea to show that the conjecture holds for $b+1$ is to argue that since it is either even or odd, it can be expressed as $2m$ or $2m+1$, respectively, where $0<m<b<b+1$, and we would be done, as $m$ has a unique binary representation ${b_i b_{i-1} b_{i-2}\ldots b_2 b_1 b_0}$ and hence both:

as needed.

However, this argument does not exactly address the uniqueness clause. The following argument resolves this.

Once again focusing on $b+1$, consider the number $2^k$, which is the largest power of two such that $2^k\leq b+1$.

If $2^k=b+1$, our work is done, as $2^k$ trivially has a binary representation which is unique as all powers of two less than $2^k$ sum up to exactly $2^{k}-1=b$. This fact is demonstrated in Proof of Binary Powers Sum, or by induction ².

Otherwise, $2^k<b+1$, and $b+1-2^k<2^k$ must be true, as the converse implies $b+1>2^{k+1}$, and we simply pick $k=k+1$.

Now, $b+1-2^k<2^k<b+1$ indicates that $b+1-2^k$ can be expressed using binary according to our inductive hypothesis that all numbers $<b+1$ have a binary representation. And $b+1-2^k+2^k=b+1$ yields back $b+1$!

This means that we can simply add a $1$ to the binary representation of $b+1-2^k$ in the $k^{\text{th}}$ index and obtain $b+1$, as needed. And this representation is also unique due to the fact we showed that it is less than $2^k$ coupled with $\sum_{i=0}^{k}2^i =2^{k+1}-1$.

Proof of Binary Powers Sum

In my opinion, this proof demonstrates why binary is such a powerful tool. Let us proceed.

Begin by calling the LHS of this lemma $P(k)$. Notice that by the definition of a binary number, $P(k)$ has a binary representation that is exactly:

as all terms are summed or ‘on’, this is a string of 1s which is of length $k+1$.

Then, it is clear that $P(k)+1$ is simply:

due to the rules of binary addition.

But notice how this can also be written as a binary number with a single 1 in the $k+1$ index. Which is exactly the definition of $2^{k+1}$!

Hence, taking 1 away from $P(k)+1=2^{k+1}$ resets the $k+1$ index and leaves us with $P(k)$, as needed.

Significance of Binary

Now that we’ve established why binary is such a powerful number system, we begin to understand its natural fit in digital circuitry: a complete numerical system built on two simple states—on or off. But how these states came to be mathematically expressed owes much to George Boole’s groundbreaking work on Boolean Algebra.

George Boole described a new subject in algebra, using 2 main values— either true (1) or false (0)— to describe logical, and not numerical operations. Instead of the familiar addition, subtraction, multiplication, and division operations, Boolean Algebra is concerned with conjunction, disjunction, and negation of variables which can be true or false. A closer look of this algebra follows.

On Boolean Algebra

We begin by defining the two values in this algebra: $1$ represents true, $0$ represents false. Now, we can define the 3 operations using $\wedge, \vee,\neg$ for conjunction, disjunction, and negation, respectively:

Thus a new subject is axiomatized³ and born, symbolizing logic (literally and metaphorically!)

However, as you may guess, this algebra was quite abstract and separate from the practical engineering world— beyond its use in philosophy and propositional calculus. This is, until Claude Shannon published his thesis “A Symbolic Analysis of Relay and Switching Circuits,” which recognized that Boolean Algebra could be used in electrical relays to perform both logical and numerical operations!

Claude Shannon’s eureka moment

Relays were electrical components which acted as switches— either preventing or allowing current flow. In his thesis, he mathematically proved that a physical system using such relays inherited all the power of a binary numerical and logical system— and could hence perform all of the same operations. Where they were previously used only for telegraphic communications, they could now be used to design precise circuitry.

Shannon noticed that the current flowing through relays could be modeled and standardized with Boolean algebra.

For example, he proposed wiring relays in a series arrangement in order to describe logical conjunction— if both relays were on or “high”, then current would be allowed to flow, and the output would also be “high”. However, if one was off, then current outside of the construction ceases to flow, perfectly matching the behavior of the boolean AND!

Similarly, disjunction would be formed by wiring the relays in parallel, yielding the exact same outcome as $\vee$ would!

Where $A.\hspace{-0.2em} B=A\wedge B$ and $A+B=A\vee B$.

This gave birth to logic gates, fundamentally transforming the digital circuit industry. No longer was circuit design a matter of trial and error; it became a precise mathematical endeavor. Engineers could now create intricate electronics with predictable, reliable outcomes.

As the 20th century marched on, 8 bits became a byte, and relays gave way to vacuum tubes, then transistors, then integrated circuits. Yet, the logic underlying all this hardware remains the same: at the heart of modern computation are the simplest of ideas—$0$s and $1$s, off and on, false and true. From this division, we’ve built machines that not only calculate but connect us— even this text you read is a bunch of $0$s and $1$s. And that, in itself, is a remarkable testament to the power of binary.

Logarithmic spirals are a class of mathematical shapes described by a simple mathematical equation. Despite the elegance & simplicity of how they are generated, they have some extremely interesting properties and characteristics which warrants a small case study on them. This blog post will look at some of those characteristics which most piqued my interest and present an attempt to prove their.

Namely, we will look at how logarithmic spirals are congruent under rotation and scaling, and hence self-similar, and the fact that their pitch angles are constant.

We begin by defining how a logarithmic¹ spiral is constructed:

Analyzing this equation qualitatively, we see that this means that as $\theta\to\infty$, the radius or distance from the origin, $r\to\infty$ too. But it does get hard to imagine exactly what this curve looks like, so let’s try plotting this on Desmos. Click on me!

Now that we can visualize what this curve looks like, we notice a few different things. First off, if you tried manipulating $k$, you would’ve noticed that increasing its magnitude increases the amount by which the spiral “opens” up. When $k$ is near 0, the curve collapses near the origin and looks more and more like a circle.

This should motivate you to notice that $k$ doesn’t just determine how fast the spiral grows, but also how much it opens up with each turn.

And as it turns out, there is indeed a beautiful link between the constant $k$ and the “openness” of the curve— a concept mathematicians call the pitch angle of the spiral.

Pitch Angles (not B...)

A pitch angle, $\alpha$, of a spiral is the angle bounded between the intersection of some circle of radius $r$, centered at the spiral's circle, and the the spiral at some point $P(x_p, y_p)$

More rigorously, we can state that the pitch angle is the angle between the tangent to the circle at point $P$ and the tangent to the spiral at that same point.

Back at our Desmos graph, open up the first folder and follow the instructons. Now, manipulate $\theta_0$ and observe the two tangents drawn to the spiral and circle. Does the angle, $\alpha$, change as you move along the spiral? Does it change for changing $k$?²

As it turns out, the relationship between $k$ and $\alpha$ is quite elegant:

This result is quite significant because it intuitively tells us that the “openness” of the curve depends only on the choice of $k$, and nothing else. And this $\alpha$ does not change for any spiral— it’s constant!

Property One: Constant Pitch Angles

Proof. We begin by observing that the pitch angle may also be redefined in terms of tangent lines to each of the spiral and circle — that is, the angle between their tangents at some point in the polar plane $(r, \theta)$. Denote each of these lines $L_s$ and $L_c$ for the spiral and circle respectively. Notice these lines can be written as (in cartesian coordinates): ³

For slopes $m$ on points $(x_s , y_s), (x_c, y_c)$ on the spiral and circle. All that is essential is finding the slopes. We know, however, that:

Since they are tangent to the same point, the spiral and circle’s intersection. And also by the product rule of differentiation. ⁴

Where the derivatives are representations of how fast the $x$ and $y$ coordinates change with respect to changes in $\theta$, respectively.

And from the chain rule ⁵ , it follows that:

For any and all polar equations and curves.

For the spiral, we have $r = e^{k\theta}$, so $r’ = ke^{k\theta}$: ⁶

For the circle, $r = r_c$, where $r_c$ is a constant radius of our liking. Hence, the derivative $r’ = 0$:

Now, recall we defined the pitch angle $\alpha$ as the angle between the tangents to the spiral and circle. Luckily, this angle depends only on the slopes of those tangents, which we have found in $(1)$ and $(2)$. The formula is as given:

At this point, we take an important step which I think younger me described well:

Now substitute back \(a\) and \(b\):

Using \(\cot(x)\sin(x) = \cos(x)\):

So finally, $\alpha = \arctan(k)$, which was to be demonstrated!

And yet again, our result makes sense, remember our Desmos graph showed that the angle between the spiral and circle tangents remained constant everywhere.

Property Two: Rotations are the same as Scaling

Open up the second folder (pt. 2), and follow the instructions listed. Slide $d$; what happens to the curve? Notice that it is being rotated, but what is intriguing is that this also looks exactly the same as zooming in or out on the curve. Try it out!

Another important property of these spirals is that rotating a spiral has the same effect as scaling (zooming in or out) it. Namely:

Proof. Consider $r(\theta-d)$.

Which was to be demonstrated, in our case $a=e^{-kd}$

Which also proves something called self-similarity.

Property Three: Self-similarity

Self-similarity occurs when a scaled shape is congruent to its original shape by rotation. That is, zooming in or out on it is the same as rotating the curve by some angle.

Luckily, we have just proven exactly that, so all logarithmic sprials are, by extension, self-similar!

For example, consider a logarithmic spiral $r(\theta)=e^{\theta/2}$ rotated $90^\circ=\frac \pi 2$. We can show, because of the previous property:

Hence, rotating it 90 degrees had the same effect as zooming out by 0.456 units.

And you can do this in the opposite direction— any scaling can be converted into a rotation.

Golden Spirals and $\varphi$: Special Logarithmic Spirals!

You’ve probably seen those viral posts on social media—gushing over the “divine” beauty of nature and the supposed presence of the golden ratio everywhere. Cue a zoomed-in image of a sunflower or a nautilus shell, complete with an overlaid golden spiral and a caption like “See? Math is the language of the universe!” I find these posts pretty pretentious: they do nothing to attempt explaining the math, and attribute any spiral looking shape to the golden spiral, which is just wrong.

The golden spiral, depicted on the left, is indeed a type of logarithmic spiral with a unique $k$, however often times in nature, spirals are not golden! That is, their $k$ is not the same as the “golden” $k$.

These posts have glamorized the golden ratio without giving any credit to the way these shapes are even generated— through logarithmic spirals. The rest of this blog will be dedicated to finding a polar form of the golden spiral— in the logarithmic spiral form we defined previously.

Mathematically, we can denote this as The Golden Spiral being $r(\theta)=e^{k\theta}$ such that:

And we need to find $k$:

This is great! Now we know $k\approx 0.306$, which means that the golden spiral has: ⁷

And plugging this back into $r(\theta)$:

And that brings it to an end! In the very last folder (pt. 3) on Desmos, you should see the Golden Spiral’s equation. Go ahead and plot it!

Hopefully this blog post has inspired a newfound love in you for logarithmic spirals— the true actors behind all of those “Nature is Magnificent!” posts you see on social media. And more importantly, I hope you learnt something new.