Irrational Numbers and the Geometry of Being…
What do you get when you replace x with c in the trigonometric identity 1 + tan²x = sec²x? Apparently, me or at least I’m sexy. That’s math, baby. But under the cheeky sheen of puns and parentheses lies something deep and slightly unruly, just like Euler’s number, just like you, and just like that slice of pizza you definitely didn’t measure with a compass.
Let’s start with Euler’s number – e, not to be confused with E! News, though both are full of surprises. Euler’s number is irrational, not in the “gets angry and throws a calculator” way, but in the mathematical sense: it cannot be written as a neat little fraction. Its digits go on forever without repeating, a kind of infinite jazz solo in the key of numerical mystery. The same goes for π (pi). People love rounding it off to 3.14 or slicing it down to 22/7, but pi laughs in the face of fractions. It’s irrational, infinite, and essential. Like your friend who talks about their dreams for 40 minutes and somehow makes a point. You don’t get geometry without it. Circles refuse to spin properly without it, that and conservation of angular momentum with a gyroscopic effect. I know that sounds like a position from the Kama Sutra but just bear with me…
Irrational numbers are the rebels of the number line. They don’t fit in boxes or grids or pie charts that make accountants sleep well at night. They’re the wild hair of mathematics: uncountable, unpredictable, infinite. You can squeeze rational numbers into a spreadsheet, but you cannot spreadsheet your way through √2. It just keeps going, kind of like a bad Tinder date story, except you want to know how this one ends. Spoiler: it doesn’t.
Now, imagine this irrationality applied to food. How do you divide a pizza into two irrational parts? First, accept that your guests may leave hungry and confused. One option: cut a slice that covers π radians—about 180 degrees. That’s irrational enough to impress a mathematician and confuse a chef. Or measure out a piece with an area of √2 square inches. Not practical, but very on brand if your dinner parties double as metaphysical debates.
You could even mark an arc on the crust equal to π/4 of the circumference and slice inward. The rest of the pizza will glare at you, uneven and mathematically flustered, as if it knows it’s been part of an intellectual exercise instead of lunch. But congratulations, you’ve made your pizza irrational. Next step: convince your friends it’s a commentary on modern existence (preferably before they call for takeout).
Which brings us to monohedral disc tilings. Fancy name, simple idea: dividing a disc (like our philosophical pizza) into identical shapes, usually wedges. You’ve seen this in action every time someone cuts a pizza into symmetrical slices and calls it dinner. But mathematicians, never content with pepperoni and predictability, began asking: what if the slices weren’t rationally divided? What if the angles or arcs were irrational?
Monohedral tiling with irrational divisions? That’s where geometry starts writing poetry. And here’s where things get philosophical. The Pythagoreans once believed that the world could be explained entirely with whole numbers and neat ratios. Then someone discovered √2, and their entire worldview cracked like an overbaked crust. Irrational numbers, like rebellious teenagers, refused to conform, and suddenly the clean universe was full of chaotic decimals.
This isn’t just math. It’s the geometry of people, of relationships, of business plans that look perfect until someone brings feelings into the meeting, of lovers who think they’re symmetrical until one of them starts dreaming in √2 and the other is still stuck on 1.618. We like things neat: “he loves me,” “the contract is signed,” “that strategy is bulletproof.” But real life tilts. It wobbles. It makes weird angles in your emotional pizza.
And let’s talk about entropy, because nothing introduces chaos quite like dating. Swipe-based courtship is the modern equivalent of throwing irrational numbers into a love equation and hoping the algorithm can make a soulmate smoothie out of it. First impressions? Oh, they matter, so much that our brains make up their minds before we’ve even finished saying “Hi.” It’s √2 all over again: things start off normal enough, and then the decimals get weird. Suddenly, you’re wondering why someone’s love language is sending you 14 TikToks before noon.
Marriage? That’s where people try to enforce symmetry. They plan honeymoons with spreadsheets, assign chores with pie charts, and hope joint bank accounts will balance emotional overdrafts. But love, like π, doesn’t loop neatly. It spirals, beautifully, occasionally with socks left on the floor and inexplicable arguments about which direction the toilet paper should roll.
Even sexual preferences are irrational in their own mathemagical way. There’s no formula for desire. What turns one person on might confuse another entirely – why feet, why latex, why library whispering? You might as well ask why some people think cilantro tastes like soap. The irrational doesn’t ask for permission; it just shows up, kicks over your logic table, and asks if you’re into roleplay.
We see these same entropic curves in business and leisure. You plan a meeting, but someone brings emotions. You plan a vacation, but chaos stows away in your carry-on. Whether it’s love or logistics, people are built more like fractals than formulas.
And yet, despite the irrationality, or maybe because of it, we keep showing up. We keep tiling our lives with little attempts at symmetry, fully aware the universe might draw its next line at an irrational angle.
All of this touches a deeper nerve: why does math work so well in the real world? Why do abstract concepts like π or e or √2 describe physical reality so precisely? Do these numbers exist independently, like invisible celestial bodies waiting to be discovered? Or are they just mental scaffolding we’ve erected to keep the universe from collapsing into complete absurdity?
Mathematical Platonists would argue that these numbers live in a timeless realm of ideal forms. Others believe we invented them because they’re useful, like post-it notes and excuses. Still others argue that math is baked into the very structure of human cognition—nature’s way of making sure we could build bridges without falling off them.
Either way, the connection between irrational numbers and philosophical tiling patterns reminds us that the world, like math and like people, isn’t as simple as we often pretend. There’s a kind of messy, elegant truth in how the infinite curls into the finite, how the unsolvable still shows up to dinner.
We are all, perhaps, monohedral tilings, trying to find symmetry in ourselves while being shaped by irrational angles. Not every part of us fits neatly. Our arcs are off. Our equations don’t always balance. But maybe that’s the point.
Author’s Note: Final Thought (Before We Fall Off the Edge)
The universe is weird. Numbers are weird. Dating is weirder. Pizza is still great. If this essay confused you, you’re not alone… I had to google half of it twice and consult my inner teenager and my 8 year old for the rest. But maybe there’s something poetic in knowing that our deepest truths, whether mathematical or emotional, aren’t easily divided. Whether you’re an irrational number, a tilted tile, a person wondering why your crush ghosted you after three perfect dates, or just someone with too many tabs open, you belong. Just maybe don’t let me cut the pizza.
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