Mathematics for people who want to understand it, not just solve it.
Joined July 2025
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Primes look random until you start watching them closely.
The sieve doesn’t just remove composites. It leaves a residue pattern that keeps repeating at every scale, even as the numbers get larger.
By the time you reach a few hundred, the gaps between primes are already bigger than the early ones, yet the density still follows the same slow decline predicted by the prime number theorem.
π(x) grows roughly like x / ln x. The formula doesn’t tell you where the next prime is, but it does tell you that the gaps cannot stay small forever.
The same process that produces twin primes also produces long stretches with nothing. Both behaviors come from the same underlying distribution.
Most of the interesting structure sits in the tension between these two facts: primes become rarer, but they never quite disappear.
Everything else > the spiral, the residues, the occasional large gaps > is just a visible trace of that tension.
Mathverse retweeted
A drunk man always finds his way home. A drunk bird only does about 34% of the time.
That's not a joke, it's a theorem. Pólya proved it in 1921.
Let someone wander randomly, one step at a time, to a random neighbour. On a line, they come back to where they started with probability 100%. On a flat grid, also 100%.
Add one more dimension and it breaks. In 3D the chance of ever coming home is 34.05%.
We ran 360 walkers in each world for 16,000 steps. On the line, 99.4% were already back. On the plane, 75.3%, still climbing slowly. In space, 36.7%, and it barely moves.
Why? In 3D the odds of standing at home after 2n steps shrink like 1/n^1.5. Add them all up and the total stays finite: 1.5164 visits on average, counting the start. So the chance of a second visit is 1 − 1/1.5164.
One extra direction to get lost in, and two thirds of the walkers never come back.
Everything else > the spinning cube, the pink dots, the sound of each return > is just that sum refusing to reach infinity.
A drunk man always finds his way home. A drunk bird only does about 34% of the time.
That's not a joke, it's a theorem. Pólya proved it in 1921.
Let someone wander randomly, one step at a time, to a random neighbour. On a line, they come back to where they started with probability 100%. On a flat grid, also 100%.
Add one more dimension and it breaks. In 3D the chance of ever coming home is 34.05%.
We ran 360 walkers in each world for 16,000 steps. On the line, 99.4% were already back. On the plane, 75.3%, still climbing slowly. In space, 36.7%, and it barely moves.
Why? In 3D the odds of standing at home after 2n steps shrink like 1/n^1.5. Add them all up and the total stays finite: 1.5164 visits on average, counting the start. So the chance of a second visit is 1 − 1/1.5164.
One extra direction to get lost in, and two thirds of the walkers never come back.
Everything else > the spinning cube, the pink dots, the sound of each return > is just that sum refusing to reach infinity.
Mathverse retweeted
Your brain can juggle about four things at once. A 7th order expansion in 10 variables has 11,440 terms.
That gap is where most of mathematics was born.
Expand anything with 10 variables. Third order: 220 cross terms. Fifth: 2002. Seventh: 11,440. Somewhere around the third, a human gives up on writing it out.
So we invented a different game. Look for symmetry. Find an invariant. Draw a picture. Prove it without ever opening the expression.
Elegance, in a way, is compression for a mind that can't hold the full sum.
AI doesn't need the compression. It can open all 11,440 terms, tune a few coefficients and watch every divergent piece cancel to zero.
The uncomfortable joke behind the recent wave of AI proofs: the answer might be right, and still come with no idea a human can hold.
Everything else > symmetry, invariants, geometric intuition > may turn out to be our workaround, not the universe's.
Your brain can juggle about four things at once. A 7th order expansion in 10 variables has 11,440 terms.
That gap is where most of mathematics was born.
Expand anything with 10 variables. Third order: 220 cross terms. Fifth: 2002. Seventh: 11,440. Somewhere around the third, a human gives up on writing it out.
So we invented a different game. Look for symmetry. Find an invariant. Draw a picture. Prove it without ever opening the expression.
Elegance, in a way, is compression for a mind that can't hold the full sum.
AI doesn't need the compression. It can open all 11,440 terms, tune a few coefficients and watch every divergent piece cancel to zero.
The uncomfortable joke behind the recent wave of AI proofs: the answer might be right, and still come with no idea a human can hold.
Everything else > symmetry, invariants, geometric intuition > may turn out to be our workaround, not the universe's.
Rewire 146 friendships out of 5000 and the average distance between strangers drops from 50 handshakes to 6.
That's the small world effect, and you can watch it happen.
Put 1000 people on a ring. Each one knows only their 10 nearest neighbours. To reach someone on the far side you need 50 handshakes on average, and the farthest person is 100 away.
Now take each friendship and, with probability p, redirect it to a random stranger. Just 2.9% of links.
Average distance: 50.45 → 6.06. Farthest person: 100 → 9.
And the neighbourhoods barely notice. Clustering, the chance that two of your friends know each other, goes from 0.667 to 0.612. Your circle still looks the same.
One random shortcut doesn't just help two people. It shortens the route for everyone who can reach either end.
Everything else > the pink arcs, the collapsing rings, the falling spikes on the rim > is just a few long links doing the work of thousands of short ones.
Mathverse retweeted
One line of school algebra, and after about 53 steps your laptop can no longer tell you the answer.
The rule is x → r·x·(1 − x). Pick a number between 0 and 1, plug it in, repeat. That's the whole thing.
Turn the dial r slowly and watch. Below 3 the numbers settle on one value. At 3 they start jumping between two. At 3.449 between four, at 3.544 between eight.
Each new split arrives about 4.669 times sooner than the last one. By r ≈ 3.569946 the splits pile up into chaos.
From there, 195 of the 216 values we tested never repeat. At r = 4 any tiny error doubles every step, so 53 bits of precision last about 53 steps.
And yet at r = 1 + √8 ≈ 3.828, a clean cycle of three walks straight out of the noise.
Everything else > the splitting tree, the tangled web, the ticking sound > is just that one line, run over and over.
One line of school algebra, and after about 53 steps your laptop can no longer tell you the answer.
The rule is x → r·x·(1 − x). Pick a number between 0 and 1, plug it in, repeat. That's the whole thing.
Turn the dial r slowly and watch. Below 3 the numbers settle on one value. At 3 they start jumping between two. At 3.449 between four, at 3.544 between eight.
Each new split arrives about 4.669 times sooner than the last one. By r ≈ 3.569946 the splits pile up into chaos.
From there, 195 of the 216 values we tested never repeat. At r = 4 any tiny error doubles every step, so 53 bits of precision last about 53 steps.
And yet at r = 1 + √8 ≈ 3.828, a clean cycle of three walks straight out of the noise.
Everything else > the splitting tree, the tangled web, the ticking sound > is just that one line, run over and over.
Mathverse retweeted
To get one more digit of π by throwing needles, you need 100 times more needles.
Buffon's needle isn't a party trick. It's a real way to measure π with nothing but lined paper and some sticks.
Drop a needle as long as the gap between the lines. It crosses a line with probability 2/π ≈ 0.6366. Count the crossings and π falls out: π ≈ 2 × drops / crossings.
We dropped 3000. 1880 crossed. That gives 3.19149, right in the first decimal, wrong in the second.
24000 darts in a quarter circle gave 3.14100. 3000 terms of 4·(1 − 1/3 + 1/5 − …) gave 3.14126, and its error is almost exactly 1/3000.
Random methods improve like 1/√n. The series improves like 1/n.
Randomness always finds π. It just never finds it quickly.
Everything else > the pink needles, the dart cloud, the zigzag > is just that square root slowing you down.
To get one more digit of π by throwing needles, you need 100 times more needles.
Buffon's needle isn't a party trick. It's a real way to measure π with nothing but lined paper and some sticks.
Drop a needle as long as the gap between the lines. It crosses a line with probability 2/π ≈ 0.6366. Count the crossings and π falls out: π ≈ 2 × drops / crossings.
We dropped 3000. 1880 crossed. That gives 3.19149, right in the first decimal, wrong in the second.
24000 darts in a quarter circle gave 3.14100. 3000 terms of 4·(1 − 1/3 + 1/5 − …) gave 3.14126, and its error is almost exactly 1/3000.
Random methods improve like 1/√n. The series improves like 1/n.
Randomness always finds π. It just never finds it quickly.
Everything else > the pink needles, the dart cloud, the zigzag > is just that square root slowing you down.
Mathverse retweeted
Primes look random until you start watching them closely.
The sieve doesn’t just remove composites. It leaves a residue pattern that keeps repeating at every scale, even as the numbers get larger.
By the time you reach a few hundred, the gaps between primes are already bigger than the early ones, yet the density still follows the same slow decline predicted by the prime number theorem.
π(x) grows roughly like x / ln x. The formula doesn’t tell you where the next prime is, but it does tell you that the gaps cannot stay small forever.
The same process that produces twin primes also produces long stretches with nothing. Both behaviors come from the same underlying distribution.
Most of the interesting structure sits in the tension between these two facts: primes become rarer, but they never quite disappear.
Everything else > the spiral, the residues, the occasional large gaps > is just a visible trace of that tension.
9 Fibonacci fractions, 833.09 cents: why does this sound draw a curve that never closes ?
• Two sine waves with frequencies an and b draw a Lissajous figure, and it closes only when a/b is a fraction. For 3/2 that is a perfect fifth, 701.96 cents, and 10 frame touches per loop. The Fibonacci fractions 5/3, 8/5, 13/8 land above and below φ in turn, and 89/55 misses it by just 0.00015.
• Precision costs complexity: 89/55 is already 288 frame touches per loop, and exact φ turns a finite figure into a curve that never repeats. It sounds like an interval of 833.09 cents, which matches neither the minor nor the major sixth. A win for the eye, an annoyance for tuning an instrument.
• Forecast: the worse a number is approximated by fractions, the longer the illusion that the curve is almost closed. The golden ratio is the record holder, because its continued fraction is all ones, so its Fibonacci approximations converge the slowest of any number.
96 of 359 multipliers: only these never lose a single point on the circle. What happens to the other 263 ?
∴ Every chord runs from point i to point k × i on a 360 point circle. Multiplier k draws a curve with k − 1 cusps: k = 2 is a cardioid, k = 3 a nephroid, k = 5 has four cusps. Past k = 12 the cusps vanish into a lattice of overlapping chords.
∴ When k shares a factor with 360, several points land on the same one and information is lost. Only φ(360) = 96 multipliers are safe. The same gcd = 1 rule decides which RSA exponents can be undone.
∴ I think this circle arithmetic stays even after RSA goes. Post quantum lattice schemes still compute modulo a small prime, 3329 in ML-KEM, just in hundreds of dimensions
A difference in the ninth decimal, and the forecast is dead by t = 20.6. What does one more day of accuracy cost ?
The Lorenz equations contain no randomness, and both copies ran the same code on the same machine. The error grows like e^(λt) with λ near 0.9, so the distance between the copies reached 1 at t = 20.6. Past that point the two runs share nothing but the equations.
I measured the horizon over 40 start points per precision level. A measurement 1000 times better adds only about 7.7 time units, and a billion times better multiplies the horizon by 4.6. Each step costs thousands of times more effort and buys the same amount of time.
I think money in chaotic systems will go to models and ensembles, not to ever sharper sensors. Doubling the horizon roughly means squaring the precision.
900 circles, 0.15% error: that is how I redrew the edge of the Mandelbrot set. How many circles would it take to reach 0% ?
→ I took 3,569 points of the edge, split them with a discrete Fourier transform and sorted the circles by radius. Two circles give a 28.2% error, sixty give 1.8%, nine hundred give 0.15% of the shape's width. Each new circle adds less, because the fine details of the edge have the smallest coefficients.
→ The edge has dimension 2, meaning infinite length, so an exact decomposition with a finite number of circles does not exist. The 0.15% describes the pixel image, it says nothing about the set itself. The true edge always hides one more smaller bump.
→ I think this approach will stay where shapes are smooth, like leaf and bone outlines in biology. On fractals it loses to methods that repeat the pattern itself. This is formula #10 for the article above.
For every complex c, the map z ↦ z² + c has a Julia set J(c).
Here c walks around the circle |c| = 0.7885.
The Fatou–Julia dichotomy: J(c) is connected if and only if c belongs to the Mandelbrot set M. Otherwise it shatters into a Cantor dust.
Along this one circle, c lies inside M for about 15% of the full turn. For the other 85%, the set is dust.
One equation. A single circle. And the shape flips between a solid and a cloud of points.
Mathverse retweeted
A double pendulum is four numbers (θ₁, θ₂, ω₁, ω₂) and two deterministic equations. I released 100 copies whose starting angles differ by at most 10⁻⁶ rad.
Their RMS spread stays under 1 mm until t ≈ 5 s, reaches 0.5 m at 7.8 s and 1.24 m at 12 s, a growth by a factor of 4.3×10⁶. The measured growth rate is 2.5 per second, so each extra decimal digit of precision buys only ln 10 / 2.5 ≈ 0.9 s.
Ten times better instruments buy less than one more second.