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Add the odd numbers one at a time: 1, 3, 5, 7, 9, 11. You always get a square. · XDL

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Add the odd numbers one at a time: 1, 3, 5, 7, 9…

Add the odd numbers one at a time: 1, 3, 5, 7, 9, 11. You always get a square.

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Pascal's triangle, keeping only whether each number is odd or even. The odd ones draw the Sierpinski triangle.

A polyrhythm is two even pulses sharing one span. One hand turns every 3 s and sounds each corner it crosses: the square's four, every 0.75 s, against a regular n-gon's n, every 3/n s. The two pulses meet gcd(n, 4) times a turn: once for 3:4, 5:4, 7:4, 9:4 and 15:4, twice for

Hilbert Curve, one path, refined six times, until it fills the whole square.

The primes have no geometry of their own. They borrow whatever 2π is lending. At (r, θ) = (n, n) the arms count the rational numbers closest to 2π, and close in that is 44/7 = 6.2857…, so 44 arms form. No prime shares a factor with 44, so the primes hold φ(44) = 20 of those

Japanese line multiplication draws each digit as that many parallel lines: 413 as groups of 4, 1 and 3 lines, and 3 as three lines across them. A group of a lines crossing b lines meets them in a·b points, so each group is one partial product: 4·3 = 12 hundreds, 1·3 = 3 tens,

Forty-seven symbols of written mathematics fall into six families: relations, logic, sets, number systems, operations and calculus. The number systems nest, ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ, and ∅ is a subset of every set. n! counts the orderings of n objects and ⁿCᵣ = n!/(r!(n −

A square wave flipping between ±1 at multiples of π is the limit of f_N(t) = (4/π) ∑ sin((2k+1)t)/(2k+1) for k < N. Only odd harmonics appear, since f(t + π) = −f(t). Term k is a circle of radius (4/π)/(2k+1) turning 2k+1 times as fast as the first; the chain ends at height

A three-digit number with every digit equal to n, divided by the sum of its digits: nnn/(n + n + n). The numerator is 111·n and the denominator 3n, so n cancels and the quotient is 111/3 = 37 for every digit from 1 to 9. With k repeated digits the quotient is Rₖ/k, where Rₖ =

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Love how you broke that down!
Square numbers never get old!