geometry

Every so often a mathy essay climbs the ladder by saying something that feels almost too obvious once said out loud: humans think in whole rotations more naturally than they think in π. Turns versus radians is one of those fights that looks pedantic until you have debugged a graphics bug at midnight and watched two coordinate systems disagree by a factor nobody can feel in their bones. Radians ar…

This is a story about geometry, algebra and many different dimensions, best read with construction paper, scissors and tape on hand.

A new paper just came out, The Maximum-Area Small Polygon Problem. The paper solves the problem of finding, for each n, the n-gon with diameter 1 and maximum area. For odd n, the solution is what you might expect: a regular n-gon. I would expect this to be the solution for even n as well, […] The post Big little hexagon first appeared on John D. Cook .

A mathematician may have solved a 250-year-old question about whether every polygon contains a path home

I found that Max Dehn (1903) proved that if a rectangle is dissected into squares, then the ratio of length of side of squares are rational. Is the following generalization true ? : If a rectangle is ...

I couldn't resist posting this problem I just came up with (see the attached figure). I consulted Google's AI several times to get an idea of what this unknown angle might be, but each time, I ...

The mathematical beauty of hyperbezier curves Raph Levien, August 8, 2026 I have for many decades been fascinated by the prospect of a curve family better suited for interactive design than cubic Béziers. In that search, I have come to a new-found respect for those Béziers. In particular, though other curves like Euler spirals are better at representing smooth curves, they fall short at represent…

Number Garden – A Brilliant Interactive Pattern System

I know that, for $\vec{u},\vec{v},\vec{w}\in V$ where $V$ is an inner product space, the distance between $\vec{u}$ and $\vec{w}$ is given by the norm $d\left( \vec{u},\vec{w}\right)=\left\| ...

In 1987, Miles Reid conjectured that a combination of contracting (resp. introducing) some isolated curves and smoothing (resp. singularizing) deformation connects all Calabi-Yau 3-folds to each other, as well as to a (non-Kähler, ) connected sum of copies of , called a rakshasa, thus connecting their individual moduli spaces into an irreducible object: The moduli space of 3-folds with may nevert…

Every finite set of points in the Euclidean plane that is not collinear has a line that passes through exactly two of the points. This proof is by Michigan State University mathematician Leroy Milton Kelly. Consider a set S of points that aren’t all collinear, and define a connecting line to be a line that contains at least two of these points. There must be some point P and connecting line ℓ tha…