Intersect n unit-radius cylinders with axes through the origin. The closest shape to the sphere is achieved when the axes lie in a plane at equal angles. mathoverflow.net/q/430504/6094 #math #geometry

*Tangled Up in Blue*: "And all the people we used to know They're an illusion to me now Some are **mathematicians** Some are carpenter's wives I don't know how they all got started I don't know what they're doing with their lives"

The Lyusternik–Schnirelmann theorem says there are at least 3 simple closed geodesics on a topological sphere. I've always thought ellipsoids had exactly 3. But Klingenberg shows in his book *Riemannian Geometry* that if the ellipsoid is pancake-like, then there are more than 3.

Observation: Every regular polygon of 𝘯 vertices has a closed billiard path (angle of incidence = angle of reflection), forming a reduced regular 𝘯-gon.

Billiard circuits in pentagons mathoverflow.net/q/455880/6094?… Oscar Lanzi showed that the pentagon need not be cyclic. #geometry

A unit-diameter disk can be covered by five disks each of radius about 0.3. Birgin, Laurain, "Shape Optimization for Covering Problems," AMS Notices, Fig.1. #Geometry #Math

Baseball homeplate math violates Pythagoras: 12^2 + 12^2 = 288 not 289 = 17^2. From 𝘈 𝘗𝘢𝘯𝘰𝘱𝘭𝘺 𝘰𝘧 𝘗𝘰𝘭𝘺𝘨𝘰𝘯𝘴, Alsina & Nelsen, 𝘈𝘔𝘚, 2023. #Math #Geometry #BaseBall

An old conjecture settled: In a mirror polygon with vertex angles rational multiples of π, a light at any point 𝘗 leaves at most a finite number of points 𝘘 dark. Samuel Lelièvre, Thierry Monteil, Barak Weiss. “Everything is illuminated.” #Geometry #Math

It is now known (since 2015) that there are 15 convex pentagons that can tile the infinite plane, and since 2017 that the inventory is complete. Here's the 15th.

A dodecahedral calendar: One month for each of 12 faces, arranged on a Hamiltonian cycle so the next month just needs one turn. #Mathematics #Geometry ms.uky.edu/~jrge/Calendar…

Euler's inequality: The circumradius 𝘙 of a triangle is at least twice its inradius 𝘳. #Mathematics #Geometry

V.Y. Protasov constructed a nonconvex polyhedron that has arbitrarily long simple closed geodesics. arxiv.org/abs/2312.10554… #Mathematics #Geometry

Fractal Christmas tree. Designed and construction by David Richeson. Fig. 1.20 in 𝘗𝘰𝘱-𝘶𝘱 𝘎𝘦𝘰𝘮𝘦𝘵𝘳𝘺. cs.smith.edu/~jorourke/PopU… #Mathematics #Geometry

Most are familiar with Morley's theorem: trisecting triangle angles leads to a central equilateral triangle. Less known is Marion Walter's theorem, discovered in 1993: the central hexagon formed by trisecting the sides always has area 1/10 the original. #Geometry #Mathematics

Stoker's 50-yr old conjecture is settled by Wang and Xie: If two combinatorially equivalent convex polyhedra have the same dihedral angles, then all corresponding face angles are equal. arxiv.org/abs/2203.09511 #geometry #math #polyhedra

Most polygonal coins are historical, but the UK's regular dodecagon (12-sided) £1 coin was minted in 2017. #Geometry #Mathematics

Curved crease origami from Vesica Piscis Image by Klara Mundilova, from "On mathematical folding of curved crease origami..." *Computer-Aided Design*, Vol. 115, Oct. 2019, pp. 34-41. #geometry #origami

A tetrahedron for 2025 Joe Malkevitch suggested exploring tetrahedra with edge lengths 3,3,3,3,5,5, in light of 3^4 5^2 = 2025. #geometry #math