Published today 18Dec2025: *The Mathematics of Origami.* Cambridge link: view.updates.cambridge.org/?… #MathArt #Mathematics #Geometry #Science #Origami

*The Mathematics of Origami*. Expected online publication date: December 2025. Print publication: 31 December 2025. Cambridge University Press. science.smith.edu/~jorourke/…

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

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

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

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

V.Y. Protasov constructed a nonconvex polyhedron that has arbitrarily long simple closed geodesics. arxiv.org/abs/2312.10554v1 #Mathematics #Geometry

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

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/in…

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.

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

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

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

Billiard circuits in pentagons mathoverflow.net/q/455880/60… Oscar Lanzi showed that the pentagon need not be cyclic. #geometry

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

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.