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/PopU… #Mathematics #Geometry

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

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/455880/6094?…
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.

*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'

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

Nice: The hypercube graph Hₙ is 2-colorable. Label the 2ⁿ nodes with n-bit binary numbers so adjacent nodes differ in only one bit. Color red all nodes with an even count of 1s, and blue all with odd count. Then two adjacent nodes differ in parity and so have different colors.

One can dent an icosahedron resulting in an isometric nonconvex polyhedron with identical facial structure. However, each configuration is rigid---no continuous nondistorting motion between them. I used this as an exercise
(cs.smith.edu/~jorourke/PopU…). #math ematics #math