What is the probability that 4 points chosen uniformly at random on surface of a sphere form a tetrahedron whose four faces are each acute? Asked on MathOverflow (mathoverflow.net/q/498296/6094) with evidence that the answer is 1/12. But not yet resolved. #MathSky #Mathematics #Geometry #Probability

A monohedral tiling of the plane by "spandrelized" squares. Each unit square includes a circular arc of a 1/2-radius circle centered at each vertex. Adams, Colin. "Spandrelized Tilings." Amer. Math. Monthly 132, no. 3 (2025): 199-217. doi.org/10.1080/0002... #MathSky #Mathematics #Geometry #Tiling

Archimedes: "Every cylinder whose base is the greatest circle in a sphere and whose height is equal to the diameter of the sphere has a volume equal to 3/2 the volume of the sphere." Cicero found Archimedes' tomb ~137 yrs later with his famous theorem represented. #Mathematics #MathSky #Geometry

New tiling results on the arXiv, one of which says that determining whether or not two connected polycubes can together tile R^3 is undecidable (Cor. 5.5). A polycube is an object built by gluing cubes face-to-face. (Unrelated fig.) arxiv.org/abs/2509.07906 #MathSky #Mathematics #Geometry #Tiling

Believe it or not, origami stents have been explored: Kuribayashi et al., "Self-deployable origami stent grafts ..." (doi.org/10.1016/j.ms...) Here I show a hexagonal design built with origami waterbomb crease patterns. cs.smith.edu/~jorourke/Ma... #Mathematics #Geometry #MathSky

The conjecture that every convex polyhedron is Rupert is settled in the negative! The convex body in the image cannot pass straight through a hole inside itself. arxiv.org/abs/2508.18475 #Mathematics #Geometry #MathSky

A surprising result: 3-space can be filled with disjoint geometric unit-radius circles. So each point of R^3 lies on exactly one circle. The circles may even be chosen to be unlinked. M. Jonsson and J. Wästlund: www.jstor.org/stable/24493.... #MathSky #Geometry #Mathematics

You might guess that the maximal volume 8-vertex polyhedron inscribed in a unit sphere is the cube. But it's not even close : cube 1.54; 8-vertex max 1.82. Proved by Berman and Hanes in 1970. V=8, E=16, F=10. #MathSky #Geometry #Mathematics

Angel-wing net (edge-unfolding) of a nearly flat prismoid, top & bottom two 40-vertex regular polygons. No mathematical significance, just an attractive image. (The two red edges are not cut.) #MathSky #Geometry #Mathematics #MathArt

The Eye-Ball Theorem: Two disjoint spheres S1 and S2. Form cone C1 tangent to S1 with apex at the center of S2, and form cone C2 similarly. Then the radii of the circles of cone/sphere intersections (red) are equal. #MathSky #Geometry

Among every set of six points in 3-space (in general position) are two linked triangles: The Conway-Gordon-Sachs theorem. General position excludes three points collinear and four points coplanar. #MathSky #Geometry

Saturn's North pole hexagon. Still not thoroughly understood. Multiple Earths could fit inside. en.wikipedia.org/wiki/Saturn%... #MathSky #Geometry #Astronomy #Planets

Happy Easter from the Stanford Bunny! (en.wikipedia.org/wiki/Stanfor...) Developed by Stanford researchers in 1994 as a test bed model for computer graphics algorithms. This version: 2,503 vertices. #MathSky #Geometry #Graphics

A cube can be reoriented so that it can pass through a hole carved in a congreunt cube: Prince Rupert's cube (1693!) "It is unknown whether this is true for all convex polyhedra"! (en.wikipedia.org/wiki/Prince_...) #MathSky #Geometry

Correcting an earlier post (thanks Sophie Huiberts) to show off this attractive image. It shows the shortest paths from the white dot to each of 260 points on the genus-6 surface: 56 vertices, and regularly spaced points on edges. With student Biliana Kaneva. #MathSky #Geometry

This open question was resolved ~20 yrs ago: Can one tie a knot with one foot of one-inch diameter rope? The answer is No: Diao, Yuanan. "The lower bounds of the lengths of thick knots." *Journal of Knot Theory and Its Ramifications* 12, no. 01 (2003): 1-16. #MathSky #Geometry #Topology

There are three simple closed geodesics on a cube. Two are planar: an equatorial band and the hexagon slice. The 3rd is a bit more unusual. It angles off the front bottom edge at arctan(2) = ~63 deg, forming a nonplanar hexagon of length 2 sqrt{5}. #MathSky #Geometry

A Collatz-like function f(n) that bifurcates on the primes, I posed a decade ago. It remains unknown if it always falls into a 2/4 cycle. For n=229, f(n) shoots off to 10^{376} before returning to that cycle after 6309 iterations. mathoverflow.net/questions/20... #MathSky

Unfoldings of the hypercube. There are 261 unfoldings of the 4D hypercube into 3D. It is known since 2021 that none self-overlap (doi.org/10.37236/9796) and each tiles 3-space (Moritz Firsching). The most famous is the Dali cross, immortalized in his painting *Corpus Hypercubus*. #MathSky #Geometry

The attractively intricate optimal tiling of an equilateral triangle by the trapezoid polyiamond (pentiamond), found in 2019 by an anonymous user ('theonetruepath') on MathOverflow. #MathSky #Geometry

There exists a minimum area convex k-gon Q circumscribing a convex n-gon P (n>k) that has at least (k-1) edges flush with P and includes the midpoint on the k-th edge. A theorem of Adlai DePano. In the example, k=4 and 3 edges are flush, with the 4th balanced on a midpoint. #geometry #mathsky

On a 𝘡𝘰𝘭𝘭 𝘴𝘶𝘳𝘧𝘢𝘤𝘦, every geodesic is simple (non-self-intersecting) and closed. Aside from the sphere, remarkable that such surfaces exist! Zoll was a 1903 student of David Hilbert. Image by Konrad Polthier and Markus Schmies. #Geometry #MathSky

The alpha-complex of ~1500 points on the surfaces of two linked tori. #Geometry #MathSky

An n=20 polar zonohedron and its edge-unfolding to a net. #geometry #mathsky

The traditional art gallery problem asks for the fewest guards to guard a polygonal art gallery. Through the work of Abrahamsen, Adamaszek, Miltzow, it is now known that if the polygon coordinates are rational, the guards may need irrational coordinates. [Wikipedia image] #geometry #mathsky

A nice theorem: Isosceles tetrahedra are the only convex polyhedra that admit arbitrarily long simple, closed geodesics. See V. Protasov, "Simple Closed Geodesics on a Polyhedron." *The Mathematical Intelligencer* (2024): 347-354. (Isosceles: Opposite edges have same length.) #geometry #mathsky

"Dudeney’s Dissection is Optimal." It took 120 yrs. I posed the question 13 yrs ago here: mathoverflow.net/q/80101/6094 #mathsky #geometry

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 #mathsky #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

"All Polyhedral Manifolds are Connected by a 2-Step Refolding": Any polyhedral manifold A can be unfolded to B, which can be refolded differently to manifold C. So the space is connected via unfold/refold operations, answering a 2007 question *Geom. Folding Algs*. arxiv.org/abs/2412.02174 #geometry

Finally settled: "Optimality of Gerver's Sofa": arxiv.org/abs/2411.19826