An unsolved problem. Given an m×n map formed of unit squares, with a given Mountain/Valley assignment for every crease, is there a subexponential algorithm to decide if it can be folded to a 1×1 stack of squares? Example: Yes. #MathSky #Mathematics #Geometry #Origami 🧪 cs.smith.edu/~jorourke/Ma...

A billiard knot is a path in a mirror polyhedron that realizes a knot by reflected lightrays. Not every knot is realizable in a cube, but every knot is a billiard knot in some convex right prism. arxiv.org/abs/1106.5600 mathoverflow.net/questions/38... #MathSky #Mathematics #Geometry #Knots 🧪

A 164 triangles version of the Stanford Bunny folded from a 4ft x 4ft thin aluminum sheet, following a crease pattern created by the *Origamizer* algorithm of E.Demaine & T.Tachi. Folded by an MIT group in 2011. cs.smith.edu/~jorourke/Ma... #MathSky #MathArt #Origami #Mathematics #Engineering 🧪

There is research on fabricating micropolyhedra using lithographic techniques via self-assembly / self-folding of nets that fold to, e.g., a cube as illustrated. Note the tiny size: 0.2mm. The goal is to minimize mis-foldings. doi:10.1371/journal.pone.0004451 cs.smith.edu/~jorourke/Ma... #MathSky 🧪

A 164 triangles version of the Stanford Bunny folded from a 4ft x 4ft thin aluminum sheet, following a crease pattern created by the *Origamizer* algorithm of E.Demaine & T.Tachi. Folded by an MIT group in 2011. cs.smith.edu/~jorourke/Ma... #MathSky #MathArt #Origami #Mathematics #Engineering 🧪

In rigid origami, the rigid faces hinge on creases. Much is unknown, but degree-4 vertices are understood. An example is the Miura Map fold. #MathSky #Mathematics #Geometry #Origami 🧪

Robert Lang's origami *White-Tailed Deer*, Opus 550. Design based on his "uniaxial bases" and the "circle/river" and "tree methods." Chapter 6 in *The Mathematics of Origami*. cs.smith.edu/~jorourke/Ma... #MathSky #Mathematics #MathArt #SciArt #Origami 🧪

Published today 18Dec2025: *The Mathematics of Origami.* Cambridge link: view.updates.cambridge.org?qs=99a0b7610... #MathSky #MathArt #Mathematics #Geometry #Science #Origami

Curved circular creases of annuli. A construction by Erik and Martin Demaine (all rights reserved). Several annuli intertwined. #MathSky #MathArt #Geometry #Origami More examples: erikdemaine.org/curved/)

*The Mathematics of Origami*. Expected online publication date: December 2025. Print publication: 31 December 2025. www.science.smith.edu/~jorourke/Ma... #MathSky #Mathematics 🧪 #Geometry #Origami #MathArt

"Louvre robbery: Could a 50-year-old maths problem have kept the museum safe?" This is a BBC article by Kit Yates about the art gallery theorem. In the figure, four red vertex guards suffice to visually cover the whole polygon. #Mathematics #MathSky #GraphTheory www.bbc.com/future/artic...

Crescent Moon. Did you ever notice that the outer convex curve of the crescent is a semicircle, but the inner concave curve is (half of) an ellipse. An ellipse because we are viewing a circle at an angle; a circle projects to an ellipse. #MathSky #Mathematics #Geometry #Pumpkin #Moon

It is *still* unknown whether or not every triangle admits a periodic billiard trajectory. Every triangle with rational angles does. And so does every obtuse triangle of at most 112.4 deg. "112.5 appears to be a natural barrier." gwtokarsky.github.io. #MathSky #Mathematics #Geometry #Billiards

Stoker's Conjecture settled by Cho & Kim positively: Every 3D polyhedron is uniquely determined by its dihedral angles and edge lengths, even if nonconvex or self-intersecting (subject to technical restrictions). doi.org/10.1007/s004... #MathSky #Mathematics #Geometry #Polyhedra

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