Joseph O'Rourke @josephorourke.bsky.social · 31/08/2026I can confirm. I pursued the 1st invitation until they wanted $200 for their “book club” to read my book. 010
Joseph O'Rourke @josephorourke.bsky.social · 27/06/2026An 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... 0111
Joseph O'Rourke @josephorourke.bsky.social · 31/05/2026Displaying the dynamic Voronoi diagram. #Geometry 060
Joseph O'Rourke @josephorourke.bsky.social · 27/05/2026You might be interested in this: Devadoss, Satyan L., and Matthew Harvey. "Unfoldings and nets of regular polytopes." Computational Geometry 111 (2023): 101977. arxiv.org/abs/2111.01359 #Geometry #Mathematicsarxiv.orgUnfoldings and Nets of Regular PolytopesOver a decade ago, it was shown that every edge unfolding of the Platonic solids was without self-overlap, yielding a valid net. We consider this property for regular polytopes in arbitrary dimensions... 051
Joseph O'Rourke @josephorourke.bsky.social · 26/05/2026Oh, I see, I missed your interest in convex polytopes. Sorry! 020
Joseph O'Rourke @josephorourke.bsky.social · 26/05/2026Nope. Easier to see for the 261 unfoldings of the 4-cube into 3-space: mathoverflow.net/a/199003/6094 #Geometry #Mathematics 010
Joseph O'Rourke @josephorourke.bsky.social · 26/05/2026Any one of the 9694 unfoldings of the 5-dimensional hypercube into 4-space. Excluding the regular tesseract! :-) #Geometry #Mathematics 220
Joseph O'Rourke @josephorourke.bsky.social · 21/05/2026Do you know Ice Nine? en.wikipedia.org/wiki/Ice-nine Apologies if you mentioned it in the (fascinating!) article and I missed it.en.wikipedia.orgIce-nine - Wikipedia 000
Joseph O'Rourke @josephorourke.bsky.social · 02/05/2026I've never considered this. Interesting question! Would want to maximize emitted light. 020
Joseph O'Rourke @josephorourke.bsky.social · 02/05/2026A 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 🧪 25910
Joseph O'Rourke @josephorourke.bsky.social · 28/04/2026"Is the boundary ∂S analogous to a derivative?" mathoverflow.net/questions/46... Answered by Terry Tao and others. #MathSky #Mathematics #MathOverflow 030
Joseph O'Rourke @josephorourke.bsky.social · 28/04/2026I loved APL, had an APL keyboard, great for data analysis using matrices. 010
Joseph O'Rourke @josephorourke.bsky.social · 13/04/2026I posed this on MathOverflow 11 years ago: mathoverflow.net/q/187677/6094 #mathematics #MathSky #Geometrymathoverflow.netThe view from inside of a mirrored tetrahedronSuppose you were standing inside a regular tetrahedron $T$ whose internal face surfaces were perfect mirrors. Let's assume $T$'s height is $3{\times}$ yours, so that your eye is roughly at the cent... 193
Reposted by Joseph O'RourkeJoseph O'Rourke @josephorourke.bsky.social · 23/03/2026A 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 🧪 1286
Joseph O'Rourke @josephorourke.bsky.social · 28/03/2026There 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 🧪 0132
Joseph O'Rourke @josephorourke.bsky.social · 23/03/2026A 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 🧪 1286
Joseph O'Rourke @josephorourke.bsky.social · 21/03/2026Manu link broken. I am a computer scientist/mathematician, sci. communicator. Website: cs.smith.edu/~jorourke/ Three recent books: The Math of Origami: cs.smith.edu/~jorourke/Ma... PopUp Geometry: cs.smith.edu/~jorourke/Po... How To Fold It: cs.smith.edu/~jorourke/Ho... Lots of Engineering/Physicsscience.smith.eduJoseph O'Rourke 130
Joseph O'Rourke @josephorourke.bsky.social · 16/03/2026If a degree-4 vertex is flat-foldable, then opposite dihedrals have the same magnitude (reducing 4 DOF to 2 DOF), and adjacent dihedrals are related via a simple half-tangent formula, reducing to 1 DOF. Animation: cs.smith.edu/~jorourke/Ma... #MathSky #Mathematics #Geometry #Origami 🧪cs.smith.eduMath Origami: Animations & TemplatesA click on a thumbnail will run an animated GIF in a new browser window/tab. To repeat the animation, refresh the browser window. 042
Joseph O'Rourke @josephorourke.bsky.social · 16/03/2026In 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 🧪 150
Joseph O'Rourke @josephorourke.bsky.social · 31/12/2025Robert 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 🧪 0122
Joseph O'Rourke @josephorourke.bsky.social · 18/12/2025Related books #MathSky #MathArt #Mathematics #Geometry #Science #Origami 061
Joseph O'Rourke @josephorourke.bsky.social · 18/12/2025Published today 18Dec2025: *The Mathematics of Origami.* Cambridge link: view.updates.cambridge.org?qs=99a0b7610... #MathSky #MathArt #Mathematics #Geometry #Science #Origami 2267
Joseph O'Rourke @josephorourke.bsky.social · 12/12/2025Forgot to link to the book: cs.smith.edu/~jorourke/Ma... #MathSky #Mathematics #Origami 🧪science.smith.eduMath Origami: Animations & TemplatesA click on a thumbnail will run an animated GIF in a new browser window/tab. To repeat the animation, refresh the browser window. 032
Joseph O'Rourke @josephorourke.bsky.social · 12/12/2025Discussed in book published 18Dec2025. #MathSky #Mathematics #Origami 🧪 170
Joseph O'Rourke @josephorourke.bsky.social · 12/12/2025The challenge is to avoid trying every possible folding. consistent with the M/V assignments to determine the answer, for there are an exponential number of such possibilities. As yet only understood for 2xn maps via a complex polynomial-time algorithm. #MathSky #Mathematics #Origami 🧪 100
Joseph O'Rourke @josephorourke.bsky.social · 14/11/2025Besides #MathSky #MathArt #Geometry #Origami, neglected to include also: #ArtMath #Mathematics and 🧪 020
Joseph O'Rourke @josephorourke.bsky.social · 14/11/2025Each annulus alternates mountain folds with valley folds. It is not yet proved that this folding "exists" in the sense that only the circular creases are necessary. Strong numerical evidence, but not formally proved. #MathSky #MathArt #Geometry #Origami 191
Joseph O'Rourke @josephorourke.bsky.social · 14/11/2025Curved 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/) 1347
Joseph O'Rourke @josephorourke.bsky.social · 05/11/2025Cambridge University Press. www.cambridge.org/core/books/m... #MathSky #Mathematics 🧪 #Geometry #Origami #MathArt 010
Joseph O'Rourke @josephorourke.bsky.social · 05/11/2025*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 2509
Joseph O'Rourke @josephorourke.bsky.social · 02/11/2025In fact in this example, 3 guards suffice. Minimal guarding is an NP-hard problem, i.e., intractable. #Mathematics #MathSky #GraphTheory www.science.smith.edu/~jorourke/bo... 040
Joseph O'Rourke @josephorourke.bsky.social · 02/11/2025"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... 2121
Joseph O'Rourke @josephorourke.bsky.social · 31/10/2025Crescent 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 190
Joseph O'Rourke @josephorourke.bsky.social · 22/10/2025These triangles are known to have a periodic billiard path: (1) All acute triangles. (2) All right triangles. (3) All rational triangles. (4) All obtuse triangles with obtuse angle smaller than 5 pi/8 (the 112.4 deg that I quoted). #MathSky #Mathematics #Geometry #Billiards 050
Joseph O'Rourke @josephorourke.bsky.social · 18/10/2025Sharp eyes to notice the two perpendicular bounces. Probably not for all triangles, I agree. 100
Joseph O'Rourke @josephorourke.bsky.social · 18/10/2025Beautiful indeed. And with recent results from the study of translation surfaces. 010
Joseph O'Rourke @josephorourke.bsky.social · 18/10/2025It 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 5559
Joseph O'Rourke @josephorourke.bsky.social · 27/09/2025Sure. Have them email me, jorourke@smith.edu. 010
Joseph O'Rourke @josephorourke.bsky.social · 26/09/2025Stoker'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 2140
Joseph O'Rourke @josephorourke.bsky.social · 25/09/2025See also: "Why can't a nonabelian group be 75% abelian?" mathoverflow.net/q/211159/6094 010
Joseph O'Rourke @josephorourke.bsky.social · 23/09/2025What 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 0223
Joseph O'Rourke @josephorourke.bsky.social · 17/09/2025A 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 0123
Joseph O'Rourke @josephorourke.bsky.social · 14/09/2025I wonder in which dimensions is the cylinder/sphere volume ratio rational? #Mathematics #MathSky #Geometry 110
Joseph O'Rourke @josephorourke.bsky.social · 14/09/2025Archimedes: "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 2101
Joseph O'Rourke @josephorourke.bsky.social · 11/09/2025New 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 0204
Joseph O'Rourke @josephorourke.bsky.social · 31/08/2025p.4 of their paper details the construction. "the Noperthedron has 3·30=90 vertices." They set three pts C1,C2,C3 and then apply the cyclic group C_30 to each. 120
Joseph O'Rourke @josephorourke.bsky.social · 30/08/2025Believe 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 1162
Joseph O'Rourke @josephorourke.bsky.social · 28/08/2025The Rupert property requires the convex polyhedron P to tunnel by translation through an isometric copy of P. I wonder if twisting while translating would permit any P---even the "Noperthedron"---to pass through itself? #Mathematics #Geometry #MathSky 120