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Joseph O'Rourke

@josephorourke.bsky.social
734 followers 484 following 97 posts

Mathematician and Computer Scientist, Smith College, USA. cs.smith.edu/~jorourke Polyhedron displayed in banner has max volume of all convex foldings from a square.

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Joseph O'Rourke @josephorourke.bsky.social · 31/08/2026
I can confirm. I pursued the 1st invitation until they wanted $200 for their “book club” to read my book.
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Joseph O'Rourke @josephorourke.bsky.social · 27/06/2026
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...
2x4 map to fold to 1x1
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Joseph O'Rourke @josephorourke.bsky.social · 31/05/2026
Displaying the dynamic Voronoi diagram. #Geometry
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Joseph O'Rourke @josephorourke.bsky.social · 27/05/2026
You 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 #Mathematics
arxiv.org
Unfoldings and Nets of Regular Polytopes
Over 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...
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Joseph O'Rourke @josephorourke.bsky.social · 26/05/2026
Oh, I see, I missed your interest in convex polytopes. Sorry!
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Joseph O'Rourke @josephorourke.bsky.social · 26/05/2026
Nope. Easier to see for the 261 unfoldings of the 4-cube into 3-space: mathoverflow.net/a/199003/6094 #Geometry #Mathematics
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Joseph O'Rourke @josephorourke.bsky.social · 26/05/2026
Any one of the 9694 unfoldings of the 5-dimensional hypercube into 4-space. Excluding the regular tesseract! :-) #Geometry #Mathematics
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Joseph O'Rourke @josephorourke.bsky.social · 21/05/2026
Do 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.org
Ice-nine - Wikipedia
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Joseph O'Rourke @josephorourke.bsky.social · 02/05/2026
I've never considered this. Interesting question! Would want to maximize emitted light.
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Joseph O'Rourke @josephorourke.bsky.social · 02/05/2026
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 🧪
Lightray in mirror cube.
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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
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Joseph O'Rourke @josephorourke.bsky.social · 28/04/2026
I loved APL, had an APL keyboard, great for data analysis using matrices.
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Joseph O'Rourke @josephorourke.bsky.social · 13/04/2026
I posed this on MathOverflow 11 years ago: mathoverflow.net/q/187677/6094 #mathematics #MathSky #Geometry
mathoverflow.net
The view from inside of a mirrored tetrahedron
Suppose 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...
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Reposted by Joseph O'Rourke
Joseph O'Rourke @josephorourke.bsky.social · 23/03/2026
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 🧪
Aluminum Stanford Bunny.
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Joseph O'Rourke @josephorourke.bsky.social · 28/03/2026
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 🧪
Cube net folding.
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Joseph O'Rourke @josephorourke.bsky.social · 23/03/2026
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 🧪
Aluminum Stanford Bunny.
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Joseph O'Rourke @josephorourke.bsky.social · 23/03/2026
Thank you!
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Joseph O'Rourke @josephorourke.bsky.social · 21/03/2026
Manu 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/Physics
science.smith.edu
Joseph O'Rourke
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Joseph O'Rourke @josephorourke.bsky.social · 16/03/2026
If 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.edu
Math Origami: Animations & Templates
A click on a thumbnail will run an animated GIF in a new browser window/tab. To repeat the animation, refresh the browser window.
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Joseph O'Rourke @josephorourke.bsky.social · 16/03/2026
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 🧪
Miura map fold is built from degree-4 vertices.
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Joseph O'Rourke @josephorourke.bsky.social · 07/02/2026
Thanks, Erick! :-)
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Joseph O'Rourke @josephorourke.bsky.social · 31/12/2025
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 🧪
Lang's White-Tailed DeerCrease pattern.
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Joseph O'Rourke @josephorourke.bsky.social · 18/12/2025
Related books #MathSky #MathArt #Mathematics #Geometry #Science #Origami
Covers of three other books on folding/unfolding.
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Joseph O'Rourke @josephorourke.bsky.social · 18/12/2025
Published today 18Dec2025: *The Mathematics of Origami.* Cambridge link: view.updates.cambridge.org?qs=99a0b7610... #MathSky #MathArt #Mathematics #Geometry #Science #Origami
Cover image.
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Joseph O'Rourke @josephorourke.bsky.social · 12/12/2025
Forgot to link to the book: cs.smith.edu/~jorourke/Ma... #MathSky #Mathematics #Origami 🧪
science.smith.edu
Math Origami: Animations & Templates
A click on a thumbnail will run an animated GIF in a new browser window/tab. To repeat the animation, refresh the browser window.
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Joseph O'Rourke @josephorourke.bsky.social · 12/12/2025
Discussed in book published 18Dec2025. #MathSky #Mathematics #Origami 🧪
Cover of *The Mathematics of Origami*.
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Joseph O'Rourke @josephorourke.bsky.social · 12/12/2025
The 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 🧪
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Joseph O'Rourke @josephorourke.bsky.social · 14/11/2025
Besides #MathSky #MathArt #Geometry #Origami, neglected to include also: #ArtMath #Mathematics and 🧪
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Joseph O'Rourke @josephorourke.bsky.social · 14/11/2025
Each 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
Concentric M/V folds.
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Joseph O'Rourke @josephorourke.bsky.social · 14/11/2025
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/)
Intertwined annuli.
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Joseph O'Rourke @josephorourke.bsky.social · 05/11/2025
Cambridge University Press. www.cambridge.org/core/books/m... #MathSky #Mathematics 🧪 #Geometry #Origami #MathArt
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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
Cover: The Mathematics of Origami
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Joseph O'Rourke @josephorourke.bsky.social · 02/11/2025
In 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...
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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...
3-coloring if a triangulated polygon
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Joseph O'Rourke @josephorourke.bsky.social · 31/10/2025
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
Crescent moon carved into pumpkin.
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Joseph O'Rourke @josephorourke.bsky.social · 22/10/2025
These 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
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Joseph O'Rourke @josephorourke.bsky.social · 18/10/2025
Sharp eyes to notice the two perpendicular bounces. Probably not for all triangles, I agree.
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Joseph O'Rourke @josephorourke.bsky.social · 18/10/2025
Beautiful indeed. And with recent results from the study of translation surfaces.
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Joseph O'Rourke @josephorourke.bsky.social · 18/10/2025
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
Triangle w complex periodic orbit.
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Joseph O'Rourke @josephorourke.bsky.social · 27/09/2025
Sure. Have them email me, jorourke@smith.edu.
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Joseph O'Rourke @josephorourke.bsky.social · 26/09/2025
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
Vertex v mapped to sphere.
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Joseph O'Rourke @josephorourke.bsky.social · 25/09/2025
See also: "Why can't a nonabelian group be 75% abelian?" mathoverflow.net/q/211159/6094
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Joseph O'Rourke @josephorourke.bsky.social · 23/09/2025
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
Tetrahedron in a sphere.
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Joseph O'Rourke @josephorourke.bsky.social · 17/09/2025
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
Scalloped square tiling.
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Joseph O'Rourke @josephorourke.bsky.social · 14/09/2025
I wonder in which dimensions is the cylinder/sphere volume ratio rational? #Mathematics #MathSky #Geometry
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Joseph O'Rourke @josephorourke.bsky.social · 14/09/2025
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
Sphere/Cylinder
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Joseph O'Rourke @josephorourke.bsky.social · 11/09/2025
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
Fig. 22(b)
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Joseph O'Rourke @josephorourke.bsky.social · 31/08/2025
p.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.
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Joseph O'Rourke @josephorourke.bsky.social · 30/08/2025
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
Hexagonal Waterbomb stent.
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Joseph O'Rourke @josephorourke.bsky.social · 28/08/2025
The 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
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