Joseph O'Rourke
@josephorourke.bsky.social
Mathematician and Computer Scientist, Smith College, USA.
https://cs.smith.edu/~jorourke/
Polyhedron displayed in banner has max volume of all foldings from a square.
https://cs.smith.edu/~jorourke/
Polyhedron displayed in banner has max volume of all foldings from a square.
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/)
#MathSky #MathArt #Geometry #Origami
More examples: erikdemaine.org/curved/)
November 14, 2025 at 1:15 AM
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/)
#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
Expected online publication date: December 2025. Print publication: 31 December 2025.
www.science.smith.edu/~jorourke/Ma...
#MathSky #Mathematics 🧪 #Geometry #Origami #MathArt
November 5, 2025 at 3:01 PM
*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
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...
November 2, 2025 at 11:03 PM
"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
October 31, 2025 at 12:58 AM
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
gwtokarsky.github.io. #MathSky #Mathematics #Geometry #Billiards
October 18, 2025 at 12:15 AM
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
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
doi.org/10.1007/s004...
#MathSky #Mathematics #Geometry #Polyhedra
September 26, 2025 at 4:47 PM
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
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
#MathSky #Mathematics #Geometry #Probability
September 22, 2025 at 11:58 PM
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
#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
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
September 17, 2025 at 6:25 PM
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
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
#Mathematics #MathSky #Geometry
September 14, 2025 at 12:29 AM
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
#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
arxiv.org/abs/2509.07906
#MathSky #Mathematics #Geometry #Tiling
September 11, 2025 at 1:14 AM
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
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
(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
August 30, 2025 at 12:05 AM
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
(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
arxiv.org/abs/2508.18475
#Mathematics #Geometry #MathSky
August 27, 2025 at 1:11 PM
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
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
#MathSky #Geometry #Mathematics
PARTITIONS OF R
3
INTO CURVES on JSTOR
M. JONSSON, J. WÄSTLUND, PARTITIONS OF R 3 INTO CURVES, Mathematica Scandinavica, Vol. 83, No. 2 (1998), pp. 192-204
www.jstor.org
June 23, 2025 at 12:44 AM
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
#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
June 10, 2025 at 1:35 AM
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
May 31, 2025 at 12:27 AM
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
Saturn's North pole hexagon. Still not thoroughly understood. Multiple Earths could fit inside.
en.wikipedia.org/wiki/Saturn%...
#MathSky #Geometry #Astronomy #Planets
en.wikipedia.org/wiki/Saturn%...
#MathSky #Geometry #Astronomy #Planets
May 6, 2025 at 11:36 PM
Saturn's North pole hexagon. Still not thoroughly understood. Multiple Earths could fit inside.
en.wikipedia.org/wiki/Saturn%...
#MathSky #Geometry #Astronomy #Planets
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
(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
April 20, 2025 at 12:16 AM
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
(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
(en.wikipedia.org/wiki/Prince_...)
#MathSky #Geometry
April 17, 2025 at 10:22 PM
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
(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
April 15, 2025 at 11:43 PM
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
April 6, 2025 at 12:18 AM
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
#MathSky
March 21, 2025 at 10:45 PM
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
#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
March 13, 2025 at 9:26 PM
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