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...

I posed this on MathOverflow 11 years ago: mathoverflow.net/q/187677/6094
#mathematics #MathSky #Geometry

Aluminum Stanford Bunny.

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
🧪

Cube net folding.

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 🧪

Aluminum Stanford Bunny.

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
🧪

Thank you!

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

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.

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 🧪

Miura map fold is built from degree-4 vertices.

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 🧪

Thanks, Erick! :-)

Lang's White-Tailed Deer

Crease pattern.

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 🧪

Covers of three other books on folding/unfolding.

Related books
#MathSky #MathArt #Mathematics #Geometry #Science #Origami

Cover image.

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

Forgot to link to the book: cs.smith.edu/~jorourke/Ma...
#MathSky #Mathematics #Origami 🧪

Cover of *The Mathematics of Origami*.

Discussed in book published 18Dec2025.
#MathSky #Mathematics #Origami 🧪

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 🧪

2x5 example with M/V assignments marked.

Unsolved problem. *Q*. Given an m×n map formed of unit squares, with a given Mountain/Valley assignment for every crease (i.e., for every edge shared by two squares), can it be folded to a 1 ×1 stack of squares?
In example: Red: M. Blue-dashed: V.
#MathSky #Mathematics #Origami 🧪

Besides #MathSky #MathArt #Geometry #Origami, neglected to include also: #ArtMath #Mathematics and 🧪

Concentric M/V folds.

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

Intertwined annuli.

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/)

Cambridge University Press.
www.cambridge.org/core/books/m...
#MathSky #Mathematics 🧪 #Geometry #Origami #MathArt

Cover: The Mathematics of Origami

*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

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...

3-coloring if a triangulated polygon

"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 carved into pumpkin.

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

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

Sharp eyes to notice the two perpendicular bounces. Probably not for all triangles, I agree.

Beautiful indeed. And with recent results from the study of translation surfaces.

Triangle w complex periodic orbit.

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

Sure. Have them email me, jorourke@smith.edu.

Vertex v mapped to sphere.

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