There is a nice photograph of some interlocking origami dodecahedra created by Dirk Eisner on the website Mathematical Origami. But it’s hard to be sure how many dodecahedra the whole model contains, since some are hidden from view. This raises a puzzle: assuming the configuration is as symmetrical as possible, how many dodecahedra are there? Here you see Greg Egan’s answer to this puzzle—and to a much more challenging puzzle.
When can you fit a tetrahedron between two nested spheres? Suppose the radius of the large sphere is $R$ and the radius of the small one is $r$. Suppose the distance between their centers is $d$. Then you can fit a tetrahedron between these spheres if and only if the Grace–Danielsson inequality $ d^2 \le (R + r)(R – 3r) $ holds. This was independently proved by Grace in 1917 and Danielsson in 1949. But Antony Milne has found a new proof of this inequality using quantum information theory!
The Penrose kite and dart are a pair of tiles that can be used to create aperiodic tilings of the plane. This image illustrates a ‘pattern-equivariant 1-chain’, a tool used by James J. Walton to study the topology of the kite and dart tiling, and other aperiodic tilings.
This is the {6,3,6} honeycomb, drawn by Roice Nelson. A 3-dimensional honeycomb is a way of filling 3d space with polyhedra or infinite sheets of polygons. Besides honeycombs in 3d Euclidean space, we can also have honeycombs in 3d hyperbolic space, a non-Euclidean geometry with constant negative curvature. The {6,3,6} honeycomb lives in hyperbolic space… and it’s special, because it’s self-dual!
This is the {6,3,5} honeycomb, drawn by Roice Nelson. A 3-dimensional honeycomb is a way of filling 3d space with polyhedra or infinite sheets of polygons. Besides honeycombs in 3d Euclidean space, we can also have honeycombs in 3d hyperbolic space, a non-Euclidean geometry with constant negative curvature. The {6,3,5} honeycomb lives in hyperbolic space, and every vertex has 12 edges coming out, just as if you drew edges from the middle of an icosahedron to its corners.
This is the {6,3,4} honeycomb, drawn by Roice Nelson. A 3-dimensional honeycomb is a way of filling 3d space with polyhedra or infinite sheets of polygons. Besides honeycombs in 3d Euclidean space, we can also have honeycombs in 3d hyperbolic space, a non-Euclidean geometry with constant negative curvature. The {6,3,4} honeycomb lives in hyperbolic space, and each vertex has 6 edges coming out of it, just as if you drew edges from the middle of an octahedron to its corners.
This is the {6,3,3} honeycomb, drawn by Roice Nelson. A 3-dimensional honeycomb is a way of filling 3d space with polyhedra. It’s the 3-dimensional analogue of a tiling of the plane. Besides honeycombs in 3d Euclidean space, we can also have honeycombs in 3d hyperbolic space. The hexagonal tiling honeycomb lives in hyperbolic space, and each vertex has 4 edges coming out, just as if we drew edges from the middle of a tetrahedron to its 4 corners.
Take a cube. Chop it into 3×3×3 = 27 small cubes. Poke holes through it, removing 7 of these small cubes. Repeat this process for each remaining small cube. Do this forever! The result is called the Menger sponge.
To make this shape, start with a cube. Chop it into 3×3×3 smaller cubes, and remove all of them except the 8 at the corners. Then do the same thing for each of these 8 smaller cubes, and so on, forever. The stuff that’s left is called ‘Cantor’s cube’.
This is the {5,3,5} honeycomb, drawn by Jos Leys. A 3-dimensional honeycomb is a way of filling 3d space with polyhedra. It’s the 3-dimensional analogue of a tiling of the plane. Besides honeycombs in Euclidean space, we can also have honeycombs in hyperbolic space, which is a 3-dimensional Riemannian manifold with constant negative curvature. The {5,3,5} honeycomb lives in hyperbolic space.
Notices of the AMS · Bulletin of the AMS
American Mathematical Society · 201 Charles Street Providence, Rhode Island 02904-2213 · 401-455-4000 or 800-321-4267
© Copyright , American Mathematical Society · Privacy Statement
