Visual Insight
http://blogs.ams.org/visualinsight
Mathematics Made VisibleThu, 17 Jul 2014 11:08:49 +0000en-UShourly1http://wordpress.org/?v=3.9.1{7,3} Tiling
http://blogs.ams.org/visualinsight/2014/07/15/73-tiling/?utm_source=rss&utm_medium=rss&utm_campaign=73-tiling
http://blogs.ams.org/visualinsight/2014/07/15/73-tiling/#commentsTue, 15 Jul 2014 01:00:22 +0000http://blogs.ams.org/visualinsight/?p=946This picture, drawn by Anton Sherwood, shows the {7,3} tiling: a tiling of the hyperbolic plane by equal-sized regular heptagons, 3 meeting at each vertex. ]]>http://blogs.ams.org/visualinsight/2014/07/15/73-tiling/feed/1Sierpinski Carpet
http://blogs.ams.org/visualinsight/2014/07/01/sierpinski-carpet/?utm_source=rss&utm_medium=rss&utm_campaign=sierpinski-carpet
http://blogs.ams.org/visualinsight/2014/07/01/sierpinski-carpet/#commentsTue, 01 Jul 2014 01:00:12 +0000http://blogs.ams.org/visualinsight/?p=952To build the Sierpinski carpet you take a square, cut it into 9 equal-sized smaller squares, and remove the central smaller square. Then you apply the same procedure to the remaining 8 subsquares, and repeat this ad infinitum. This image by Noon Silk shows the first six stages of the procedure.]]>http://blogs.ams.org/visualinsight/2014/07/01/sierpinski-carpet/feed/3Origami Dodecahedra
http://blogs.ams.org/visualinsight/2014/06/15/origami-dodecahedra/?utm_source=rss&utm_medium=rss&utm_campaign=origami-dodecahedra
http://blogs.ams.org/visualinsight/2014/06/15/origami-dodecahedra/#commentsSun, 15 Jun 2014 07:45:58 +0000http://blogs.ams.org/visualinsight/?p=908There 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.]]>http://blogs.ams.org/visualinsight/2014/06/15/origami-dodecahedra/feed/0Grace–Danielsson Inequality
http://blogs.ams.org/visualinsight/2014/06/01/grace-danielsson-inequality/?utm_source=rss&utm_medium=rss&utm_campaign=grace-danielsson-inequality
http://blogs.ams.org/visualinsight/2014/06/01/grace-danielsson-inequality/#commentsSun, 01 Jun 2014 01:00:57 +0000http://blogs.ams.org/visualinsight/?p=843When 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!]]>http://blogs.ams.org/visualinsight/2014/06/01/grace-danielsson-inequality/feed/19Pattern-Equivariant Homology of a Penrose Tiling
http://blogs.ams.org/visualinsight/2014/05/15/pattern-equivariant-homology-of-a-penrose-tiling/?utm_source=rss&utm_medium=rss&utm_campaign=pattern-equivariant-homology-of-a-penrose-tiling
http://blogs.ams.org/visualinsight/2014/05/15/pattern-equivariant-homology-of-a-penrose-tiling/#commentsThu, 15 May 2014 01:00:13 +0000http://blogs.ams.org/visualinsight/?p=817The 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. ]]>http://blogs.ams.org/visualinsight/2014/05/15/pattern-equivariant-homology-of-a-penrose-tiling/feed/0{6,3,6} Honeycomb
http://blogs.ams.org/visualinsight/2014/05/01/636-honeycomb/?utm_source=rss&utm_medium=rss&utm_campaign=636-honeycomb
http://blogs.ams.org/visualinsight/2014/05/01/636-honeycomb/#commentsThu, 01 May 2014 01:00:01 +0000http://blogs.ams.org/visualinsight/?p=801This 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!]]>http://blogs.ams.org/visualinsight/2014/05/01/636-honeycomb/feed/0{6,3,5} Honeycomb
http://blogs.ams.org/visualinsight/2014/04/15/635-honeycomb/?utm_source=rss&utm_medium=rss&utm_campaign=635-honeycomb
http://blogs.ams.org/visualinsight/2014/04/15/635-honeycomb/#commentsTue, 15 Apr 2014 01:00:53 +0000http://blogs.ams.org/visualinsight/?p=789This 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.]]>http://blogs.ams.org/visualinsight/2014/04/15/635-honeycomb/feed/2{6,3,4} Honeycomb
http://blogs.ams.org/visualinsight/2014/04/01/634-honeycomb/?utm_source=rss&utm_medium=rss&utm_campaign=634-honeycomb
http://blogs.ams.org/visualinsight/2014/04/01/634-honeycomb/#commentsTue, 01 Apr 2014 01:00:21 +0000http://blogs.ams.org/visualinsight/?p=782This 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.]]>http://blogs.ams.org/visualinsight/2014/04/01/634-honeycomb/feed/0{6,3,3} Honeycomb
http://blogs.ams.org/visualinsight/2014/03/15/633-honeycomb/?utm_source=rss&utm_medium=rss&utm_campaign=633-honeycomb
http://blogs.ams.org/visualinsight/2014/03/15/633-honeycomb/#commentsSat, 15 Mar 2014 10:01:05 +0000http://blogs.ams.org/visualinsight/?p=745This 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.]]>http://blogs.ams.org/visualinsight/2014/03/15/633-honeycomb/feed/0Menger Sponge
http://blogs.ams.org/visualinsight/2014/03/01/menger-sponge/?utm_source=rss&utm_medium=rss&utm_campaign=menger-sponge
http://blogs.ams.org/visualinsight/2014/03/01/menger-sponge/#commentsSat, 01 Mar 2014 01:00:57 +0000http://blogs.ams.org/visualinsight/?p=529Take 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.]]>http://blogs.ams.org/visualinsight/2014/03/01/menger-sponge/feed/2