Visual Insight
http://blogs.ams.org/visualinsight
Mathematics Made VisibleSat, 03 Oct 2015 17:58:57 +0000en-UShourly1http://wordpress.org/?v=4.3.1Balaban 10-Cage
http://blogs.ams.org/visualinsight/2015/10/01/balaban-10-cage/
http://blogs.ams.org/visualinsight/2015/10/01/balaban-10-cage/#commentsThu, 01 Oct 2015 01:00:25 +0000http://blogs.ams.org/visualinsight/?p=1755Balaban 10-cage, the first known (3,10)-cage. An \((r,g)\)-cage is graph where every vertex has \(r\) neighbors, the shortest cycle has length at least \(g\), and the number of vertices is maximal given these constraints.
]]>http://blogs.ams.org/visualinsight/2015/10/01/balaban-10-cage/feed/2McGee Graph
http://blogs.ams.org/visualinsight/2015/09/15/mcgee-graph/
http://blogs.ams.org/visualinsight/2015/09/15/mcgee-graph/#commentsTue, 15 Sep 2015 01:00:59 +0000http://blogs.ams.org/visualinsight/?p=1813McGee graph. It is 3-regular graph, meaning that every vertex has 3 neighbors. It also has girth 7, meaning that the shortest cycles have length 7. What makes the McGee graph special is that it has the least number of vertices of any 3-regular graph of girth 7. ]]>http://blogs.ams.org/visualinsight/2015/09/15/mcgee-graph/feed/0Hypercube of Duads
http://blogs.ams.org/visualinsight/2015/09/01/hypercube-of-duads/
http://blogs.ams.org/visualinsight/2015/09/01/hypercube-of-duads/#commentsTue, 01 Sep 2015 01:00:41 +0000http://blogs.ams.org/visualinsight/?p=1879Greg Egan shows a hypercube with all vertices except the bottom labelled by duads, that is, 2-element subsets of a 6-element set. There are 15 duads, while the hypercube has 16 vertices. ]]>http://blogs.ams.org/visualinsight/2015/09/01/hypercube-of-duads/feed/0Tutte–Coxeter Graph
http://blogs.ams.org/visualinsight/2015/08/15/tutte-coxeter-graph/
http://blogs.ams.org/visualinsight/2015/08/15/tutte-coxeter-graph/#commentsSat, 15 Aug 2015 01:00:53 +0000http://blogs.ams.org/visualinsight/?p=1752Tutte--Coxeter graph. This graph was discovered by the famous graph theorist William Thomas Tutte in 1947, but its remarkable properties were studied further by him and the geometer H. S. M. Coxeter in a pair of papers published in 1958.]]>http://blogs.ams.org/visualinsight/2015/08/15/tutte-coxeter-graph/feed/0Heawood Graph
http://blogs.ams.org/visualinsight/2015/08/01/heawood-graph/
http://blogs.ams.org/visualinsight/2015/08/01/heawood-graph/#commentsSat, 01 Aug 2015 01:00:54 +0000http://blogs.ams.org/visualinsight/?p=1747Heawood graph. This graph can be drawn on a torus with no edges crossing in such a way that it divides the torus into 7 hexagons, each pair of which shares an edge. In 1890, Percy John Heawood proved that for any map drawn on a torus, it takes at most 7 colors to ensure that no two countries sharing a common boundary have the same color. The Heawood graph proves that the number 7 is optimal.]]>http://blogs.ams.org/visualinsight/2015/08/01/heawood-graph/feed/2Dyck Words
http://blogs.ams.org/visualinsight/2015/07/15/dyck-words/
http://blogs.ams.org/visualinsight/2015/07/15/dyck-words/#commentsWed, 15 Jul 2015 01:00:17 +0000http://blogs.ams.org/visualinsight/?p=1719Tilman Piesk shows the 14 Dyck words of length 8. A Dyck word is a balanced string of left and parentheses. In the picture, a left parenthesis is shown as upward-slanting line segment, and a right parenthesis as a downward-slanting one. ]]>http://blogs.ams.org/visualinsight/2015/07/15/dyck-words/feed/0Petersen Graph
http://blogs.ams.org/visualinsight/2015/07/01/petersen-graph/
http://blogs.ams.org/visualinsight/2015/07/01/petersen-graph/#commentsWed, 01 Jul 2015 01:00:44 +0000http://blogs.ams.org/visualinsight/?p=1702Petersen graph. ]]>http://blogs.ams.org/visualinsight/2015/07/01/petersen-graph/feed/0Lattice of Partitions
http://blogs.ams.org/visualinsight/2015/06/15/lattice-of-partitions/
http://blogs.ams.org/visualinsight/2015/06/15/lattice-of-partitions/#commentsMon, 15 Jun 2015 01:00:11 +0000http://blogs.ams.org/visualinsight/?p=1673Tilman Piesk shows the 15 partitions of a 4-element set, ordered by refinement. Finer partitions are connected to coarser ones by lines going down. In the finest partition, on top, each of the 4 elements is in its own subset. In the coarsest one, on bottom, all 4 elements are in the same subset.]]>http://blogs.ams.org/visualinsight/2015/06/15/lattice-of-partitions/feed/2Harmonic Orbit
http://blogs.ams.org/visualinsight/2015/06/01/harmonic-orbit/
http://blogs.ams.org/visualinsight/2015/06/01/harmonic-orbit/#commentsMon, 01 Jun 2015 01:00:49 +0000http://blogs.ams.org/visualinsight/?p=1637Kepler problem concerns a particle moving under the influence of gravity, like a planet moving around the Sun. Newton showed the orbit of such a particle is an ellipse, assuming it doesn't fly off to infinity. There are many ways to prove this, but the most illuminating is to reparametrize time and think of the orbit as a circle in 4 dimensions. When the circle is projected down to 3-dimensional space, it becomes an ellipse. The animation in this post, created by Greg Egan, shows how this works. ]]>http://blogs.ams.org/visualinsight/2015/06/01/harmonic-orbit/feed/2Dodecahedron With 5 Tetrahedra
http://blogs.ams.org/visualinsight/2015/05/15/dodecahedron-with-5-tetrahedra/
http://blogs.ams.org/visualinsight/2015/05/15/dodecahedron-with-5-tetrahedra/#commentsFri, 15 May 2015 01:00:12 +0000http://blogs.ams.org/visualinsight/?p=1611Greg Egan shows 5 ways to inscribe a regular tetrahedron in a regular dodecahedron. The union of all these is a nonconvex polyhedron called the compound of 5 tetrahedra, first described by Edmund Hess in 1876.]]>http://blogs.ams.org/visualinsight/2015/05/15/dodecahedron-with-5-tetrahedra/feed/0