Roice's G+ Posts

It feels silly to reshare John’s posts because anyone following me is surely following him too, but I can’t help myself when he posts about things I’ve worked on! Originally shared by John Baez The beauty of hyperbolic heptagons Check out Roice Nelson's new picture! This picture lives in hyperbolic space, which been squashed down to a ball. The 'dents' are hyperbolic planes tiled by regular heptagons, each subdivided into 7 red and 7 blue triangles. These triangles don't look like they have the same size - but in 3d hyperbolic space they do! The problem is that we've squashed hyperbolic space down to a ball. It's impossible to fit hyperbolic space in ordinary Euclidean space without doing violence to it. Hyperbolic space has a group of symmetries called the Lorentz group . This is famous in special relativity: it's the group containing rotations and also Lorentz transformations. The Lorentz group has symmetries that can map any of the red or blue tr...

I enjoyed this talk, especially the latter part.  It was interesting to hear about all the thought that has gone into hyperbolic manifolds with infinite volume and "exotic manifolds" that are "geometrically infinite" (a distinction I do not yet understand).  Thanks for your post Refurio Anachro! Originally shared by Refurio Anachro The Royal Institution was founded in 1799, and since the beginning it has been supporting public engagement with science. In their gorgeous premises they've been hosting the "Friday Evening Discourses" pop science lectures since 1825, nowadays recorded on video for you to watch. Those are for everyone and it's not unusual for them to have many kids in the audience. A recent one is "Topology, Geometry and Life in Three Dimensions" given by Caroline Series . She is striving hard to explain William Thurston 's geometrization conjecture for the uninitiated, and it's the first recording where i saw the au...

Originally shared by John Baez A sunflower at infinity This picture by Roice Nelson shows the 'view at infinity' of a honeycomb in hyperbolic space. A honeycomb is a way of chopping space into polyhedra.  For example, we can chop ordinary 3d space into cubes.  This is called the {4,3,4} honeycomb .  Why? • a square has 4 sides so its symbol is {4} • a cube has 3 squares meeting at each corner so its symbol is {4,3} • the cubical honeycomb has 4 cubes meeting at each edge so its symbol is {4,3,4} The picture here is a view of the {3,3,7} honeycomb .  This is defined in the same sort of way, but it doesn't fit into ordinary Euclidean space.  It fits into a curved space called hyperbolic space!    The honeycomb extends forever, and it forms this pattern where it meets the 'plane at infinity' of hyperbolic space. For links to related pictures, visit my American Mathematical Society blog Visual Insight : http://blogs.ams.org/visualinsight/2014/09/01/inters...

It feels great contributing to Visual Insight images! Originally shared by John Baez {7,3,3} meets the plane at infinity Maryam Mirzakhani won the Fields medal largely for her work on hyperbolic geometry.  The hyperbolic plane is something we can all enjoy: it's not flat, but curved like a saddle, and triangles have angles that add up to less than 180 degrees.  We can draw it crushed down to a disk, as in this picture by Roice Nelson.  All the light blue circles should really be the same size - but the ones near the edge have been squashed. Instead of learning Euclidean geometry in school, you could have learned hyperbolic geometry - it's almost the same, with some big differences.  The main thing is that the parallel postulate is false in hyperbolic geometry: there are many ways to draw a line through a point parallel to another line!  This is how Lobachevsky  discovered hyperbolic geometry in 1823: by seeing what would happen if you changed the rules t...

Originally shared by John Baez A honeycomb in hyperbolic space Hyperbolic space is a 3d space where the angles of a triangle add up to less than 180 degrees.  A new kind of space lets you imagine new kinds of patterns - mathematical art that doesn't quite fit into our universe! This drawing by Roice Nelson shows what you'd see if you lived in hyperbolic space and filled this space with a {6,3,3} honeycomb . Hyperbolic space is curved, but each hexagon here lies on a flat plane inside hyperbolic space.  Each of these plane is tiled with hexagons in the usual way, 3 meeting at each corner.  However, each edge in this picture has 3 different planes like this going through it!   Planes of hexagons, 3 hexagons meeting at a corner, with each edge lying on 3 planes - that's the reason for the symbol {6,3,3}. For much more about this honeycomb and its relatives, visit my Visual Insight blog: http://blogs.ams.org/visualinsight/2014/03/15/633-honeycomb/ It contains a p...

Originally shared by Erno Rubik A 5-dimensional Analogue of the Rubik's Cube You can now try to solve a 5-dimensional Rubik's Cube. How many permutations of the 3x3 cube in 5-dimensions?  Approx. 7.0 x 10^560 Good luck. Visit here to download the program and learn more: http://www.gravitation3d.com/magiccube5d/index.html #mathematics   #geometry   #rubikscube