Greg's Math Stuff

GIF:  YUKI’S Ilustration Works on Tumblr (Momoimo514.tumblr.com)

Other images: I made it with math ideas from Formulae of the year 2016 on Math.stackexchange.

Moebius Menger Sponge 3D Stereogram.

The coordinates of vertexes of Regular Dodecahedron and Icosahedron is formulated very simply with Golden Ratio(φ).

Teake Nutma, The Exceptional Symmetry of E8.

This movie attempts to show the beautiful symmetry of the exceptional Lie group E8. Out of all the known Lie groups, E8 stands out as the largest and most complex exceptional group. It has 248 generating elements, which by themselves have an astounding degree of symmetry. This symmetry can only be fully grasped in 8-dimensional space. But luckily it is also possible to project E8 onto a two-dimensional plane, chosen such that the resulting image preserves a small fraction of its total symmetry. There are different choices for these projections, some preserving more symmetry than others. The movie rotates through a selection of projections in succession.

Dijkstra’s algorithm - Simple Path Finding

Dijkstra’s algorithm, conceived by computer scientist Edsger Dijkstra in 1956 and published in 1959, is a graph search algorithm that solves the single-source shortest path problem for a graph with non-negative edge path costs, producing a shortest path tree. This algorithm has applications in game design, artificial intelligence (AI), and robotics.

This first gif is an llustration of Dijkstra’s algorithm for finding path from a start node (lower left, red) to a goal node (upper right, green) in a robot motion planning problem. Open nodes represent the “tentative” set. Filled nodes are visited ones, with color representing the distance: the greener, the farther. Nodes in all the different directions are explored uniformly, appearing as a more-or-less circular wavefront as Dijkstra’s algorithm uses a heuristic identically equal to 0.

The second gif a graphical example of Dijkstra’s algorithm. It picks the unvisited vertex with the lowest-distance, calculates the distance through it to each unvisited neighbor, and updates the neighbor’s distance if smaller.

(if you would like to see and example of Dijksta’s algorithm written in C, C++, or Java click here!)

In an important field of mathematics called topology, two objects are considered to be equivalent, or “homeomorphic,” if one can be morphed into the other by simply twisting and stretching its surface; they are different if you have to cut or crease the surface of one to reshape it into the form of the other.

Consider, for example, a torus — the dougnut-shape object shown in the intro slide. If you turn it upright, widen one side and indent the top of that side, you then have a cylindrical object with a handle. Thus, a classic math joke is to say that topologists can’t tell their doughnuts from their coffee cups.

On the other hand, Moebius bands — loops with a single twist in them — are not homeomorphic with twist-free loops (cylinders), because you can’t take the twist out of a Moebius band without cutting it, flipping over one of the edges, and reattaching.

Topologists long wondered: Is a sphere homeomorphic with the inside-out version of itself? In other words, can you turn a sphere inside out? At first it seems impossible, because you aren’t allowed to poke a hole in the sphere and pull out the inside. But in fact, “sphere eversion,” as it’s called, is possible.

Incredibly, the topologist Bernard Morin, a key developer of the complex method of sphere eversion shown here, was blind.

Eigenvectors and Eigenvalues.

Transform the unit Hue disc.

Pick the same colors. It’s eigenvectors!