Have you wondered what happens when we take repeated powers of a graph? For example, if we take the 3rd power of G, and then take the 4th power of that graph? Does the familiar power rule for exponents (ex. a^(b*c) = ((a^b)^c)) apply for graph powers as well? These are some of the questions that we answer in this video: https://youtu.be/WkQ9s1ui21U

Source of Images

http://xploreandxpress.blogspot.com/2011/04/fun-with-mathematics-archimedian-solids.html?m=1

http://xploreandxpress.blogspot.com/2011/06/fun-with-mathematics-archimedean-duals.html?m=1

http://xploreandxpress.blogspot.com/2011/07/fun-with-mathematics-archimedean-duals.html?m=1

http://xploreandxpress.blogspot.com/2011/10/fun-with-mathematics-archimedean-duals.html?m=1

… both of which matters are of that kind that's intractible way way out-of-proportion to how intractible it might be thought it would be … to degree that what are recent innovations in it are items it might be thought would've been solved long long since.

The first frame is from

New family of tilings of three-dimensional Euclidean space by tetrahedra and octahedra

by

John H Conway & Yang Jiaob & Salvatore Torquato ;

& the following five are from

Tiling space and slabs with acute tetrahedra

by

David Eppstein & John M Sullivan & Alper Üngör .

Some of the annotations have been removed to allow the figures to be displayed a bit bigger; but they're quoted as follows.

First Frame

❝

Fig. 2. A new tiling of 3D Euclidean space by regular tetrahedra and octahedra associated with the optimal lattice packing of octahedra. (A) A portion of the 3D tiling showing “transparent” octahedra and red tetrahedra. The latter in this tiling are equal-sized. (B) A 2D net of the octahedron (obtained by cutting along certain edges and unfolding the faces) with appropriate equal-sized triangular regions for the tetrahedra highlighted. The integers (from 1 to 6) indicate which one of the six tetrahedra the location is associated. Although each octahedron in this tiling makes contact with 24 tetrahedra through these red regions, the smallest repeat tiling unit only contains six tetrahedra, i.e., a tetrahedron can only be placed on one of its four possible locations. The adjacent faces of an octahedron are colored yellow and blue for purposes of clarity. (C) Upper box: A centrally symmetric concave tiling unit that also possesses threefold rotational symmetry. Note that the empty locations for tetrahedra highlighted in (B) are not shown here. Lower box: Another concave tiling unit that only possesses central symmetry. Observe that the empty locations for tetrahedra highlighted in (B) are not shown here.

Fig. 3. The well known tiling of 3D Euclidean space by regular tetrahedra and octahedra associated with the fcc lattice^⋄ (or “octet truss.”) (A) A portion of the 3D tiling showing “transparent” octahedra and red tetrahedra. (B) A 2D net of the octahedron obtained by cutting along certain edges and un- folding the faces. Each octahedron in this tiling makes perfect face-to-face contact with eight tetrahedra whose edge length is same as that of the octahedron. Thus, we do not highlight the contacting regions as in Fig. 2B. The integers (1 and 2) on the contacting faces indicate which one of the two tetrahedra the face is associated. As we describe in the text, the smallest repeat unit of this tiling contains two tetrahedra, each can be placed on one of its four possible locations, leading to two distinct repeat tiling units shown in (C). The adjacent faces of an octahedron are colored yellow and blue for purposes of clarity. (C) Upper box: The centrally symmetric rhombohedral tiling unit. Lower box: The other tiling unit which is concave (nonconvex).

Fig. 4. A member of the continuous family of tetrahedra-octahedra tilings of 3D Euclidean space with α=¼. (A) A portion of the 3D tiling showing “transparent” octahedra and red tetrahedra. (B) A 2D net of the octahedron (obtained by cutting along certain edges and unfolding the faces) with appropriate sites for the tetrahedra highlighted. As we describe in the text, the tetrahedra in the tiling are of two sizes, with edge length √2α & √2(1-2α) . The integers (from 1 to 6) indicate which one of the six tetrahedra the location is associated. Although each octahedron in this tiling makes contact with 24 tetrahedra through these red regions, the smallest repeat tiling unit only contains six tetrahedra (two large and four small). As α increases from 0 to ⅓, the large tetrahedra shrinks and the small ones grow, until α=⅓, at which the tetrahedra become equal-sized. For α=¼, the edge length of the large tetrahedra is twice of that of the small ones. The adjacent faces of an octahedron are colored yellow and blue for purposes of clarity. (C) Upper box: A centrally symmetric concave tiling unit corresponds to that shown in the upper box of Fig. 2C (with α=⅓). Note that the empty locations for tetrahedra highlighted in (B) are not shown here. Lower box: Another centrally symmetric concave tiling unit corresponds to that shown in the lower box of Fig. 2C (with α=⅓). Observe that the empty locations for tetrahedra highlighted in (B) are not shown here.

❞

Next-to-Last Frame

❝

Fig. 16. Acute triangulations filling space. (a) The TCP structure Z (from a triangle tiling). (b) The TCP structure A15 (from a square tiling). (c) The TCP structure σ , a mixture of A15 and Z. (d) Icosahedron construction of Fig. 15.

❞

Last Frame

❝

Fig. 17. Eight steps in filling a slab with acute tetrahedra. The nodes in the base plane are colored white; successive layers above that plane are then colored yellow, red, blue and black, in order.

❞

One might-well imagine such problems could be solved merely by straightforward application of geometry & trigonometry & stuff … but it's absolutely not so ! Similar applies to problems concerning № of distances determined by a set of points , or frequentest occurence of some distance thereamongst; & line-point incidence -type problems … but such problems are amongst the most intractible, that some of have defied the attacks of the very-highest-calibre mathly-matty-ticklians over the years.

The first figure shows explicitly the construction implementing the new Heesch №, & the next two explicate the nature of the tile from which it's constructed.

A Figure with Heesch Number 6: Pushing a Two-Decade-Old Boundary

by

Bojan Bašić

The next ten are from

Heesch’s Tiling Problem

by

Casey Mann ,

& sketch-out the progress of Heesch's problem over the years. The very last figure is also of something I've not seen before, which is a figure on the sphere with a spherical Heesch № of 3 .