SUMMARY / RELATED TOPICS

7-cubic honeycomb

The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation in Euclidean 7-space. It is analogous to the cubic honeycomb of 3-space. There are many different Wythoff constructions of this honeycomb; the most symmetric form is regular, with Schläfli symbol. Another form has two alternating 7-cube facets with Schläfli symbol; the lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol 7. The, Coxeter group generates 255 permutations of uniform tessellations, 135 with unique symmetry and 134 with unique geometry; the expanded 7-cubic honeycomb is geometrically identical to the 7-cubic honeycomb. The 7-cubic honeycomb can be alternated into the 7-demicubic honeycomb, replacing the 7-cubes with 7-demicubes, the alternated gaps are filled by 7-orthoplex facets. A quadritruncated 7-cubic honeycomb, contains all tritruncated 7-orthoplex facets and is the Voronoi tessellation of the D7* lattice. Facets can be identically colored from a doubled C ~ 7 ×2, alternately colored from C ~ 7, three colors from B ~ 7, 4 colors from D ~ 7, symmetry.

List of regular polytopes Coxeter, H. S. M. Regular Polytopes, Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 H. S. M. Coxeter and Semi-Regular Polytopes III

Aperiodic tiling

An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic patches. A set of tile-types is aperiodic; the Penrose tilings are the best-known examples of aperiodic tilings. Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subsequently won the Nobel prize in 2011. However, the specific local structure of these materials is still poorly understood. Several methods for constructing aperiodic tilings are known. Consider a periodic tiling by unit squares. Now cut one square into two rectangles; the tiling obtained in this way is non-periodic: there is no non-zero shift that leaves this tiling fixed. But this example is much less interesting than the Penrose tiling. In order to rule out such boring examples, one defines an aperiodic tiling to be one that does not contain arbitrarily large periodic parts. A tiling is called aperiodic; the hull of a tiling T ⊂ R d contains all translates T+x of T, together with all tilings that can be approximated by translates of T.

Formally this is the closure of the set in the local topology. In the local topology two tilings are ε -close if they agree in a ball of radius 1 / ε around the origin. To give an simpler example than above, consider a one-dimensional tiling T of the line that looks like...aaaaaabaaaaa... where a represents an interval of length one, b represents an interval of length two. Thus the tiling T consists of infinitely many copies of a and one copy of b. Now all translates of T are the tilings with one b as else; the sequence of tilings where b is centred at 1, 2, 4, …, 2 n, … converges - in the local topology - to the periodic tiling consisting of as only. Thus T is not an aperiodic tiling, since its hull contains the periodic tiling...aaaaaa.... For well-behaved tilings holds: if a tiling is non-periodic and repetitive it is aperiodic; the first specific occurrence of aperiodic tilings arose in 1961, when logician Hao Wang tried to determine whether the Domino Problem is decidable — that is, whether there exists an algorithm for deciding if a given finite set of prototiles admits a tiling of the plane.

Wang found algorithms to enumerate the tilesets that cannot tile the plane, the tilesets that tile it periodically. In 1964 Robert Berger found an aperiodic set of prototiles from which he demonstrated that the tiling problem is in fact not decidable; this first such set, used by Berger in his proof of undecidability, required 20,426 Wang tiles. Berger reduced his set to 104, Hans Läuchli subsequently found an aperiodic set requiring only 40 Wang tiles. An smaller set of six aperiodic tiles was discovered by Raphael M. Robinson in 1971. Roger Penrose discovered three more sets in 1973 and 1974, reducing the number of tiles needed to two, Robert Ammann discovered several new sets in 1977; the aperiodic Penrose tilings can be generated not only by an aperiodic set of prototiles, but by a substitution and by a cut-and-project method. After the discovery of quasicrystals aperiodic tilings become studied intensively by physicists and mathematicians; the cut-and-project method of N. G. deBruijn for Penrose tilings turned out to be an instance of the theory of Meyer sets.

Today there is a large amount of literature on aperiodic tilings. There are a few constructions of aperiodic tilings known; some constructions are based on infinite families of aperiodic sets of tiles. Those constructions which have been found are constructed in a few ways by forcing some sort of non-periodic hierarchical structure. Despite this, the undecidability of the Domino Problem ensures that there must be infinitely many distinct principles of construction, that in fact, there exist aperiodic sets of tiles for which there can be no proof of their aperiodicity. To date, there is not a formal definition describing; as with the term "aperiodic tiling" itself, the term "aperiodic hierarchical tiling" is a convenient shorthand, meaning something along the lines of "a set of tiles admitting only non-periodic tilings with a hierarchical structure". Each of these sets of tiles, in any tiling they admit, forces a particular hierarchical structure.. No tiling admitted by such a set of tiles can be periodic because no single translation can leave the entire hierarchical structure invariant.

Consider Robinson's 1971 t