![]() ![]() The Big List could not be written out and explored without these two changes. Second, the shift in notation from descriptive strings to sets of single-letter symbols. First, the shift of focus from the 14 vertices of regular polygons to the 14 dual polygons generated by those vertices. The Big List System is made possible by two simple shifts. ![]() What would happen if we tested every one of these 16383 combinations, in order to find out which combinations will tile the plane, without gaps or overlaps, in at least one way?ĪS FAR AS I KNOW, THIS QUESTION HAS NEVER BEFORE BEEN ASKED, LET ALONE ANSWERED. How many ways can these 14 letters be combined? The answer is: 16383 combinations in 14 Groups consisting of combinations of from 1 to 14 letters respectively. Let’s say that we label each of these 14 dual polygons (or, if you prefer, each of these 14 vertices of regular polygons- it amounts to the same thing) with a letter of the alphabet, A through N. For every tiling of regular polygons there is a tiling of dual polygons and vice versa. In dual tilings, these 14 vertices become 14 distinct dual polygons. Leaving out the octagon (which will tile only with the square), there are 14 ways that the 4 useable regular polygons (triangle, square, hexagon, and dodecagon) will fit around a vertex. There are 20 such tessellations (Krotenheerdt)” “A more precise term of demi-regular tessellation is 2-uniform tessellation (Grunbaum and Shepard). Caution is therefore needed in attempting to determine what is meant by ‘demi-regular tessellation.’ However, not all sources apparently give the same 14. “The number of demi-regular tessellations is commonly given as 14 (Critchlow, Ghyka, Williams, Steinhaus). Some authors define it as an orderly composition of the 3 regular tessellations and the 8 semiregular tessellations (which is not precise enough to draw any conclusions from), while others define it as a tessellation having more than one transivity class of vertices (which leads to an infinite number of possible tilings). “A demi-regular tessellation, also called a polymorph tessellation, is a type of tessellation whose definition is somewhat problematic. (See Section 6, below.)įrom Wolfram Alpha (Eric Weisstein), slightly condensed: (Both drawings shows the diameters.) Sometimes I used different drawings on different kind of shapes.AND, BELIEVE IT OR NOT, SOME COMBINATIONS YIELD ONLY TILINGS WHICH ARE NEITHER PERIODIC NOR APERIODIC. In the example the drawing of the two kind of shapes are identical - just they are adopted to the different ratios of the different rhombuses. In my drawings I used only straight lines, and the lines are always connected at the edges of the rhombuses. The original shapes of the Penrose-tiling can be replaced with any drawings. The small drawings shows the diameters of the shapes. On the figure there is the simplest example. The small drawings have a special property, the lines always intersect the edges of the rhombuses on the same place. (There is only one Penrose-tiling.) I changed the rhombuses of the Penrose-tiling with small drawings, so the original structure of the tiling is disappears. All of the 100 drawings represents the same part of the same Penrose-tiling. The Penrose-tiling has many exciting properties, but in the drawings I used one of them: the order of the shapes never repeats itself. The rule is: the light parts has to be fit to light parts, dark to dark ones. Gaps and horns could be used on their side (inherited from the Wang-dominoes), but the pattern is more clear. Easy to find a pattern which forces to fit them on the right way. There are a simple rule for them, they can not be fitted by any way. As it can be seen on the figure it is constructed from two kind of rhombuses. Penrose-tiling is aperiodic, so the order of the shapes never repeats itself. Penrose-tiling the pattern forces the aperiodic tiling Tiles can tile only nonperiodically called aperiodic tiling. Nobody could imagine, that there are shapes can tile only nonperiodically. Until the beginning of the eighties mathematicians used only this two classes. But one of the important properties seems very well: The shapes can tile both periodically and nonperiodically. Periodic and non-periodic tiling from two kind of quadrangles Randomly rotating the squares, the tiling is nonperiodic. They tile periodically on the first figure. There is a variation of the tiling with the squares: each square intersected by a straight line, which is not rectangular to the edge of the shape. In fact very simple to create a nonperiodic tiling. This does not mean, that their pattern should be chaotic or should not follow any rules. The outlined part can be refitted into the tilingĪn other class of tilings are nonperiodic. ![]()
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