A tiling of pentagons, ten at a vertex, is {5,10/3} or x5o10/3o.

The vertices correspond to the cyclotomic numbers CZ5, which is of some interest to those of us who play around with the connection between bases and polytopes. I discuss a corresponding calculation of the CZ15, the 15-gon being the smallest polygon of odd sides that one can reach the centre of. (Any polygon, that is not prime or a prime power will do).

See eg

https://www.tapatalk.com/groups/twelfty ... s-t69.htmlThese tilings are "piecewise discrete", which means that you can completely build any surtope and anything attached to it. The intersections are not 'seen' if there is no direct connection. It's taken to be on different floors or planes of existance. If you stood in a cell of these, you would see pentagons and bits of pentagons, but you can't "see around corners", that is, you can't see for example, the complete vertex-figure, just 360 of the 1080 degrees on offer.

The cyclotomic numbers CZ5 are C2D2 numbers, class-2 2-dimensional numbers. This equates to for example, folding a four-dimensional lattice onto two dimensions. The tilings {8,8/3} and {12,12/5} can actually be represented as for example {4,4}+i{4,4} and {3.6}+i{3,6}, that is, an ordinary square lattice, and one rotated 45 degrees. Every point of {4,3,3,4} appears as a different spot, and the petrie polygon of a tesseract, if in one projection is an octagon, in the other is an octagram.

The quasic-periodic nature comes when one is restricted to one floor (ie no intersections). You still get the various pentagonal defraction-lines, but it's not a pretty lattice like the tiling of squares. You can create rules as to what is allowable at a vertex, such as in Klitzing's thesis. For the class-2 systems, the general pattern is for closed holes of varying size. For class-3 and above, (such as {7,14/5} and {15,30/13}, the holes never close up generally.