Harriss Tiles

A generalization of a rhomb tiling for arbitrary n>3 was introduced by E.O. Harriss. It is based on an edge sequence of (0 ,1 ,-1 ,0). Using our treatment and notation  the basic substitution tile matrix for this edge sequence is

(1)   \begin{equation*}S_s^n=\begin{bmatrix}T_s^n &T_{s+1}^n & T_{s-1}^n & T_{s}^n  \\T_{s-1}^n &T_{s}^n  & T_{s-2}^n & T_{s-1}^n  \\T_{s+1}^n &T_{s+2}^n & T_{s}^n  & T_{s+1}^n  \\T_s^n &T_{s+1}^n & T_{s-1}^n & T_{s}^n  \\\end{bmatrix}\end{equation*}

To reduce the size of the prototile set for odd n, Harriss used a substitution rule in which the s\pm1 tiles are replaced by the n-s\mp1 tiles or

(2)   \begin{equation*}S_s^n=\begin{bmatrix}T_s^n &T_{n-s-1}^n & T_{n-s+1}^n & T_{s}^n  \\T_{n-s+1}^n &\underline{T_{s}^n}& \underline{T_{s-2}^n}& T_{n-s+1}^n  \\T_{n-s-1}^n &\underline{T_{s+2}^n }&\underline{T_{s}^n} & T_{n-s-1}^n  \\T_s^n &T_{n-s-1}^n & T_{n-s+1}^n & T_{s}^n  \\\end{bmatrix}\end{equation*}

Note that the pair of s-tiles in the center have to be rotated over \pi in order to fit in for higher generations. The twofold rotation is signified by underlining the matrix entry.

In the figure below the substitution tiles for n=5 are shown for both cases.

Fig. H1. Comparison of our basic substitution rule for n=5 and an edge sequence (0 ,1 , -1, 0) and Harriss’s substitution rule for odd n and even s.

To apply the basic substitution rule eq. (1) all possible prototiles, the positive as well as the negative ones, have to be included. The tiles in the s=4 tile can be rearranged to avoid the negative s=6 tile, but it is not possible to circumvent the s=7 negative tile in the s=5 substitution tile. And although not visible the s=5 tile (a straight line) has to be used in the s=4 substitution tile as will become apparent in the next generation supertiles.

For odd n, Harriss’s substitution rule separates the prototiles into two groups, the odd and even s prototiles. Formally, the zero area and negative tiles are also involved. Nevertheless, the set of even tiles may be reduced to two positive tiles, the s=2 and the s=4 tiles, by neglecting the s=0 tile, and by  letting the s=6 negative tile annihilate a s=4 tile and a rearrangement of the remaining tiles. In the set of odd s tiles, s=1 , 3, 5, the s=5 tile has to be used. Because this tile sonsists of congruent negative and positive parts, its role is to move a patch of tiles from a n overlapping to an empty part of the substitution tile. Harriss’s solution was to add substitution tiles with additional or missing prototiles. Our treatment gives the same result, but in a simpler, more straightforward way.

There are many other Harriss-like tilings without negative tiles for edge sequences with one or more \pm1 pairs and starting with a 0 to avoid crossing of neighboring edges.

(0 ,1,-1) edge with substitution tile matrix:

(3)   \begin{equation*}S_s^n=\begin{bmatrix}T_s^n &T_{n-s-1}^n & T_{n-s+1}^n\\T_{n-s+1}^n &\underline{T_{s}^n}& \underline{T_{s-2}^n} \\T_{n-s-1}^n &\underline{T_{s+2}^n }&\underline{T_{s}^n} \\\end{bmatrix}\end{equation*}

The right column and last row  of the Harriss tile have been removed. The inflation factor is 1+2\cos{\pi/n}

(0 ,1 ,0 ,-1 ,0) edge with substitution tile matrix:

(4)   \begin{equation*}S_s^n=\begin{bmatrix}T_s^n &T_{n-s-1}^n & T_{s}^n & T_{n-s+1}^n & T_{s}^n  \\T_{n-s+1}^n &\underline{T_{s}^n}& T_{n-s+1}^n & \underline{T_{s-2}^n}& T_{n-s+1}^n  \\T_s^n &T_{n-s-1}^n &T_s^n & T_{n-s+1}^n & T_{s}^n  \\T_{n-s-1}^n &\underline{T_{s+2}^n }&T_{n-s-1}^n &\underline{T_{s}^n} & T_{n-s-1}^n  \\T_s^n &T_{n-s-1}^n &T_s^n & T_{n-s+1}^n & T_{s}^n  \\\end{bmatrix}\end{equation*}

An extra row and column of prototiles have been added in the middle of the  Harriss tile. In this case a rearrangement of the prototiles s= 4 is not even needed. The inflation factor is 3+2\cos{\pi/n}.

A second set of substitution tiles with a different edge shape (or substitution tile edge sequence) may be obtained by rotating all prototilesover \pi in the above Harriss-like substitution tiles.

Harriss-like tilings
Fig. H2. Harriss-like tilings