Tile Color Matching Using Simple Universal Cycles

Year: 2010 Authors: Anna Virágvölgyi

Core claim

Universal cycles can generate visually appealing tile matching rules that allow many valid rotations and interconnections in square tilings.

Topics

edge-colored tilings, universal cycles, parquet patterns, tile matching

Domains

combinatorics, discrete mathematics, word sequences, tiling design, pattern design, visual art

Methods

constructive combinatorics, sequence enumeration, pattern visualization

Media

square tiles, diagonal stripes, edge colors, bead-like symbols

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2010: Mathematics, Music, Art, Architecture, Culture

Tile Color Matching Using Simple Universal Cycles

Anna Virágvölgyi

Sip u. 6. Budapest H-1075, HUNGARY

E-mail: viragvolgyi.anna@gmail.com

Abstract

In a square tiling, one can mark squares using edge-colored matching rules. I describe a set of matching rules based on universal cycles. These arise when one studies arrangements of different letters from a small alphabet into a single sequence in which all possible permutation of a given length can be found. The results are interesting visually. They may have applications in creating parquet or other two dimensional tiling patterns.

Elements

Consider an alphabet of $k$ letters, where $k$ ( k > 2 ) is always odd. Using this alphabet, create a set of words where each word is of even length $2n$ , but where no adjacent letters in the word are the same. For each $n$ , if we ignore the direction of reading, the number of possible words is $S_n=k(k-1{)}^{2(n-1)}$ .

A universal cycle is a compact listing of a class of combinatorial objects [1]. One can prove that for the above sets of words exist universal cycles. An unwrapped universal cycle for $n=3$ , $S_3=3\times 2^4=48$ with alphabet $\{\mathbf{a},\mathbf{b},\mathbf{c}\}$ :

abcabababcbcbacabcacbacabacbcaacabcacabcb

This cyclic string contains $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{c}$ equally 16 times. Each above defined word of length 6 occurs exactly once on this cycle:

ababac

babaca

abacac…etc.

Each letter $(\mathbf{a},\mathbf{b},\mathbf{c})$ occurs in the entire set of words the same number of times $-2n(k-1{)}^{2n-2}=2\times 3\times 2^4=96$ With other symbols $(\mathbf{a}=\bullet ,\mathbf{b}=\bullet ,\mathbf{c}=◯)$ the above chain is:

By substituting stripes for beads due to the nature of universal cycles each elements of $S_3$ one can get as diagonal striped square tiles.

Figure 1: Unwrapped universal cycle with the $S_3=48$ elements.

Areas of the different colors in the Figure 1 are equal to each other. The picture shows how a great number of possible interconnection are between this tiles. This feature enable the tiles to be matched in many ways. The rotated tiles can be matched as well.

Virágvölgyi

Patterns

a

b

c Figure 2

Each half-tile can fit seven $(2^3-1)$ others. Tiles in an arrangement are different and none of them earn distinction. Nevertheless the form of arrangements contiguous to the position of the tile.

The white circles show localization of a certain tile in different arrangements. A tile’s rotation and the combination of the eight fitted tiles (framed with dark line on c picture of Figure 2) determine the form of the pattern.

The number of the edges of tiles on the circumferences of the pictures are different. Figure 2’s arrangements have less exterior and more interior tiles than Figure 3’s ones. The structures of the b pictures in Figure 2 and 3 are on the right.

On the left the rectangular arrangement has 17 “eyes” $(\equiv \bullet )$, while in the cross (structure on the right) are only 16 ones.

a

b

c Figure 3

In spite of no square is identical to any others and there are no special symmetries or groups in the pattern of pictures the form of pictures are symmetrical.

Reference

[1] G. Brockman, B. Kay, E. E. Snively, On Universal Cycles of Labeled Graphs, The Electronic Journal of Combinatorics 17 (1), r4. <http: volume_17="" pdf="" v17i1r4.pdf="" www.combinatorics.org="">

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