Aesthetics of Aperiodic Tilings

Wang tilings seem to me to be visually more interesting than their plain or decorated aperiodic tiling generators. The various versions of the same Penrose P2 tiling below all look very uniform, with their randomness hardly registering, whilst the randomness of their Wang equivalents is much more apparent.

It also seems that the mathematical desire to reduce the number of tiles and colours to a minimum reduces the visual interest of Wang tilings.

The first three Wang tilings below are derived from Penrose P2 and Ammann A2 aperiodic tilings as described in “Wang Tiles and Aperiodic Tiling” where the Culik / Kari method of deriving the fourth tiling is also illustrated.

I am not sure if, in Cecil Belmond’s terms, the second tiling is more “alive” than the first, which is smoother and more evenly distributed. There is a tension in the second tiling caused by adjacent blocks and runs of similar colours that is not present in the first.

It is perhaps worth saying the obvious here, that Wang tiles can be seen as a means of delivering random sets of coloured square tiles because their edge colours have to match, thereby creating coloured squares.

With this in mind it can be seen that Wang tilings bear a close resemblance to a number of artists’ work.  The resemblances probably relate to the use of chance in the composition of these works. An early example is Ellsworth Kelly and his “Colors for a large Wall” 1951. Oil on canvas mounted on 64 wood panels; overall 7′ 10¼” x 7′ 10¼”.

Ellsworth Kelly had met John Cage and Merce Cunningham in Paris after the Second World War and the title of the painting below “spectrum colors arranged by chance II” 1951 clearly acknowledges the use of chance.

Bridget Riley has also produced a number similar works. “From Here” 1994  has connected blocks and runs of similar colour, that closely resemble the Wang version of the Ammann A2 24 tiles over 24 colours tiling.

Ignoring their superficial diagonality, Wang tilings also seem to bear a striking similarity to some of Gerhard Richter’s Colour Chart paintings such as his “4900 Colors” of 2007 that uses randomly selected colours and also appears to have clumps and runs of colours as one should expect with a truly random composition method (see “Find Your Own Space”).

Various versions consist of different numbers of panels but Version I  consists of  196 panels, each of which consists of 25 squares. Each individual panel is made up of spray-painted enamel squares that measure 9.7 x 9.7 cm.

In architecture Sauerbruch and Hutton’s High Bay Warehouse uses 16,000 25 x 160 cm metal panels in 20 colours.

High Bay Warehouse Dogern: Sauerbruch and Hutton 2002

Dr. Bernhard E. Kallup, Chairman of Sedus Stoll AG, comments: “We offer huge product variety for all potential applications. Our broad portfolio for administration, communication and regeneration scenarios includes millions of technical and visual options, to ensure that no restrictions are put on architects and special building project managers. The warehouse architects have symbolised this freedom in their facade design.”

Scale seems to be an important attribute of this scheme, projects with larger panels and or fewer colours seem much cruder, as in Morrison’s huge warehouse at Bridgewater just off the M5 with its enormous panels in just four colours. It looks much better in this photograph than it does in reality.

Apart from the physical size of the panels I think this relates directly to an understanding of the different number of tiles and colours in the Wang tiling examples above. That is 13 tiles over 5 colours or 16 tiles over 6 colours is too crude and 32 tiles over 16 colours is probably too bland and smooth. Leaving 24 tiles over 24 colours as offering the liveliest set that is not too crude or bland.

Bibliography

Grünbaum B., Shephard G. C. (1987)
Tilings and Patterns
W.H. Freeman and Company
New York
 
Kari J., 1996
A small aperiodic set of Wang Tiles
Discrete Mathematics Volume 160 Issue 1-3 Pages 259-264
 
Culik K., 1996
An Aperiodic Set of 13 Wang Tiles
Discrete Mathematics Volume 160 Issue 1-3 Pages 245-251
 
Belmond C., Smith J. (2007)
Informal, Pages 189-264
Prestel Munich, Berlin, London, New York

About Graham Shawcross

Architect PhD student at Edinburgh University Interested in order, rhythm and pattern in Architectural Design
This entry was posted in Aesthetics, Aperiodic Tiling, Architecture, Tiling and tagged , . Bookmark the permalink.

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