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Parquet is a common type of flooring that uses rectangular wooden tiles, which are usually arranged in a herringbone or parallel style.
In the 1960s an American architecture professor, William Huff, coined the term ‘parquet deformation’ to mean a regular pattern of tiles that transforms as you go from left to right whilst maintaining the regularity of the tiling.
Here’s an example:
Huff never made any floors like this – he was interested in the way that the pattern must be ‘read’ from one side to the other.
In this way, it makes the pattern a ‘temporal’ composition – and possibly the nearest that geometry comes to music, which is also appreciated temporally.
In recent years the parquet deformation has been rediscovered by Craig Kaplan, professor of computer science at the University of Waterloo, Canada, and a well-known mathematical artist.
“I am fascinated by geometric designs that depict processes of growth, evolution, or metamorphosis,” he says. “When the objects being transformed are tiles, the process must overcome an additional constraint: the tiles must jostle against each other to depict the overall process of evolution, while simultaneously meeting without any gaps or overlaps.”
The rules for parquet deformations are:
- the change happens in one dimension
- the tiling is always regular. (For more explanation see my post on tessellations and the mutt’s nuts.)
There are many strategies for making deformations. For example, the tile can evolve in a way that adds squares from a base grid:
The right side of the original tile is a line, three squares are added to make a new tile, then two squares and so on. The diagram also shows how the tiles fit together.
Here’s a pattern that was constructed like this:
And here are some more:
Another way is to use an iterative system, in which each line segment is replaced by a new path that includes smaller line segments, and then each new line segment is replaced by the same path on a smaller scale:
Deformations like this will create ‘fractal’ tiles.
A third strategy is to turn lines into “organic labyrinthine curves”, using an algorithm based on Brownian motion and developed by Hans Pedersen and Karan Singh.
Craig has also made an Islamic parquet deformation.
One that goes in two dimensions:
And one that is shaped in a circle:
Speaking of circles, he turned this design…
…into a ring, which is on sale at the 3-D printers Shapeways.
Further reading:
Craig Kaplan: Metamorphosis in Escher’s Art
Craig Kaplan: Curve Evolution Schemes for Parquet Deformations
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