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:

Illustration: Craig Kaplan

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:

Diagram of a grid-based evolving path. Illustration: Craig Kaplan

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:

Grid-based parquet deformation. Illustration: Craig Kaplan

And here are some more:

Funky tiles. Illustration: Craig Kaplan

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:

Iteration rule. Illustration: Craig Kaplan

Deformations like this will create ‘fractal’ tiles.

Iteration deformations. Illustration: Craig Kaplan

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.

Organic labyrinth growth. Illustration: Craig Kaplan

Craig has also made an Islamic parquet deformation.

Islamic tiling. Illustration: Craig Kaplan

One that goes in two dimensions:

2D parquet deformation. Illustration: Craig Kaplan

And one that is shaped in a circle:

Circular deformation. Illustration: Craig Kaplan

Speaking of circles, he turned this design…

Illustration: Craig Kaplan

…into a ring, which is on sale at the 3-D printers Shapeways.

Deformed ring. Illustration: Craig Kaplan

Further reading:

Craig Kaplan: Metamorphosis in Escher’s Art



Craig Kaplan: Curve Evolution Schemes for Parquet Deformations



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