Technique · Fields
Hyperbolic tiling
A tiling of the hyperbolic plane in the Poincare disk, tiles shrinking toward the edge, slowly sliding through the disk.
Hyperbolic tiling is a field drawn for every pixel by the graphics chip. Every piece on Synapse Drip is made from a recipe that layers techniques like this one, and you can change any of it in the Studio.
Good to know
- Henri Poincare described this model of the hyperbolic plane in 1882; in it straight lines are drawn as circle arcs that meet the edge of the disk at right angles.
- The hyperbolic plane has constant negative curvature: the circumference of a circle there grows exponentially with its radius, which is why the tiles crowd up so fast toward the rim.
- In 1958 Escher made Circle Limit I, the first of four woodcuts in which an endless pattern shrinks toward the edge of a disk without ever reaching it.
- M. C. Escher made his four Circle Limit woodcuts between 1958 and 1960 after seeing a diagram by the geometer H. S. M. Coxeter, and Circle Limit III is built on a tiling like this one.
- The mathematician Daina Taimina crocheted the first model of the hyperbolic plane in 1997; a ruffle of kale, lettuce or coral has this geometry, because there is more edge than flat space can hold.
The mathematics
- The tiling is written {7, 3}: 7-sided tiles, 3 at each corner; it only fits the hyperbolic plane because the angles of a triangle with corners pi/2, pi/7 and pi/3 add up to less than pi.
- Each of the infinitely many triangles has the same area, pi (1/2 - 1/7 - 1/3), which is about 0.075 in the units where the curvature is minus one; on a flat plane the same triangle would have an area of zero.
- Folding by mirrors is exact: every reflection in a circle that meets the disk at right angles is a symmetry of the hyperbolic plane, so one fundamental triangle and three mirrors generate the whole tiling.
In art
- Escher's Circle Limit prints are the best-known hyperbolic art; in Circle Limit III the fish of one colour swim along arcs from edge to edge of the disk.
- Hyperbolic geometry has also been crocheted, knitted and folded in paper, in craft projects that began with Daina Taimina's crocheted models.
More ways to use it
Related techniques
Nebula and stars · Liquid · Droplets · Network · Terrain and maps · Neuron · Order to chaos · Fractal tree