Kepler Triangle and Tiling

Kepler Triangle and Tilings Dr N K Srinivasan

Introduction Kepler triangle is a unique right-angle triangle with the sides in the ratio: 1 : sqrt(phi) : phi, where phi is the Golden Ratio GR = 1.618...... We can build tiling patterns with this triangle which is aperiodic , like the well-known Roger Penrose tiling.

Kepler triangle and tiling patterns were explored by the famous astronomer Johannes Kepler and later several artist-mathematicians developed this topic.

TILINGS Let us briefly discuss the tiling patterns that are familiar to all of us...in floor tiling, wall hangings, quilts and so on. Simple tiling can be made with squares and rectangles as



most tilings are. The tiling pattern should satisfy the criteria that the tiles fit well without gaps or overlap between the tiles. Moving away from the square and the rectangle, it is easy to build tiling using regular polygons of side 3 and 6---that is, using equilateral triangles and hexagons of equal sides. These tilings are periodic , in the sense that they have translational, rotational and reflective symmetries, over the enitire area of tiling. What about tilings made out of regular pentagon with all the five sides of equal length and the internal angle of (n-2)pi/n = (3/5)180 = 108. It turns out that we cannot make a periodic tiling with the regular pentagon, but only "aperiodic tiling" in which we have local five -fold symmetry, but that does not extend over larger areas. While many early artists used periodic tilings---for instance Albrecht Durer and M S Escher, with remarkable pictures, Roger Penrose investigated the construction of tilings using regular pentagons.



Roger Penrose [ a professor of mathematics and astro-physicist at Oxford University] developed aperiodic tiling with five fold symmetry using isosceles triangles of the kind: 72-36-72 angles and 36-108-36 angles. Two such triangles can be combined into rhombuses,called thick and thin rhombuses.

Penrose and Martin Gardener also showed how these tilings can be made using 'kite' and 'dart' configuration as "prototiles" with the triangle 72-36-72. [Note that cos {pi/5] = cos 72 is close to phi/2=0.806.]

Kepler triangles for tiling Kepler triangles embody the Golden ratio and can be easily used for aperiodic tiling. I have been studying this and one small tiling is shown in the figure.





Kepler Triangle The tiles are made of the right triangle with the ratio: 1 : sqrt(phi) : phi. Note that this ratio satisfies the Pythagorian theorem and the sides are in geometric ratio.





Kepler Tiling Further, the area of this triangle is just (1/2)phi = 1.272/2 = 0.636 One can select different colors for the tiling and I have used three primary colors of Yellow,Red and Blue for aesthetic reasons. This tiling could be of greater interest because of simplicity and ease of construction, compared to other aperiodic tilings.



I hope artists and architects make wider use of these tilings. Contact: [email protected] -----------------------------------------------------