# Douglas G. Burkholder

Professor of Mathematics

Lenoir-Rhyne University

Hickory, North Carolina

For years, my love of geometry and my desire for visualization for the sake of mathematical understanding have been central to my teaching and to my comprehension of mathematical concepts. My recent work explores hidden beauty within Penrose Tilings. Penrose tiling is both mathematically and aesthetically interesting due to its non-periodic nature. My current pieces explore tiling of the small stellated dodecahedron with Penrose tiles. Although a Penrose kite exactly fits a pair of isosceles triangular faces of this surface, to complete the tiling we must use half-kites. Since this solid is artistically interesting, I chose to proceed by including half-kites and half-darts and call these Pseudo-Penrose tiling.

Stellated Penrose

50 x 50 cm

Digital Art

2017

Stellated Penrose is a pseudo-Penrose tiling of the small stellated dodecahedron containing 95,844 half-kites and 59,220 half-darts colored either red or black. The tiling starts from 60 half-kites each covering one isosceles triangle on the surface. The tiles are then subdivided into half-kites and half-darts for eight iterations. The final step is to label every tile A, B, C, D, or E based upon its location in the subdivision of the larger tile and then use paint-by-number to paint the tiles on each of the twelve pyramid points. Ten of the pyramid points have two of the five tile types painted red and two pyramid points have only one type painted red. For example, every tile labeled either B or E in the top pyramid point is painted red.