PENROSE TILING By Meghan Niswander

Original question  Is it possible for there to exist a non-periodic set of infinite or finite tiling.  Non- periodic means lacks transitional symmetry, so that the copies will never fully match the original.  For many years people didn’t think it was possible.

Hao Wang  He was the first to ask the question  This is when he came up with the Wang dominoes. He placed the same colors together. Reflections and rotations were not allowed. Wang conjectured that any set of tiles which can tile the plane can tile it periodically and showed that if this is the case, there is a decision procedure for such tiling.  He said that a set of non-periodic tiles could possible exist.

Wang dominoes

Then the critics came  In 1964 Robert Berger was the first to disprove Mr. Wang. He found a non- periodical set of 20,426 with the dominoes. He then was able to simplify it to 104  Next Donald Knuth reduced the number to a whopping 92  Raphael Robinson used what he called “jigsaw pieces” that worked the same as the colors and created a non-periodic prototile.

Roger Penrose enter the game  1974 Roger Penrose found tiling with six tiles That were non symmetrical in a plane.

Continued…  He then was able to lower the number to four.  With the help of other math mathematicians he was able to find a set of two!  First was a kite and a dart

Penrose Tiling

Note the angle measures of the kite and dart

And then came the Rhombus  Penrose then discovered he could also make the aperiodical set of tiles using skinny and fat rhombs  The thin rhombus has two angles measured at 36˚ and the other two measuring at 144˚  The fat rhombus has two angles measuring at 72˚ and the other two at 108˚