Is it possible to tile a floor with regular pentagons

A tessellation , or tiling, is the covering of the plane by closed shapes, called tiles, without gaps or overlaps [17, page ]. Tessellations have many real-world examples and are a physical link between mathematics and art.

Artists are interested in tilings because of their symmetry and easily replicated patterns. Darrick Koitsch Explainer. What are two shapes that have 5 sides? A five-sided shape is called a pentagon. A six-sided shape is a hexagon, a seven-sided shape a heptagon, while an octagon has eight sides…. Siegfried Anchel Explainer.

What is the sum of angles in a pentagon? Here's a pentagon , a 5-sided polygon. From vertex A we can draw two diagonals which separates the pentagon into three triangles. We multiply 3 times degrees to find the sum of all the interior angles of a pentagon , which is degrees. Danguole Klefoth Explainer. Do all triangles tessellate? Since all types of triangles contain angles that are divisors of , all types of triangles can tessellate.

Ferdinando Beloqui Pundit. What does tile the plane mean? A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles , with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Weam Quer Pundit. Why do some shapes tessellate? Some shapes cannot tessellate because they are not regular polygons or do not contain vertices corner points.

They therefore cannot be arranged on a plane without overlapping or leaving some space uncovered. Due to its rounded edges and lack of vertices, the circle is normally not tessellated. Yurii Karger Pundit. Can a circle Tessellate yes or no? The answer is no , circles will not tessellate. Sanjuana Rouze Pundit. Does a trapezium Tessellate? Squares, rectangles, parallelograms, trapezoids tessellate the plane; each in many ways. Each of these can be arranged into an infinite strip with parallel sides, copies of which will naturally cover the plane.

A parallelogram is cut by either of its diagonals into two equal triangles. So far, we know of 15 types of pentagons which could fill the tile — many described by Reinhardt himself, several identified by other mathematicians , even amateurs. In , the 15th type was described, 30 years after the previous. But there was no definitive answer as to whether others also remained.

Rao started with a computer algorithm which generated all possible pentagonal shapes. In his new computer-assisted proof , he used a computer algorithm and found a total of families of pentagons. Out of these, only 19 were convex and could successfully tile a plane.

As it turned out, four of these are particular cases of these 15 types, so lo and behold, 15 and only 15 types of pentagons can fill a tile. The einstein is a hypothetical shape that can only tile the plane nonperiodically, in a never-repeating orientation pattern. He sees this study not as a goal in itself, but rather as a milestone in a much larger quest. Join the ZME newsletter for amazing science news, features, and exclusive scoops.

More than 40, subscribers can't be wrong. As you can imagine, this only adds to its allure. From to , various contributors added to the list of tiling pentagons until there were fourteen known varieties. Those fourteen stood alone until a recent breakthrough at the University of Washington Bothell that added a fifteenth. David Von Derau arrived at the University of Washington Bothell seeking an undergraduate degree, but brought with him years of experience as a professional software developer.

McLoud-Mann and Mann recruited him to their project, provided him with their algorithm, and Von Derau programmed a computer to do the necessary calculations.