Is it possible to tile a regular pentagon

He discovered five classes of pentagon that can each be described by an equation. For the curious, the equations are here. Most people assumed Reinhardt had the complete list until half a century later in when R. Kershner found three more. Richard James brought the number of types of pentagonal tile up to nine in An amateur mathematician, Rice developed her own notation and method and over the next few years discovered another four types of pentagon that tile the plane.

In Rolf Stein found a fourteenth. Way to go! But then the hunt went cold. Do you have to show this with induction evidence? Toby Mak Toby Mak Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The unofficial elections nomination post.

Visit chat. Linked Related 1. The teachers cut out multiple copies of their pentagon and arranged them on tables. Some of the pentagons proved easy to tessellate. In other cases, the participants reached the decision that the shape would not tessellate. As the teachers struggled to get some of the shapes to tessellate, we gave some additional information about the sizes of the angles, in the hope that this would allow them to see arrangements of angles that would add to degrees.

Some of the groups found the additional information helpful and were then able to create a tessellation, but some groups still struggled with the more difficult shapes. After each group had time to explore and share their experiences, they learned that all of the pentagons provided would cover the plane. This lead to a discussion of the history of tessellating with convex pentagons. At this point, the teachers were informed that Shape E had been discovered only a few months prior to our meeting and was the fifteenth type of polygon that would tessellate.

This led to an exciting discussion of the methods used to discover this new type. To end the meeting, we shared links to recent press items. Working with newly discovered mathematics was a thrilling experience for all. There were no textbooks or daunting formulas, just observations, patterns, discussions, and lots of inquiry. This session could be part of a larger series of sessions on tilings. The exhaustive proof also helps direct the search for the hypothetical einstein — that coveted puzzle piece that locks together with itself in an ever-changing sequence of tile orientations.

Since all 15 of the tessellating convex pentagons and all other convex polygons tile the plane periodically, meaning in a sequence of tile orientations that regularly repeats, the einstein, if it exists, must be concave, with jagged corners that bend both inward and outward like the corners of a star.

That such an elusive shape would be needed to tessellate the plane nonperiodically only adds to its allure. Nonperiodic tilings exist when you have tiles of at least two different shapes to play with —an example is the famous Penrose tiling — or when using a bizarre tile consisting of parts that are not connected, called the Socolar-Taylor tile.

But whether a single connected tile exists that can do the job, and what its properties might be, remains unknown. Researchers have already proved that no algorithm exists that can decide if an arbitrary collection of different shapes tiles the plane. In a backward kind of way, this would imply the existence of the einstein tile. The existence of the einstein tile and the hardness of the single-tile decision problem go hand in hand. Recently, Rao and a collaborator proved a different result about nonperiodic tilings of Wang tiles — squares whose colored edges can only be placed side by side if the colors match.

Previous work had demonstrated that collections of Wang tiles exist that only give rise to nonperiodic tilings.