15º pentágono que cubre el plano

Hi rocketdan65,

My daughter brought this thread to my attention, and since I was already writing up a short background summary of the problem, I thought I'd post what I have so far here. It is hard to say what any of this means or why it is important. Mathematical facts have a funny way of being "important" to science and engineering in unexpected ways, but as a mathematician, I'd say our discovery is analogous to finding a new type of subatomic particle (maybe not that big of a deal, though!). It tells us a little more about the complexity with which shapes can fit together. I think the properties of how shapes fit together is going to have really important applications one day (for example, in future self-assembling nanobots that will enslave or destroy human kind).

Here’s a little background on the discovery we made last week. The problem has a very interesting history, so it is very exciting to make a small contribution to its solution. This is a bit long, but still not detailed enough to make complete sense to the uninitiated.

In 1900, the German mathematician David Hilbert presented a set of 23 mathematical problems that would greatly influence the direction of mathematical research in the 20th century. The 18th of Hilbert's problems asks if there exists a 3-dimensional shape (i.e. a tile), copies of which can tessellate all of 3-dimensional space, but the tile itself is never the “fundamental region” of these tilings. In mathematical terminology, Hilbert asked if there exists an anisohedral 3-D tile. Hilbert assumed that such a tile did not exist among 2-dimensional tiles; that is, he thought that all 2-dimensional tiles that can tessellate the plane must admit at least one “isohedral" tiling.

In 1928, Karl Reinhardt, a former assistant of Hilbert, solved the 18th problem by demonstrating the existence of a 3-dimensional anisohedral tile. Previous to this, Reinhardt had studied the problem in 2 dimensions, and in doing so, he classified all convex pentagons that admit isohedral tilings. There turned out to be 5 distinct types of isohedral pentagons, and these are now referred to as Types 1 - 5. It is reasonable to think that Reinhardt believed that his 5 types represented a complete classification of the convex pentagons that tile the plane since at the time it was thought that 2-D anisohedral did not exist. Interestingly, the 2-dimensional version of Hilbert’s 18th problem was solved by H. Heesch in 1935, after the 3-dimensional version was solved. Heesch demonstrated that 2-D anisohedral tiles do in fact exist. Heesch's discovery opened the door to the possibility of additional types of convex pentagons being found.

It was 1968 before further progress was made on the problem of classifying the convex pentagons that tile admit tilings of the plane. At that time, R. B. Kershner published the article On Paving the Plane in Mathematics Magazine in which he presented three new types of (anisohedral) convex pentagons that tile the plane (Types 6 - 8). Further, Kershner claimed in that article that he had completed the classification and that there were no more convex pentagons that tile the plane to be found. In 1975, in his Scientific American column, Martin Gardner reported on Kershner’s claims about the classification of convex pentagons that tile the plane. This brought the problem to a very wide audience.

Among those who read Gardner’s column were Marjorie Rice, a homemaker from California with only a minimum of mathematical training, and Richard James III, a computer scientist from California. Rice found 4 more types and James separately found another (Types 9 - 13). An interesting account of this part of the history of the problem can be found in D. Schattschneider’s 1978 Mathematics Magazine article Tiling the Plane with Congruent Pentagons. Later, in 1985, R. Stein, a graduate student from Germany, found a 14th type.

Since 1985, while good some progress has been made on the problem of classifying convex pentagons that tile the plane, no new types had been found until 2015 with our newly identified example. Hirschhorn and Hunt proved that all equilateral convex pentagons that tile the plane are among the Types 1-14 (1985), and more recently, O. Bagina has proven that all convex pentagons that admit edge-to-edge tilings of the plane are accounted for among types 1-14. The result of Hirschhorn and Hunt and that of Bagina directed our search; from these results, we knew we needed to search for non-equilateral convex pentagons that admit only non-edge-to-edge tilings. Our new type of pentagon was identified using a computer algorithm. Using some theoretical results we have proven that limits the parameters of the search to a finite (but large) number of possibilities, we were able to develop an algorithm that can identify, for each number i, all possible convex pentagons that admit i-block transitive tilings. Our UW Bothell student, David Von Derau, worked over the course of the past year to refine and implement our algorithms to be executed on the HYAK computing cluster at the UW. We were just in the process of debugging and optimizing the code when our new example was found. Because we are in the early stages of the computational experiments, we were surprised to find this example so quickly. We are hopeful of finding more new examples as we proceed.

Regards,

Casey

Hola rocketdan65,

Es difícil decir qué significa o para qué importa esto. Los hechos matemáticos tienen una manera extraña de ser útiles para la ciencia o las ingenierías, pero para las matemáticas este descubrimiento es tan importante como el de una nueva partícula subatómica (quizás no tanto!). Nos dice un poco más sobre la complejidad con la que las formas pueden encajar. Creo que las propiedades de cómo encajan las formas tendrán aplicaciones importantes algún día (por ejemplo, en un futuro nanorobots se podrían automontar de esta manera esclavizando o destruyendo la humanidad) (N. del T.: Automontan suena mal, además los nanobots no tienen por qué destruir la humanidad!)

Un poco de trasfondo sobre el descubrimiento que hicimos la semana pasada. El problema tiene una historia muy interesante así que es emocionante hacer una pequeña contribución a su solución.

En 1900, el matemático alemán David Hilbert presentó un conjunto de 23 problemas que influirían la dirección de la investigación matemática en el siglo 20. El 18avo problema pregunta si existe una forma tridimensional (i.e. un "tile"), copias de la cual puedan teselar todo el espacio, pero el "tile" (N. del T. a partir de ahora lo llamaré pieza) en sí mismo nunca es la "región fundamental" de estas teselaciones. En lenguaje matemático, Hilbert preguntó si existe una pieza anisohédrica de 3 dimensiones. Hilbert asumió que una pieza así no existe en 2 dimensiones, es decir, pensaba que todas las piezas que puedan teselar el plano necesitan admitir al menos una teselación "isohedrica" (isohédrica significa con ciertas simetrías).

En 1928, Karl Reinhardt, un antiguo ayudante de Hilbert, resolvió este problema demostrando la existencia de una pieza anisohédrica 3-dimensional. Previamente, Renhardt estudió el problema en 2 dimensiones y clasificó todos los pentágonos convexos que admiten teselaciones isohédricas.