Historias de pi: la cuadratura del círculo (à la Tarski)

La cuadratura del círculo con regla y compás, tal y como lo entendían los griegos, tiene otra interesante lectura que ha recibido un sorprendente avance en estos últimos años.

AlfredTarski

La idea ya no es construir el cuadrado partiendo del círculo de la manera tradicional, sino descomponerlo en trozos para con ellos, agrupados de la manera conveniente, construir un cuadrado.

Este cambio de reglas se debe al matemático polaco Alfred Tarski. Su nombre original era Alfred Teitelbaum. Nacido en Varsovia, en 1901, era de origen judío, de una familia acomodada. Cambió su apellido al convertirse al catolicismo.En 1939 emigró a los Estados Unidos de América, mientras que la mayor parte de su familia que permaneció en Polonia fueron asesinados por los nazis. Tarski es un matemático muy relevante, conocido sobre todo por sus resultados en teoría de conjuntos y lógica matemática, pero también en otras áreas.

Tarski se interesó por el problema de la cuadratura del círculo cambiando las reglas. Este tema de dividir y luego reunir de otra foma no era nuevo para Tarksi. En 1924, él y Stefan Banach demostraron que una bola puede cortarse en un número finito de trozos y volver a ensamblarse en una bola de mayor tamaño o, alternativamente, puede volver a ensamblarse en dos bolas cuyo tamaño sea igual al de la original. Este resultado se llama ahora la paradoja de Banach-Tarski.

Así que en 1925 Alfred Tarski (TARSKI, A. Probléme 38.Fund. Math. 7(1925), 381) reformuló la cuadratura del círculo preguntándose si se podía llevar a cabo la tarea dividiéndolo en un número finito de piezas que se pudieran mover dentro de un plano y volver a ensamblar en un cuadrado de igual área (es decir, que las dos figuras son equidescomponibles). Miklós Laczkovich demostró en 1990 que esto era posible en 1990; y estimó el número de piezas de su descomposición en aproximadamente 1050. Pero su demostración no era constructiva: se podía hacer pero no se sabía como srían las piezas.

Miklós Laczkovich

Łukasz Grabowski, András Máthé y Oleg Pikhurko, en 2016, consiguieron encontrar una demostración constructiva. Era posible excepto en un conjunto de medida cero (una idea intuitiva de un conjunto de medida cero la puede dar la medida de una colección finita de puntos o de segmentos en una plano). Y en 2017, Andrew Marks y Spencer Unger (2017) dieron una solución completamente constructiva utilizando alrededor de 10200 trozos.

La última vuelta de tuerca en la cuadratura del círculo se debe a los matemáticos Andras Máthé y Oleg Pikhurko, de la Universidad de Warwick, y Jonathan Noel, de la Universidad de Victoria. En un reciente preprint en arxiv han probado el resyultado pero con piezas de formas más sencillas y fáciles de visualizar.

Łukasz Grabowski
András Máthé
Oleg Pikhurko

 

Jonathan Noel

Los autores siguen trabajando y creen que pueden disminuir el número de piezas de forma considerable. Veremos lo que nos deparan los próximos años inmersos en estas sutilezas de la teoría de la medida y la combinatoria.

______________

Manuel de León (CSIC, Fundador del ICMAT, Real Academia de Ciencias, Real Academia Canaria de Ciencias, Real Academia Galega de Ciencias).

Compartir:

Entradas relacionadas

3.361 comentarios

  1. May I simply just say what a comfort to find someone who really understands what they’re discussing on the internet. You certainly know how to bring a problem to light and make it important. More and more people ought to look at this and understand this side of your story. I was surprised you are not more popular since you surely have the gift.

  3. At this time it seems like WordPress is the top blogging platform available right now. (from what I’ve read) Is that what you’re using on your blog?

  4. Hey I know this is off topic but I was wondering if you knew of any widgets I could add to my blog that automatically tweet my newest twitter updates. I’ve been looking for a plug-in like this for quite some time and was hoping maybe you would have some experience with something like this. Please let me know if you run into anything. I truly enjoy reading your blog and I look forward to your new updates.

  5. Aw, this was an extremely good post. Finding the time and actual effort to produce a good article… but what can I say… I procrastinate a whole lot and never manage to get anything done.

  6. Aw, this was a very nice post. Taking a few minutes and actual effort to create a very good article… but what can I say… I hesitate a lot and don’t manage to get anything done.

  13. Aw, this was an incredibly good post. Taking a few minutes and actual effort to create a great article… but what can I say… I put things off a whole lot and never manage to get nearly anything done.

  21. F168.com được nhiều người tìm kiếm khi muốn tham gia các trò chơi giải trí trực tuyến. Chỉ cần mở đường dẫn, bạn đã có thể bước vào thế giới activity đa dạng và sôi động.

  28. F168 mang đến một thế giới video game trực tuyến phong phú cho cộng đồng người chơi. Với nhiều thể loại video game khác nhau, người dùng có thể thoải mái lựa chọn để giải trí.

  29. Casino F168 là nơi tập hợp nhiều trò chơi giải trí trực tuyến dành cho người chơi yêu thích Online casino. Với nhiều lựa chọn khác nhau, người dùng có thể trải nghiệm game theo sở thích cá nhân.

  33. F168 cung cấp nhiều tựa sport giải trí đa dạng phù hợp với nhiều người chơi. Tại đây, người dùng có thể khám phá và lựa chọn trò chơi theo sở thích của mình.

  34. Hey I know this is off topic but I was wondering if you knew of any widgets I could add to my blog that automatically tweet my newest twitter updates. I’ve been looking for a plug-in like this for quite some time and was hoping maybe you would have some experience with something like this. Please let me know if you run into anything. I truly enjoy reading your blog and I look forward to your new updates.

  35. Wow that was strange. I just wrote an incredibly long comment but after I clicked submit my comment didn’t appear. Grrrr… well I’m not writing all that over again. Regardless, just wanted to say superb blog!

  36. Wow that was unusual. I just wrote an very long comment but after I clicked submit my comment didn’t show up. Grrrr… well I’m not writing all that over again. Anyhow, just wanted to say excellent blog!

  37. Wow that was strange. I just wrote an really long comment but after I clicked submit my comment didn’t appear. Grrrr… well I’m not writing all that over again. Anyhow, just wanted to say superb blog!

  38. Aw, this was an exceptionally good post. Finding the time and actual effort to make a very good article… but what can I say… I procrastinate a lot and don’t manage to get anything done.

  39. At this time it appears like WordPress is the preferred blogging platform out there right now. (from what I’ve read) Is that what you’re using on your blog?

  42. Right now it appears like Expression Engine is the top blogging platform out there right now. (from what I’ve read) Is that what you are using on your blog?

  48. Cược thể thao uy88 mang đến trải nghiệm giải trí với nhiều trò chơi quen thuộc trong thế giới casino. Người chơi có thể tham gia và tận hưởng những phút giây thư giãn cùng nhiều tựa sport hấp dẫn.

  50. Currently it seems like WordPress is the preferred blogging platform out there right now. (from what I’ve read) Is that what you’re using on your blog?

Deja un comentario