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.
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.
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.
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).
excellent points altogether, you just received a brand new reader. What may you recommend about your publish that you simply made some days in the past? Any certain?
At this time it sounds like Drupal is the best blogging platform available right now. (from what I’ve read) Is that what you’re using on your blog?
Currently it seems like Drupal is the preferred blogging platform available right now. (from what I’ve read) Is that what you’re using on your blog?
Fascinating how the quadrature of the circle evolved from ancient Greek constraints to Tarski’s decomposition approach. The Banach-Tarski paradox always blows my mind. I wonder if these constructive methods could have practical applications in geometry or even in game design—speaking of which, I recently found Gameilo – Free Online Browser Games and it’s a fun way to explore spatial puzzles.
This website truly has all the information and facts I wanted about this subject and didn’t know who to ask.
Greate article. Keep posting such kind of information on your blog. Im really impressed by it.
Nhà cái MB66 cam kết trải nghiệm người chơi.
MB66 phù hợp với xu hướng game online.
MB66 được xây dựng hướng tới trải nghiệm người chơi.
Nice post. I was checking constantly this blog and I am impressed!
Very helpful information specifically the last part :
) I care for such info a lot. I was seeking this particular
info for a long time. Thank you and best of luck.
Greetings! Very helpful advice in this particular article!
It’s the little changes that will make the largest changes.
Many thanks for sharing!
Nhà cái MB66 là nền tảng game được nhiều người lựa chọn.
MB66 được xây dựng nhằm phục vụ nhu cầu trải nghiệm game của người chơi.
MB66 tối ưu trải nghiệm người dùng.
Simply desire to say your article is as astounding. The clearness in your post is just nice and i can assume you are an expert on this subject. Fine with your permission allow me to grab your feed to keep up to date with forthcoming post. Thanks a million and please keep up the enjoyable work.
Nhà cái MB66 giúp người chơi trải nghiệm nhiều trò chơi.
MB66 là nền tảng game được nhiều người lựa chọn.
MB66 tối ưu nền tảng giải trí.
MB66 nổi bật với nhiều game giải trí phong phú.
Nhà cái MB66 nổi bật với kho trò chơi đa dạng.
Fantastic beat ! I wish to apprentice even as you amend your web site, how
can i subscribe for a weblog site? The account helped me a applicable
deal. I have been tiny bit familiar of this your broadcast provided vivid clear concept
Nhà cái MB66 giúp người chơi trải nghiệm nhiều trò chơi.
Nhà cái MB66 là nền tảng giải trí trực tuyến được nhiều người chơi biết đến.
Hi there, yup this article is truly fastidious and I have
learned lot of things from it about blogging. thanks.
Thanks for one’s marvelous posting! I really enjoyed reading it, you are a great
author.I will be sure to bookmark your blog and will come back in the foreseeable future.
I want to encourage you continue your great posts, have a nice afternoon!
MB66 mang lại trải nghiệm game cho người chơi.
MB66 mang lại giải trí trực tuyến mượt mà.
MB66 đáp ứng nhiều nhu cầu giải trí.
MB66 liên tục nâng cấp nền tảng.
Nhà cái MB66 xây dựng hệ sinh thái game đa dạng.
I’m gone to tell my little brother, that he should also visit this
web site on regular basis to take updated from hottest information.
Hello just wanted to give you a quick heads up and let you know a few of the images aren’t loading properly.
I’m not sure why but I think its a linking issue.
I’ve tried it in two different web browsers and both
show the same results.
I was recommended this website by my cousin. I’m not sure whether this post is written by him as
no one else know such detailed about my trouble. You’re
amazing! Thanks!
I visit every day a few blogs and information sites
to read content, except this web site provides feature based content.
Hi there would you mind sharing which blog platform you’re using?
I’m going to start my own blog in the near future but I’m having
a tough time deciding between BlogEngine/Wordpress/B2evolution and Drupal.
The reason I ask is because your design and style seems different then most blogs and I’m looking for
something completely unique. P.S My apologies for getting off-topic
but I had to ask!
Way cool! Some extremely valid points! I appreciate you penning this post plus the rest
of the site is also really good.
hey there and thank you for your info – I have definitely picked up anything new from
right here. I did however expertise several technical points using this web site,
since I experienced to reload the website lots of times previous to I could get it to load correctly.
I had been wondering if your hosting is OK? Not that I am complaining, but slow loading
instances times will very frequently affect your placement
in google and can damage your quality score if ads and marketing with Adwords.
Well I am adding this RSS to my e-mail and could
look out for much more of your respective interesting content.
Ensure that you update this again soon.
I believe what you published was actually very logical.
However, consider this, what if you added a little content?
I ain’t suggesting your information is not solid., however suppose you added
a title to possibly get a person’s attention? I mean Historias de pi: la cuadratura del círculo (à la Tarski) – Matemáticas y sus fronteras is
a little plain. You might glance at Yahoo’s home page and note
how they create article titles to get viewers to open the links.
You might add a video or a pic or two to get readers excited
about everything’ve written. Just my opinion, it would bring your posts a little bit more
interesting.
Good day! I know this is kinda off topic however
I’d figured I’d ask. Would you be interested in trading links or maybe guest writing
a blog post or vice-versa? My website discusses
a lot of the same topics as yours and I think we could greatly benefit from each other.
If you might be interested feel free to send me an email.
I look forward to hearing from you! Great blog by the way!
My family members every time say that I am wasting my time here at
net, however I know I am getting knowledge
everyday by reading thes nice content.
Excellent post. I am going through a few of these issues as well..
I’d like to find out more? I’d love to find out more details.
Thanks for sharing your thoughts. I really appreciate your efforts and I am waiting for
your next post thanks once again.
Yesterday, while I was at work, my cousin stole my apple ipad and tested to see if it can survive a twenty five foot drop, just so she can be a youtube sensation. My iPad is now broken and she has 83 views. I know this is entirely off topic but I had to share it with someone!
I was suggested this website by way of my cousin. I am now not sure whether or not this
publish is written by way of him as no one else understand such certain approximately my trouble.
You are amazing! Thanks!
certainly like your web site however you have to test the spelling on several of your posts.
Many of them are rife with spelling problems and I in finding
it very troublesome to inform the truth however I will certainly come again again.
I have fun with, result in I discovered exactly what
I used to be looking for. You’ve ended my 4 day long hunt!
God Bless you man. Have a great day. Bye
Currently it appears like Expression Engine is the preferred blogging platform out there right now. (from what I’ve read) Is that what you’re using on your blog?
I’d like to thank you for the efforts you’ve put in penning
this site. I’m hoping to view the same high-grade
content from you later on as well. In truth, your
creative writing abilities has encouraged me to get my own, personal site
now 😉
Hệ thống sản phẩm game của CM88 được tối ưu theo công nghệ hiện đại.