{"id":1565,"date":"2018-10-31T21:37:19","date_gmt":"2018-11-01T01:37:19","guid":{"rendered":"https:\/\/people.clas.ufl.edu\/boyland\/?page_id=1565"},"modified":"2026-03-19T08:10:27","modified_gmt":"2026-03-19T12:10:27","slug":"spencer-unger","status":"publish","type":"page","link":"https:\/\/people.clas.ufl.edu\/boyland\/spencer-unger\/","title":{"rendered":"Spencer Unger"},"content":{"rendered":"\r\n<section class=\"fullwidth-text-block\">\r\n\t<div class=\"container px-0 pt-5\">\r\n\t\t<div class=\"row align-items-start\">\r\n\t\t\t<div class=\"col-12\">\r\n\t\t\t\t\n<h2 class=\"wp-block-heading\"><span style=\"font-size: small\">A constructive solution to Tarski&#8217;s circle squaring problem<\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><span style=\"font-size: small\">Abstract: In 1925, Tarski asked whether a disk in R^2 can be partitioned into finitely many pieces which can be rearranged by isometries to form a square of the same area.\u00a0 The restriction of having a disk and a square with the same area is necessary.\u00a0 In 1990, Laczkovich gave a positive answer to the problem using the axiom of choice.\u00a0 We give a completely explicit (Borel) way to break the circleand the square into congruent pieces.\u00a0 This answers a question of Wagon.\u00a0 Our proof has three main components.\u00a0 The first is work of Laczkovich in Diophantine approximation.\u00a0 The second is recent progress in a program of descriptive set theory to understand the complexity of actions of amenable groups.\u00a0 The third is the study of flows in networks.<br>\n<\/span><\/p>\n\n\n\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t<\/div>\r\n<\/section>\r\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":146,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"featured_post":"","footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-1565","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages\/1565","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/users\/146"}],"replies":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/comments?post=1565"}],"version-history":[{"count":2,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages\/1565\/revisions"}],"predecessor-version":[{"id":3951,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages\/1565\/revisions\/3951"}],"wp:attachment":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/media?parent=1565"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}