{"id":209926,"date":"2025-03-28T01:31:03","date_gmt":"2025-03-28T06:31:03","guid":{"rendered":"https:\/\/lifeboat.com\/blog\/2025\/03\/klein-bottle"},"modified":"2025-03-28T01:31:03","modified_gmt":"2025-03-28T06:31:03","slug":"klein-bottle","status":"publish","type":"post","link":"https:\/\/lifeboat.com\/blog\/2025\/03\/klein-bottle","title":{"rendered":"Klein bottle"},"content":{"rendered":"<p><\/p>\n<p><iframe style=\"display: block; margin: 0 auto; width: 100%; aspect-ratio: 4\/3; object-fit: contain;\" src=\"https:\/\/www.youtube.com\/embed\/I3ZlhxaT_Ko?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope;\n   picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p>Suppose for clarification that we adopt time as that fourth dimension. Consider how the figure could be constructed in xyzt-space. The accompanying illustration (\u201cTime evolution\u2026\u201d) shows one useful evolution of the figure. At t = 0 the wall sprouts from a bud somewhere near the \u201cintersection\u201d point. After the figure has grown for a while, the earliest section of the wall begins to recede, disappearing like the Cheshire Cat but leaving its ever-expanding smile behind. By the time the growth front gets to where the bud had been, there is nothing there to intersect and the growth completes without piercing existing structure. The 4-figure as defined cannot exist in 3-space but is easily understood in 4-space.<sup id=\u201d cite_ref-FOOTNOTEAllingGreenleaf1969_4-1\u201d class=\u201d reference\u201d> [ 4 ]<\/sup><\/p>\n<p>More formally, the Klein bottle is the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Quotient_space_(topology)\" title=\"Quotient space (topology)\">quotient space<\/a> described as the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Square_(geometry)\" class=\"\" title=\"Square (geometry)\">square<\/a> [0,1] \u00d7 [0,1] with sides identified by the relations (0, <i>y<\/i>) ~ (1, <i>y<\/i>) for 0 \u2264 <i>y<\/i> \u2264 1 and (<i>x<\/i>, 0) ~ (1 \u2212 <i>x<\/i>, 1) for 0 \u2264 <i>x<\/i> \u2264 1.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Suppose for clarification that we adopt time as that fourth dimension. Consider how the figure could be constructed in xyzt-space. The accompanying illustration (\u201cTime evolution\u2026\u201d) shows one useful evolution of the figure. At t = 0 the wall sprouts from a bud somewhere near the \u201cintersection\u201d point. After the figure has grown for a while, [\u2026]<\/p>\n","protected":false},"author":709,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[385,8],"tags":[],"class_list":["post-209926","post","type-post","status-publish","format-standard","hentry","category-evolution","category-space"],"_links":{"self":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts\/209926","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/users\/709"}],"replies":[{"embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/comments?post=209926"}],"version-history":[{"count":0,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts\/209926\/revisions"}],"wp:attachment":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/media?parent=209926"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/categories?post=209926"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/tags?post=209926"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}