{"id":33323,"date":"2025-06-23T07:56:34","date_gmt":"2025-06-23T07:56:34","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=33323"},"modified":"2025-06-23T07:56:35","modified_gmt":"2025-06-23T07:56:35","slug":"describe-all-of-the-rotational-symmetries-of-a-cube","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/describe-all-of-the-rotational-symmetries-of-a-cube\/","title":{"rendered":"Describe all of the rotational symmetries of a cube"},"content":{"rendered":"\n<p>Describe all of the rotational symmetries of a cube. <\/p>\n\n\n\n<p>a. Find four things that are permuted when the cube undergoes a rotational symmetry. (the things can\u2019t be the vertices, edges, or faces. But, they could involve combinations of these elements.) <\/p>\n\n\n\n<p>b. Label the four things a, b, c, and d. For each rotational symmetry of the cube, express the permutation of the four things produced by the rotation.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p><strong>Correct Answer:<\/strong><\/p>\n\n\n\n<p>a. One possible set of four things that are permuted under cube rotations are the four <strong>space diagonals<\/strong> of the cube. A space diagonal connects two opposite corners (vertices) of the cube. A cube has exactly <strong>four<\/strong> space diagonals, and these are distinct from edges, faces, or vertices.<\/p>\n\n\n\n<p>b. Label the four space diagonals as follows:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>a<\/strong>: Diagonal from vertex 1 to 7<\/li>\n\n\n\n<li><strong>b<\/strong>: Diagonal from vertex 2 to 8<\/li>\n\n\n\n<li><strong>c<\/strong>: Diagonal from vertex 3 to 5<\/li>\n\n\n\n<li><strong>d<\/strong>: Diagonal from vertex 4 to 6<\/li>\n<\/ul>\n\n\n\n<p>Each <strong>rotational symmetry<\/strong> of the cube results in a <strong>permutation<\/strong> of the four diagonals. The group of all rotational symmetries of the cube is isomorphic to the symmetric group <strong>S\u2084<\/strong>, meaning that each rotation corresponds to a permutation of the four diagonals {a, b, c, d}.<\/p>\n\n\n\n<p>There are <strong>24 distinct rotational symmetries<\/strong> of a cube. These include:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Identity rotation<\/strong> (no movement): (a)(b)(c)(d)<\/li>\n\n\n\n<li><strong>Rotations of 90\u00b0, 180\u00b0, 270\u00b0 around face axes<\/strong><\/li>\n\n\n\n<li><strong>Rotations of 120\u00b0, 240\u00b0 around body diagonals<\/strong><\/li>\n\n\n\n<li><strong>Rotations of 180\u00b0 around edge axes<\/strong><\/li>\n<\/ol>\n\n\n\n<p>Each rotation permutes the diagonals in a different way. Some sample permutations:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A 120\u00b0 rotation around the space diagonal a sends b \u2192 c, c \u2192 d, d \u2192 b, a \u2192 a. So the permutation is (bcd)<\/li>\n\n\n\n<li>A 180\u00b0 rotation about a line through midpoints of opposite edges may swap two diagonals and fix the others, for example: (a b)(c d)<\/li>\n\n\n\n<li>A 90\u00b0 rotation around the vertical axis could result in (a b c d)<\/li>\n<\/ul>\n\n\n\n<p>These permutations can be listed as elements of the group <strong>S\u2084<\/strong>, showing how the cube\u2019s rotational symmetries act on the set {a, b, c, d}. Since the cube\u2019s symmetry group is isomorphic to S\u2084, all 24 permutations arising from its rotations correspond to the 24 elements in S\u2084. Thus, each rotation results in a unique rearrangement of the four labeled space diagonals.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Describe all of the rotational symmetries of a cube. a. Find four things that are permuted when the cube undergoes a rotational symmetry. (the things can\u2019t be the vertices, edges, or faces. But, they could involve combinations of these elements.) b. Label the four things a, b, c, and d. For each rotational symmetry of [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-33323","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/33323","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=33323"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/33323\/revisions"}],"predecessor-version":[{"id":33326,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/33323\/revisions\/33326"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=33323"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=33323"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=33323"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}