{"id":2866,"date":"2022-11-13T13:19:37","date_gmt":"2022-11-13T17:19:37","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=2866"},"modified":"2024-10-25T15:13:04","modified_gmt":"2024-10-25T19:13:04","slug":"rotational-symmetry-of-platonic-solids","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=2866","title":{"rendered":"Rotational Symmetry of Platonic Solids"},"content":{"rendered":"\n<p>In 3D geometry, a Platonic Solid is a convex polyhedron with all its faces are congruent regular polygons. There are only 5 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Platonic_solid\">Platonic Solids<\/a>, Tetrahedron, Cube, Octahedron, Dodecahedron, and Icosahedron.<\/p>\n\n\n\n<p><a href=\"https:\/\/en.wikipedia.org\/wiki\/Rotational_symmetry\">Rotational Symmetry<\/a> is the property of a geometric shape has when it looks the same after some rotation by a partial turn. <\/p>\n\n\n\n<p>The number of the rotational symmetries of a  is easily obtained by any of several methods:<\/p>\n\n\n\n<ul><li>Number of faces times number of sides per face<\/li><li>Number of vertices times number of edges per vertex<\/li><li>Number of edges times 2 (number of faces per edge)<\/li><\/ul>\n\n\n\n<p>The full number of symmetries is twice the number of rotational symmetries as each possible rotation has a unique reflection across any suitable plane.<\/p>\n\n\n\n$$\\begin{array}{|l|c|c|c|l|c|c|}\n\\hline\nPlatonic&amp;\\# of &amp;\\# of&amp;\\# of&amp;Face&amp;Rotational&amp;Full\\\\\nSolids&amp;Faces&amp;Vertics&amp;Edges&amp;Shape&amp;Symmetry&amp;Symmetry\\\\\n\\hline\nTetrahedron&amp;4&amp;4&amp;6&amp;Triangle&amp;12&amp;24\\\\\n\\hline\nCube&amp;6&amp;8&amp;12&amp;Square&amp;24&amp;48\\\\\n\\hline\nOctahedron&amp;8&amp;6&amp;12&amp;Triangle&amp;24&amp;48\\\\\n\\hline\nDodecahedron&amp;12&amp;20&amp;30&amp;Pentagon&amp;60&amp;120\\\\\n\\hline\nIcosahedron&amp;20&amp;12&amp;30&amp;Triangle&amp;60&amp;120\\\\\n\\hline\n\\end{array}$$\n\n\n\n<p><strong>Question 1:<\/strong> How many different way to paint the surface of a Tetrahedron with at-most of 4 different colors?<\/p>\n\n\n\n<p>Answer: With 1 color, there is only 1 way. With 2 colors, either 1 face is painted with one color, the other 3 faces in the other color, i.e. 2 ways; or 2 faces painted in one color, the other 2 in the other color, i.e. 1 ways; therefore, there are $2+1=3$ ways. With 3 colors, 3 faces are painted in the different colors, and the 4th one painted in one of the 3 colors, i.e. 3 ways. With 4 colors, considering the rotational symmetry, there are $\\dfrac{4!}{12}=2$ ways. Therefore, the answer to the question is $1+3+3+2=\\boxed{9}$.<\/p>\n\n\n\n<p><strong>Question 2<\/strong>: How many different ways to lable 1, 2, 3, 4, 5, and 6 on a regular cubic dice?<\/p>\n\n\n\n<p>Considering the rotational symmetry, the answer is $\\dfrac{6!}{24}=\\boxed{30}$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In 3D geometry, a Platonic Solid is a convex polyhedron with all its faces are congruent regular polygons. There are only 5 Platonic Solids, Tetrahedron, Cube, Octahedron, Dodecahedron, and Icosahedron. Rotational Symmetry is the property of a geometric shape has &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=2866\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[10],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2866"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2866"}],"version-history":[{"count":15,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2866\/revisions"}],"predecessor-version":[{"id":2886,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/2866\/revisions\/2886"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2866"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2866"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2866"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}