{"id":1611,"date":"2015-05-15T01:00:12","date_gmt":"2015-05-15T01:00:12","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=1611"},"modified":"2017-05-07T08:04:04","modified_gmt":"2017-05-07T08:04:04","slug":"dodecahedron-with-5-tetrahedra","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2015\/05\/15\/dodecahedron-with-5-tetrahedra\/","title":{"rendered":"Dodecahedron With 5 Tetrahedra"},"content":{"rendered":"
\n
\"Dodecahedron<\/a>

Dodecahedron With 5 Tetrahedra – Greg Egan<\/p><\/div>\n<\/div>\n

This image by Greg Egan<\/a> shows 5 ways to inscribe a regular tetrahedron in a regular dodecahedron. More precisely, it shows 5 ways to choose 4 vertices of the dodecahedron that are also vertices of a regular tetrahedron.<\/p>\n

The union of all these tetrahedra is a nonconvex polyhedron called the compound of 5 tetrahedra<\/b><\/a>, first described by Edmund Hess in 1876.<\/p>\n

This polyhedron is chiral<\/b><\/a>: it has no reflection symmetries. In rough terms, it has an inherent ‘left-handedness’: if you start at a point where the 5 tetrahedra meet, you’ll see the lines of intersection ‘swirl counterclockwise’ as they go out. <\/p>\n

In fact there are 10 ways to inscribe a regular tetrahedron in a regular dodecahedron. 5 form the left-handed polyhedron above, and the other 5 give the right-handed version of this polyhedron. The union of all 10 tetrahedra is a polytope called compound of 10 tetrahedra<\/b><\/a>, which has reflection symmetries:<\/p>\n

\n
\"Dodecahedron<\/a>

Dodecahedron With 10 Tetrahedra – Greg Egan<\/p><\/div>\n<\/div>\n

The two kinds of tetrahedra are colored yellow and cyan. Regions belonging to both are colored magenta. It’s pretty—but it’s hard to see the individual tetrahedra, because they overlap a lot.<\/p>\n

We can do something similar starting with a cube:<\/p>\n

\n
\"Cube<\/a>

Cube With 2 Tetrahedra – Greg Egan<\/p><\/div>\n<\/div>\n

There are 2 ways to inscribe a regular tetrahedron in a cube. Taken together they form a nonconvex polyhedron called the stellated octahedron<\/b><\/a>.<\/p>\n

There are 5 ways to inscribe a cube in a dodecahedron. Their union is called the compound of 5 cubes<\/b><\/a>:<\/p>\n

\n
\"Dodecahedron<\/a>

Dodecahedron With 5 Cubes – Greg Egan<\/p><\/div>\n<\/div>\n

Inscribing 2 tetrahedra in each of these cubes, we obtain the 10 tetrahedra in the dodecahedron.<\/p>\n

What is the mathematics underlying these relationships between the tetrahedron, cube and dodecahedron? The finite subgroups of the rotation group $\\mathrm{SO}(3)$ can be classified, up to conjugation:<\/p>\n

• for each $n \\ge 1$, the cyclic group<\/a> $\\mathbb{Z}\/n$,<\/p>\n

• for each $n \\ge 2$, the dihedral group<\/a> $\\mathrm{D}_n$,<\/p>\n

• the rotational symmetry group of the tetrahedron, which is isomorphic to the alternating group<\/a> $\\mathrm{A}_4$, the group of even permutations of the tetrahedron’s 4 vertices.<\/p>\n

• the rotational symmetry group of the cube, which is isomorphic to the symmetric group<\/a> $\\mathrm{S}_4$, the group of permutations of the cube’s 4 diagonal axes. <\/p>\n

• the rotational symmetry group of the dodecahedron, which is isomorphic to the symmetric group $\\mathrm{A}_5$, the group of even permutations of the 5 cubes inscribed in the dodecahedron.<\/p>\n

So, what’s at work here is a relation between $\\mathrm{A}_4, \\mathrm{S}_4$ and $\\mathrm{A}_5$, both as abstract groups and as subgroups of $\\mathrm{SO}(3)$.<\/p>\n

• Tetrahedron and cube.<\/b> $\\mathrm{A}_4$ is a subgroup of $\\mathrm{S}_4$. So, every symmetry of an inscribed tetrahedron gives a symmetry of the cube. Since $\\mathrm{A}_4$ is a subgroup of $\\mathrm{S}_4$ in a unique way<\/i>, both tetrahedra in the cube have the same<\/i> subgroup of $\\mathrm{S}_4$ as symmetries—not merely conjugate<\/i> subgroups.<\/p>\n

• Tetrahedron and dodecahedron.<\/b> $\\mathrm{A}_4$ is a subgroup of $\\mathrm{A}_5$. So, every symmetry of an inscribed tetrahedron gives a symmetry of the dodecahedron. $\\mathrm{A}_4$ is a subgroup of $\\mathrm{A}_5$ in 5 different ways that are all conjugate. For each of these 5 subgroups, 2 of the 10 tetrahedra in the dodecahedron have this subgroup as symmetries.<\/p>\n

• Cube and dodecahedron.<\/b> $\\mathrm{S}_4$ is not a subgroup of $\\mathrm{A}_5$. So, not every symmetry of an inscribed cube gives a symmetry of the dodecahedron. <\/p>\n

Puzzle 1:<\/b> Describe a tetrahedron in a cube, tetrahedron in a dodecahedron or cube in a dodecahedron using the language of group theory. Use this to explain why there are 2 tetrahedra in the cube, 10 tetrahedra in the dodecahedron and 5 cubes in the dodecahedron.<\/p>\n

Puzzle 2:<\/b> $\\mathrm{A}_4$ is a normal subgroup of $\\mathrm{S}_4$ but not of $\\mathrm{A}_5$. How does this make tetrahedra in the cube different than tetrahedra in the dodecahedron?<\/p>\n

For more, see:<\/p>\n

• Alternating group $\\mathrm{A}_4$<\/a>, Groupprops<\/i>.<\/p>\n

• Symmetric group $\\mathrm{S}_4$<\/a>, Groupprops<\/i>.<\/p>\n

• Alternating group $\\mathrm{A}_5$<\/a>, Groupprops<\/i>.<\/p>\n

• Classification of finite subgroups of $\\mathrm{SO}(3)$<\/a>, Groupprops<\/i>.<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

This image by Greg Egan<\/a> shows 5 ways to inscribe a regular tetrahedron in a regular dodecahedron. The union of all these is a nonconvex polyhedron called the compound of 5 tetrahedra<\/a>, first described by Edmund Hess in 1876.<\/p>\n

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