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The purpose of this Honors Contract is to investigate the groups of rigid motions of the five platonic solids and to analyze which permutation group each is isomorphic to. The five platonic solids are the tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron. It is important to note the concept of duality for polyhedra. Duality is defined as such:[1] For each regular polyhedron, the dual polyhedron is defined to be the polyhedron constructed by

(1) placing a point in the center of each face of the original polyhedron,

(2) connecting each new point with the new points of its neighboring faces, and

(3) erasing the original polyhedron

As such, the tetrahedron is dual to itself, the hexahedron is dual to the octahedron, and the dodecahedron is dual to the icosahedron. The explicit rotations for the tetrahedron, hexahedron and the octahedron will be provided in the appendix; the dodecahedron and the icosahedron will be excluded because of the size of both rotational groups and will have an implicit count.

Two theorems will be used to aid in the discussion:

Theorem 1 (Cayley’s Theorem): Every group is isomorphic to a group of permutations.[2]

Theorem 2: If the group of rigid motions of a regular solid, Gr, acts faithfully on some set of features of the regular solid X, i.e. vertices, diagonals, inscribed solids etc, then Gr is a subgroup of .[3]

Proof: Let Gr be a group of rigid motions of a given regular solid and let X be the set of features of this regular solid. Assume that Gr acts faithfully on X. Then, define the map , where r is a rotation element of Gr and is a permutation in S|X|. This map is a homomorphism because permutation multiplication is function composition. Furthermore, this homomorphism is well-defined because it involves the permutations on the set X. Since is a homomorphism, we have that the image of is a subgroup of S|X|, namely . By the Fundamental Isomorphism Theorem, we have that . Since Gr acts faithfully on X, the only element of Gr that keeps X fixed is the identity element; hence, ker is trivial. Thus, implies that .

The Tetrahedron

Claim: The group of rigid motions of the tetrahedron is isomorphic to A4.

Let T denote a tetrahedron and let Gr(T) be the group of rigid motions of the tetrahedron and let X = {1, 2, 3, 4} be the set of vertices of the tetrahedron. Let Gr(T) act on the set X.Appendix A shows the rotations of the vertices of the tetrahedron. From Appendix A, it is clear that the only element that fixes the four vertices of the tetrahedron is the identity element;thus, by Theorem 2 . From Appendix A, the order of Gr(T) is 12 while the order of S4 is 24.Hence, (S4: Gr(T)) = 2 implies that Gr(T) is a normal subgroup of S4. Since the only other normal subgroup of order 12 in S4 is A4, the group of even permutations, Gr(T) must be A4. This can be noted since all elements in Gr(T) are even permutations of the vertices of the tetrahedron. Hence, .

The Hexahedron and Octahedron

Claim: The group of rigid motions of both the hexahedron and octahedron are isomorphic to S4.

Let H denote the hexahedron and Gr(H) be the group of rigid motions of the hexahedron. Similarly, let O denote the octahedron and Gr(O) be the group of rigid motions of the octahedron. Appendix B and Appendix C give the explicit count to the number of elements in both Gr(H) and Gr(O); note that both have the same order. Similarly, since the hexahedron is dual to the octahedron, both groups of rigid motions will be isomorphic to the same group. This proof will be done for the group of rigid motions of the hexahedron.

Let X = {A, B} be the set of two inscribed tetrahedrons inside of the hexahedron. A tetrahedron is inscribed by using the diagonals of each face of the hexahedron; the other tetrahedron is inscribed by using the remaining diagonals. A picture is provided in Appendix B. Let Gr(H) act on the set X. There are four different ways the elements of Gr(H) can act on X: leaving the elements of X fixed with the identity, through rotation through the center of the face, through rotation of opposite vertices, and through rotation through the midpoints of opposite edges. All of the elements of Gr(H) are shown in Appendix B. Because the rigid motions of the cube preserve the isometry, the diagonals of the faces are mapped to diagonals of other faces; also, faces that are right next to each other are still next to each other after a rotation. As such, the inscribed tetrahedrons have their isometries preserved as well through the action of Gr(H); each element of Gr(H) uniquely rotates each of the inscribed tetrahedrons and preserves the isometry. As such, the only element that leaves the inscribed tetrahedrons fixed is the identity element. Note that the order of Gr(H) is 24. Since it was shown that , and since Gr(H) preserves the isometry of both inscribed tetrahedrons, Gr(H) is isomorphic to a group of order . Namely, . Since the hexahedron is dual to the octahedron, we also have that .

The Dodecahedron and the Icosahedron

Claim: The group of rigid motions of both the dodecahedron and icosahedronhave order 60.

Let D denote the dodecahedron and Gr(D) be the group of rigid motions of the dodecahedron. Similarly, let I denote the icosahedron and Gr(I) be the group of rigid motions of the icosahedron.The rigid motions of the dodecahedron are as follows:

(1)Identity

(2)6 sets of rotations about the center of a face. Since each face is a pentagon, it can be rotated 4 times before returning to its fixed position. Hence, there are 24 elements.

(3)10 sets of rotations about opposite vertices. These rotations can be done twice per set of opposite vertices. Hence, there are 20 elements.

(4)15 sets of rotations about midpoints of opposite edges. This can be done only once per set of opposite edges. Hence, there are 15 elements.

Therefore, Gr(D) has order 1 + 24 + 20 + 15 = 60. Similarly, the rigid motions of the icosahedron are as follows.

(1)Identity

(2)10 sets of rotations about the center of a face. Since each face is a triangle, it can be rotated 2 times before returning to its fixed position. Hence, there are 20 elements.

(3)6 sets of rotations about opposite vertices. These rotations can be done four times per set of opposite vertices. Hence, there are 24 elements.

(4)15 sets of rotations about midpoints of opposite edges. This can be done only once per set of opposite edges. Hence, there are 15 elements.

Therefore, Gr(I) has order 1 + 20 + 24 + 15 = 60.

Claim: The group of rigid motions of both the dodecahedron and icosahedron are isomorphic to A5.

Since the dodecahedron is dual to the icosahedron, this proof will be exercised for the dodecahedron.

Proof: Let Gr(D) be the group of rigid motions of a dodecahedron and let X = {a, b, c, d, e} be the set of inscribed hexahedrons inside the dodecahedron. A hexahedron is inscribed by taking the “diagonal” of each of the pentagonal faces and connecting them to each other. An image is provided for one such hexahedron in Appendix D. Let Gr(D) act on the set X. Similar to how Gr(H) preserved the isometry of the two inscribed tetrahedrons, Gr(D) preserves the isometry of the five inscribed hexahedrons. This is because each of the twelve edges of one hexahederon lies on exactly one diagonal of each pentagonal face. Since the diagonals of the pentagonal faces are rotated to other diagonals of pentagonal faces, the only element in Gr(D) that leaves the five inscribed hexahedrons fixed is the identity. As such, by Theorem 2 . This can also be observed in a similar way. Since the and since there are five inscribed hexahedrons, the action of Gr(D) on X produces elements; so again, . By the counting argument proposed above, |Gr(D)| = 60 implies that (S5: Gr(D)) = 2. As such, Gr(D) is a normal subgroup of S5. Since the only normal subgroup of S5 is A5, the group of even permutations, we have that Gr(D) = A5. Hence, . Again, since the dodecahedron is dual to the icosahedron we have that .

Appendix

A: The rotations of the Tetrahedron

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Rotation about Vertex 1

  1. (234)
  2. (243)

Rotation about Vertex 2

  1. (134)
  2. (143)

Rotation about Vertex 3

  1. (124)
  2. (142)

Rotation about Vertex 4

  1. (123)
  2. (132)

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Rotation through Midpoint a

  1. (12)(34)

Rotation through Midpoint b

  1. (14)(23)

Rotation through Midpoint c

  1. (13)(24)

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B1: The rotations of the Hexahedron

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Rotation about Vertices 1 and 7

  1. (245)(386)
  2. (254)(368)

Rotation about Vertices 2 and 8

  1. (163)(457)
  2. (136)(475)

Rotation about Vertices 3 and 5

  1. (186)(274)
  2. (168)(247)

Rotation about Vertices 4 and 6

  1. (183)(257)
  2. (138)(275)

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Rotation about x-axis

  1. (1265)(4378)
  2. (16)(25)(47)(38)
  3. (1562)(4873)

Rotation about y-axis

  1. (1485)(2376)
  2. (18)(45)(27)(36)
  3. (1584)(2673)

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Rotation about z-axis

  1. (1234)(5678)
  2. (13)(24)(57)(68)
  3. (1432)(5876)

Rotation through Midpoint a

  1. (28)(35)(14)(67)

Rotation through Midpoint b

  1. (35)(46)(12)(78)

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Rotation through Midpoint c

  1. (17)(46)(23)(58)

Rotation through Midpoint d

  1. (17)(28)(34)(56)

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Rotation through Midpoint e

  1. (28)(46)(15)(37)

Rotation through Midpoint f

  1. (17)(35)(26)(48)

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B2: Tetrahedron inscribed in the hexahedron

C: The rotations of the Octahedron

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Rotation about Vertices 1 and 6

  1. (2345)
  2. (24)(35)
  3. (2543)

Rotation about Vertices 2 and 4

  1. (1365)
  2. (16)(35)
  3. (1563)

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Rotation about Vertices 3 and 5

  1. (1264)
  2. (16)(24)
  3. (1462)

Rotation through Midpoint a

  1. (24)(15)(36)

Rotation through Midpoint b

  1. (35)(12)(46)

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Rotation through Midpoint c

  1. (24)(13)(56)

Rotation through Midpoint d

  1. (35)(14)(26)

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Rotation through midpoints of triangle 154 and 236

  1. (152)(346)
  2. (125)(364)

Rotation through midpoints of triangle 123 and 546

  1. (134)(265)
  2. (143)(256)

Rotation through midpoints of triangle 143 and 526

  1. (154)(263)
  2. (145)(236)

Rotation through midpoints of triangle 152 and 436

  1. (123)(456)
  2. (132)(465)

D: Hexahedron inscribed in a Dodecahedron

References

[1]Lim, Yongwhan. Symmetry Groups of Platonic Solids. MIT, Massachusetts. December 4, 2008.

[2]Fraleigh, John B. A First Course in Abstract Algebra 7th Edition. University of Rhode Island, Rhode Island: Pearson Education, Inc., 2003. pp. 82.

[3]Siegel, Adam. Regular Solids and their Rotation Groups. Spring 2007.