symmetry groups of the platonic solids mathematics, statistics and computer science department...
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Symmetry groups of the platonic solids Mathematics, Statistics and Computer Science Department Xiaoying (Jennifer) Deng
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Outline 0Introduction0Properties0Rotational symmetry groups of some platonic solids0Related groups0Future work0Exam question
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Introduction0Definition: A platonic solid is a convex polyhedron that
is made up of congruent regular polygons with the same number of faces meeting at each vertex.
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OctahedronHexahedron (Cube)
Tetrahedron
IcosahedronDodecahedron
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Euler’s formula
Name F E V
Tetrahedron 4 6 4
Cube 6 12 8
Octahedron 8 12 6
Dodecahedron 12 30 20
Icosahedron 20 30 12
F + V - E = 2
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Duality0Definition: A dual of a polyhedron is formed by 0place points on the center of every faces0 connect the points in the neighbouring faces of the original
polyhedron to obtain the dual
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Cube IcosahedronTetrahedron
Lemma: Dual polyhedra have the same symmetry groups.
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Symmetry group0Definition:0Let X be a platonic solid.0Rotational(Direct) symmetry group of X is a symmetry
group of X if only rotation is allowed.0Full symmetry group of X is a symmetry group of X if
both rotation and reflection are allowed.
0For a finite set A of n elements, the group of all permutations of A is the symmetric group on n letters.
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The Tetrahedron
Rotational symmetry Permutations of 4 numbers
P(1) = 1 P(2) = 2P(3) = 3 P(4) = 4
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01200 ; Two new symmetries for each vertex.04 × 2 = 8 new symmetries. 0Vertex 1; (2, 4, 3) (2, 3, 4)0Vertex 2; (1,3, 4) (1, 4, 3)0Vertex 3; (1, 2, 4) (1, 4, 2)0Vertex 4; (1, 2, 3) (1, 3, 2)
P(1) = 1 P(2) = 2P(3) = 3 P(4) = 4
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01800 ; One symmetry for each axis.03 × 1 = 3 new symmetries.0 (1, 2)(3, 4) (1, 3)(1, 4) (1, 4)(2, 3)
P(1) = 1 P(2) = 2P(3) = 3 P(4) = 4
0 (1, 2)(1, 2)01 + 8 + 4 = 12 rotational symmetries.
0The alternating group: A4
https://www.youtube.com/watch?v=qAR8BFMS3Bc ( 2:01)
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The cube
http://www.youtube.com/watch?v=gBg4-lJ19Gg (1:38)
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0 1200 ; Two new symmetries for each axis.0 4 × 2 = 8 new symmetries.0 1800 ; One symmetry for each axis.0 6 × 1 = 6 new symmetries.
0 1 + 9 + 8 + 6 = 24 rotational symmetries.
0 S4
0 900 ; Three new symmetries for each axis.0 3 × 3 = 9 symmetries
d1
d4
d3
d2
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The Octahedron
Name Rotationalsymmetries
Rotation Group Dual
Tetrahedron 12 A4 Tetrahedron
Cube 24 S4 Octahedron
Octahedron 24 S4 Cube
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Future work 0Reflection group of platonic
solids
0Reflection group of the tetrahedron
0Full symmetry group of the tetrahedron
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0The rotational symmetry group of the dodecahedron and the Icosahedron
Name Rotational symmetries
Rotation Group Dual
Dodecahedron 60 A5 Icosahedron
Icosahedron 60 A5 Dodecahedron
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Name Orbit(vertices)
Stabilizer (faces at each vertex)
|G+|
Tetrahedron 4 3 12
Cube 8 3 24
Octahedron 6 4 24
Dodecahedron 20 3 60
Icosahedron 12 5 60
0Stabilizer ; The Orbit-Stabilizer Theorem
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Exam QuestionHow many rotational symmetries of the cube?
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0900 ; 3 × 3 = 9 symmetries.01200 ; 4 × 2 = 8 new symmetries.01800 ; 6 × 1 = 6 new symmetries.01 + 9 + 8 + 6 = 24 rotational symmetries.
Solution:
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Reference 0 Kappraff, J. (2001). Connections: The geometric bridge between art and
science. Singapore: World Scientific.0 Hilton, P., Pedersen, J., & Donmoyer, S. (2010). A mathematical tapestry:
Demonstrating the beautiful unity of mathematics. New York: Cambridge University Press.
0 Senechal, M., Fleck, G. M., & Sherer, S. (2012). Shaping space: Exploring polyhedra in nature, art, and the geometrical imagination. New York: Springer.
0 Berlinghoff, W. P., & Gouvêa, F. Q. (2004). Math through the ages: A gentle history for teachers and others. Washington, DC: Mathematical Association of America.
0 Richeson, D. S. (2008). Euler's gem: The polyhedron formula and the birth of topology. Princeton, N.J: Princeton University Press.
0 In Celletti, A., In Locatelli, U., In Ruggeri, T., & In Strickland, E. (2014). Mathematical models and methods for planet Earth.
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