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Leonardo Da Vinci’sVitruvian Man (1490)Canon of Properties
Symmetry, Art & Science
4. Symmetry &Group Theory
often used as an implied symbol of the essential symmetryof the human body, and by extension, to the universe as a whole.
4. Symmetry &Group Theory
Symmetry & Nature
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4. Symmetry &Group Theory
Symmetry & Nature
4. Symmetry &Group Theory
Symmetry & Nature
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4. Symmetry &Group Theory
Symmetry & Nature
Symmetry & Nature (Spiral Symmetry)
4. Symmetry &Group Theory
Apex episcopus Nautilus pompilus
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4. Symmetry &Group Theory
Symmetry & Nature (Spiral Symmetry)
Symmetry & Architecture (Spiral Symmetry)
Great Mosque of Samarra (Iraq)
Genome (Larry Young)
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4. Symmetry &Group Theory
Symmetry & Nature (Bilateral / Mirror Sym)
Symmetry & Architecture
Giza Pyramid Field (Cairo, Egypt)
4. Symmetry &Group Theory
Taj Mahal (Agra, India)
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Symmetry & Architecture (Mirror Symmetry)
Eiffel Tower (Paris, 325 m, 10.1 tons)
4. Symmetry &Group Theory
Trocadero Palace (Paris, from Eiffel t)
Symmetry & Landscaping (Mirror Symmetry)
Toukouji (Japan)
4. Symmetry &Group Theory
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4. Symmetry &Group Theory
Super-High Symmetry in Super-Low “Life Forms”
4. Symmetry &Group Theory
Super-High Symmetry in Super-Low “Life Forms”
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4. Symmetry &Group Theory
Super-High Symmetry in Super-Low “Life Forms”
4. Symmetry &Group Theory
2-foldaxes
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4. Symmetry &Group Theory
3-foldaxes
4. Symmetry &Group Theory
5-foldaxes
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4. Symmetry &Group Theory
2-foldaxes
4. Symmetry &Group Theory
3-foldaxes
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4. Symmetry &Group Theory
5-foldaxes
4. Symmetry &Group Theory
2-foldaxes
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4. Symmetry &Group Theory
3-foldaxes
4. Symmetry &Group Theory
5-foldaxes
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4. Symmetry &Group Theory
The Icosahedral Symmetryof Viruses
some nomen-clature…
4. Symmetry &Group Theory
138.1896852°Dihedral Angleregular, convexPropertiesdodecahedronDual polyhedron
Icosahedral(Ih), order 120
Symmetrygroup
3Vertices per face5Faces per vertex12Vertices30Edges20FacestriangleFace polygonPlatonicType
The Icosahedron
Connecting the centers of all the pairs of adjacent faces of any platonic solid produces another (smaller) platonic solid. The number of faces and vertices is interchanged, while the number of edges of the two is the same.
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4. Symmetry &Group Theory
The Icosahedron
138.1896852°Dihedral Angleregular, convexPropertiesdodecahedronDual polyhedron
Icosahedral(Ih), order 120
Symmetrygroup
3Vertices per face5Faces per vertex12Vertices30Edges20FacestriangleFace polygonPlatonicType
total # ofsymmetry operations!
4. Symmetry &Group Theory
regular, convexProperties
icosahedronDual polyhedron
icosahedral (Ih)order 120Symmetry group
5Vertices per face
3Faces per vertex
20Vertices
30Edges
12Faces
pentagonFace polygon
PlatonicType
The Dodecahedron
Despite appearances, when a dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedroninscribed in the same sphere (60.54%).
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4. Symmetry &Group Theory
regular, convex, zonohedronProperties
octahedronDual polyhedron
octahedral (Oh)order 48Symmetry group
4Vertices per face
3Faces per vertex
8Vertices
12Edges
6Faces
squareFace polygon
PlatonicType
The Cube (Hexahedron)
regular, convexProperties
cubeDual polyhedron
octahedral (Oh)order 48Symmetry group
3Vertices per face
4Faces per vertex
6Vertices
12Edges
8Faces
triangleFace polygon
PlatonicType
4. Symmetry &Group Theory
The Octahedron
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4. Symmetry &Group Theory
regular, convexProperties
70° 32' = arccos(1/3)Dihedral angle
tetrahedron (self-dual)Dual polyhedron
tetrahedral (Td)order 24Symmetry group
3Vertices per face
3Faces per vertex
4Vertices
6Edges
4Faces
triangleFace polygon
PlatonicType
The Tetrahedron
4. Symmetry &Group Theory
Ih53123020icosahedron
Ih35203012dodecahedron
Oh436128octahedron
Oh348126cube(hexahedron
Td33464tetrahedron
Symmetry group
Faces meetingat each vertex
Edges per faceVerticesEdgesFacesPictureName
The Platonic Solids
