beauty, form and function: an exploration of symmetry asset no. 20 lecture ii-6 the platonic solids...
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Beauty, Form and Function: An Exploration of SymmetryBeauty, Form and Function: An Exploration of Symmetry
Asset No. 20
Lecture II-6
The Platonic Solids
PART IIPlane (2D) and Space (3D) Symmetry
By the end of this lecture, you will be able to:
• see that tessellation in three dimensions creates polyhedra
• label polyhedra according to Schäfli notation
• recognize the 5 Platonic solids
• describe 1 Archimedean solid
Objectives
Polyhedra
Polyhedra are closed figures with polygonal faces.
Regular polyhedra have all vertices related by symmetry and all faces congruent.
Convex polyhedra have all dihedral angles (angles between faces) less than 180o when viewed from the outside.
A cube (43) is a regular polyhedron with:
• 8 vertices• 12 edges• 6 faces
3 squares (4-gons)around each vertex
1
2
3
4
5
6
The Tetrahedron and Octahedron - Platonic Solids
1
2
3
4
A tetrahedron (33) has:• 4 vertices• 6 edges• 4 faces
A octahedron (34) has:• 6 vertices• 12 edges• 8 faces
1 2
3 4
5 6 7
8The five convex regular polyhedrons - tetrahedron, octahedron, cube, icosahedron, dodecahedron - known collectively as Platonic Solids.
The Icosahedron and Dodecahedron
A dodecahedton (53) has:• 20 vertices• 30 edges• 12 faces
An icosahedron (35) has:• 12 vertices• 30 edges• 20 faces
12
34
56 789
1011
12
10
12 3 4 5
8 96 7 11
151413
12
20
1816
1917
Properties of Platonic Solids
Triangle
Square
Pentagon
All the Platonic Solids have vertices on the surface of a sphere and are constructed with a single type of polygon - triangle, square, pentagon.
33 34 43 53 35
Singular Pluraltetrahedron* tetrahedraoctahedron octahedracube (hexahedron) cubes (hexahedra) icosahedron icosahedradodecahedron dodecahedra
Greektetra 4octa 8hexa 6icosa 20dodeca 12
Cuboctahedron - An Archimedean Solid
Polyhedra with equivalent (i.e. symmetry related) vertices but more than one kind of regular polygonal face are the semi-regular or Archimedean Solids.
An important Archimedean solid in crystallography is the cuboctahedron.
The Euler Rule of Platonic and Archimedean Solids states that V-E+F = 2 which relates the number of vertices (V), edges (E) and faces (F).
A cuboctahedron (3.4.3.4) has: • 12 vertices
• 24 edges• 14 faces
1
23 4 5 67 9
810
1112 13
14
There are 5 regular polyhedra known as the Platonic solids - tetrahedron, octahedron, cube, icosahedron, dodecahedron
Regular polyhedra are composed of one type of polygon and every vertex is identical
Semi-regular polyhedra are composed to two or more polygons with every vertex regular
The cuboctahedron is an Archimedean solid
Summary