Beauty, Form and Function: An Exploration of Symmetry Asset No. 20 Lecture II-6 The Platonic Solids...

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