The Platonic SolidsIn Perspective

Symmetry and Regularity• Objects that are symmetrical look the same from

several different views, or two sides are mirror images of each other.

• Symmetric solids are referred to as regular, or Platonic solids.

Symmetry in the Plane

3D Analogies?

The Mystical Allure• For centuries, the Platonic solids were associated

with mystical powers.• Euclid wrote about these regular solids. However,

they are named after Plato because he tried to relate them to the five elements: fire, earth, wood, metal, water.

• The Pythagoreans knew there were only 5 and held them in awe.

Johannes Kepler• Remembered best for laws of planetary motion.• In his time there were 6 known planets and he

showed that it is possible to take the 5 regular solids, put one inside the other, and have the sizes of inscribed and circumscribed spheres about these solids reveal the sizes of the orbits of the planets.

• Once Uranus was discovered in 1781 his theory fell apart.

The Tetrahedron• Made up of 4 identical equilateral triangles.• Four identical faces, four vertices, and six edges.

The Cube• The most famous of the Platonic solids, we see it

as dice most frequently.• Six faces, 12 edges, 8 vertices

The Octahedron• Constructed from 8 identical, equilateral triangles. • Eight faces, 12 edges, 6 vertices

The Dodecahedron• The only solid with pentagonal faces.• Twelve faces (do = 2, dec = 10, dodec = 2+10),

20 vertices, 30 edges.

The Icosahedron• Constructed from 20 identical equilateral

triangles.• Twenty faces, 12 vertices, 30 edges.

From Qualitative to Quantitative

# Vertices

# Edges # Faces # Faces at each vertex

# sides at each face

Tetrahedron

4 6 4 3 3

Cube 8 12 6 3 4

Octahedron

6 12 8 4 3

Dodecahedron

20 30 12 3 5

Icosahedron

12 30 20 5 3

Observations• The number of faces of the cube equals the

number of vertices of the octahedron; the number of vertices of the cube equals the number of faces of the octahedron.

• The number of edges of both figures is the same.• Why is that?

Dualing Solids• From the cube we can construct an octahedron

using midpoints of the cube faces. This is called a dual.

• The process of creating one solid from another is duality.

• Cube – octahedron, dodecahedron – icosahedron, tetrahedron is self dual.

Golden Promise

Perspective• Have you ever looked down a long road, or train

tracks, and noticed that it looked like it was getting “thinner” in a way?

• This is an optical illusion created by perspective.• Perspective is defined as the way in which objects

appear to the eye.• To draw an image in perspective, we need to have

a horizon and a vanishing point.• The horizon is an imaginary line in which the

ground “ends” and the sky “begins” in relation to your image.

• The vanishing point is an imaginary point in which everything seems to disappear, or vanish.