In Perspective
• 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.
• 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.
• 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.
• Made up of 4 identical equilateral triangles.
• Four identical faces, four vertices, and six edges.
• The most famous of the Platonic solids, we see it as dice most
frequently.
• Six faces, 12 edges, 8 vertices
• Constructed from 8 identical, equilateral triangles.
• Eight faces, 12 edges, 6 vertices
• The only solid with pentagonal faces.
• Twelve faces (do = 2, dec = 10, dodec = 2+10), 20 vertices,
30 edges.
• Constructed from 20 identical equilateral triangles.
• Twenty faces, 12 vertices, 30 edges.
# 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
Dodecahedr
on
20
30
12
3
5
Icosahedron
12
30
20
5
3
• 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?
• 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.
• 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.