Geometry

1
Geometry

• Polyhedra

2
Goals

• Know terminology about solids.
• Identify solids by type.
• Use Eulers Theorem to solve problems.

3
Polyhedron

• A solid that is bounded by polygons.
• The polygons are faces.
• An edge is the intersection of two faces.
• A vertex is the intersection of three or more
  faces.

Face
Face
Face
4
Polyhedron
Many Faces
5
Polyhedron Views
Solid
Wire Frame
All three views will be used in these
presentations, the text and other materials.
Hidden Line
6
Which of these are Polyhedrons?
NO
YES
YES
NO
YES
7
Concave Polyhedra
A diagonal, or part of a diagonal, is outside the
figure.
8
Regular Polyhedra

• All of the faces are congruent, regular polygons.

9
Cross Section

• The intersection of a solid and a plane.

Cross section is a circle.
10
Cross Section

• What is the intersection now?

Cross section is a rectangle.
11
What would the cross section be?
A Square
12
Leonard Euler

• 1707 1783
• Probably the greatest mathematician of all time.
• Worked in, and made enormous contributions to,
  every branch of mathematics.

13
Eulers Formula
Count F, the number of faces.
4
3
5
1
6
F 6
2
14
Eulers Formula
7
6
5
8
2
3
Count V, the number of vertices.
4
1
V 8
15
Eulers Formula
7
6
10
5
8
11
12
3
9
2
1
4
Count E, the number of edges.
E 12
16
Eulers Formula
Faces 6 Vertices 8 Edges 12
V F E 2
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Eulers Formula
Faces 6 Vertices 8 Edges 12
6 8 12 2
18
Eulers Formula
Faces 6 Vertices 8 Edges 12
6 8 12 2 14 14
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Eulers Formula
V F E 2
20
Try another figure
Faces Vertices Edges F V E 2 5 5
8 2 10 10
21
Eulers Formula
V F E 2
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Solve

• A polyhedron has 8 faces and 12 vertices. How
  many edges does it have?
• 18
• V F E 2
• 12 8 E 2
• 20 E 2
• E 18

23
Solve

• A polyhedron has 24 vertices and 36 edges. How
  many faces does it have?
• 14
• V F E 2
• 24 F 36 2
• 24 F 38
• F 14

24
Solve

• A polyhedron has 32 faces and 60 edges. How many
  vertices does it have?
• 30
• V F E 2
• V 32 60 2
• V 32 62
• V 30

25
The Platonic Solids

• There are only five of them.
• They are regular, convex polyhedra.
• First described ca. 350 BC by Plato in Timaeus.
• Have been found in many ancient cultures.

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The Five Platonic Solids
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Tetrahedron
Has four triangular sides. Associated with fire.
28
Hexahedron (cube)
Has six square sides. Associated with earth.
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Octahedron
Has eight triangular sides. Associated with air.
30
Dodecahedron
Has 12 pentagonal faces. Associated with the
heavens.
31
Icosahedron
Has 20 triangular faces. Associated with water.
32
Johannes Kepler

• In 1596 Kepler published a tract called The
  Cosmic Mystery in which he envisioned the
  universe as consisting of nested Platonic Solids
  whose inscribed spheres determine the orbits of
  the planets, all enclosed in a sphere
  representing the outer heavens.

33
Dungeons and Dragons
34
Public Toilets in South Korea
This is not a Platonic Solid. It is a compound
polyhedron. Can you find out its correct name?
35
Platonic Solid Links
Mathworld
GSP Icosahedron
36
Summary

• A polyhedron is a solid object.
• The sides are faces.
• Regular polyhedra have congruent faces.
• There are 5 regular polyhedra (the Platonic
  Solids).
• Eulers Formula F V E 2

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Homework