Polyhedra - BakerMath.org

Download Report

Transcript Polyhedra - BakerMath.org

Geometry
Polyhedra
Goals
Know terminology about solids.
 Identify solids by type.
 Use Euler’s Theorem to solve problems.

April 13, 2015
2
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.
April 13, 2015
Face
Face
Face

3
Polyhedron
April 13, 2015
4
Polyhedron Views
Solid
Wire Frame
All three views will be used
in these presentations, the
text and other materials.
Hidden Line
April 13, 2015
5
Which of these are Polyhedrons?
NO
YES
April 13, 2015
YES
NO
YES
6
Concave Polyhedra
A diagonal, or part of a
diagonal, is outside the figure.
April 13, 2015
7
Regular Polyhedra

All of the faces are
congruent, regular
polygons.
April 13, 2015
8
Cross Section

The intersection of a solid and a plane.
Cross section is a circle.
April 13, 2015
9
Cross Section

What is the
intersection now?
Cross section is a rectangle.
April 13, 2015
10
What would the cross section be?
A Square
April 13, 2015
11
Leonard Euler



1707 – 1783
Probably the greatest
mathematician of all
time.
Worked in, and made
enormous
contributions to, every
branch of
mathematics.
April 13, 2015
12
Euler’s Formula
Count F, the
number of faces.
12
April 13, 2015
4
3 5
6 F=6
13
Euler’s Formula
7
8
Count V, the
number of
vertices.
April 13, 2015
2
1
5
6
3
4
V=8
14
Euler’s Formula7
6
9
Count E, the
number of edges.
April 13, 2015
10 5
3 12
2
1
8 11
4
E = 12
15
Euler’s Formula
Faces =
6
Vertices =
8
V+F=E+2
Edges =
12
April 13, 2015
16
Euler’s Formula
Faces =
6
Vertices =
8
6 + 8 = 12 + 2
Edges =
12
April 13, 2015
17
Euler’s Formula
Faces =
6
Vertices =
8
6 + 8 = 12 + 2
14 = 14
Edges =
12
April 13, 2015
18
Euler’s Formula
April 13, 2015
19
Try another figure…
Faces =
Vertices =
Edges =
F+V=E+2
5+5=8+2
10 = 10
April 13, 2015
20
Euler’s Formula
April 13, 2015
21
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

April 13, 2015
22
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

April 13, 2015
23
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

April 13, 2015
24
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.

April 13, 2015
25
The Five Platonic Solids
April 13, 2015
26
Tetrahedron
Has four triangular sides.
Associated with fire.
April 13, 2015
27
Hexahedron (cube)
Has six square sides.
Associated with earth.
April 13, 2015
28
Octahedron
Has eight triangular sides.
Associated with air.
April 13, 2015
29
Dodecahedron
Has 12 pentagonal faces.
Associated with the heavens.
April 13, 2015
30
Icosahedron
Has 20 triangular faces.
Associated with water.
April 13, 2015
31
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.
April 13, 2015
32
Dungeons and Dragons
April 13, 2015
33
Public Toilets in South Korea
This is not a
Platonic Solid. It is a
compound
polyhedron. Can you
find out its correct
name?
April 13, 2015
34
Platonic Solid Links
Mathworld
GSP Icosahedron
April 13, 2015
35
Summary
A polyhedron is a solid object.
 The sides are faces.
 Regular polyhedra have congruent faces.
 There are 5 regular polyhedra (the
Platonic Solids).
 Euler’s Formula: F + V = E + 2

April 13, 2015
36
Homework
April 13, 2015
37