Patterns + Women =
Figures for Mathematics!
Alissa S. Crans
Loyola Marymount University
Expanding Your Horizons
James Madison University
March 24, 2012
The above image is from the Wikipedia article on Grace Chisholm Young. It qualifies as fair use under U.S. Copyright law.
Wednesday, May 30, 2012
1
Some Famous Mathematicians
Pascal
Einstein
Euclid
Fibonacci
Newton
Pythagoras
These images are from the Wikipedia articles on Blaise Pascal, Isaac Newton, Albert Einstein, and Fibonacci.
All images belong to the public domain.
Wednesday, May 30, 2012
2
Grace Chisholm Young
• English mathematician
1868 - 1944
The above image is from the Wikipedia article on Grace Chisholm Young. It qualifies as fair use under U.S. Copyright law.
Wednesday, May 30, 2012
3
Grace Chisholm Young
• English mathematician
• First Ph.D. awarded to a
woman in Germany
1868 - 1944
The above image is from the Wikipedia article on Grace Chisholm Young. It qualifies as fair use under U.S. Copyright law.
Wednesday, May 30, 2012
3
Grace Chisholm Young
• English mathematician
• First Ph.D. awarded to a
woman in Germany
•married, 6 children--son and
granddaughter both have
Ph.D.’s in mathematics
1868 - 1944
The above image is from the Wikipedia article on Grace Chisholm Young. It qualifies as fair use under U.S. Copyright law.
Wednesday, May 30, 2012
3
Grace Chisholm Young
• English mathematician
• First Ph.D. awarded to a
woman in Germany
•married, 6 children--son and
granddaughter both have
Ph.D.’s in mathematics
1868 - 1944
• A First Book of Geometry,
written with her husband
The above image is from the Wikipedia article on Grace Chisholm Young. It qualifies as fair use under U.S. Copyright law.
Wednesday, May 30, 2012
3
Leonhard Euler
• German mathematician
1707 - 1783
This image is from the Wikipedia article on Leonhard Euler. Permission to use it was granted under the GNU Free Documentation License.
Wednesday, May 30, 2012
4
Leonhard Euler
• German mathematician
• numerous contributions to many
areas of mathematics and
physics: geometry, trigonometry,
calculus
1707 - 1783
This image is from the Wikipedia article on Leonhard Euler. Permission to use it was granted under the GNU Free Documentation License.
Wednesday, May 30, 2012
4
Leonhard Euler
• German mathematician
• numerous contributions to many
areas of mathematics and
physics: geometry, trigonometry,
calculus
1707 - 1783
• most prolific mathematical
writer of all time!
This image is from the Wikipedia article on Leonhard Euler. Permission to use it was granted under the GNU Free Documentation License.
Wednesday, May 30, 2012
4
Leonhard Euler
• German mathematician
• numerous contributions to many
areas of mathematics and
physics: geometry, trigonometry,
calculus
1707 - 1783
• most prolific mathematical
writer of all time!
• married with 5 children
This image is from the Wikipedia article on Leonhard Euler. Permission to use it was granted under the GNU Free Documentation License.
Wednesday, May 30, 2012
4
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
5
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
5
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
5
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
6
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
6
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
6
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
6
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
6
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
6
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Vertices: 5
Edges:
Vertices: 8
Faces:
Edges: 5
Faces:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
6
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
7
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
7
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
7
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
7
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
7
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
7
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
7
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
7
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
7
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
Vertices: 5
Edges: 8
Faces: 5
Vertices:
Edges:
Faces:
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
7
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
8
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
Vertices:
Edges:
Faces:
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
8
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
Vertices:
Edges:
Faces:
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
9
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
Vertices:
Edges:
Faces:
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
10
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
Vertices:
Edges:
Faces:
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
11
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
Vertices:
Edges:
Faces:
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
12
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Text
Text
Vertices:
Edges:
Faces:
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
13
Euler’s Formula
This formula gives a relationship between the
number of vertices, edges, and faces of certain
3d figures called polyhedra.
Example:
Vertices: 5
Edges: 8
Faces: 5
Text
Text
The image above is from the Wikipedia article on pyramid. It was taken by Mr. Jon Bodsworth, who gave permission to use it.
Wednesday, May 30, 2012
14
Euler’s Formula
Example:
Vertices:
Edges:
Faces:
Rubik’s Cube
This image is taken from WP Clipart and belongs to the public domain.
Wednesday, May 30, 2012
15
Euler’s Formula
Example:
Vertices: 8
Edges: 12
Faces: 6
Rubik’s Cube
This image is taken from WP Clipart and belongs to the public domain.
Wednesday, May 30, 2012
15
Euler’s Formula
Example:
Vertices:
Edges:
Faces:
prism
These images are from the Wikipedia article on prism and triangular prism. Permission to use the first was granted under the Creative Commons
Agreement and permission to use the second was given by Robert Webb who created the image using his Stella software.
Wednesday, May 30, 2012
16
Euler’s Formula
Example:
Vertices:
Edges:
Faces:
prism
These images are from the Wikipedia article on prism and triangular prism. Permission to use the first was granted under the Creative Commons
Agreement and permission to use the second was given by Robert Webb who created the image using his Stella software.
Wednesday, May 30, 2012
16
Euler’s Formula
Example:
Vertices: 6
Edges: 9
Faces: 5
prism
These images are from the Wikipedia article on prism and triangular prism. Permission to use the first was granted under the Creative Commons
Agreement and permission to use the second was given by Robert Webb who created the image using his Stella software.
Wednesday, May 30, 2012
16
Euler’s Formula
Example:
Text
Vertices:
Edges:
Faces:
Te
dreidel
This image is taken from WP Clipart and belongs to the public domain.
Wednesday, May 30, 2012
17
Euler’s Formula
Example:
Text
Vertices: 9
Edges: 16
Faces: 9
Te
dreidel
This image is taken from WP Clipart and belongs to the public domain.
Wednesday, May 30, 2012
17
Euler’s Formula
tetrahedron
octahedron
The animated pictures are from the Wikipedia article on Platonic Solid.
They were created by Cyp, who granted permission to use them under the GNU Free Documentation License.
Wednesday, May 30, 2012
18
Euler’s Formula
dodecahedron
icosahedron
The animated pictures are from the Wikipedia article on Platonic Solid.
They were created by Cyp, who granted permission to use them under the GNU Free Documentation License.
Wednesday, May 30, 2012
19
Euler’s Formula
Name
pyramid
Rubik’s Cube
prism
dreidel
tetrahedron
octahedron
dodecahedron
icosahedron
Wednesday, May 30, 2012
Vertices Edges Faces
5
8
6
9
4
6
20
12
8
12
9
16
6
12
30
30
5
6
5
9
4
8
12
20
20
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
Name
Vertices
Edges
Faces
pentagonal prism
hexagonal prism
octagonal prism
cubeoctahedron
truncated cube
truncated octahedron
square frustum
pentagonal frustum
pentagonal trapezohedron
10
12
16
12
24
24
8
10
12
15
18
24
24
36
36
12
15
20
7
8
10
14
14
14
6
7
10
Wednesday, May 30, 2012
21
Euler’s Formula
What’s the pattern??? What is Euler’s Formula?
Verti
16
ces:
32
Edge
16
s:
Faces
:
Text
Text
Wednesday, May 30, 2012
22
Euler’s Formula
What’s the pattern??? What is Euler’s Formula?
V-E+F=2
Text
Verti
16
ces:
32
Edge
16
s:
Faces
:
Text
Wednesday, May 30, 2012
22
Euler’s Formula
What’s the pattern??? What is Euler’s Formula?
V-E+F=2
Text
But...Does this always hold?
Verti
16
ces:
32
Edge
16
s:
Faces
:
Text
Wednesday, May 30, 2012
22
Euler’s Formula
What about a shape like this:
Vertices:
Edges:
Faces:
Text
Verti
16
ces:
32
Edge
16
s:
Faces
:
Text
Wednesday, May 30, 2012
23
Euler’s Formula
What about a shape like this:
Vertices: 16
Edges: Verti
32
Faces: ces:
16
Edge
s:
Faces
:
Text
Text
Wednesday, May 30, 2012
24
Euler’s Formula
What about a shape like this:
Text
Text
Wednesday, May 30, 2012
Vertices: 16
Edges: 32
Faces: Verti
16
ces:
Edge
s:
Faces
:
25
Euler’s Formula
What about a shape like this:
Vertices: 16
Edges: 32
Faces: 16
Text
Text
Wednesday, May 30, 2012
26
Euler’s Formula
What about a shape like this:
Vertices: 16
Edges: 32
Faces: 16
Text
Euler’s formula only holds for nice shapes!
(ones without holes)
Text
Wednesday, May 30, 2012
26
Don’t Forget...
Text
Text
Wednesday, May 30, 2012
27