The Fourth Dimension
… and Beyond
Les Reid, Missouri State University
What is the fourth dimension?
What is the fourth dimension?
• Time?
What is the fourth dimension?
• Time?
• Speed?
What is the fourth dimension?
• Time?
• Speed?
• Color?
What is the fourth dimension?
• Time?
• Speed?
• Color?
• Any fourth number (in addition to length,
width, and height) that describes an object
The Geometry of Four Dimensions
• How do we do geometry with four
coordinates?
The Geometry of Four Dimensions
• How do we do geometry with four
coordinates?
• How do we do geometry with two or three
coordinates?
The Geometry of Four Dimensions
• How do we do geometry with four
coordinates?
• How do we do geometry with two or three
coordinates?
• In 2D, the distance between (x,y) and (a,b) is
given by
 x  a   y  b
2
2
• In 4D, we define the distance between
(x,y,z,w) and (a,b,c,d) to be
 x  a    y  b   z  c   w  d 
2
2
2
and given this, we can use a little
trigonometry to compute angles.
2
Squares, Cubes, and Hypercubes
• The points (0,0), (1,0), (0,1), and (1,1) are the
vertices of a square.
Squares, Cubes, and Hypercubes
• The points (0,0), (1,0), (0,1), and (1,1) are the
vertices of a square.
• The vertices (0,0,0), (1,0,0), (0,1,0), (0,0,1),
(1,1,0), (1,0,1), (0,1,1), and (1,1,1) are the
vertices of a cube.
Squares, Cubes, and Hypercubes
• The points (0,0), (1,0), (0,1), and (1,1) are the
vertices of a square.
• The vertices (0,0,0), (1,0,0), (0,1,0), (0,0,1),
(1,1,0), (1,0,1), (0,1,1), and (1,1,1) are the
vertices of a cube.
• The vertices (0,0,0,0), (1,0,0,0), … , (1,1,1,1)
[all possible combinations of 0’s and 1’s] are
the vertices of a hypercube or “tesseract”.
• (More on visualizing this later)
Circles, Spheres, and Hyperspheres
Circles, Spheres, and Hyperspheres
• An equation for a circle of radius r is
x y r
2
2
2
Circles, Spheres, and Hyperspheres
• An equation for a circle of radius r is
x y r
• An equation for a sphere of radius r is
x2  y 2  z 2  r 2
2
2
2
Circles, Spheres, and Hyperspheres
• An equation for a circle of radius r is
x y r
• An equation for a sphere of radius r is
x2  y 2  z 2  r 2
• An equation for a hypersphere of radius r is
2
2
2
x  y z w r
2
2
2
2
2
Area/Volume of Circles/Spheres, Etc.
Area/Volume of Circles/Spheres, Etc.
• Area of a circle:  r
2
Area/Volume of Circles/Spheres, Etc.
• Area of a circle:  r
2
4 3
• Volume of a sphere:
r
3
Area/Volume of Circles/Spheres, Etc.
• Area of a circle:  r
2
4 3
• Volume of a sphere:
r
3
• Hypervolume of a (4D) hypersphere:
Area/Volume of Circles/Spheres, Etc.
• Area of a circle:  r
2
4 3
• Volume of a sphere:
r
3
• Hypervolume of a (4D) hypersphere:

2
2
r
4
Area/Volume of Circles/Spheres, Etc.
• Area of a circle:  r
2
4 3
• Volume of a sphere:
r
3
• Hypervolume of a (4D) hypersphere:
• Hypervolume of a (5D) hypersphere:

2
2
r
4
Area/Volume of Circles/Spheres, Etc.
• Area of a circle:  r
2
4 3
• Volume of a sphere:
r
3
• Hypervolume of a (4D) hypersphere:

2
r
4
2
2
8

• Hypervolume of a (5D) hypersphere:
r5
15
How to Visualize Four Dimensions
How to Visualize Four Dimensions
Edwin A. Abbott
Flatland (A Romance of Many Dimensions)
Flatland (A Romance of Many Dimensions)
• 1884
Flatland (A Romance of Many Dimensions)
• 1884
• Partly a satire of Victorian society (citizens
were polygons, the more sides the higher the
rank; priests were circles; women were line
segments)
Flatland (A Romance of Many Dimensions)
• 1884
• Partly a satire of Victorian society (citizens
were polygons, the more sides the higher the
rank; priests were circles; women were line
segments)
• Our hero: A Square
Flatland (A Romance of Many Dimensions)
• 1884
• Partly a satire of Victorian society (citizens
were polygons, the more sides the higher the
rank; priests were circles; women were line
segments)
• Our hero: A Square
• The visitor: A Sphere
Flatland (A Romance of Many Dimensions)
• 1884
• Partly a satire of Victorian society (citizens
were polygons, the more sides the higher the
rank; priests were circles; women were line
segments)
• Our hero: A Square
• The visitor: A Sphere
• Visualization by analogy
Methods of Visualization
• Slicing
• Unfolding
• Projection
Slicing
Slicing
• A Sphere
Slicing
• A Sphere
Slicing
• A Cube
Slicing
• A Cube
Slicing
• A Triangular Pyramid
Slicing
• A Triangular Pyramid
A Drawback of Slicing
• What is it?
A Drawback of Slicing
• What is it?
• A cube!
Unfolding
Unfolding
• A cube
Unfolding
• A cube
Unfolding
• A triangular pyramid
Unfolding
• A triangular pyramid
A Drawback of Unfolding
• What is it?
A Drawback of Unfolding
• What is it?
• A Buckyball
Projection
• A cube
Projection
• A cube
Projection
• A triangular pyramid
Projection
• A triangular pyramid
Slices of 4D Objects
• Hypersphere
Slices of 4D Objects
• Hypersphere
Slices of 4D Objects
• Hypercube
Slices of 4D Objects
• Hypercube
Slices of 4D Objects
• Hyperpyramid (simplex)
Slices of 4D Objects
• Hyperpyramid (simplex)
Unfolding 4D Objects
• Hypercube
Unfolding 4D Objects
• Hypercube
Two Asides
• Robert A. Heinlein’s short story “-And He Built
a Crooked House
Two Asides
• Robert A. Heinlein’s short story “-And He Built
a Crooked House
• Salvador Dali’s “Corpus Hypercubus”
Corpus Hypercubus
Unfolding 4D Objects
• Hyperpyramid
Unfolding 4D Objects
• Hyperpyramid
Projections of 4D Objects
• Hypercube
Projections of 4D Objects
• Hypercube
Projections of 4D Objects
• Hyperpyramid
Projections of 4D Objects
• Hyperpyramid
Rotating Hypercube
Regular Polyhedra (3D)
• Every face is a regular polygon
• All faces are congruent
• There are the same number of faces at each
vertex.
Regular Polyhedra (3D)
• Possible Faces: equilateral triangle, square,
regular pentagon, regular hexagon, …
Regular Polyhedra (3D)
• Possible Faces: equilateral triangle, square,
regular pentagon, regular hexagon, …
• Equilateral triangle angle: 60 degrees, so we
could fit 3, 4, or 5 around a vertex (6 gives 360
degrees which is flat).
Regular Polyhedra (3D)
• Possible Faces: equilateral triangle, square,
regular pentagon, regular hexagon, …
• Equilateral triangle angle: 60 degrees, so we
could fit 3, 4, or 5 around a vertex (6 gives 360
degrees which is flat).
• Square angle: 90 degrees, so we could fit 3
Regular Polyhedra (3D)
• Possible Faces: equilateral triangle, square,
regular pentagon, regular hexagon, …
• Equilateral triangle angle: 60 degrees, so we
could fit 3, 4, or 5 around a vertex (6 gives 360
degrees which is flat).
• Square angle: 90 degrees, so we could fit 3
• Regular pentagon angle: 108 degrees yields 3
Regular Polyhedra (3D)
• Possible Faces: equilateral triangle, square,
regular pentagon, regular hexagon, …
• Equilateral triangle angle: 60 degrees, so we
could fit 3, 4, or 5 around a vertex (6 gives 360
degrees which is flat).
• Square angle: 90 degrees, so we could fit 3
• Regular pentagon angle: 108 degrees yields 3
• Regular hexagon angle: 120 degrees (can’t do)
Regular Polyhedra (3D)
• There are five possibilities and they all occur
Regular Polyhedra (3D)
• There are five possibilities and they all occur
• From left to right they are the tetrahedron,
the cube, the octahedron, the dodecahedron,
and the icosahedron
Regular Polytopes (4D)
• Every “face” is a regular polyhedron
• All faces are congruent
• The same number of polyhedra meet at each
edge
Aside on Dihedral Angles
• If you slice a polyhedron perpendicular to an
edge, the angle obtained is called the dihedral
angle at that edge.
Aside on Dihedral Angles
• If you slice a polyhedron perpendicular to an
edge, the angle obtained is called the dihedral
angle at that edge.
• For example, the dihedral angle of any edge of
a cube is 90 degrees.
Regular Polytopes (4D)
polyhedron
dihedral angle
number at an edge
tetrahedron
cube
octahedron
dodecahedron
icosahedron
70.5 degrees
90 degrees
109.5 degrees
116.6 degrees
138.2 degrees
3, 4, or 5
3
3
3
not possible
Regular Polytopes (4D)
• This gives a total of six possibilities and they
all occur.
Regular Polytopes (4D)
Regular Polytopes (4D)
• The 120-cell consists of 120 dodecahedra with
3 at each edge
• The 600-cell consists of 600 tetrahedra with 5
at each edge
Polytope Sculptures
Other Polytopes
Alicia Boole Stott
1860-1940
Alicia Boole Stott
1860-1940
• The daughter of George Boole (who created
Boolean algebra used in logic and computer
science)
Alicia Boole Stott
1860-1940
• The daughter of George Boole (who created
Boolean algebra used in logic and computer
science)
• As a child she was trained by the amateur
mathematician Charles Hinton to think fourdimensionally
Alicia Boole Stott
1860-1940
• The daughter of George Boole (who created
Boolean algebra used in logic and computer
science)
• As a child she was trained by the amateur
mathematician Charles Hinton to think fourdimensionally
• She coined the term “polytope”, discovered
the 6 regular ones, and helped the geometer
H.S.M. Coxeter in his research
Higher Dimensions
• In three dimensions there are 5 regular
polytopes
• In four dimensions there are 6 regular
polytopes
• What happens in five dimensions?
Higher Dimensions
• In five dimensions and higher there are only
three regular objects, the analogs of the
tetrahedron, the cube, and the octahedron
(the simplex, the hypercube, and the 16-cell in
four dimensions)
Higher Dimensions
Kissing Number
• The number of n-dimensional spheres of
radius 1 that can simultaneously touch a
central sphere of radius 1 is called the kissing
number in that dimension
Kissing Number
• The number of n-dimensional spheres of
radius 1 that can simultaneously touch a
central sphere of radius 1 is called the kissing
number in that dimension
• For example, when n=2
Kissing Number
• The number of n-dimensional spheres of
radius 1 that can simultaneously touch a
central sphere of radius 1 is called the kissing
number in that dimension
• For example, when n=2, the kissing number is
6
Kissing Number
• Isaac Newton and David Gregory argued about
the kissing number in three dimensions.
Newton thought it was 12, while Gregory
thought it might be 13
Kissing Number
• Finally, in 1953 it was proven that Newton was
correct.
Kissing Number
• Finally, in 1953 it was proven that Newton was
correct.
• In 2003, it was proven that the kissing number
in four dimensions is 24 (the 24-cell is used).
Kissing Number
• Finally, in 1953 it was proven that Newton was
correct
• In 2003, it was proven that the kissing number
in four dimensions is 24 (the 24-cell is used)
• It is known that the kissing number in five
dimensions is between 40 and 44 (inclusive)
Kissing Number
• Finally, in 1953 it was proven that Newton was
correct
• In 2003, it was proven that the kissing number
in four dimensions is 24 (the 24-cell is used)
• It is known that the kissing number in five
dimensions is between 40 and 44 (inclusive)
• It has long been known that kissing number in
8D is 240
Kissing Number
• Finally, in 1953 it was proven that Newton was
correct
• In 2003, it was proven that the kissing number
in four dimensions is 24 (the 24-cell is used)
• It is known that the kissing number in five
dimensions is between 40 and 44 (inclusive)
• It has long been known that kissing number in
8D is 240 and in 24D is 196,460
Keller’s Conjecture
• Every tiling of n-dimensional space by unit
cubes must have at least two cubes that share
an (n-1)-dimensional face.
Keller’s Conjecture
• Every tiling of n-dimensional space by unit
cubes must have at least two cubes that share
an (n-1)-dimensional face.
• True in 2D
Keller’s Conjecture
• Every tiling of n-dimensional space by unit
cubes must have at least two cubes that share
an (n-1)-dimensional face.
• True in 2D
• And in 3D
Keller’s Conjecture
• The conjecture is known to be true in
dimensions 1 through 6.
Keller’s Conjecture
• The conjecture is known to be true in
dimensions 1 through 6, but is false in
dimension 8 and higher
• The status of the conjecture in dimension 7 is
still open
Star Polyhedra
• There are four of them
Star Polyhedra
Star Polytopes (4D)
• There are six of them
{5/2,3,5}
{5,5/2,5}
{5,3,5/2}
{5,5/2,3}
{3,3,5/2}
{5/2,3,3}
Questions?
Thank you