The Fourth Dimension … and Beyond Les Reid, Missouri State University

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

In 4D, we define the distance between (x,y,z,w) and (a,b,c,d) to be and given this, we can use a little trigonometry to compute angles.

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

An equation for a circle of radius r is

Circles, Spheres, and Hyperspheres An equation for a circle of radius r is An equation for a sphere of radius r is

Circles, Spheres, and Hyperspheres An equation for a circle of radius r is An equation for a sphere of radius r is An equation for a hypersphere of radius r is

Area/Volume of Circles/Spheres, Etc.

Area of a circle:

Area/Volume of Circles/Spheres, Etc. Area of a circle: Volume of a sphere:

Area/Volume of Circles/Spheres, Etc. Area of a circle: Volume of a sphere: Hypervolume of a (4D) hypersphere:

Area/Volume of Circles/Spheres, Etc. Area of a circle: Volume of a sphere: Hypervolume of a (4D) hypersphere:

Area/Volume of Circles/Spheres, Etc. Area of a circle: Volume of a sphere: Hypervolume of a (4D) hypersphere: Hypervolume of a (5D) hypersphere:

Area/Volume of Circles/Spheres, Etc. Area of a circle: Volume of a sphere: Hypervolume of a (4D) hypersphere: Hypervolume of a (5D) hypersphere:

How to Visualize Four Dimensions

Edwin A. Abbott

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

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

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) polyhedrondihedral angle number at an edge tetrahedron70.5 degrees3, 4, or 5 cube90 degrees3 octahedron109.5 degrees3 dodecahedron116.6 degrees3 icosahedron138.2 degreesnot possible

Regular Polytopes (4D) This gives a total of six possibilities and they all occur.

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

The daughter of George Boole (who created Boolean algebra used in logic and computer science)

Alicia Boole Stott 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 four- dimensionally

Alicia Boole Stott 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 four- dimensionally 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