Journey Through the Fourth Dimension

Tom Downey

SED ME 559

Abstract

This paper presents an overview of higher-dimensional geometry, focusing on the fourth dimension. It relies on concepts developed in one, two, and three dimensions to help explain the corresponding ideas in higher dimensions. Topics covered include distance and measurement, regular polytopes (particularly the hypercube), extensions to Euler’s polyhedron forumla, and visualization of four-dimensional shapes via unfolding, projection, and slicing. The paper concludes with a discussion of how to use higher-dimensional geometry in the high school classroom, with suggestions for activities, explorations, projects, and computer lab work.

1         Introduction

We live in a three-dimensional world, surrounded by objects that have length, width, and height. In school we study plane geometry (two dimensions), including lines (one dimension) and points (no dimensions). But what is a dimension, that our world has three of them, and a plane has only two? Could there be more than three, and if so where are they and what do they look like?

One way to think about dimensions is in terms of perpendicular ways of moving, borrowing concepts and worlds from Flatland and Sphereland. On a one-dimensional line world, a creature made up of a segment can move only left or right. If we think of their line world as a number line, we only need one number to specify the creature’s location on the line. On a two-dimensional plane world, a polygonal creature can move left or right along a line, but it can now also move perpendicular to this line (i.e., back and forth). Putting Cartesian coordinates on the plane, it takes two numbers to specify the location of the creature. In a three-dimensional space, a polyhedral creature can move left or right, back and forth, but can also move along a line that is perpendicular to both of the two previous lines (i.e., up and down). It takes three numbers to specify location in space.

Figure 1 One-, Two-, and Three-Dimensional Worlds

What happens if we extend this one more time? In a four-dimensional space, a four-dimensional creature could move left and right, back and forth, up and down, but now can move along a line that is perpendicular to the other three lines (these new directions are sometimes referred to as ana and kata). It will take four numbers to specify a four-dimensional creatures location. But how can a line possibly be perpendicular to three mutually perpendicular lines? No such thing exists in our three-dimensional world; no one has seen such a line. However, we can use mathematics and analogies to explore some of the properties of four-dimensional space and the objects in it, building on two- and three-dimensional geometry.

Why study higher dimensions, if our world has only three? One answer is that the mathematics and insights developed as part of exploring and understanding higher dimensions have applicability in certain areas of science. There, even though our universe is three-dimensional, the models and equations look more like those of a higher dimensional world. The best-known example is Einstein’s space-time, where time is treated as a fourth dimension.

Even more interesting, several modern physics theories that attempt to unite Einstein’s theories of relativity with those of quantum mechanics require more than three spatial dimensions in our universe! These theories model the universe using nine, ten, eleven, or even twelve dimensions. This paper will touch on these ideas briefly.

2         Terminology

Before we begin in earnest, it is useful to define some additional terms to extend those of familiar plane and three-dimensional geometry.

Axes: In two-dimensional geometry, we typically refer to the two perpendicular axes as x and y, while the third dimension adds a z axis. In four-dimensional space, it is conventional to refer to the fourth axis as w.

Space: In general, an n-dimensional space is referred to as an n-space (e.g., 3-space, 4-space, etc.). A 1-space is also called a line, while a 2-space is also known as a plane. Similarly, 4-space is often referred to as hyperspace, although this term is sometimes used to refer to any space with more than three dimensions. Hyperplane is also sometimes used to refer to an (n-1)-space embedded in an n-space (e.g., in 4-space, a 3-space is sometimes called a hyperplane, since it is related to the 4-space in the same way that a two-dimensional plane is related to 3-space).

Polytopes: The generic name for a shape in n-space that is bounded by (n-1)-dimensional shapes is a polytope. A two-dimensional polytope is more usually called a polygon (bounded by one-dimensional segments) and a three-dimensional polytope is called a polyhedron (bounded by two-dimensional polygons). A four-dimensional polytope is sometimes referred to as a polychoron (plural is polychora).

Polytope boundaries: The elements of polyhedral boundaries are called vertices, edges, and faces for the zero-, one-, and two-dimensional boundaries, respectively. The boundaries of a four-dimensional polytope are three-dimensional polyhedra and are commonly referred to as cells, although some authors refer to them as hyperfaces. In this paper, I will also use cells to refer to any higher-dimensional boundary of a higher dimensional polytope.

Hypervolume or Content: The extension of area in two dimensions and volume in three dimensions to higher dimensions.

3         Basic Concepts

3.1      Dimensions

Let’s revisit the description of dimensions provided in the introduction. To begin, imagine a single point. It has no length, width, depth, or any size at all; it is zero-dimensional. Now move this point one unit of distance. The result is a segment, and one dimension has been defined, namely the line that contains this segment. This segment has length, but no width or any other size. It has two points as vertices, one at each end.

Next, imagine moving this entire segment one unit in a new direction that is perpendicular to the line containing the segment. The result is a square, and a two-dimensional plane has been defined, namely the one that contains the square. The square is bounded by four one-dimensional edges: the segment that we started with, the same segment after it stops moving, and two segments that are created by the motion of the segment’s two vertices.

Again, imagine moving this entire square one unit in a new direction that is perpendicular to the plane containing the square. The result is a cube, and a three-dimensional space has been defined, namely the one that contains the cube. The cube is bounded by six two-dimensional faces: the square that we started with, the same square after it stops moving, and four squares that are created by the motion of the square’s four edges.

Figure 2 Moving Through Perpendicular Dimensions

Now for the leap: imagine moving this entire cube one unit in a new direction that is perpendicular to the space containing the cube. The result is a tesseract, or hypercube, and a four-dimensional space has been defined, namely the one that contains the hypercube. Since we live in a three-dimensional world, imagining this fourth perpendicular direction is very difficult, if not impossible. However, if we assume that it did exist and we could move the cube through this fourth dimensional direction as described, we can draw some conclusions about the hypercube that results. This hypercube is bounded by eight three-dimensional cells: the cube that we started with, the same cube after it stops moving, and six cubes that are created by the motion of the cube’s six faces.

Figure 3 Moving in the Fourth Dimension

3.2      Distance

How far apart are two points in higher dimensional spaces? First, let’s review distances in more familiar dimensions. Assume that we’ve defined Cartesian coordinates (i.e., perpendicular axes) in each space that we’ll examine, and by distance we mean the familiar concept of distance in such a coordinate system. For a one-dimensional line, the distance between two points with coordinates (x1) and (x2) is typically given by |x1- x2|. However, let’s define it instead using the equivalent formula:

                                         Distance in one dimension =                                    (1)

Distance in two dimensions can be derived using the Pythagorean theorem. The difference in the x values of the two points constitutes one side of a right triangle, and the difference in the y values constitutes the other. Then the distance between (x1, y1) and (x2, y2) is the length of the hypotenuse, which is given by:

                               Distance in two dimensions =                          (2)

Figure 4 Distances in One, Two, and Three Dimensions

Distance in three dimensions can also be derived using the Pythagorean theorem twice, as illustrated in Figure 4. The formula is:

                     Distance in three dimensions =                (3)

By this point, the pattern should be obvious. The distance between two points in 4-space is given by:

           Distance in four dimensions =      (4)

Getting slightly more complicated, what is the distance between a point and an (n-1)-space in an n-space? For a 2-space, this is simply the distance between a point and a line, which is measured perpendicular to the line (i.e., given a point, A, and a line, m, draw a second line through A that is perpendicular to m; the distance sought is the distance between point A and the point of intersection of the two lines). In 3-space, we want the distance between a point and a plane, which is measured in a similar manner (i.e., along a line perpendicular to the plane that runs through the point).

Figure 5 Distance from a Point

The extension to 4-space should be obvious (although hard to think about visually!). Draw a line through the point, A, perpendicular to the 3-space, and measure the distance between A and the point where the drawn line intersects the 3-space. The hard part to deal with, conceptually, is the idea that a line can intersect three-dimensional space at a single point, since we normally think of a line as fully contained within three-dimensional space. To help think about this, first realize that in 3-space there are an infinite number of planes, as illustrated in Figure 6. A line can either be contained in a specific plane p, be parallel to p and never touch it (i.e., it lies in a parallel plane), or can intersect p at a single point. When we add a fourth dimension, we now have an infinite number of 3-spaces, just as in 3-space we have many planes. Hence in 4-space, a line can either be wholly contained in a 3-space, parallel to the 3-space and never touch it (i.e., it lies in a parallel 3-space), or can intersect it at a single point.

Figure 6 Lines and Planes

3.3      Parallelism

The concept of parallel 3-spaces is strange, so let’s explore that a bit more before moving on. In Euclidean geometry in a plane, two lines are parallel if they are everywhere the same distance from each other, with distance defined as in the previous section. This idea is easy to extend to three dimensions, where two planes are parallel if they are also everywhere the same distance from each other. Note that in three dimensions, it is possible to have two lines that never meet but are not parallel; these are called skew lines.

Figure 7 Parallel Lines and Planes

In four dimensions, two 3-spaces are parallel if they are the same distance from each other everywhere (hence never meet). Note that in 4-space, you could have two planes that never meet, but are not parallel (i.e., skew planes). This idea generalizes to parallel (n-1)-spaces in an n-space, just as the distance formula generalizes to higher dimensions.

3.4      Handedness

In our three-dimensional world, we are familiar with the concept of handedness, as in left-handed screw threads or right-handed screws, or cars with steering wheels on the left or right sides. This concept of handedness, and its relationship with higher dimensions, is explored in Sphereland. In this book, the residents of Flatland keep pet dogs, which come in two varieties: mongrel and pedigreed, as shown in Figure 8. No amount of rotation or motion in the plane of Flatland can turn a mongrel dog into a pedigreed dog. However, to us three-dimensional creatures, it is obvious how such a transformation could be accomplished: simply lift the mongrel dog out of the Flatland plane, flip it over, and return it to Flatland. Voila! A pedigreed dog!

Figure 8 Dogs in Flatland

Figure 9 Changing a Mongrel into a Pedigreed Dog via the Third Dimension

A four-dimensional being could perform the same feat in our world by taking a left-handed screw, “lifting” it out of our 3-space via the fourth dimension, “flipping’ it over through the fourth dimension, and then returning it to our 3-space. Voila! A right-handed screw! Our four-dimensional friend could perform the same feat with books, resulting in text that needs a mirror to read. One character in a science fiction story, after being “flipped” left-for-right via a time-machine that used the fourth dimension, came back with his heart on the right side of his chest instead of on the left (which, of course, saves his life later in the story, when the villain shoots him in the left side of his chest).

3.5      Inside and Outside

We think of the inside of an object (like our own bodies) as something that can’t be gotten at without cutting through the object’s boundaries (or skin). However, higher dimensional creatures have another way inside: through the fourth dimension.

Both Flatland and Sphereland talk about removing things from locked cabinets. Imagine a cabinet in Flatland as a square with one side hinged so that you can swing it open, place two-dimensional objects inside, and then close it. There is no way a Flatland resident can remove an object from the closed cabinet without opening the door. However, a being from the third dimension can see right into the cabinet! What’s more, the three-dimensional being can reach into the closed cabinet, lift the object out of the Flatland plane and out of the cabinet, and place it back into Flatland, but outside the cabinet. From the point of view of the Flatlanders, the object appears out of thin air, and when they look inside the cabinet, it’s empty.

Figure 10 Looking Inside a Closed Cabinet via the Third Dimension

By analogy, imagine a cabinet in our three-dimensional world, such as a china cupboard with wooden doors containing glasses. There is no way for us to remove a glass from the cupboard without opening the door, but a being in the fourth dimension has no such restriction. This being can see inside the cupboard, reach in via the fourth dimension, lift out a glass from our 3-space (and from the cupboard), and replace it back into our 3-space but outside the cupboard. If the cupboard had a glass front, we’d see a glass vanish, and then reappear outside the cupboard!

You can imagine all sorts of possible uses for the ability to move objects in our world through a fourth dimension. Magicians would have an easy time with their shows, if they had a little help from a fourth dimension assistant. Bank robbers could pilfer the contents of uncrackable safes at will. Surgeons could operate on your vital organs without ever cutting your skin, or even any intervening tissue. What’s more, doctors could look at your insides in full view and detail without x-rays or endoscopes. Of course, until we learned to build walls in the fourth dimension, privacy would be a thing of the past…

4         Polytopes

As discussed in the terminology section, polytopes are n-dimensional objects bounded by (n-1)-dimensional polytopes. Polytopes are also made of straight segments, and include polygons and polyhedra for the two- and three-dimensional cases. We’ll first look at the hypercube and its cousins, then explore other regular polygons in the fourth dimension, and finally return to attempt to visualize a hypercube.

4.1      Hypercubes

As described previously, if we move a point we get a segment; if we move the segment in a perpendicular direction we get a square; if we move the square we get a cube; and if we move the cube we get a tesseract (also called a hypercube). By thinking about how the vertices, edges, etc. of each object move to create the hypercube in the next higher dimension, we can build a table showing the number of vertices, edges, faces, etc. for a cube in each dimension.

 

Polytope

Dimensionality

Vertices (0 dim.)

Edges (1 dim.)

Faces (2 dim.)

Cells (3 dim.)

Point

0

1 point

-

-

-

Segment

1

2 points

1 segment

-

-

Square

2

4 points

4 segments

1 square

-

Cube

3

8 points

12 segments

6 squares

1 cube

Tesseract

4

16 points

32 segments

24 squares

8 cubes

5-cube

5

32 points

80 segments

80 squares

40 cubes

6-cube

6

64 points

192 segments

240 squares

160 cubes

7-cube

7

128 points

448 segments

672 squares

560 cubes

Table 1 Boundaries of Hypercubes

The number of boundaries of different dimensions in an n-cube can be found from the coefficients of (2x+1)n. Using the binomial theorem, we can find a formula for the number of k-cubes in an n-cube:

                                                                                          (5)

The four-dimensional cube, called a tesseract or hypercube, is bounded by eight cubes. This may seem a bit odd, but remember that we have one additional dimension to deal with. In the fourth dimension we can have many three-dimensional spaces that do not overlap, hence we require pieces of these spaces (i.e., polyhedra) to act as the boundaries of our four-dimensional polytope. Note that there should actually be more columns in Table 1 for the 5-, 6- and 7-cubes, since these are bounded by 4-, 5- and 6-cubes, respectively.

4.2      Measures of Content

Before moving on to other polytopes, let’s consider how to measure the content of a polytope. By content we mean what’s “inside” the polytope, or within its boundaries. For an n-dimensional polytope, it is bounded by (n-1)-dimensional polytopes. Thus, for a tesseract, we want to find out the content, or hypervolume, that lies within 4-space “inside” the eight boundary cubes.

As before, we’ll work our way up to this from lower dimensions. On a one-dimensional line, the “inside” of a line segment bounded by two points is simply the length of the segment. For simplicity, let’s use meters as our unit of measure. Hence a segment would be a certain number of meters long. For a two-dimensional plane, the inside of a square is the length times the width; since these are the same, we can just compute the second power of the length of a side to get the area of the square, in meters2.

Similarly, the content of a cube in three dimensions is the length times width times height. Again, since these are all equal, the content or volume is the third power of the length of a side, measured in meters3. In the fourth dimension, the content or hypervolume of a tesseract is just the fourth power of the length of a side, measured in meters4. This can be extended to calculate the volume of an n-dimensional hypercube whose side is s meters long:

                                                                                                        (6)

The content of a prism in higher dimensions can also be found via a similar analogy. In 3-space a prism has a 2-dimensional base and a height (in the third dimension). Like we moved a square perpendicular to its plane to create a cube, you can move any two-dimensional shape perpendicular to its plane to create a prism. The volume (in meters3) of the prism is found by multiplying the area of the base (in meters2) times the height (in meters). In four dimensions, the base of the hyperprism is a three-dimensional solid, which is moved through the fourth dimension to create the hyperprism. The hypervolume (in meters4) of the hyperprism is found by multiplying the volume of the base (in meters3) times the height (in meters).

In the three-dimensional case, the prism can be “slanted” and its volume will be the same as a right-angled prism, as long as the heights are the same. Height is defined as the distance from the plane containing the top face of the prism to the plane containing the base (by definition of a prism, these planes are parallel). Similarly, a four-dimensional hyperprism can be slanted without changing its hypervolume, as long as its height remains the same. Here, height is the distance from the 3-space containing its top to the 3-space containing its base (by definition, these two 3-spaces are parallel).

4.3      Regular Polychora

Students study the Platonic solids in geometry, learning that they are the only regular, convex polyhedra (regular means their sides are all the same and are made of regular polygons). In 4-space, there are six regular polychora, as listed in Table 2. Remember that, in 3-space, a regular polyhedron is bounded by regular polygons; hence in 4-space a regular polychoron is bounded by regular polyhedra.

 

Name

Schläfli

Cell type

# Vertices

# Edges

# Faces

# Cells

Pentatope

{3,3,3}

Tetrahedron

5

10

10

5

Tesseract

{4,3,3}

Cube

16

32

24

8

Cross-polytope (16-cell)

{3,3,4}

Tetrahedron

8

24

32

16

24-cell

{3,4,3}

Octahedron

24

96

96

24

120-cell

{5,3,3}

Dodecahedron

600

1200

720

120

600-cell

{3,3,5}

Tetrahedron

120

720

1200

600

Table 2 Regular Polychora

As there are also concave regular polygons (four of them, the Kepler-Poinsot polyhedra), there are also concave regular polychora. In particular, there are ten regular stellated polychora.

A uniform polyhedron is one made up of more than one type of regular polygon, and for which the configuration at each vertex point is the same. While there are 75 uniform polyhedra made up of two different kinds of polygons (80 if you count pentagonal prisms), there are currently over 8,000 known uniform polychora.

4.4      Special Polytopes

Spaces with different numbers of dimensions seem to have different numbers of regular polytopes, but each dimensional space has at least three regular polytopes, generically referred to as the hypercube, the simplex, and the cross-polytope.

Hypercube: The generic hypercube is an n-cube with 2n vertices and bounded by 2n hypercubes of dimension (n-1). The Schläfli symbol is a 4 followed by a series of (n-1) 3’s. A hypercube in n-space with sides of length one is called the measure polytope, because it defines how distance is measured in that space.

Simplex: The simplex is so-named because it is the simplest polytope in any space. It is made from the fewest number of vertices, one per dimension, and includes the triangle in 2-space, the tetrahedron in 3-space, and the pentatope in 4-space. The simplex is essentially a pyramid in the nth dimension, with an (n-1)-dimensional simplex as its base. The Schläfli symbol for an n-dimensional simplex is a series of n 3’s.

Cross-polytope: The cross-polytope is so-named because it has two vertices on each of the n Cartesian axes in n-space, with the vertices located equidistant from the origin. The octahedron is the three-dimensional cross-polytope, while the 16-cell is the four-dimensional cross-polytope. In two dimensions, a square, rotated to look like a diamond, is the cross-polytope (hence in two dimensions, the square does double-duty as both the hypercube and the cross-polytope). A cross-polytope has 2n vertices and is bounded by 2n simplexes of dimension (n-1). While the simplex is a pyramid, the cross-polytope is a dipyramid (i.e., points in both directions along a line perpendicular to its base) and has an (n-1)-dimensional cross-polytope as its base. For example the octahedron has a 2-dimensional cross-polytope (the square) as its base. The Schläfli symbol is a series of (n-1) 3’s followed by a 4.

 

 

2-space version

3-space version

4-space version

Schläfli

# of Vertices

# of bounding cells

Bounding cell type

Hypercube

Square

Cube

Tesseract

{4,3,3,…}

2n

2n

(n-1)-cube

Simplex

Triangle

Tetrahedron

Pentatope

{3,3,3…}

n

n

(n-1)-simplex

Cross-polytope

Square

Octahedron

16-cell

{3,3,…,4}

2n

2n

(n-1)-simplex

Table 3 Basic Polytopes

4.5      Euler’s Polyhedron Forumla

Euler’s polyhedron forumla states that, for V vertices, E edges, and F faces,

                                              V-E+F=2, for polyhedra in 3-space.                                        (7)

What about other dimensions? A little exploration shows that, in two dimensions, we need a slightly different equation, since we have no faces. Also, polygons have the same number of vertices as edges, which leads us to Euler’s polyhedron forumla for 2-space:

                                                                   V-E=0.                                                             (8)

A four-dimensional hypercube has 16 vertices, 32 edges, 24 faces, and 8 cells. Since 16+32-24+8=0, we can conjecture that the formula for 4-space, where C is the number of three-dimensional cells, is:

                                                               V-E+F-C=0.                                                         (9)

Similarly, a five-dimensional hypercube has 32 vertices, 80 edges, 80 faces, 40 3-dimensional cells, and 10 4-dimensional cells. Since 32+80-80+40-10=2, we can conjecture that the formula for 5-space, where C3 is the number of three-dimensional cells and C4 is the number of four-dimensional cells, is:

                                                           V-E+F-C3+ C4=2.                                                   (10)

In fact, it has been shown that this pattern continues, alternately adding and subtracting the number of cells of various new dimensions, and the total equals 0 for spaces with an even number of dimensions and equals 2 for spaces with an odd number of dimensions. In fact, for the one-dimensional case, the formula is simply V=2, saying that every segment has exactly two vertices.

4.6      Visualization via Unfolding

Now that you have some mathematical feel for polychora, what do they look like? Since we live in a three-dimensional world, it is quite difficult to visualize what a four-dimensional object, such as a hypercube, looks like. However, we can use several techniques to represent four-dimensional objects in three dimensions, to gain some understanding of their shape and structure. One tool is unfolding. A cube unfolds into six squares in a variety of ways, as shown in Figure 11. In the unfolded version, two vertices with the same labels are actually the same vertex in the original cube.

Figure 11 Unfolding a Cube

We can use the data on the number of faces, vertices, and cells to unfold a tesseract into eight cubes, as shown. Obviously, there’s no way to re-fold the cubes in 3-space to get these matching vertices to touch, without distorting the cubes. However, if there was a fourth dimension, we can fold this shape up through that dimension, without distorting the cubes, and have the matching vertices coincide.

Figure 12 Unfolding a Tesseract

4.7      Visualization via Projections

Another way to look at four-dimensional shapes is via projecting them into three dimensions. If we project a three-dimensional cube onto a two-dimensional plane using isometric projection (i.e., without a vanishing point), we get the images shown in Figure 13. The form of the image depends on the direction used for the projection. In the figure, the different images show what happens to the projection when the cube is rotated in the xy plane (in which this piece of paper lies), the xz plane, or the yz plane. These images were generated with a Java applet available at: http://darkwing.uoregon.edu/~koch/java/FourD.html.

           A Cube                Rotated in xy plane         Rotated in xz plane           Rotated in yz plane

Figure 13 Projections of a Cube, with Rotations

We can use this same method to project a four-dimensional tesseract onto a 3-space, resulting in a three-dimensional object that looks something like Figure 14 through Figure 16. For presentation purposes on this two-dimensional sheet of paper, we are actually projecting these three-dimensional objects onto a two-dimensional plane.  The figures show various rotations about planes that include the fourth dimension. A particularly valuable way to view these projections is via one of several computer programs that both allow dynamic manipulation of the tesseract and viewing in stereo via colored glasses.

Figure 14 Hypercube Projections, Rotation in xw Plane

Figure 15 Hypercube Projections, Rotation in yw Plane

Figure 16 Hypercube Projection with Both xw and yw Rotation

This next series of projections shows a given angle of view, but highlights the three-dimensional cells of the tesseract, one at a time. This program is quite useful in exploring the tesseract, as you can more easily follow the shape changes in the three-dimensional projection as you rotate the tesseract in the fourth dimension. Also included are projections of the other regular polychora. We can also use projections with vanishing points, as shown in Figure 19.


Figure 17 Hypercube Projections with Cells Highlighted

                   Pentatope (simplex)    Cross-polytope (16-cell)                 24-cell      


                                   120-cell                                                     600-cell

Figure 18 Projections of Regular Polychora, with One Cell Highlighted

  

generated with http://dogfeathers.com/java/hyprcube.html

Figure 19 Hypercube, Pentatope, and Cross-Polytope with Perspective

4.8      Visualization via Slicing

A third tool for visualizing higher-dimensional objects is slicing or cross-sections. If we drag a cube in 3-space through a plane, the plane “slices” the cube into differently shaped polygons, depending on the orientation of the cube as it passes through the plane. Similarly, if we drag a four-dimensional polychoron through a 3-space, we get a polyhedron that changes shape as the polychoron passes through the 3-space. Figure 20 (from http://www.dogfeathers.com/java/hyperslice.html ) shows what happens when we drag a hypercube by one of its vertices through a 3-space, using the analogy of dragging a cube through a 2-space (i.e., a plane).

Figure 20 3-D Analogy for Slicing a Hypercube

Figure 21 (from http://www.uccs.edu/~eswab/hcubsect.htm ) shows the progression of cross-sections of the tesseract as it is dragged through a 3-space by one of its vertices. The web page from which these are drawn also supports a dynamic animation of this sequence (http://cips02.physik.uni-bonn.de/~scheller/index.html), which adds to the visualization.

Figure 21 Vertex-First Slices of a Hypercube

The following figures show selected slices of the hypercube and the regular polychora. These were all done with HyperSlice, a Java applet that provides many useful features for visualizing polychora via slicing.

Figure 22 Hypercube Slices, Vertex First

Figure 23 Hypercube Slices, Edge First

Figure 24 Hypercube Slices, Face First

Figure 25 Hypercube Slices, Cell First

Figure 26 Simplex Slices, Vertex First

Figure 27 Simplex Slices, Edge First

Figure 28 Cross Polytope, Vertex First

Figure 29 Cross Polytope, Edge First

Figure 30 24-cell, Vertex First

Figure 31 120-Cell, Vertex First

Figure 32 600-Cell, Vertex First

 

5         Geometry and Modern Physics

Flatland helped 19th-century readers gain some understanding of what the fourth dimension might be like, if it existed. Dutch mathematician Dionys Burger wrote a sequel, Sphereland, in 1965 to use the characters and concepts of Flatland to help explain some of the geometrical ideas that arose from Einstein’s relativity theories and current understandings of cosmology. In particular, Burger explained the concept of a closed universe by having the inhabitants of Flatland discover that, instead of living on a plane, they actually lived on a very large sphere. He then extended this to explain the idea of an expanding universe by having the Flatlander’s sphere expand.

Since then, physics and cosmology have moved on towards the difficult goal of uniting relativity theory with the other great 20th-century physics theory, quantum mechanics. The existing formulations of these two theories are incompatible, but both have been shown via experiments to be very good theories of how the universe works. Modern superstring and supersymmetry theories attempt to resolve these incompatibilities. However, one of the requirements of these theories is that the universe in which we live has more than three spatial dimensions. How many more? The theories differ, but they are all have around nine spatial dimensions, give or take a few. We can clearly see and experience three of these, but where are the other six?

An answer was provided by Theodor Kaluza in 1919 and refined by Oscar Klein in 1926 and is referred to as Kaluza-Klein theory. The basic idea is that the dimensions that we don’t see are small, curled-up, closed dimensions, with the curl so small we can’t even measure them (at least with any technology known today). Brian Greene, author of The Elegant Universe, borrows from Flatland to illustrate this concept. Imagine a garden hose with small creatures inhabiting its surface. Creatures can move left or right in Hoseland, and can also move around the hose, hence its surface is two-dimensional. Now, imagine that the diameter of the hose shrinks to a very, very small size. For all practical purposes, the Hoselanders now would think that they live in a one-dimensional world, since they have no way to experience or measure the second dimension.

Figure 33 Hose World

Similarly, Kaluza-Klein theory states that our universe has three “uncurled” dimensions, plus multiple curled-up dimensions that are too small to see. It’s easy to imagine one dimension curled up into a small circle, or two dimensions curled up into a small sphere or a small torus. What about six dimensions? Physicists don’t know how they are curled up, but the current front-runner is a class of shapes called Calabi-Yau manifolds (unfortunately I couldn’t find a picture of a 3D projection of one…).

6         Higher Dimensional Geometry in the Classroom

6.1      Motivation

Why teach a mini-unit on higher dimensional geometry in high school? For several reasons:

Review: Many of the concepts already studied in plane and solid geometry come up again in higher-dimensional geometry. Taught towards the end of the year, this can provide a vehicle for reviewing material already covered in an interesting way.

Reinforcement: Many concepts are also covered in a way that increases the student’s feel for the topic. For example, Euler’s polyhedron forumla is interesting in itself, but students may not remember it. Developing the progression from one dimension up through five or more gives students a way to think about it that will help them when they need to recall the three-dimensional version. Similarly, attempting to visualize four-dimensional objects will help the students better visualize and understand three-dimensional solids.

Fun: The topic is also quite fascinating for many students. It’s connection to science fiction and to modern physics makes it an intriguing area of study for the students, and one that’s interesting to teach.

6.2      Activities

A mini-unit on higher dimensional geometry could be structured over one or two weeks, towards the later part of the year in a geometry class or integrated course with substantial geometry content. This section describes a variety of classroom activities to help students explore this topic.

6.2.1      Dimensions

Figure 34 4-Dimensional Tic tac Toe

                                                                                                     (11)

For example, it’s easy to count that there are eight ways to get three in a row when d=2 and n=3 (i.e., standard tic-tac-toe). In 3D, there are now n=3 copies of this 2D grid in each of the d=3 dimensions, for a total of 3x3x8=72 ways to win. But, you also need to add in the 3D space diagonals through the cube between opposite corners. Since there are half as many of these as there are vertices in the 3D cube, there are 8/2=4 of these. Hence the total is 72+4=76. In the 4D case, there are n=3 copies of the 3D grid in each of the d=4 dimensions, for a total of 3x4x76=912. Add in the 4D space diagonals of the hypercube, which total 16/2=8, and you get 920.

6.2.2      Distance and Parallelism

6.2.3      Handedness, Inside and Outside

6.2.4      Polytopes

6.2.5      Visualization

6.3      Larger Projects

6.4      Computer Lab

The computer and the World Wide Web are extremely valuable tools for visualizing polytopes in higher dimensions. Many of the images in this paper came from web-accessible software. Most of the software listed here are Java applets and will run on Mac’s or PC’s, except as noted.

6.4.1      Projections

Two very useful websites provide tools for viewing projections of polytopes into three dimensions.  The first is http://darkwing.uoregon.edu/~koch/java/FourD.html. This Java applet shows a wire-frame image of one of a variety of 3D and 4D polytopes, and allows you to rotate it in various planes. It can highlight one of the faces of the polytope and cycle through the faces when you click the Next or Previous buttons. It can also provide views in stereo via colored glasses. A screen shot is shown below.

Figure 35 http://darkwing.uoregon.edu/~koch/java/FourD.html

Several useful Java applets can be found at http://dogfeathers.com/java. These applets rotate a 4D hypercube, simplex, cross-polytope, or 24-cell via animation. The applets provide a variety of stereo viewing modes, and also enables the degree of projection perspective to be varied (as opposed to the previous site, where the projections are all isometric). A screen shot is shown below.

Figure 36 http://dogfeathers.com/java/hyprcube.html

Not quite as useful, but yet valuable for its simplicity and color, is http://www.magenet.com/~julie/java/hyper/. This applet rotates a hypercube in any direction, and uses different colors for two of the cells and their connection edges to help you visualize the hypercube.

Figure 37 http://www.magenet.com/~julie/java/hyper/

6.4.2      Slicing

The most useful tool for slicing that I found was HyperSlice. This tool enables you to drag 4-D polytopes along the w axis through a 3-space, and shows an image of the resulting three-dimensional slice. You can select one of the regular polychora or one of several face-centered polychora, and you can choose to drag the polytope through the 3-space vertex first, edge first, face first, or cell first. You can also chose one of several stereo or mono viewing modes. It can be found at http://dogfeathers.com/java/hyperslice.html. I had trouble running this under Internet Explorer 5, but it worked fine with Netscape Communicator 4.

Figure 38 HyperSlice

http://cip.physik.uni-bonn.de/ScienceSite/hypercubus/animator/ provides an animated view of a vertex-first series of slices of a hypercube.

Figure 39 http://cip.physik.uni-bonn.de/ScienceSite/hypercubus/animator/

6.4.3      Games

http://www1.tip.nl/~t515027/hypercube.html provides a very difficult game, where you move around in 4-space. I find it impossible, but I’m sure video-game veterans will figure it out…

Figure 40 http://www1.tip.nl/~t515027/hypercube.html

http://www.kbs.cs.tu-berlin.de/~jutta/swd/hyper-game.html A game you can play that involves the fourth dimension, but doesn’t require a computer.

http://www.cs.caltech.edu/~frechet/ 3D and 4D tic tac toe Java applet, very flexible

http://www.rose-hulman.edu/~berglunb/Rubik/index.html If you thought the regular Rubik’s cube was hard; wait until you try it in 4D! This is PC software, not a Java applet.

Figure 41 4D Rubik's Cube

6.4.4      Other Useful Sites

http://www.cs.mu.oz.au/~amb/4d/ a large collection of links to 4-D-related sites

http://www.uccs.edu/~eswab/hyprspac.htm information on the regular polychora

http://w3.one.net/~monkey/mathematics/simplex/ provides a wealth of info on the simplex

http://mathworld.wolfram.com/Polychoron.html much information about many different polytopes (and anything else mathematical, too!)

http://www.geom.umn.edu/docs/forum/polytope/ lots of info on polytopes

http://www.theory.caltech.edu/people/patricia/lctoc.html and http://www.theory.caltech.edu/people/jhs/strings/intro.html contain some info on modern physics

http://www.ics.uci.edu/~eppstein/junkyard/highdim.html another good list of links on higher dimensional geometry

7         Bibliography

Abbott, Edwin A. Flatland: A Romance of Many Dimensions. New York: HarperCollins Publishers, Inc., 1983.

Burger, Dionys. Sphereland: A Fantasy About Curved Space and an Expanding Universe. New York: HarperCollins Publishers, Inc., 1965.

Greene, Brian. The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. New York: W. W. Norton & Company, 1999.

Koch, Richard. Java Examples. Available: http://darkwing.uoregon.edu/~koch/java/FourD.html.

Manning, Henry Parker. The Fourth Dimension Simply Explained. New York: Munn & Company, Inc., 1910.

Newbold, Mark. Hyperspace Polytope Slicer. Available: http://www.dogfeathers.com/java/hyperslice.html, 1999.

Newbold, Mark. Stereoscopic Animated Hyperspace Objects. Available: http://www.dogfeathers.com/java/, 1999.

Olshevsky, George. Uniform Polytopes in Four Dimensions. Available:  http://members.aol.com/Polycell/uniform.html list of lots of polytopes, 1997.

Towle, Russell. Regular Polytopes in Higher Space. Available: http://www.mathsource.com/MathSource/Applications/Mathematics/0210-643/RegularPolytopes.nb, November 1999.

Weisstein, Eric. Eric Weisstein’s World of Mathematics. Available: http://mathworld.wolfram.com/, 2000.