Tesseract

From Math Images

Tesseract

Field: Geometry
Image Created By: Jason Hise
Website: wikipedia.org

Tesseract

The animation shows a three dimensional projection of a rotating tesseract, the four dimensional equivalent of a cube.

Basic Description

The tesseract, or tetracube, is a shape inhabiting four spatial dimensions. More specifically, it is the four dimensional hypercube. The sides of the four dimensional tesseract are three dimensional cubes. Instead of a cube’s four corners, or vertices, a tesseract has sixteen. If you find this hard to picture, don’t worry. As inhabitants of a three dimensional world, we cannot fully visualize objects in four spatial dimensions. But if we use our imaginations, we can come close.

Constructions

Sweeping

The tesseract is analogous to the cube in the same way that the cube is analogous to the square, the square to the line, and the line to the point.

To begin thinking about the relationship between tesseracts and cubes, it is helpful to consider the relation of cubes to squares, squares to lines, and lines to points. Let’s start from the zero dimensional point and build our way up to the four dimensional tesseract.

We form a one dimensional line from a point by sweeping, or stretching, the point straight out in some direction. This is the first step shown in Image 1.

Now imagine taking hold of this line and sweeping it out in a direction perpendicular to its length. If you sweep out a distance equal to the length of the line, you will form a two dimensional square. This is shown in the second step of the diagram.

Image 1

We can do the same sort of thing with our square to form a cube. Imagine pulling the square outward in a direction perpendicular to its surface. You will have swept out a three dimensional cube, shown in the third step.

Now we know the procedure to use in constructing a tesseract from a cube. At each step so far, we took the original object and swept it out in a new direction perpendicular to every direction in the original object. We only have three spatial dimensions, and a cube inhabits all three, but try to imagine a new direction perpendicular all of the up-down, left-right, and back-forth directions of the cube. Stretch the cube out a distance equal to the length of one of its sides into this new, fourth direction and you will have swept out a tesseract. This is shown in the last panel of Image 1. In the diagram, the yellow w direction is not actually perpendicular to the other three, but it is the best we can do on a two dimensional screen.

Image 2

Image 3

Folding

Imagine folding the six squares in Image 2 into a closed, hollow object.

This is another approach that we can use to construct a cube from squares. You take two dimensional squares and use the third dimension to fold them into a cube.

We can follow the same approach to construct a tesseract. Consider Image 3. It looks a lot like Image 2, except instead of a flat collection of six squares we have a three dimensional collection of eight cubes. It is far more difficult to imagine than for the squares, but using the fourth dimension we could fold these eight cubes together to form a tesseract. Each cube would become one of the the eight faces of the tesseract, such that there are two parallel faces in each of the four spatial directions.

Visualizing the Tesseract

Visualizing four dimensions isn’t easy when you live in three. In order to better understand the tesseract and interpret images like the one at the top of the page, it is helpful to consider how inhabitants of a two dimensional world would go about understanding objects in three dimensions. Edwin Abbott’s book Flatland presents such an analogy. Inhabitants of Flatland see and move about in just two dimensions. In their world, three dimensional shapes cannot be seen all at once, just as we cannot fully visualize a tesseract. There are two main ways an inhabitant of a flat world could perceive the structure of a three dimensional object. We can use analogous methods to picture the tesseract.

Slices

Image 4	Image 5
Image 6	Image 7

If a three dimensional shape were to pass through the two dimensional world of flatland, the inhabitants would perceive a series of its slices.

For the sphere, first a point would appear, then a gradually growing circle until the sphere was half-way through, and finally a circle that shrinks until it disappears altogether.

An object like a cube would be more confusing to a flatlander, since its slices look different depending on how it is tilted as it passes through a flat plane.

Consider Images 4 and 5, which show a cube passing through a two dimensional plane and the corresponding slices at two different tilts. The second half of each image is all a flatlander would be able to experience. As you can see, the slices in the two images look fairly different. This doesn't seem too strange to us, since we can see the cube in the first half of each image. But for a flatlander it might be hard to tell that the slices are from the same object.

Just as flatlanders can only perceive two dimensional slices of three dimensional objects, we are limited to visualizing three dimensional “slices” of the four dimensional tesseract. Images 6 and 7 show such slices of a tesseract passing through three dimensional space at two different tilts. The perspective is closely analogous to the flatlander’s view of a passing cube in the second halves of Images 4 and 5.

Similar to Image 4’s depiction of a square being sliced parallel to one of its faces as it passes through a two dimensional plane, image 6 shows a tesseract being sliced in three dimensions parallel to one of its cubical faces. As illustrated in the animations, slicing a cube this way yields a square while slicing a tesseract this way yields a cube.

In Image 7 the tesseract is being sliced corner to corner as it passes through our three dimensional view, analogous to how the tilted cube is being sliced in Image 5 as it passes through a two dimensional plane.

Projections

Image 8

The second way Flatlanders could try to understand a three dimensional object is by looking at its projections onto their two dimensional world. The easiest way to think about projections is probably as shadows. Shadows can look different depending on the distance between the object casting a shadow and the source of light. There are two main types of projections.

For objects held close to a light source, features that are farther away appear smaller in the shadow than those that are nearby. This kind of shadow depicting objects in perspective is called a Stereographic Projection.

For objects very far away from a light source, the light rays are so close to parallel that farther away features cease to be reduced in size in the shadow. A shadow cast by the parallel light of an infinitely or approximately infinitely distant source is called an Orthographic Projection. This type of projection makes for more symmetric images, but lacks the sense of depth provided by stereographic projection. Even if a flatlander was told which type of shadow they were looking at, it would still be quite a challenge for them to mentally translate the two dimensional projection into a three dimensional shape . They would need to be told what motion and positioning in three dimensions looks like in a projection.

Image 9

Consider the shadow cast by the rotating cube beneath a nearby light source in Image 8, an example of stereographic projection. What is really a cube with six square faces appears in the shadow as a small square inside of a large square with four highly distorted squares in between. As the cube rotates, the side lengths and internal angles in the projection change; the distorted sides morph into squares and back as the inner and outer faces change places. We know these distortions are not actually occurring, and that as a part of the shadow grows the corresponding cube face is just rotating closer to the light source. The important features of the cube like the number of faces and vertices stay true to the actual three dimensional object even in projection. These would all be important things for an inhabitant of Flatland trying to understand a cube to know.

By analogy we can use these lessons about shadows to better visualize and understand the tesseract. The four dimensional equivalent of a shadow is a three dimensional projection, like the one shown in the animation at the top of the page and here in the still Image 9. As with the animation of the rotating cube, these are stereographic projections. While a cube with a face directed towards a nearby light in three dimensions casts a shadow of a square within a square, a tesseract with its face directed towards a nearby light source in four dimensions casts a three dimensional “shadow” of a cube within a cube. Instead of the cube’s close face projected as an outer square, the far face projected as an inner square, and the four other sides as highly distorted squares in between, we have the tesseract’s close face projected as an outer cube, the farthest face as an inner cube, and the other six cubic faces highly distorted spaces in between. Just as the inner and outer squares in cube shadow are really just faces of the cube on opposite sides, the inner and outer cubes of the tesseract projection are just faces of the tesseract on opposite sides. In the flat shadow of the square we saw the inner square replace the outer square as the cube rotated through a third dimension. In the animation at the top of the page we observe the inner cube, really just a face of the tesseract farther away in the fourth dimension, unfold to replace the outer cube as the tesseract completes a half turn through the fourth dimension.

These images help us to interpret stereographic projections of a tesseract from one perspective, but we could always change the tesseract's orientation so that, say, a corner were facing the light source. The projection would look quite different. To explore what the projection of a differently oriented tesseract would look like, try out the interactive feature below. To view stereographic projections like the one in image 9 or this page's main animation, check the Perspective box. This provides a greater sense of depth, albeit with greater distortion of the true dimensions of the tesseract than the with the default orthographic projection setting.

Your browser does not support iframes. Visit http://hypercube.milosz.ca/to view the embedded site.

Additional Resources

A gradual explanation of the tesseract with lots of interactive features: http://www.learner.org/courses/mathilluminated/interactives/dimension/
A series of videos explaining higher dimensional shapes with a focus on visualizations using a form of stereographic projection: http://www.dimensions-math.org/
More great animations: http://www.math.union.edu/~dpvc/math/4D/welcome.html
Carl Sagan discussing Flatland and hypercubes on an episode of Cosmos:http://www.youtube.com/watch?v=KIadtFJYWhw
An online version of Edwin Abbott’s classic Flatland: http://www.ibiblio.org/eldritch/eaa/FL.HTM

A More Mathematical Explanation

Four Dimensions

Mathematically, four dimensions is a natural extension of the more familiar two [...]

Four Dimensions

Mathematically, four dimensions is a natural extension of the more familiar two or three dimensions. While three dimensional space $\mathbb{R}^3$ is the set of all ordered triples $(x_1, x_2, x_3)$ or $(x, y, z)$ of real numbers, four dimensional space $\mathbb{R}^4$ is the set of all ordered quadruples $(x_1, x_2, x_3, x_4)$ or $(x ,y, z, w)$ of real numbers. Formally, we have:

$\mathbb{R}^4 = \{(x,y,z,w)|x,y,z,w\in R\}$

The coordinate system in $\mathbb{R}^4$ consists of four axes, with points of the form $(x,0,0,0), (0,y,0,0), (0,0,z,0)$ , and $(0,0,0,w)$ respectively, and which intersect at the origin point $(0,0,0,0)$ . Any two of these four coordinate axes determine a coordinate plane, and any three of the coordinate axes determines a coordinate hyperplane. For example, the x, z, and w coordinate axes determine the coordinate hyperplane $\{(x,0,z,w) | x,z,w\in R\}$ . In general, a hyperplane is a three dimensional set of vectors $u = (x, y, z, w)$ satisfying an equation of the form

$ax+by+cz+dw = 0$

Just as there are infinitely many planes in $\mathbb{R}^3$ , there are an infinite number of hyperplanes in $\mathbb{R}^4$ .

Other aspects also carry over from lower dimensions. The standard basis in $\mathbb{R}^4$ is the four vectors $e_1 = (1, 0, 0, 0), e_2 = (0, 1, 0, 0), e_3 = (0, 0, 1, 0), e_4 = (0, 0, 0, 1)$ where every vector in four space can be uniquely expressed as a linear combination of these basis vectors, the zero-vector is $(0, 0, 0, 0)$ , and the distance between two points $(x_1, y_1, z_1, w_1)$ and $(x_2, y_2, z_2, w_2)$ is found using a straightforward analogue of the Pythagorean Theorem:

$d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2+(w_2-w_1)^2}$

The Geometry of the Tesseract

The tesseract is a type of regular polytope. Since its sides are mutually perpendicular, it is further classified as an orthotope, the generalization of a rectangle or box to higher dimensions. More specifically, the tesseract is the four dimensional case of an orthotope with all its edges of equal length. The tesseract is formally denoted $\gamma_4$ .

Below is a summary of the basic geometric features of n-cubes, or hypercubes, for n = 0 to n = 4.

Dimension	Name	Number of Vertices	Number of Edges	Number of Faces
0	Point	1	0	0
1	Line Segment	2	1	2 (points)
2	Square	4	4	4 (line segments)
3	Cube	8	12	6 (squares)
4	Tesseract	16	32	8 (cubes)

Coordinates of the Tesseract

In two dimensional space, a unit square can be specified by the coordinates of its four vertices: $(0, 0), (0, 1), (1, 0), (1, 1)$ , which are all the possible pairs of the numbers 0 and 1. A unit cube can similarly be specified in three dimensions by its eight vertices occupying all the possible coordinate triples composed of 0s and 1s. The unit tesseract can be specified in the same way. Its sixteen vertices are given by the points

(0, 0 , 0, 0)	(0, 0, 0, 1)	(0, 0, 1, 0)	(0, 0, 1, 1)
(0, 1, 0, 0)	(0, 1, 0, 1)	(0, 1, 1, 0)	(0, 1, 1, 1)
(1, 0, 0, 0)	(1, 0, 0, 1)	(1, 0, 1, 0)	(1, 0, 1, 1)
(1, 1, 0, 0)	(1, 1, 0, 1)	(1, 1, 1, 0)	(1, 1, 1, 1)

Which represent all possible quadruples of the numbers 0 and 1.

Schläfli Symbol

The Schläfli symbol for the tesseract is $\{4,3,3\}$ . Each number in this list refers to how a component of the tesseract is arranged to form the overall shape. Schläfli notation is recursive, since each progressive number refers to the arrangement of the component indicated by the last number. The first number, 4, indicates that the fundamental two dimensional component of the tesseract is the four sided square. The second number, 3, indicates that there are three squares folded together around every vertex, forming cubes. The third and last number, 3, indicates that there are three cubes folded together around every edge in the tesseract.

In general Schläfli notation starts by indicating the underlying two dimensional component of the shape, then indicates how many of these two dimensional components come together at each zero dimensional vertex, then how many of the resulting three dimensional components meet at each one dimensional edge, and so on until all the lower dimensional components of a shape are accounted for. This means that the Schläfli symbol for any n dimensional shape will include n-1 numbers, the last number always referring to the arrangement of the shape's n-1 dimensional faces.

Hypervolume

A line segment is said to have length $m$ if we can cover it with exactly $m$ line segments of unit length. Likewise, the surface of a square with sides of length $m$ can be covered with $m^2$ unit squares, a quantity called its area, and a cube with edges of length $m$ can be filled by exactly $m^3$ unit cubes, defining its volume. The equivalent feature of the Tesseract is hypervolume. A tesseract with edges of length $m$ has a four dimensional interior which can be filled by $m^4$ unit tesseracts. Therefore the hypervolume of a tesseract is equal to $m^4$ .

Diagonals

The unit square has only one diagonal, which can easily be found to be of length $\sqrt{2}$ using the Pythagorean Theorem. As we form the unit cube from the unit square, a new longer diagonal cutting across the inside of the cube is formed. This diagonal can be viewed as the hypotenuse of the right triangle formed by the shorter diagonal of length $\sqrt{2}$ cutting across the square base of the cube and a perpendicular edge of the cube extending 1 unit in the vertical z direction. Once again using the Pythagorean Theorem, this longer diagonal is found to be of length $\sqrt{3}$ . When we form the unit tesseract from the unit cube, a third even longer diagonal is formed, which like the longest of the cube diagonals can be viewed as the hypotenuse of a right triangle. This time the bases of the triangle are the longest of the cube diagonals from a face of the tesseract and a perpendicular edge of the tesseract extending 1 unit in the w direction. Therefore the length of the Tesseract’s third and longest diagonal is

$\sqrt{\sqrt{3}^2+1^2} = \sqrt{3+1} = \sqrt{4} = 2$

The lengths of the longest diagonal of a non-unit square, cube, or tesseract is proportional to the length of a side. A tesseract with sides length $m$ has diagonals of length $\sqrt{2}m$ , $\sqrt{3}m$ , and $2m$ .

Why It's Interesting

Higher Dimensions

A two dimensional orthographic projection of the 10 dimensional hypercube.

All the visualization techniques and much of the geometry we have been discussing for tesseracts can be extended to help us understand other four dimensional objects and even higher dimensional shapes.

In three dimensions there are five regular polytopes, known as the Platonic Solids, which as the name suggests have been studied since the time of the ancient greeks. One of the first mathematicians to take four dimensions seriously was Ludwig Schläfli. In the mid-1800s, Schläfli figured out that in four dimensional space there are six regular polytopes^[1]. The tesseract is one, as is an enormous shape with 600 faces, each one a three dimensional tetrahedron. And why stop at four? We can mathematically analyze and even form visual projections of objects in five, six, or more dimensions. Instead of sets of triples or quadruples of real numbers, we can consider a space of arbitrarily many dimensions, consisting of all n-tuples of the form $(x_1,x_2,...,x_n)$ . Generally, for n dimensions, there are three regular polytopes, including the n dimensional hypercube denoted $\gamma_n$ .

[Show more about Hypercubes][Hide more about Hypercubes]

Basic Features of Hypercubes

The same progressions from squares to cubes to tesseracts which we used to examine the geometry of the tesseract apply more generally to hypercubes. Every time we form an n+1 dimensional hypercube from an n dimensional hypercube, we are in effect taking hold of the shape’s vertices and sweeping the whole thing out in a new direction perpendicular to all the directions in the original hypercube. The resulting hypercube has twice as many vertices as the old hypercube. Therefore, building up from a zero dimensional point with 1 vertex, the number of vertices in an n dimensional hypercube is $2^n$ . This sweeping into a new direction also doubles the number of edges in the hypercube, plus one new edge for each vertex in the original hypercube. In general, an n dimensional hypercube has $2^{n-1}n$ edges. Below is a summary of the geometric features of n dimensional hypercubes with edges of length m.

Number of Vertices	$2^n$
Number of Edges	$2^{n-1}n$
Number of Faces	$2n$
Content	$m^n$
Length of Longest Diagonal	$m\sqrt{n}$

These are just the regular polytopes, shapes with all identical faces. Higher dimensions are home to innumerable irregular polytopes as well. Our newfound knowledge of the tesseract and how to wrap our heads around its four dimensional structure will be essential in tackling those bizarre creatures on some other occasion.

Historically. Development; preceded later expansions of concept of dimensions in physics
Discussion of physical and philosophical limitations to visualizing tesseracts

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References

↑ Rehmeyer, J. (2008). Seeing in Four Dimensions. Science News.

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Tesseract

From Math Images

Contents

Basic Description

Constructions

Sweeping

Folding

Visualizing the Tesseract

Slices

Projections

Additional Resources

A More Mathematical Explanation

Four Dimensions

Four Dimensions

The Geometry of the Tesseract

Coordinates of the Tesseract

Schläfli Symbol

Hypervolume

Diagonals

Why It's Interesting

Higher Dimensions

Basic Features of Hypercubes

Teaching Materials

References

Views

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