Tesseract
From Math Images
Tesseract |
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Tesseract
- The animation shows a three-dimensional projection of a rotating tesseract, the four-dimensional equivalent of a cube.
Contents |
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 eight 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. However, we can develop a general understanding of the tesseract by learning its structure, examining representations of the shape in lower dimensions, and exploring the math behind it.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.
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 to construct 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 orange w direction is not actually perpendicular to the other three, but it is the best we can do in a three-dimensional world. In fact, even the blue z direction isn't actually perpendicular to the flat x and y directions in the diagram. We just know to interpret the fact that these directions are perpendicular in three dimensions from how they are drawn on the two-dimensional computer screen.
This is an important fact to keep in mind when discussing the 4-D tesseract. We can't directly draw 4-D objects, but we can't directly draw 3-D objects either since drawings are two dimensional. So whenever we talk about a 3-D visualization of a 4-D object, we really mean a 2-D representation of a 3-D representation of the real 4-D object.
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. Each of the squares becomes a flat face of the 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 becomes a side of the tesseract, analogous to the square faces of the cube, except three-dimensional. These eight cubic facets are oriented such that two parallel facets lie on opposing sides of the tesseract in each of the four spatial directions.
Visualizing the Tesseract
Visualizing four dimensions isn’t easy when you live in three and use computer screens in two. 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 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
For more animations like these, check out this site. |
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 a 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 right-side panel 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 the 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 facets. 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. Note that only the light blue parts of these animations are the actual slices. The static backgrounds are just shadows or projections of the shapes being sliced.
Projections
The easiest way to think about projections is probably as shadows. There are two main types of shadows, depending on the distance between the object casting a shadow and the source of light. These correspond to the two main types of projection that can be used to visualize objects in fewer dimensions than they inhabit.
For objects held close to a light source, features that are farther away from the light appear smaller in the shadow than those that are near the light source. 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 features farther from the source cease to be reduced in size in the shadow. The limiting situation, a shadow cast by the exactly parallel light of an 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.
Cube Shadows
Consider the shadow cast by the rotating cube beneath a nearby light source in the animation on the right, an example of stereographic projection. Note that although the cube may look solid in the animation, it casts a shadow as if its sides were semi-transparent, perhaps made of red glass, with edges made of some solid like wire. 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 squares 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.
Tesseract "Shadows"
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 facet directed towards a nearby light in three dimensions casts a shadow of a square within a square, a tesseract with a facet 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 facet closest to the light source projected as an outer square, we have the facet of the tesseract closest to the four-dimensional light source projected as an outer cube. Similarly, instead of the facet of a cube farthest from the light projected as an inner square, the facet of a tesseract farthest from the light is projected as an inner cube.
In the shadow of a cube, the four sides appear as highly distorted squares in between these inner and outer squares. In the 3-D tesseract projection in image 9, six of the tesseract's facets appear as highly distorted cubes occupying the space between the inner and outer cubes.
In the 2-D shadow of the rotating cube 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 facet of the tesseract farther away from the light source 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 with the default orthographic projection setting.
This applet was created by Milosz Kosmider.
For more visualizations of the tesseract, see the Related Links section at the bottom of the page.
A More Mathematical Explanation
In the language of geometry, the tesseract is a type of regular polytope. [...]
In the language of geometry, 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 a hypercube, an orthotope with all its edges of equal length.
Coordinates of the Tesseract
To understand how to mathematically represent the Tesseract, it is helpful to look at how we represent the square and the cube.One of the most powerful mathematical methods for describing these kinds of shapes is coordinate geometry. In two-dimensional space, coordinates are represented by pairs of numbers, usually labeled x and y, with each pair specifying a point in the xy plane. Within this framework, a unit square can be defined with the coordinates of its four vertices, (0, 0), (0, 1), (1, 0), (1, 1), all the possible pairs of the numbers 0 and 1.
Creating the unit cube by sweeping the unit square out in a new direction requires us to use three numbers to specify our points, usually called x, y, and z. The z coordinates for the vertices of the original unit square, now the base of the unit cube, are all 0. As a result of sweeping, we now have four more vertices in the same x and y positions but raised 1 unit in the z direction. Their z coordinates are thus all 1. In total the unit cube has eight vertices, occupying all the possible coordinate triples composed of 0s and 1s
(0, 0 , 0) | (0, 1, 0) | (1, 0, 0) | (1, 1, 0) |
(0, 0, 1) | (0, 1, 1) | (1, 0, 1) | (1, 1, 1) |
We follow the same pattern when sweeping out the unit tesseract with the unit cube. Once again we add a new coordinate, so that each point is now represented as (x ,y, z, w), and once again we double the number of vertices. Half of the unit tesseract's vertices have the same coordinates as the unit cube's vertices except for 0s in the new w place, while the eight newly swept out vertices all have a w value of 1. All told, the sixteen vertices of the unit tesseract are given by the points
(0, 0 , 0, 0) | (0, 1, 0, 0) | (1, 0, 0, 0) | (1, 1, 0, 0) |
(0, 0, 1, 0) | (0, 1, 1, 0) | (1, 0, 1, 0) | (1, 1, 1, 0) |
(0, 0 , 0, 1) | (0, 1, 0, 1) | (1, 0, 0, 1) | (1, 1, 0, 1) |
(0, 0, 1, 1) | (0, 1, 1, 1) | (1, 0, 1, 1) | (1, 1, 1, 1) |
Which represent all possible quadruples of the numbers 0 and 1.
Within this framework of coordinate geometry, four dimensions is a natural extension of the more familiar two or three dimensions. To move up to a higher dimension, we just add a new coordinate to each of our points. Even though we can't visualize a four-dimensional point, it is perfectly legitimate and quite helpful to represent it mathematically. For example, the animations on this page were probably programmed using coordinate representations.
Using Coordinates to Find the Edges of the Tesseract
An edge of a hypercube is a two-dimensional line segment that connects two vertices which differ by one coordinate. To figure out how many edges a tesseract has, we can use our newly developed method of coordinate representation. But first let's apply our coordinates approach to the cube and check our answer against what we already know about the shape.
Each vertex of a cube is represented by three coordinates. So how many edges meet at each vertex? Well, three coordinates per vertex means there are three different ways we can vary a single coordinate of any given vertex. So three perpendicular edges meet at each vertex. We saw earlier that the cube had eight vertices in total. That suggests 3×8 = 24 edges. However, each edge connects two vertices, so the number 24 counts each edge twice. Therefore, the cube has 3×8/2 = 12 edges. This matches what we know: four edges for the square base, four for the top, and four more connecting the base to the top.
Now for the tesseract. We found earlier that the tesseract has sixteen vertices, each represented by four coordinates. Therefore there are four ways we can vary one coordinate of any given vertex, each way corresponding to a perpendicular direction. So four mutually perpendicular edges meet at each vertex. Again each edge corresponds to two of the tesseract's sixteen vertices, so the total number of edges in a tesseract is 4×16/2 = 32.
Using Coordinates to Find the Facets of the Tesseract
We can follow a similar approach to find how many cubic facets a tesseract has. Once again, let's first try our hand at using coordinates to find the facets of a cube and see if we get the correct result.
The facets of the 3-D cube are 2-D squares. How many square facets meet at each vertex of the cube? Well, each vertex is represented by three coordinates. Each possible alteration of two of these coordinates corresponds to a square with one corner at that vertex. There are three possible choices for changing just two coordinates, so three square facets meet at each vertex of the cube.
The cube has eight vertices, suggesting 3×8 = 24 square facets, but this would be counting each facet four times since all four corners of each square facet correspond to a vertex of the cube. Therefore the cube has 3×8/4 = 6 square facets: the top, the bottom, and the four sides.
The facets of the Tesseract are 3-D cubes. Each vertex has four coordinates, which makes for four possible alterations of three coordinates, each corresponding to a cube with a corner at that vertex. So four cubes meet at each vertex. With the tesseract's sixteen vertices, this gives us an initial count of 4×16 = 64 cubic facets. Accounting for the fact that each cubic facet touches eight vertices with its corners, we find that the tesseract actually has 4×16/8 = 8 cubic facets.
More Geometry of the Tesseract
The regular progression of the properties of squares to the properties of cubes to the properties of tesseracts extends beyond these basic features. Let's now examine the extension of the cube's three-dimensional volume and diagonals to their four-dimensional equivalents in the tesseract.
Hypervolume
Lines have length, squares have area, cubes have volume, so what does a tesseract have? To answer this question, it's once again useful to look closer at the tesseract's counterparts in lower dimensions. 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
Tesseracts have diagonals just as squares and cubes do, and they can be found in much the same way that we find the diagonals of these lower dimensional shapes.The unit square has two identical diagonals, which can easily be found to be of length √2 using the Pythagorean Theorem.
When we form the unit cube from the unit square, these become diagonals of the square faces, and a new, longer type of diagonal cutting across the inside of the unit cube is created. One of these longer diagonals can be viewed as the hypotenuse of the right triangle shown in purple in the diagram. One leg of this right triangle is the shorter diagonal of length √2 contained in the unit cube's square base. The other leg is an edge of the unit cube extending 1 unit in the z direction. Applying the Pythagorean Theorem, this longer, internal type of diagonal is found to be of length √3.
When we form the unit tesseract from the unit cube, a third, even longer type of diagonal is formed. The old diagonals are still there, in the cubic facets of the unit tesseract and their square faces, but only this third type of diagonal spans the four-dimensional interior of the unit tesseract. As before, this diagonal can be viewed as the hypotenuse of a right triangle. This time the legs of the triangle are the longest of the diagonals contained within the cubic facets of the unit tesseract and an edge of the unit tesseract extending 1 unit in the w direction. Therefore the length of the unit tesseract’s third and longest diagonal is
The lengths of the diagonals of a non-unit square, cube, or tesseract are proportional to the length of a side. In other words, a tesseract with sides length m has three types of diagonals, of length m√2, m√3, and m√4 = 2m.
Now that we have developed a mathematical representation of the tesseract and used it to find the basic properties of the shape, we can summarize the basic geometric features of n-cubes, or hypercubes, for n = 0 to n = 4. Note that m represents the length of an edge.
Dimension | Name | Number of Vertices | Number of Edges | Number of Facets | Content | Length of Longest Diagonal |
0 | Point | 1 | 0 | 0 | 0 | 0 |
1 | Line Segment | 2 | 1 | 2 (points) | m | 0 |
2 | Square | 4 | 4 | 4 (line segments) | m^{2} | m√2 |
3 | Cube | 8 | 12 | 6 (squares) | m^{3} | m√3 |
4 | Tesseract | 16 | 32 | 8 (cubes) | m^{4} | 2m |
Why It's Interesting
Today the idea of more than three dimensions is fairly common. You can read about hyperspace in science fiction stories, four-dimensional space-time in physics textbooks, and a mind boggling 10 to 26 “curled up” dimensions in the writings of many modern scientists. But prior to the development of four-dimensional geometry and the popularization of the idea of dimensions by books like Abbott’s Flatland in the 1800s, the public, the physicists, and even most mathematicians did not pay much attention to the idea of four dimensions, much less 26.^{[1]}The door to higher dimensionality opened when people started studying strange geometric shapes like the tesseract. The tesseract is probably the best known higher dimensional shape^{[2]}, and as such represents a kind of symbol of the expansion of the human imagination into higher dimensions.
As Many Dimensions as You Like
The visualization techniques and geometry we have been developing 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 dimensional geometry seriously was Ludwig Schläfli. In the mid-1800s, Schläfli figured out that in four-dimensional space there are six regular polytopes^{[3]}. 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 using triples or quadruples of coordinates, we can consider a space of arbitrarily many dimensions, consisting of all n-tuples of the form . For n > 4 dimensions, there are three regular polytopes, one of which is the n-dimensional hypercube.
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 one vertex, the number of vertices in an n-dimensional hypercube is 2^{n}.
Other properties of tesseracts that we found using analogies to lower dimensions, like the number of edges and and the length of diagonals, can be found for n-dimensional hypercubes in a similar fashion. Below is a summary of the geometric features of n-dimensional hypercubes with edges of length m.
Number of Vertices | |
Number of Edges | |
Number of Faces | |
Content | |
Length of Longest Diagonal |
These are just the regular polytopes, shapes with all identical faces. Higher dimensions are home to innumerable irregular polytopes as well. Knowledge of relatively simple higher-dimensional shapes like the tesseract and how to wrap one's head around its four-dimensional structure would be essential for anyone interested in tackling those far stranger creatures.
Higher Dimensions in Physics
Physics and mathematics borrow from each other all the time. Sometimes the mathematicians develop an idea that the physicists find useful later, and sometimes the physicists discover a phenomenon and end up developing exciting new mathematics to describe it. When the geometry of higher dimensions first started to be studied in the 1800s, it was generally regarded as purely abstract and mathematical. But with the development of Einstein's theories of relativity in the early 1900s and more recent developments in superstring theory, physicists have been taking the idea very seriously. As mathematician Ian Stewart says, "The potential importance of high-dimensional geometry for real physical space and time has suddenly become immense".^{[4]}
Space-time
Albert Einstein's theories of relativity treat space and time together, as a single four-dimensional entity called space-time. Space-time coordinates are quadruples of the form , like those used to describe the tesseract, except that the fourth spatial dimension w is replaced by the time dimension t. Any "point" specified by these four coordinates is called an event, a specific place at a specific time. Every object in space-time has a world line consisting of the space-time coordinates of all the events in the life of the object. All the coordinates specifying every event in the whole universe make up the total four-dimensional space-time continuum, which has a curved geometry.Dimensions in String Theory
In modern String Theory, the fundamental components of the universe are not particles but tiny vibrating strings. Within the framework of the theory, the large variety of different types of particles we observe are composed of the same fundamental strings vibrating at distinct frequencies. While this model is quite successful in many respects, it requires our space-time universe to be either 10-dimensional or 26-dimensional, implying there are either six or 22 spatial dimensions that we don't know about.^{[1]}
While this may sound absurd, there is no discrepancy with our everyday experience if these extra dimensions are "curled up" too small for us to detect. Imagine ants confined to walking along a thin piece of thread. For all practical means and purposes, their little world is one-dimensional. But imagine that the thread were thicker, like a large rope. Suddenly besides just going backwards and forwards the ants can move side to side along the curvature of this thick rope. According to String Theory, we are like the ants on the thread, living in a world with extra dimensions too small to be noticed.
Some physicists are considering an even stranger possibility, and suggest that the extra dimensions are quite large, so large that our four-dimensional world exists inside of a higher-dimensional reality.^{[2]} Like inhabitants of Flatland who can't move in the third dimension, we would be prevented from moving in these hidden directions by our laws of physics.
Other Applications
The mathematics of higher dimensions has applications beyond just spatial and temporal dimensions. An n-dimensional space consists of a bunch of points, each of which is list of n numbers. We don’t have to think of these numbers as coordinates for positions in physical space. They could represent any variables.
Consider a bicycle. We can describe the state of all the bicycle’s crucial components with six numbers: the angle between the handlebars and the frame, the angular positions of each of the two wheels, the positioning of the pedals’ axle, and the angular positions of each of the two pedals. A bicycle is of course a three-dimensional object. But we can describe any configuration of the bicycle with six numbers, numbers we can view as generalized coordinates, meaning that the state of the bicycle exists in an abstract, six-dimensional space. To get the hang of riding a bicycle, you need learn how these six numbers interact, not to mention the extra variables for motion and interaction with the road. This can be thought of as learning the six-dimensional geometry of “bicycle space”.
This way of visualizing the state of a system in an imaginary space of as many dimensions as you have variables turns out to be quite useful, and is used by mathematicians, physicists, and even economists and biologists^{[4]}.
Virology
For example, virologists find it useful to think of specific viruses as “points” in a space of DNA sequences. Each virus has a DNA sequence composed of a series of smaller molecular components called bases, represented by the four letters A, C, G, and T. The bases are the coordinates, and the DNA sequences are the points. Each coordinate is limited to being one of the four bases, just like how the coordinates for the vertices of a unit square, cube, or tesseract are limited to being either 0 or 1. The numbers 0 and 1 are like DNA bases, which makes unit hypercubes like DNA if it had only two bases.
This allows scientists to think of the possible viral DNA sequences as vertices of a very high-dimensional hypercube-like object, a sort of hypercube with its interior filled with smaller hypercubes. Each vertex corresponds to a specific virus, and since edges connect two vertices which differ by exactly one position, each edge can be thought of as a point mutation changing one base in a virus’ DNA sequence. Because this hypercube-like shape has such a high dimension, each vertex connects to quite a lot of other vertices, meaning that the virus has the potential to mutate in an enormous number of different ways. Using geometry of higher dimensional objects, virologists can understand how quickly the variation in possible mutations grows with longer sequences of viral DNA.^{[4]}
Teaching Materials
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Related Links
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
References
- ↑ ^{1.0} ^{1.1} Michio, K. (1994). Hyperspace: a scientific odyssey through parallel universes, time warps, and the 10th dimension. New York: Oxford University Press.
- ↑ ^{2.0} ^{2.1} Rucker, R. (1984). The Fourth Dimension: toward a geometry of higher reality. Boston: Houghton Mifflin Company.
- ↑ Rehmeyer, J. (2008). "Seeing in Four Dimensions". Science News.
- ↑ ^{4.0} ^{4.1} ^{4.2} Stewart, I. (2002). "The Annotated Flatland". Cambridge: Perseus Publishing
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