4d visualization

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4D Visualization

Introduction

Higher-Dimensional Space

The world around us exists in 3-dimensional (3D) space. There are 3 pairs of cardinal directions:

left and right, forward and backward, and up and down. All other directions are simplycombinations of these fundamental directions. Mathematically, these pairs of directions

correspond with three coordinate axes, which are conventionally labelled X, Y, and Z,respectively.

The arrows in the diagram indicate which directions are considered numerically positive and

which are negative. By convention, right is positive X, left is negative X, forward is positive Y, backward is negative Y, and up is positive Z, and down is negative Z. We shall refer to these

directions as +X, -X, +Y, -Y, +Z, and -Z, respectively. The point where the coordinate axesintersect is called the origin.

As far as we know, the space we inhabit consists of these 3 dimensions, and no more. We may

think that space has to be 3-dimensional, that it can't possibly be anything else. Physically, thismay be true, but mathematically, there is nothing special about the number 3 that makes it theonly possible number of dimensions space can have. It is possible to have dimensions lower

than 3: for example, 1D space consists of a single straight line stretching off to infinity at either end; and 2D space consists of a flat plane, extending in length and width indefinitely. However,

nothing about geometry restricts us to 3 dimensions or less. It is quite possible²andmathematically straightforward²to deal with geometry in more than 3 spatial dimensions. In

particular, we can have a 4th spatial dimension that lies perpendicular to all 3 of the familiar cardinal directions in our world. The space described by these 4 dimensions is called 4-

dimensional space, or 4D space for short.

In a 4D world, there is another directional axis which is perpendicular to the X, Y, and Z axes.We shall label this axis W , and call the direction along this axis the fourth direction. This new

axis also has positive and negative directions, which we shall refer to as +W and -W.



It is important to understand that the W-axis as depicted here is perpendicular to all of the other

coordinate axes. We may be tempted to try to point in the direction of W, but this is impossible because we are confined to 3-dimensional space.

Why Bother?

Why bother trying to visualize a higher-dimensional space that we can neither experience nor

access directly? Besides pure curiosity, 4D visualization has a wide variety of usefulapplications.

Mathematicians have long wondered how to visualize 4D space. In calculus, a very usefulmethod of understanding functions is to graph them. We can plot a real-valued function of one

variable on a piece of graph paper, which is 2D. We can also plot a real-valued function of twovariables using a 3D graph. However, we run into trouble with even the simplest complex-valued

function of 1 complex argument: every complex number has two parts, the real part and theimaginary part, and requires 2 dimensions to be fully depicted. This means that we need

4 dimensions to plot the graph of the complex function. But to see the resulting graph, one must be able to visualize 4D.

Einstein's theory of Special Relativity postulates that space and time are interrelated, forming a

space-time continuum of 3 spatial dimensions and 1 temporal dimension. While it is possible tovisualize space-time simply by treating time as time and examining ³snapshots´ of space-time

objects at various points in time, it is also useful to treat space-time geometrically. For example,the distance between two events is the distance between two 4D points. The light-cone also has a

particular shape that can only be adequately visualized as a 4D object.

Furthermore, Einstein's theory of General Relativity describes curvature in space-time. While itmay not actually be a curvature into a physical spatial dimension, it is helpful to visualize it as

such, so that we can see how space curves in 4D as a 3-manifold. If space in the universe had positive curvature, for example, it would be in the shape of a 4D hypersphere²but what exactly

does that look like?

Many other interesting mathematical objects also require 4D visualization to be appreciated

fully. Among them are 4D polytopes (4D equivalents of polyhedra), topological objects such asthe 3-torus and the Real Projective Plane which can only be embedded in 4D or higher, and the

quaternions, which are useful for representing 3D rotations. It is difficult to fully appreciate theseobjects without being able to see them in their native space.

Is it possible to visualize 4D?

Some believe that it is impossible for us to visualize 4D, since we are confined to 3D and

therefore cannot directly experience it. However, it is possible to develop a good idea of what 4D



objects look like: the key lies in the fact that to see N dimensions, one only needs an (N-1)-dimensional retina.

Even though we are 3D beings who live in a 3D world, our eyes actually only see in 2D. Our retina has only a 2D surface area with which it can detect light coming into our eye. What our

eye sees is in fact not 3D, but a 2D projection of the 3D world we are looking at.

In spite of this, we are quite able to grasp the concept of 3D. Our mind is quite facile atreconstructing a 3D model of the world around us from the 2D images seen by our retina. It does

this by using indirect information in the 2D images such as light and shade, parallax, and previous experience. Even though our retina doesn't actually see 3D depth, we instinctively infer

it. We have a very good intuitive grasp of what 3D is, to the point that we are normally quiteunconscious of the fact we're only seeing in 2D.

Similarly, a hypothetical 4D being would have a 3D retina, and would see the 4D world as 3D

projections.

It would not directly see the 4th dimension, but would infer it using indirect information such aslight and shade, parallax, and previous experience.

The key here is that what the 4D being sees in its retina is 3-dimensional, not 4-dimensional. The

4th dimension is inferred . But since we have a good intuitive grasp of 3D, it is not that difficultto understand what a 4D being sees in its retina. From there, we just need to learn how to infer

4D depth.



The rest of this document will describe in detail the basic principles of 4D visualization, as wellas provide a number of examples of 4D objects. We shall take a purely geometrical approach andtreat all 4 dimensions as spatial dimensions.

Dimensional Analogy

A very useful tool in exploring 4D, or higher dimensions in general, is dimensional analogy.

Dimensional analogy is the process of examining how a particular geometric feature in a lower dimension relates to an equivalent geometric feature in our dimension, and then applying thesame principle to relate our dimension to a higher one.

Let's examine a few examples to see how this works.

Boundaries of Objects

Let's begin with the very basics. Let's start in a 1D world. The 1D world is like a piece of string.

There is only one axis along which one may move along this string, the X-axis. The onlydimension any object can have is length, because there aren't any other dimensions to

accomodate width or height. So the only possible objects in 1D are points, which are 0D, andlines, which are 1D.

In order to completely specify a line segment, it is enough to specify its starting point and itsending point. In other words, the boundary of an object in 1D consists of points, which are 0D.

Now, let's move to the next higher dimension. The 2D world is a plane, like the surface of a piece of paper, but extending indefinitely in width and length. The 2D world is much more

interesting than the 1D world, because a much larger variety of objects are possible. For

example, we can have polygons and circles, in addition to points and lines:

What is the boundary of a polygon? A polygon is bounded by line segments, which are 1D. A

circle also has a 1D boundary: even though it is a curved boundary, it is fully specified by asingle parameter: angle. So 2D objects are bounded by 1D lines and curves.



Now let's move on to the 3D world. Objects in the 3D world are bounded not by lines or curves, but by 2D surfaces. For example, a cube is bounded by 6 squares, and a ball is bounded by aspherical surface. The spherical surface is 2D, because any point on the sphere is fully specified

by only two parameters: longitude and latitude.

We can see a pattern emerge here. Objects in 1D are bounded by 0D points; objects in 2D are

bounded by 1D lines (or curves); and objects in 3D are bounded by 2D surfaces. In other words, points in 1D are analogous to lines and curves in 2D: they form the boundaries of objects in therespective dimensions. Similarly, bounding lines and curves in 2D are analogous to surfaces in

3D. So, by applying dimensional analogy, we see that in N dimensions, objects are bounded by(N-1)-dimensional boundaries.

This leads us to conclude that in 4D, objects are bounded not by points, lines, nor even surfaces, but volumes. It would be rather difficult to realize this without applying dimensional analogy.

For example, as we shall see later, a 4D cube is bounded by 8 cubes. We call these boundingvolumes the cells of the 4D cube.

Vision

Another application of dimensional analogy has been mentioned before is the dimension of the

retina in the eye of an N-dimensional creature. We are 3D beings, yet our eyes only see in 2D because our retina is only a 2D array of light-sensitive cells.

Why are our retinas only 2D? Surely it would be much better for us to have a 3D retina, so thatwe can see every part of our 3D world simultaneously?

The reason is that in order for us to see something, light must have an unobstructed path from the

object being seen to the cells in our retina. A 2D retina works, because there is a 3rd dimension

in which the light can travel unobstructed from the object onto the retina. However, if our retinawere 3D, we would not see anything more, because light must pass the cells on the outer surfacein order to reach the inner cells, so that what the inner cells see has already been seen by the

outer cells. Since we are confined to 3D, there is no additional dimension in which light maytravel to reach these inner cells by an independent path, which might have given us additional

visual information.



This fact leads us to conclude, by dimensional analogy, that a 2D creature must only have a 1Dretina. As the diagram above shows, when a creature is confined to 2D, there is no unobstructed

path for light to travel from a 2D object to an inner cell in a 2D retina. Any light that reaches aninner cell has already passed an outer cell, so having a 2D retina would not help the creature to

see more.

We can also conclude by dimensional analogy that in 4D, it is possible to have a 3D retina, because there is now an extra dimension in which light can travel unobstructed from the object

being seen to any point on the retina.

We shall make much use of dimensional analogy to understand 4D in the subsequentdiscussions.

Cross-sections

Using Cross-sections

Since we are creatures confined to 3D, we have no way of directly exploring higher-dimensional

objects. We can, however, employ various indirect means to study and understand them. Onemethod is to intersect a higher-dimensional object with our world to see what its various cross-

sections look like.

To illustrate this, let's apply dimensional analogy again. Suppose we are only 2D beings, livingin a 2D world, and unable to see into the 3rd dimension. Suppose we're trying to understand what

a cylinder is. We know what circles and squares are, because these objects exist in our 2D worldand we can directly handle and see them. But we haven't the slightest idea what a cylinder might

be. We have no way of directly seeing such an object, because our retina is only 1D, and a 2Dretina is needed to adequately perceive a 3D object. What we can do, is to examine what happens

when a cylinder passes through our 2D world:

As the cylinder does this, we can observe its cross-sections with our planar world. For example,

if the cylinder descends through our world vertically, we would see a series of circular cross-sections, all of a constant size.



From these cross-sections, we can conclude that the cylinder must be something circular.

We can also observe the cylinder passing through our world in a different orientation:

This time, the cylinder passes through round-side first. What we see is a series of rectangular cross-sections that seems to grow and shrink in length but remain constant in width. From this,

we conclude that the cylinder must have something rectangular about it.

Thus far, we learned that a 3D cylinder is something that is both circular and rectangular. This is

obviously correct. Examining cross-sections has yielded valuable information about the shape of a cylinder.

A Fundamental Weakness

Now, still supposing we are only 2D creatures, we have a hard time understanding how

something can be both circular and rectangular at the same time, as we've just learned. As 2Dcreatures, we only have experience with 2D shapes, and none of them are simultaneously a circle

and a rectangle. We may try to investigate this further by observing the cylinder pass through our 2D world at a 45-degree angle:

The cross-sections are now rather puzzling:



They consist not of regular circles or rectangles but of ellipses in various states of truncation.

Unless we knew beforehand, we would probably not be able to deduce the shape of a cylinder from these cross-sections. We may even think that these are cross-sections of a different object

altogether.

This illustrates a fundamental weakness of cross-sections method: although it does yield some

useful information, it is difficult to synthesize this information into a coherent model of what theobject is really like. For example, if we only knew the above sequence of cross-sections of the

cylinder, it would be rather hard for us to deduce that a cylinder has two circular lids and a

curved side.

The following sequence of cross-sections further illustrate this weakness:

Can you guess what 3D object would produce this sequence of cross-sections?

Probably not, unless you knew it beforehand.

These are, in fact, cross-sections of the 3D cube, which are produced when the cube passesthrough the 2D world corner-first. It is rather difficult to know this just by examining these

cross-sections alone; information about the object such as the number of vertices (corners) andthe number and shape of its faces are not readily apparent. Most people probably don't even

know that a cube can make a hexagonal cross-section with a plane!



Now, as 3D beings, we at least have some knowledge of 3D geometry to be able to imagine thecross-sections stacked on top of each other, and perhaps deduce a vaguely cube-like shape fromthem. But consider the following sequence of cross-sections of a 4D object with 3D space:

Can you figure out what the 4D object is?

Likely not, unless you knew it beforehand, since you would have a hard time understanding justhow these cross-sections could fit together. Without an intuitive grasp of 4D, it is very hard to

reconstruct the original object from them.

The fundamental problem with cross-sections is that we are examining the object piecemeal.Important features such as the number and shape of facets, the number of vertices (corners), and

the overall shape of the object, are only implied, not explicit.

A better approach is to use projections, as we will discuss in the next chapter.

Projections (1)

Projection is a method of representing an N-dimensional object using only (N-1) dimensions.

There are several different methods of projection, but they all have the same underlying idea:imaginary rays called projectors are emanated from the object towards an (N-1)-dimensional

projection plane. The intersections of these rays with the projection plane produce an image of

the object. This is like taking a picture with a camera: light from the object travels in straightlines (rays) and strike the film (the projection plane), producing an image (the photograph) of the

object.

The above figure illustrates this process. The object is an icosahedron, residing in 3D space. Thegreen dotted lines show some of the projectors, mapping points of the icosahedron to the

projection plane. The resulting 2D image is a ³photograph´ of the icosahedron.



Projections are an easier method of exploring higher dimensions, because it gives us anintegrated view of the object, rather than isolated bits of information such as in the cross-sectionmethod. In fact, our own eyes work this way: the retina captures a projection of a 3D object

outside of us, producing a 2D image which we then reconstruct into a 3D model in our mind. Theimage captured this way retains valuable information about the 3D object: such features as

corners, edges, and the shape and number of faces, are represented as an integrated whole.Consider the following projection of the cube:

We can immediately see where the corners of the cube are, and that its faces are tetragonal. Wecan see that each corner has three faces meeting at it. Now look again at the sequence of cross-

sections we saw in the previous chapter:

Can you pick out where the corners of the cube are in these cross-sections? What about the shapeof the cube's faces? How many faces meet at each corner? We would have to analyse these cross

sections very carefully in order to derive this information. However, using projections thisinformation is immediately available. Although the cross-section method does give us valuable

information, the projection method is much easier to understand.

Types of Projections

There are two main categories of projections: parallel projection and perspective projection. In

parallel projection, the projection rays are parallel to each other. In perspective projection, therays converge on a single point behind the projection plane. Both types of projection are useful

in examining higher-dimensional objects.

Parallel Projection

Parallel projection may be further classified into orthographic projection, where the rays arealways perpendicular to the projection plane, and oblique projection, where the rays intersect the

projection plane at an angle.



For example, here is a 3D cube projected into 2D face-on using orthographic projection:

The image of the cube under orthographic projection is a square, because we are looking at it

face-on.

Here is the same cube projected using oblique projection:

Now the other faces of the cube are visible. We have colored the front face red, and the back face blue, and remaining faces yellow. The front face is connected to the back face by 4 edges.

For comparison, here is a 4D hypercube in orthographic projection:

Just as the image of the cube under orthographic projection is a 2D square, so the image of thehypercube in orthographic projection is a 3D cube.

Here's the 4D hypercube in oblique projection:



We have colored the nearest cell of the hypercube red, and the farthest cell blue, and the

remaining cells yellow. Note the similarity between this image and the oblique projection of the3D cube. Just as the oblique projection of the cube looks like two squares connected by 4 lines,

so the oblique projection of the hypercube looks like two cubes connected by 8 lines. You cansee dimensional analogy at work here.

Some of the properties of images produced by parallel projection are:

y The size of the image does not depend on the distance of the object. No matter how far away the object is, the parallel rays will always produce the same image size.

y Parallel lines in the object remain parallel in the image. For example, in the oblique projection of the cube, the vertical edges of the cube all appear vertical in the image. The

front-to-back edges appear slanted, but they are still parallel to each other, just as in thereal cube.

Projections (2)

Perspective Projection

Parallel projections, while useful, lack an important visual cue: distance. Since parallel projection preserves size, the same object looks exactly the same no matter how far or near it is.

But distance is how our mind recovers the 3rd dimension, depth, from the 2D image in our retina, so distance is a very important part in visualizing 4D. It helps us infer 4D depth from a 3D

image.

P erspective projection is how we can include information about distance in the projected image.

In perspective projection, the projection rays converge to a point behind the film:



Notice how the same object will appear smaller when it's farther away. By the relative size of the

image, we can infer its distance. This is what we've come to expect in our own experience in thereal world, because our own eyes also see in perspective projection.

Foreshortening

Perspective projection also has a side-effect: parallel lines in the object are no longer parallel in

the image.

Notice in the above image that parallel edges in the cube are not parallel in the image. This phenomenon is called foreshortening, and is a consequence of the size of the image depending

on the distance of the object.

R otating a Cube through 4D

We shall now use dimensional analogy to investigate the perspective projection of a 3D cube as

it gets rotated through 4D. This will greatly help us understand projections of 4D objects later on.

We'll start by taking a look at a 2D square rotating in 3D:



Notice how, from a purely 2D perspective, the image of the square only appears as a squarewhen viewed face-on. When viewed at from an angle, it appears not as a square but a trapezoid.

Its internal angles appear to be changing, and its outer edge appears to be lengthening andshortening as it rotates through 3D. However, we know that the square isn't actually changing its

internal angles or the length of its edges; it just appears that way because of foreshortening in perspective projection.

Now let's take a look at the analogous situation of a 3D cube rotating in 4D, and see if we can

make sense of it:

The cube appears to be turning itself inside-out and outside-in. One of its faces appears to begrowing and shrinking, and its side faces appear to be distorting into trapezoids. However, the

cube isn't actually being distorted; it only appears that way because it is rotating through the 4thdimension. More precisely, it is rotating in the XW plane (the plane defined by the X and W

coordinate axes).

We shall explain 4D rotations in more detail later on. But for now, as a help to understanding thisodd effect, consider how a purely 2D creature would think of the above animation of a square

rotating through 3D. The only rotations it knows are rotations in the plane, and none of them turnthings inside-out like that. Since it has no experience of 3D rotations, it would perceive the

square as being distorted and turning inside-out in an impossible way. But in reality, the squareisn't turning inside out; it's just doing a perfectly normal rotation through 3D. The inside-out

effect is merely an artifact of projecting 3D into 2D.

Likewise, 4D rotations to us 3D beings appear to involve incredible feats of turning inside-out;

but they are really perfectly ordinary rotations through 4D. The inside-out effect is merely anartifact of projecting 4D into 3D.

Projections (3)

Projection Envelopes



The envelope of a projected object is the outer boundary of its image. For example, the followingview of the 3D cube has a hexagonal envelope:

The same object may have different envelopes; for example, if a cube is viewed face-on, it has asquare envelope:

A cube can never have a triangular or pentagonal envelope. If you know what envelopes anobject can have, it narrows down what type of object it can be.

However, one should keep in mind that the envelope of an object really only gives limitedinformation about the object. One should not falsely think that knowing the envelope of an

object's projection is sufficient to uniquely identify the object. For example, an octahedron alsohas a square envelope, when viewed vertex-on:

Hence, if you only knew that an object has a square envelope, that doesn't tell you which object

it is. The important thing about a projected image is not the envelope, but its internal structure. Itis the internal structure that gives insight into the structure of the object. For example, knowing

that a cube has a hexagonal envelope isn't very useful in itself; it is more insightful to see howthe 3 projected faces of the cube are laid out inside that hexagon.

Look at the projection of the 3D cube again. What part of the image do you automatically focuson? Your attention naturally falls on the central region of the image, where 3 of the cube's edges

meet at the corner that's facing you. In fact, your attention so spontaneously centers itself on this



central region, that you are usually unaware that this view of the cube has a hexagonal envelope!But if you were a 2D creature, your viewpoint would be rather different: what catches your attention first would be the hexagonal envelope, and it would be tempting to identify the

hexagonal envelope with the cube. However, the real point of interest lies inside the envelope.

All this may seem obvious, but it is very important to keep in mind when we start examining projections of complex 4D objects into 3D. These projections often have fascinating envelopes,

such as rhombic dodecahedra, cuboctahedra, and other interesting polyhedra. It is tempting tounconsciously identify these polyhedral envelopes with the 4D object itself, because we are used

to identifying 3D objects by the shape of their surfaces. However, most of the information aboutthe structure of the 4D object lies inside the envelope.

A Projection of the 4D Hypercube

Consider the vertex-first projection of the 4D hypercube. Its envelope is a rhombic

dodecahedron, a polyhedron bounded by 12 rhombuses. It is tempting to only regard thisrhombic dodecahedral envelope, which is interesting in its own right:

However, if this is all we focus on, we would not know where the hypercube's cells are located

in the image. In fact, we would not even see the hypercube vertex that the 4D viewer is looking

at! The hypercube vertex in fact projects to the center of this dodecahedron, not to any of itsexternal vertices. It is the central vertex highlighted in yellow below:

Knowing this internal structure of the projection also helps us locate the 4 cubical cells of the

hypercube that are currently visible:



These cells appear to be distorted cubes, but they are actually perfectly regular cubes. They onlyappear distorted because they are foreshortened by perspective projection.

When a 4D being looks at the hypercube, its attention falls primarily on this layout of cubicalcells in the image, not on the envelope itself. The envelope is only peripheral; the inside of the

image is what is of interest. When examining projections of 4D objects, we should always focuson its internal structure rather than be distracted by its envelope.

Note that in the above views of the hypercube, only 4 of its 8 cells are visible. The reason for this

is that the other 4 cells are behind these four, and are therefore obscured. We shall explain this indetail in the next chapter.

Interpreting 4D Projections (1)

Thus far, we have seen a few examples of 4D objects projected into 3D, as well as variousmethods of enhancing the images so that they are easier to understand. But how exactly does one

interpret these images? What do the various features we find in these images really mean? Weshall discuss this now.

Thinking in Terms of Volumes

When we first introduced dimensional analogy, we discussed the boundaries of objects in

different dimensions. In 1D, objects are bounded by points, which are 0D constructs. In 2D,objects are bounded by lines and curves, which are 1D constructs. Similarly, in 3D, objects are

bounded by surfaces, which are 2D constructs. This pattern led us to conclude, via dimensionalanalogy, that 4D objects must be bounded by volumes. Let's see how this helps us interpret a 4D

projection.

We'll begin by looking at the following perspective projection of the 3D cube:



This image consists of a number of parts: the outer boundary, which is a large square; a smaller

blue square in the center; and four green trapezoid pieces connecting the outer square to the inner square.

Now, we know from our experience with 3D that the outer square corresponds with the near face

of the cube, and the inner square corresponds with the far face of the cube. Although the outer square is larger than the inner square, in actuality the near and far faces of the cube are the same

size. The far face only appears to be smaller because it is farther away in the 3rd direction.

Similarly, we know from our experience that the four trapezoid pieces between the inner andouter squares are really square faces of the cube. They only appear in trapezoid shape becausethey are being viewed at from an angle.

We also know that all these 6 faces of the cube are only on its outside. This is obvious to us, but

it is very hard for a 2D being to understand. In the image, the inner square appears to becompletely inside the outer square, so how can it be only on the boundary of the cube? Keep this

in mind as we now turn to examine the analogous situation in 4D.

The following image is a cell-first perspective projection of the 4D hypercube.

Let's compare this image bit by bit with the previous one, using dimensional analogy.

Firstly, we see that it contains two cubes: an outer cube, and a blue inner cube. The inner cube is

the equivalent of the inner square in the previous image. It is the far cell of the hypercube, just asthe inner square is the far face of the cube. Similarly, the outer cube is the equivalent of the outer

square in the previous image. It is the near cell of the hypercube. These two cubes are in fact the same size; the blue cube only appears to be smaller because it is farther away in the 4th

direction.

Secondly, there are 6 green frustum-shaped volumes that connect the outer cube to the inner cube: one on top of the inner cube, one underneath, and four around the sides. These frustums are



the equivalents of the trapezoids in the previous image. By dimensional analogy, we realize thatthey are actually not frustums; they are also perfect cubes identical to the inner and outer cubes.They only appear as frustums because they are being viewed at from an angle.

Finally, all 8 cubes, inner, outer, top, bottom, left, right, front, and back, are only on the outside

of the hypercube. They are the equivalents of the faces of the cube. This is not easy tocomprehend at first. From our perspective as 3D beings, these cubes have already completely

filled up the cubical space occupied by the image; how can they only be on the outside of thehypercube?

To gain more insight into this perplexing question, let's take a deeper look at another feature of

these images.

Where is the Inside?

Look again at the projection of the 3D cube. As 3D beings, we know that the inside of the cube

lies in the volume between the 6 faces of the cube. But if a 2D being were to look at this projection of the cube, this ³inside´ would be rather elusive. If you only said that the inside of

the cube lies between the inner and outer squares, the 2D being might incorrectly think that youare referring to the area covered by the 4 trapezoids. However, as we know, this is not the case;

the 4 trapezoids are also only the boundary of the volume inside the cube. From the 2D perspective, this is hard to understand. The 6 faces already fill up the entire square area of the

image; where else is there space for the inside of the cube?

The answer, of course, lies in the fact that there is 3D depth involved here. The inner square isdeeper in the 3rd direction than the outer square; so there is lots of room between them for the

inside of the cube. In fact, the amount of room available is the sum total of all possible squaresthat lie between two opposite faces, as shown by the following animation:

The magenta square is moving back and forth between the left and right faces of the cube. Can

you see how it traces out the volume enclosed inside the cube?



Now look at the 4D case.

Where is the inside of the hypercube? Dimensional analogy tells us that it must lie ³between´ all

8 cubes that form the boundary of the hypercube. However, when we look at the image abovefrom our 3D perspective, we can see nowhere for this ³inside´ to fit. The 8 cubes have already

filled up the entire cubical volume of the image; where else is there space for the inside of thehypercube?

The answer should be clear if we apply dimensional analogy: there is 4D depth involved here.The inner cube is deeper in the 4th direction than the outer cube; so there is ample room in between for the inside of the hypercube. In fact, the amount of room available is the sum total of

all possible cubes lying between two opposite cubes, as shown by the following animation:

The magenta frustum is actually a cube moving back and forth between the left and right cells of

the hypercube. It traces out the 4-dimensional hyper-volume that is enclosed inside thehypercube. Notice that it appears to be turning inside-out as it crosses the middle of the

hypercube. As we've explained before, this is an artifact of the perspective projection used tocreate this animation; the cube is not actually turning inside-out. This does mean, however, that

3D volumes are ³flat´ in 4D, so a 3D cube behaves like the equivalent of a plane.

Try to compare this animation with the previous one carefully, and see if you can see the analogy between the two. It is important to understand that each different apparent shape of the magenta

cube represents a distinct slice of the hypercube. The volumes occupied by the magenta cube atdifferent displacements do not overlap! As 3D beings who have little knowledge of 4D, this can

take a while before it ³clicks´. But it is worthwhile to make the effort to grasp this. It will yieldmuch insight into visualizing 4D.



Cells, R idges, Edges

The upshot of all this is that in 4D, objects have a much richer structure than in 3D. In 3D, a polyhedron like the cube has vertices, edges, and faces, and fill a 3D volume. The cube is

bounded by faces, which are 2D. Every pair of faces meet at an edge, which is 1D, and edgesmeet at vertices, which are 0D.

In 4D, objects like the hypercube not only has vertices, edges, and faces, but also cells. A 2D boundary is insufficient to bound a 4D object. Instead, 4D objects are bounded by 3D cells. Each

pair of cells meet not at edges, but at 2D faces, also called ridges. The ridges themselves meet atedges, and edges meet at vertices.

The point is that in 4D, 3D volumes play the role analogous to surfaces in 3D, and 2D ridges

play the role analogous to edges. Because of this, it is important to visualize 4D objects bythinking in terms of bounding volumes, and not 2D surfaces. A 2D surface only covers the

equivalent area of a thin string in 4D! When you see a 2D surface in the projection of a 4Dimage, you should understand that it is only a ridge, and not a bounding surface.

R otations (1)

We continue our exploration of the 4D world by taking a look at rotations.

R otation being a Planar Phenomenon

One peculiarity of the 3D space that we inhabit is that every plane of rotation has a unique axis

around which every other point revolves. The points that lie on this axis are stationary under therotation; so we may call the axis the stationary line of the rotation.

Since our experience is limited to 3D space, we may be tempted to think that rotational axes is an

inherent property of rotation. However, this is actually not true of other dimensions in general.Consider the following square rotating in the 2D plane:

Where is the axis for this rotation? From our 3D viewpoint, it lies perpendicular to the 2D plane, protruding outward from the screen. However, if we were to look from the 2D perspective, thereis no such axis; there is only a single stationary point at the center of the rotation. All other

points on the supposed axis lie outside the 2D universe!

As we shall see, rotations in 4D do not have unique axes of rotation. Instead, they have stationary planes which remain stationary under the rotation. From the 4D perspective, a rotation

has not just an axis, but a plane around which everything else rotates. As 3D beings, we may findit difficult to understand just how something could rotate ³around´ a plane.



Instead of thinking about stationary points, axes, or planes, it is much better to understandrotations directly, as a planar phenomenon. Consider 2D and 3D rotations, which we areacquianted with. There is one common characteristic among them: every point of the rotating

object traces out circles that lie on parallel planes. In 2D, there is only one plane, the 2Duniverse itself, whereas in 3D, there may be many parallel planes. Since they must always be

parallel (otherwise it is not a rotation, but a deformation of the object), we can pick one suchrepresentative plane and call it the plane of rotation. Thus, we may think of rotation as

happening in a plane rather than around an axis.

This understanding of rotation is easier to generalize to higher dimensions. It also helps usunderstand why 2D rotations only have stationary points, whereas 3D rotations have axes: being

a planar phenomenon, a rotation ³needs´ two dimensions to move in. In 2D, these are the onlytwo dimensions available; hence what is left is only a 0D stationary point. In 3D, there are 3

available dimensions, so there is still 1 dimension left when the other two are employed for therotation. This extra dimension corresponds with a single line, the rotational axis. In 4D, there are

2 dimensions left over, and so we have stationary planes.

Number of Principal R otations

Those of us who are geometrically-inclined would no doubt have learned that there are three principal rotations in 3D: those in the XY, YZ, and XZ planes. Every rotation in 3D can be

reduced to a combination of these principal rotations. The fact that there are precisely 3 principalrotations in 3D, however, is a coincidence: the number of principal rotations is not, in general,

equal to the number of dimensions. For example, in 2D, there is only one plane of rotation,which is the 2D plane itself. As we shall see, the number of principal rotations in 4D is not four,

but six.

This is a simple matter of combinatorics: we have already seen that rotations are a planar

phenomenon, and therefore ³use up´ two dimensions. The number of different principal

rotations, therefore, is equal to the number of distinct pairs of dimensions. For example, rotationis not possible in 1D, because there is only one axis, the X-axis, and no pairs are possible. In 2D,there are two axes: X and Y, and they are the only possible pair of dimensions; therefore, there is

precisely 1 plane of rotation. In 3D, there are three axes: X, Y, and Z; this gives us 3 possible pairs: XY, XZ, and YZ. Hence, there are 3 principal rotations in 3D. In 4D, we have four axes,

X, Y, Z, and W; so there are six possible pairs: XY, XZ, XW, YZ, YW, and ZW. (It is left as anexercise to the reader to check that these are the only possibilities.)

The reader may feel a little overwhelmed at this point: there are six principal rotations in 4D to

be learned, twice the number in 3D! However, this is actually not as hard as it may seem at first.The key here is to visualize 4D rotations using projections.

The Appearance of Projected R otations

Consider how one might teach 3D rotations to a 2D creature. One useful way would be to use a

projection along the Z axis, so that the X and Y axes of 3D space map to the X and Y axes of 2D,and the Z axis is collapsed into a point. Here is what a square rotating in the XZ plane might look

like under this projection:



For us 3D beings, it is easy to immediately interpret this in the 3D sense; but think for a momentfrom a purely 2D perspective. Instead of a square rotating in the plane (which is the only type of

rotation a 2D being would be acquianted with), it appears that the square is turning itself inside-out in an impossible contortion. Of course, we know from our 3D perspective that this isn't really

the case; the rotation just appears that way because it is being viewed at sideways.

What about the appearance of a square rotating in the YZ plane?

Again, we see the ³turning inside-out´ effect. This rotation appears to be identical to the firstone, except that the ³inside-out´ effect takes place along the Y-axis rather than the X-axis.

Finally, let's look at the last principal 3D rotation, the one in the XY plane:

Notice how this rotation looks exactly like a 2D rotation. This should not be surprising, sincerotation in the XY plane happens to be the same as the only principal rotation in 2D.

So here we see that projections of 3D rotations into 2D can appear in two possible ways: as a

perfectly normal 2D rotation, or as an apparent ³turning inside-out´. The normal-looking rotationis the one in the XY plane, which happens to be the same as the principal 2D rotation. The

³inside-out´ rotations are the ones that happen in planes involving the Z-axis, the axis thatextends outside the 2D plane. Furthermore, these ³inside-out´ rotations are identical in

appearance to each other except for orientation.

Visualizing 4D R otations

The same thing happens with 4D rotations: 3 of the 6 possible principal rotations appear exactlylike the 3 principal 3D rotations that we are already familiar with. They are the rotations in the

XY, XZ, and YZ planes, which happen to be the same as the 3 principal rotations in 3D. The



remaining rotations are the ones involving the W-axis: the rotations in the XW, YW, and ZW planes. They have an appearance of ³turning inside-out´. The following animations show a cubeundergoing each respective rotation:

Notice that all three rotations appear the same, except that the ³turning inside-out´ happens along

the X-, Y-, and Z-axes respectively. Hence, we really only need to learn one new type of rotation, and we will have fully grasped all 6 principal rotations in 4D.

It is important to keep in mind that in all of these animations, the cube is not actually distorting

and turning inside-out. This is merely an artifact of projecting from 4D to 3D. The cube retainsits shape unchanged throughout all of the rotations.

Clifford R otations

We said earlier that rotation requires 2 dimensions, and mentioned that the reason 4D rotations

have stationary planes is because there are 2 dimensions left over. The perceptive reader mayhave wondered if it was possible to have a second, distinct rotation in this stationary plane, since

there are 2 leftover dimensions for another rotation to happen in. That is to say, is it possible for an object to rotate simultaneously around the XY plane and the ZW plane, possibly with two

different rates of rotation, since the rotational plane of each lies precisely in the stationary plane

of the other?

The answer is, in fact, yes. It is quite possible for an object in 4D to rotate simultaneously in theXY and ZW planes²or, for that matter, the XZ and YW planes, or the YZ and XW planes.

These complex gyrations are known mathematically as Clifford rotations (or double rotations).To distinguish these from the rotations we considered earlier, we shall refer to the former as

plane rotations. Clifford rotations can be decomposed into two independent, simultaneous planerotations (hence the name double rotation), and therefore have two independent rates of rotation.

Mathematicians have known of Clifford rotations for a long time, although not many can easily

visualize them, especially since they are not possible in dimensions below 4. With what we have

learned so far, however, we are ready to actually see a 4D Clifford rotation in action. It is simplya rotation where, in projection, a 3D-like rotation happens at the same time as an ³inside-out´rotation. For example, here is a cube rotating simultaneously in the XY and ZW planes:



The ³inside-out´ rotation is happening along the vertical Z-axis, which corresponds with the ZW plane, and the 3D-like rotation is happening in the XY plane. Notice how the ZW plane projectsonto the Y-axis, which is the axis of the XY rotation in 3D.

A 4D Clifford rotation does not have a stationary plane, since all 4 dimensions are being used for

the rotation. Instead, it has a stationary point, which is the intersection of the stationary planes of its two constituent plane rotations. (In 4D, two planes will generally intersect only at a point.) In

this particular case, the stationary point of the cube lies at the center of the cube.

4d visualization

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