Produced by Andy Burbanks
The Hypercube
Spinning hypercube
(one complete revolution)
Spinning hypercube
(alternative projection method)
One of the simplest four-dimensional structures that we can imagine is the hypercube. It is the four-dimensional analogue of an ordinary cube.
To get an idea of what we mean by a "hypercube'', try the following exercise:
• Imagine a single point in space. We think of a point as being a zero-dimensional object.
• Now, imagine moving this point in one direction, say Eastward. It will sweep-out part of a line in space. We think of this line segment as being a one-dimensional object.
• If we take the line, and move it in a direction perpendicular to itself, say Northward, then the moving line will sweep-out part of a plane (in fact, a square or rectangle). A square region of the plane is a 2-dimensional object.
• Now, take the square, and move it in a direction perpendicular to itself (i.e. at right-angles to the plane), say Upward. The moving square can sweep-out the region of three-dimensional space that we would call a cube.
• Now, we just take the analogy further.
Move the cube in a direction perpendicular to the three-dimensional space it occupies. It's a bit difficult to imagine where this direction lies, but there seems to be no fundamental reason why we can't extend the process again. So let's suppose we can move the cube in this way. We'll call this new direction Tripward.
As the cube moves Tripward, it sweeps out a region of four-dimensional space which we call a hypercube.
Of course, there is no reason to stop there. There are no mathematical impediments to concieving of higher-dimensional spaces than four.
When you have seen this method of constructing a hypercube, it seems a relatively simple extension of the zero-, one-, two-, and three-dimensional versions.
But even this apparently simple structure can seem to behave in a complicated and counter-intuitive way. The movies here demonstrate what happens if we view a projection the hypercube into three-dimensional space as it is rotated around.
Rotation takes place about a point in 2-d and about a line (or `axis') in 3-d. Both affect only two of the coordinates present.
It seems only reasonable, then, that rotations in 4-d should be about a plane, and should also affect only two of the four coordinates. Among other things, this means that a reflection in a mirror in three-dimensional space can actually be achieved by a rotation of the object through the fourth dimension.
The movies show what the 3-d projection looks like as we rotate a model hypercube slowly through a complete revolution, revolving about three orthogonal planes at different rates.
Just as a 3-cube may be constructed by folding six squares together, so a 4-cube may be made by folding eight cubes into each other. In the second movie, one such cube is marked by a different colour.
Note: The projection into 3-d space is achieved for the first movie by simply ignoring the 4-th coordinate of the vertices. For the second movie, the 4th coordinate is used to project the vertices towards or away from the origin, so that objects closer to the viewpoint in the 4th coordinate bulge outward toward the viewer.
Inflated Hypercube
Produced by Keith Beardmore
Spinning "inflated'' hypercube (one complete revolution)
"I suppose Bucky Fuller would have called it a two-frequency spherical 4-cube."
The 2D analogue is the octagon. You start with a square and create a new vertex at the mid-point of each edge, then inflate the new shape. ''Inflation'' is achieved by moving all new vertices away from the centre of the square until all verticies lie on one circle. The resulting regular polygon - the octagon, has eight verticies and eight equal edges.
For the 3D analogue, you start with a cube and create a new vertex at the mid-point of each edge and also at the centre of each face, then inflate the new shape. ''Inflation'' is achieved by moving all new vertices away from the centre of the square until all verticies lie on one sphere. The resulting semi-regular polyhedron has 24 faces, 48 edges and 26 verticies.
For the 4D analogue, you start with a hypercube and create a new vertex at the mid-point of each edge, at the centre of each face and also at the centre of each 3-cube, then inflate the new shape. ''Inflation'' is achieved by moving all new vertices away from the centre of the square until all verticies lie on one hypersphere.
The movie here demonstrates what happens if we view a projection the inflated hypercube into three-dimensional space as it is rotated around.
The movie shows what the 3-d projection looks like as we rotate a model inflated hypercube slowly through a complete revolution, revolving about three orthogonal planes at different rates.
Note: The projection into three dimensional space was achieved here by selecting a point in 4-space and "casting shadows'' from that point onto a 3-dimensional slice of the space. The fourth dimension is then made more apparent as extra depth in the image (objects further away in the 4th direction appear to shrink away). The edges of the shape are coloured according to their initial distance (in 4-D) from the camera.
Hypertorus
Produced by Andy Burbanks
Spinning hypertorus (movie)
If we take a vector from the origin to a point, and rotate that vector about the origin, then the point traces a circular path. If we do the same to another vector, about a different axis this time, and add the two together, then the resulting end-point will spiral around a donut or "torus'', which has a two-dimensional curved surface in 3-d space.
By adding a third vector, rotating about yet a different axis, we find that the end-point spirals around the 3-d surface of a "hypertorus'' in four-dimensional space.
Note: The projection into three dimensional space was achieved here by selecting a point in 4-space and "casting shadows'' from that point onto a 3-dimensional slice of the space. The fourth dimension is then made more apparent as extra depth in the image (objects that are closer to the camera in 4-space appear to bulge outward towards us, whereas objects further away in the 4th direction appear to shrink away).
The models of the hypercube and hypertorus were produced with the aid of some simple Mathematica routines.
The "inflated'' hypercube was produced by a FORTRAN program.
The vertices of the 4-d objects are represented by vectors, each holding four coordinates
(x,y,z,t)
Rotation matrices are simple extensions of the 2-d and 3-d versions: for example, to rotate in the planes xy, and yt, we would use the matrices:
/ Cos(theta) -Sin(theta) 0 0 \
| Sin(theta) Cos(theta) 0 0 |
| 0 0 1 0 |
\ 0 0 0 1 /
and
/ 1 0 0 0 \
| 0 Cos(theta) 0 -Sin(theta) |
| 0 0 1 0 |
\ 0 Sin(theta) 0 Cos(theta) /
(These are basically just the 4x4 identity matrix, with elements that lie in the row and column corresponding to coordinates that will be altered by the rotation replaced by the elements of a 2x2 rotation matrix.)
Several techniques may be used to plot the structures. One way is to just rotate to the desired orientation and then ignore the 4-th coordinate of each vertex, giving a set of vertices in 3-d space to be joined together (this method was used for the first hypercube movies). This gives the effect of a simple orthogonal projection.
An alternative method is to choose a point in 4-space, and imagine that this is a light-source, we then take the shadow that is cast by light rays from this point to define our 3-d projection (this method was used for the spinning hypertorus). The 4-th dimension is then more apparent as extra depth appears in the image.
Ray-tracing was done using the Persistence of Vision (POV) and Rayshade packages with the edges connecting the vertices represented by cylinders, and the vertices themselves shown as spheres.
The movies were generated from the resulting still-frames using mpeg_encode.
http://www.t0.or.at/msguide/hyper.htm
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