If you’re anything like me, you’ve probably spent a fair amount of time wondering what is the deal with tesseracts? Just exactly what the heck is a “four-dimension cube” anyway? No doubt you’ve stared curiously at one of those 2D images (like the one here) that fakes a 3D image of an attempt to render a 4D tesseract.
Recently I spent a bunch of wetware CPU cycles, and made lots of diagrams, trying to wrap my mind around the idea of a tesseract. I think I made some progress. It was an interesting diversion, and at least I think I understand that image now!
FWIW, here’s a post about what I came up with…
No promises that this will be coherent, useful, or even interesting, but it is long. For myself, I find writing (or talking) about a topic helps clarify it, so this is mostly an exercise for the writer.
The inspiration for this came from a Greg Egan book (Diaspora) that mentions tesseracts (you run into them in science fiction sometimes; one of my childhood SF short story collections had a story featuring a tesseract house).
More to the point, Egan mentions that a tesseract is composed of 8 cubes, 24 squares, 32 lines, and 16 points.
That got me wondering what the count table looked like for all those regular square shapes. (In this case, “square” has more the “right-angle” meaning than the four-sided shape, although that shape is one of the shapes involved.)
The table below lists the square shape objects along the left and their component parts across the top. Each row indicates how many instances of the component shape are in a given object:
|tesseract||1 ||8 [8(1)]||24 [6(4)]||32 [4(8)]||16 [2(8)]|
|cube||1 ||6 [6(1)]||12 [4(3)]||8 [2(4)]|
|square||1 ||4 [4(1)]||4 [2(2)]|
|line||1 ||2 [2(1)]|
(The numbers in square brackets are factors of the bold numbers.)
I looked at that table for a while trying to figure out a formula describing the mathematical progressions.
Points were easy. They just double each row. But lines? What formula gives you 1, 4, 12, 32? Squares are even worse: 1, 6, 24? I’m not an expert mathematician, so I never came up with a simple formula that explains the column sequences.
The color coding shows a last attempt. I noticed the diagonal of identities (light blue). Obviously it takes one square to make a square. When I factored the numbers as shown I found another diagonal of identities (light green). That also seemed to give each column a base number (2, 4, 6, 8). Made me think I was on to something!
But the progression in the next diagonal (light yellow) is 2, 3, 4, which is nice and regular, but how did we jump from 1, 1, 1, 1 to that? The next diagonal (light red) was worse: 4, 8. Regular sequences, sure, but not well-related.
I gave up, because it was clear the sequences were due to geometry and increasing dimensions, so maybe there wasn’t a simple formula describing the sequences. Turns out there is, but I didn’t find it until much later.
Next, I considered how to get from a point to a line, from a line to a square, from a square to a cube, and from a cube to a tesseract (and on to higher-dimensional objects!).
So start with a point. It has no dimensions and, thus, no coordinates. The idea of an axis or center point has no meaning.
To create a 1D line, “sweep” (that is, move) the 0D point through a new dimension (call it x).
Sweep it a specific distance, call it L (for length).
This sweep also generates a second point in addition to the original starting point.
So, as the table shows, a line is: 1 line and 2 points.
This idea of sweeping a shape through a new dimension is the basis of creating all these “square” higher-dimensional shapes.
Sweeping a zero-dimensional point results in a one-dimensional line.
To create a 2D square, sweep the 1D line — along with its 2 points — a distance of L through a new dimension (call it y).
The new position of the line gives us a new line.
Sweeping the original line’s 2 points through y generates 2 new lines (green) and two new points (bottom).
(This is the same as before; sweeping a point through x made a line. Here the sweep is through y.)
So, as the table shows, a square is: 1 square, 4 lines, and 4 points.
Important: Sweeping a shape includes all of its component parts. Sweeping each part produces new higher-dimension component shapes in the final shape.
All sweep shapes have components of each lower-dimensional shape. A line has points, and a square has both lines and points.
A cube, therefore, will have squares, lines, and points.
To create a 3D cube, sweep the 2D square — and component parts — a distance of L through a new dimension (call this one z).
The new position of the square gives us a new square.
Sweeping the original square’s 4 lines through z creates 4 new squares (on top, bottom, and both sides).
Sweeping the square’s 4 points creates 4 new lines (in blue) as well as four new points.
So, as the table shows, a cube is: 1 cube, 6 squares, 12 lines, and 8 points. (As expected, the cube contains all the lower-dimensional shapes.)
The process continues to higher-dimensional shapes, but the dimensions become imaginary since there are only three axes of freedom in three dimensions. That’s what three-dimensional means!
So the diagrams are going to get challenging; some imagination is required to see exactly what they try to depict. For example:
To make a 4D tesseract, sweep the 3D cube — and parts — through a fourth new dimension (call it w, for weird).
The new position of the cube gives us a new cube.
Sweeping the cube’s 6 squares through w gives 6 new cubes. These cubes are weird! One of their three dimensions is in w!
Sweeping the original cube’s 12 lines creates 12 new squares (which are the side faces of the weird cubes), and sweeping the 8 points creates 8 new lines (plus 8 new points).
So, as the table shows, a tesseract is: 1 tesseract, 8 cubes, 24 squares, 32 lines, and 16 points.
Understanding a tesseract requires some imagination. The diagrams above try to illustrate the sweep through the purple dimension w, but what really happens?
One observation is that, with regard to the sweep object, both the old and new share the same dimensional coordinates. The only difference is that they differ in the new coordinate — which is fixed to a single, different, value in each.
For example, a line has a set of x-coordinates. When swept through y to make a square, the new line has the same x-coordinates. But the original line has one y-coordinate (for all its points) and the new line has another.
Likewise the cube-creating square has a set of xy-coordinates. Both the original square and the ending square share all of them, but each has a different, fixed, z coordinate.
Therefore, in a tesseract, the old and new cubes share xyz-coordinates. They differ only in their w coordinate.
That means, from a 3D perspective, the sweep through w doesn’t move the cube!
The traditional diagram of a tesseract shows a smaller cube inside a larger cube. The large cube is the original cube, the smaller inner cube is the new cube.
(Or vice versa; works either way.)
The name, tesseract, which comes from Greek and means “four rays,” comes from the fact that each point has four lines connecting to it. (Each point of a cube has three; each point of a square has two; each point of a line has one.)
These more traditional diagrams are similar to the larger diagrams above, except the red cube is shown inside the blue cube. Both diagrams “lie” about the red cube!
The reality is that both the outer and inner cube are the same size and share the same xyz-coordinates!
In fact, just as all the squares of a cube are the same shape and size, all the cubes of a tesseract are the same shape and size.
This applies to the six new “weird” cubes created by sweeping the six faces (squares) of the original cube through w.
These six cubes connect a face of the outer cube to the matching face of the inner cube. The traditional diagram shows these as truncated pyramid shapes connecting outer and inner faces, but they are actually square cubes (with the same size as the original cube)!
They are created by sweeping a 2D square to make a 3D cube. The difference is that one of the new cube’s dimensions is in w!
That means, from a 3D perspective, those six cubes have no thickness in one dimension!
Start with that traditional diagram and expand the inner cube to make it the same size as the outer cube.
In the process, the six cubes formed by sweeping the original cube’s squares decrease until they are completely flat in one of the 3D dimensions.
The top and bottom cubes become flat in the up-down (z) dimension. The front and back cubes become flat in y, and the left and right become flat in x.
But all six are full-sized cubes with length in w accounting for the missing x, y, or z, dimension.
It makes it very interesting to speculate what might happen if a tesseract actually existed as a house-sized (hollow!) object.
If there were portals (4D doors!) between the 8 cubes, what would happen upon stepping from either the inner or outer cubes (which exist in the x, y, z dimensions we know) to one of the “flat” sweep cubes?
Would the portal have to convert the missing 3D dimension to the w dimension? (Perhaps that’s a natural translation of 4D doors?)
Or would it be flat in some weird way (like Abbott’s Flatland)?
From the 3D perspective, occupants of the sweep cubes would certainly look flat. It’s anyone’s guess what it would feel like to the occupants!
Even weirder, proceeding from the inner cube, straight through a sweep cube, to the outer cube (or vice versa) returns to the same 3D space. Remember that both the inner and outer cubes occupy the same 3D points! All points inside the tesseract occupy the same 3D space as the cube!
Another odd thing to ponder is what doors to outside space would be like for the eight tesseract cubes. It’s especially strange with regard to the six “flat” cubes!
Another observation about these square shapes is that whatever their dimensional, two points of that dimension, on diagonally opposite corners, bound the shape:
- two x points bound a 1D line
- two xy points bound a 2D square
- two xyz points bound a 3D cube
- two xyzw points bound a 4D tesseract
The first point is found on the original shape, while the second point is found on that shape in its new position. Further, the second point crosses all available diagonals of that shape.
For example, in a cube, if the first point is on a corner of the original square, the second point is on the diagonal point of that square in its new position.
The bounding points for a tesseract would look similar to the diagram shown here for a cube, except that the lower right point would actually be on the inner cube. The black diagonal line crosses w as well as x, y, and z.
Mathematically, these points represent the minimum and maximum spacial extent of the shape (exactly what we mean by bounding). The interesting thing is that (with square shapes) it takes only two points, regardless of the number of dimensions!
It’s possible now to construct a formula for the number of component shapes in an object.
The process of sweeping means this count of components is affected by more than just the previous amount.
For example, the number of cubes in an object depends on the previous number, plus any new cubes resulting from moving an existing cube, plus any new cubes resulting from sweeping squares.
The count of squares, likewise, depends on the previous number, plus new ones from moving those, plus new squares from sweeping lines.
Effectively, as with points, each component type doubles its members in the new position of the main shape. Each component type also increases by the next lower component (in dimension) creating new instances of the higher one through sweeping (lines create squares, etc).
The result is this:
2Sn + Sn-1
Where S refers to the current count of a given component shape. The subscripts (n) index the shapes as shown in the Square Shapes table. The exception is with points, where n=0. In this case the formula is just 2S0.
Which shows why the number of points just grows by a nice factor of two, while the others have more complex progressions!
For example, given a line (1 line, 2 points), creating a new square involves:
- 1 square
- 2(1 line) + (2 points) = 2 + 2 = 4 lines
- 2(2 points) = 4 points
As the table says. For a cube (given a square):
- 1 cube
- 2(1 square) + (4 lines) = 2 + 4 = 6 squares
- 2(4 lines) + (4 points) = 8 + 4 = 12 lines
- 2(4 points) = 8 points
As the table says. The tesseract is left as an exercise for the reader!
For extra credit, extend the shape-creating process into the fifth dimension (call it u for unusually weird). Apply all aspects of the this discussion to the new shape.
This isn’t the end of the discussion, but the post has gotten awfully long. The next topics, rotations and reflections, very likely require about as much discussion, so I’ll leave them for another post.