Contents
Tesserawalk
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Highlight 'face' 1 |
Oh, so you followed the link? Hmm. I don't know if you'll like it in here. Anyway, I would have a model hypercube for you, but the applet I'm using can only handle three dimensions. Odd that. If I ever get a good enough understanding of how it works, I may extend it to four, but that's some way off yet.
Oh, and in case you don't know, would anyone like me to explain tesseracts to them?
So without further ado, let me tell you about Tesserawalk. Firstly, it's not as simple as before. Before we used faces, and the word faces was well defined. Now it is less so. If anyone works out a good way of using the cubic faces of the polytope itself, or the faces of the faces, I would welcome suggestions. Until then, I'm using the vertices, and their connecting edges.
Just out of interest, how many right angles do you think you can see in the diagram above? I mean the picture itself, rather than the actual figure (since all of the angles in the hypercube are right).
There are 16 vertices, with 4 connecting edges each. This gives fewer players than Icosawalk, but with more candidates per vote. One version of the table is as follows:
| Vertex | Opposite | Adjacents | |||
|---|---|---|---|---|---|
| 1 | 16 | 6 | 9 | 13 | 15 |
| 2 | 15 | 5 | 10 | 14 | 16 |
| 3 | 14 | 8 | 11 | 13 | 15 |
| 4 | 13 | 7 | 12 | 14 | 16 |
| 5 | 12 | 2 | 9 | 11 | 13 |
| 6 | 11 | 1 | 10 | 12 | 14 |
| 7 | 10 | 4 | 9 | 11 | 15 |
| 8 | 9 | 3 | 10 | 12 | 16 |
| 9 | 8 | 1 | 5 | 7 | 14 |
| 10 | 7 | 2 | 6 | 8 | 13 |
| 11 | 6 | 3 | 5 | 7 | 16 |
| 12 | 5 | 4 | 6 | 8 | 15 |
| 13 | 4 | 1 | 3 | 5 | 10 |
| 14 | 3 | 2 | 4 | 6 | 9 |
| 15 | 2 | 1 | 3 | 7 | 12 |
| 16 | 1 | 2 | 4 | 8 | 11 |
Go on then, explain it...
A Hypercube (or Tesseract, or Measure Polytope) is a cube extended into four dimensions. It has sixteen vertices, 28 edges, eight cubic 'hyperfaces' and goodness-knows-how-many square faces (count them if you like; I can't be bothered).
If you don't believe that there are eight faces (it is quite hard to visualise, and it took me some time to find them all), go back up to the top, and look at the image up there. When you roll the cursor over the writing to the right, it should highlight the different faces. Bear in mind that most of them don't look cubic; that's a two dimensional representation of a four dimensional figure, so it's bound to look funny.
It is generated by, well, let's assume for a moment that we don't know what a cube is. In no dimensions, the only possible figure is a point. Get hold of an extra dimension from somewhere (don't ask where; if I knew that I would be selling the knowledge of how to walk 'through' walls right about now). Extend your point through a finite distance in that dimension.
It might be helpful if your distance was one unit, but I don't mind. If you're trying to be awkward, I'll just define a new unit equal to the length of your line segment. So anyway, now you have a unit [one unit long] line segment. Get another dimension, perpendicular to the one you already have.
Extending the line through a unit in this new direction gives a square. Get another dimension, perpendicular to both of the others. Extending in this new one gives a cube. Notice the way that on the diagram, the third dimension doesn't seem perpendicular. That's because you only have a two-dimensional display, and if I were to use a mutually orthogonal [perpendicular] direction it would have to go into or come out of your screen.
Now we have a problem. There isn't a dimension perpendicular to all three of the ones we have. Not in our space at least. But mathematicians care not for such problems, so suppose there is such a direction, and extend through one unit in it.
We now have an object that can't be drawn, can't be built, and is of no use to us whatsoever. Except perhaps for playing Tesserawalk...