Cubes PageMarkus Schmidmeier |
Suppose someone is dropping a 4D-cube through our beautiful three dimensional world. The problem is that (unless you study this page) you may not even notice that they dropped their 4D-cube. Because the 4D-cube does not exactly look like a cube if it's dropped vertex first into three space. The dropped 4D-cube
Here is how it looks like. For illustration, I use zometool models.
First comes the vertex. Attached to the vertex are 4 edges (since we are in 4D), they appear next. That is to say, what we see are four points on each of the four edges, all four have the same distance from the vertex. And they have the same distance from each other. In three space, we can see the lines connecting those four points. Since they are all the same length, they form a regular tetrahedron.
You can see the tetrahedron in green. The vertex on the left is the one which came first. It is the starting point of the four edges which connect it with the vertices of the tetrahedron. Three of the edges are in blue, one is in yellow. Note that the tetrahedron is exactly as we see it. But what is in blue and in yellow is the shadow of the part of the 4D-cube which has already passed through our three space. As usual with shadows, it is distorted a bit. What is unusual is that it's a three dimensional shadow of a four dimensional object.
When the end points of the edges attached to the first vertex appear then we are one fourth done. Each of those end points is connected (in four space) with three more edges. They appear as the cube moves on to the left. Thus each of the vertices of the tetrahedron branches out into a triangle --- we are dealing with a truncated tetrahedron!
In the picture the truncated tetrahedron is in green. In the part on the left of it, you can see the four edges which came through first and attached to each three shorter edges (in blue and in yellow) which connect to the vertices of the truncated tetrahedron.
Note that the truncated part keeps growing as the cube moves on, so one would think that the green solid gets smaller. But strangely, it keeps growing! Until we reach the end points of the twelve edges that are second to the initial vertex. Then we are half way done. Fun fact: Now the solid with the green edges is four times as big as the largest tetrahedron.
It keeps getting stranger. At the midpoint, there is still no cube in sight. If you take a tetrahedron of edge length 2 and you cut off the corners and with each corner parts of the edges of length one, then what you are left with is a regular octahedron. It is in green because this is what we see in three space. In blue and yellow we see 16 of the 32 edges of the four dimensional cube. This is the shadow of the part of the cube which has already passed through three space.
What comes next? Well, the part in three space must be something we have seen before. Because we can reverse the process of moving the four cube through three space. So we are dealing with the truncated tetrahedron. Here it is.
The truncated tetrahedron is in green, as we see it. Above it is the shadow of the four cube. We see the initial vertex on the left, then the four attached edges, three in blue, one in yellow, then for each of those edges the three neighboring edges which are completely visible. Their end points meet in pairs, from each intersection two more edges emanate. In the picture they are shorter because the four cube is between one half and three quarters on the ``left'' of our three space. The right hand side end points of the the short edges are exactly the vertices of the truncated tetrahedron.
Now you can guess how the four cube will complete his trip through our three space.
To get back to the question from the beginning: How do I notice if someone drops their 4D-cube through our three space? Remeber the sequence of figures:
vertex - growing tetrahedron - truncated tetrahedron - octahedron - truncated tetrahedron - shrinking tetrahedron - vertex
Isn't it strange that during the whole process we never saw a cube or even a square? But then, if we go one dimension lower, we should not have expected that.
Because if you take a sugar cube and dissolve it, vertex first, in coffee, then the surface at the bottom is first a triangle, then a hexagon, then again a triangle.
In the picture, the sugar cube is in blue, but the surface of the coffee is green. Going around clockwise, you see how the sugar cube dissolves in the coffee. Well, they really should produce some green lines in black!
Okay, so much for now. Here is your homework problem. Suppose somebody bites off part of the four dimensional cube. This is how it looks like. You see, biting off a corner of a square results in a missing triangle. Biting off a corner from a cube gives a missing tetrahedron. On the right hand side of the picture, you see what is left over when a corner is biten off a four cube --- of course, this is only the shadow of a four dimensional object.
How does it look like if this object is dropped through our tree dimensional world? Would you recognize it?
Last modified: by Markus Schmidmeier |