Prashantkrishnan wrote:Even by all the methods that we try to visualise the fourth dimension, will our brains actually adapt to it? It has been an year and a half since I started learning the concepts about the fourth dimension, but I managed to register here only tonight. I am nowhere near visualising a duocylinder or a glome - these are the most difficult rotatopes for me.
I couldn't visualize a duocylinder for a long time -- the mathematical descriptions of it were not helpful to me -- until I "accidentally" re-discovered it as the limiting shape of duoprisms. Of course, I had a hard time visualizing duoprisms until I "accidentally" re-discovered them by noticing certain features about the tesseract.
Consider the following usual projection of a tesseract into 3D space:
There are 8 cells depicted here: the outer cube, the inner cube, and 6 flat frustums that connect the outer cube to the inner cube. Of course, in 4D these are actually all regular cubes; the frustums only appear that way because of perspective distortion (just like looking at a square edge-on makes it appear as a thin trapezoid).
First, notice that 4 of the flat frustums form a 4-membered ring: take the front, back, left, and right frustums, and you see that they are cubes joined to each other by their opposite faces, and bent around the vertical axis to form a ring. The remaining cells then, form a second ring: the inner cube is connected to the top frustum, the top frustum is connected to the outer cube, the outer cube is connected to the bottom frustum, and the bottom frustum connects back to the inner cube. The trick here is to notice that this second ring is actually
exactly the same as the first ring: each cube is connected to its neighbours by opposite faces. Of course, here the ring appears to be very different, but that's actually just an artifact of the projection. In 4D space, these two rings of cubes are actually exactly the same shape. Thus, the tesseract's surface can be cut into two rings of 4 cubes each.
Now begins the fun part. Suppose we ignore the second ring for the moment, and look at the first (horizontal) ring. That is, the front/back, left/right frustums. What happens if we insert a 5th cube into this ring? It would become a ring of 5 cubes (which would appear, in projection, as 5 frustums), which we can adjust the angles of, so that they form a pentagonal ring. The original second ring would no longer fit in with this ring, of course, because now the "hole" in this ring is pentagonal, whereas the cells of the second ring are cubes. But a simple modification makes the two rings fit together once more: by changing the cubes of the second ring into pentagonal prisms! So then we now have the 1st ring consisting of 5 cubes, and the second ring consisting of 4 pentagonal prisms, and these two rings will now close up in 4D to make ... the 5,4-duoprism.
Now suppose instead of having just 4 pentagonal prisms in the second ring, we have 6 pentagonal prisms? Well, then the 5 cubes of the first ring would no longer fit into the second ring. But again, another simple modification fixes this: just replace the 5 cubes with 5 hexagonal prisms. Thus, we have a ring of 5 hexagonal prisms and a second ring of 6 pentagonal prisms. These rings will now close up to form a closed 4D shape, that we call a 5,6-duoprism:
You probably noticed a pattern here. Given a ring of n-gonal prisms and another ring of m-gonal prisms, if there are m prisms in the first ring and n prisms in the second ring, they will close up into an m,n-duoprism. This construction works for any m≥3 and n≥3. It just so happens that when m=n=4, we get a tesseract. For all other values, we get various duoprisms.
Of course, it may not be immediately obvious that the above construction of duoprisms is equivalent to taking the Cartesian product of two polygons, but some careful study of the matter, which I'll leave as an exercise for the reader, should settle this.
So now we know how to visualize duoprisms. Let's take things one step further. Nobody says m and n are restricted to finite numbers. For example, we can construct an m,∞-duoprism by constructing m,n-duoprisms for larger and larger values of n, and then take the limit as n approaches infinity. So we get the following sequence of duoprisms (taking m=4 as an example case):
...
As n approaches infinity, then, the n-gonal prisms in the vertical ring of the m,n-duoprisms become closer and closer to cylinders. At the same time, the cubes of the horizontal ring become flatter and flatter -- at the limit, then, the vertical ring becomes a ring of 4 cylinders, and the cubes of the horizontal ring become a
torus with a square cross-section. This 4,∞-duoprism, it turns out, is none other than the cubinder:
But we aren't restricted to m=4. The following, for example, shows the case of m=6 (with a slightly rotated 4D viewpoint so that the vertical/horizontal rings are interchanged, but this is just a simple rotation in 4D):
These fascinating new shapes that have cylinders as cells are none other than the Cartesian products of a circle with various polygons.
By now, it should be obvious that there's no reason we restrict m to finite values; let's see what happens when m also approaches infinity:
...
As you can see, as m increases, the cross-section of the torus becomes a higher and higher polygon that approximate a circle; while the cylinders in the ring of cylinders become flatter and flatter, and approach an infinitely flat disk. At the limit, an interesting thing happens: the torus becomes circular torus, and the ring of cylinders also become a circular torus. The two rings of the m,n,duoprisms have therefore converged into two interlocking tori:
This, we call a duocylinder.
It's none other than the Cartesian product of two circles. The two tori are identical to each other; they only appear different in the above image because they're being seen from different angles in 4D. A simple 4D rotation would transform them into each other.