So, take a tesseract and place its vertex at the origin of 4 axes. Because you can break up a 4D space into 4 3d subspaces.

That's where 4D is vastly different from 3D. Stacking cubes doesn't fill up a 4D space. But stacking 3d things in 4D, apparently, can make up a 4d object?A cube is flat in 4D, it has zero length on the W axis. No matter how many planes we stack on top of each other in 3D, they still will amount to 0 thickness. This is entirely different from planes in 3D. quickfur Pentonian Posts: 2873 Joined: Thu 11:20 pm Location: The Great White North This is about as helpful in understanding what a cube is, as a 2D image of a 3D projection of a 4D tesseract.

When we draw a diagram of the cube, we may draw something like this (say on a piece of paper):īut, seen from the side, it looks like this: A 2D image of a 3D projection of a 4D object is about as helpful in visualizing 4D as a 1D projection of a 2D projection of a 3D cube is in visualizing 3D. It's the same as looking edge-on at the drawing of a cube.Īnd the blame partly rests on me, for not making it clear that when we look at a (2D image of a) 3D projection of a 4D object, we aren't seeing the projection from a 'native' 4D point of view at all, but from the side, as it were. There's no way to directly see the 4-volume when looking at a projection of a 4D object, since your POV is in the 3-plane of the projection. Don't think that you are somehow seeing in 4D when you look at a projection of a 4D object. Ovo wrote: I believe you have been partly misled by the commonly made mistake/missing step in the 3D->2D analogy that I have pointed out here. 4Dspace Trionian Posts: 103 Joined: Tue 2:51 pm Location: USA So, in this particular (and unique, lol) case where the edge = 8 units, the 8 3-volumes = 1 4-volume. Since we have 8 of them, we multiply 512 by 8 and get 4096 as well. And 8 cubed = 512 = the 3-volume of one of the cubes. If we cheat a bit and take 8 equal cubes with edge=8 units, then 8 to the 4th = 4096 and this is the 4-volume of a tesseract with edge = 8units. But stacking 3d things in 4D, apparently, can make up a 4d object?įor example. How can it be? Only because they are arranged in 4D? 5 3d things, make up one simplex, a 4d thing. In any rate, the opposite pairs of cubes (right-most and left-most, or bottom and top) do not intersect each other and neither are adjacent to each other.Īnd so, the other very interesting thing to notice about the tesseract is that this 4d object apparently consists of 8 3d cubes. Their intersection on the projection reflects the fact that one hyperplane is in front of another to our POV. So, what we see are 2 hyperplanes, each consisting of 4 cubes. In fact, they are adjacent to each other. And while on this projection the 2 sets intersect each other, the 4 cubes that make up each set do not intersect with each other in 3D. Note that there are 2 sets of 4 cubes each: one set is seen if you follow the blue central edges and the other set is seen if you follow red. From which cube you can get into which and in what sequence.Īlso I was looking at this stereo projection which ones are adjacent to each other is interesting. It is their relationship to each other, i.e. And so, from inside the 3-space, the cubes look like regular cubes. Note the small model with 8 colored cubes in the upper right of the screen. You can explore the tesseract from inside. Doing projections onto 3D can be misleading as well. This forum shows that analogies do not actually help people visualize 4D. I believe this is a superior method than using the analogies. To this effect I bumped into a paper today describing unfolding of the tesseract: So, the task becomes to arrange the 8 cubes in such a way that they make up a tesseract. Then, a cube made of 8 cubes (2 per edge) becomes interesting too, because a tesseract is made of 8 cubes. This may take some time, but once visualizing of a cube is accomplished, one can see a cube made of, say 27 cubes (3 per edge), which I described on one of the current threads. But not completely flat, like a square is flat. The surprising thing one notices right away is that the 3d objects appear rather flat in 4D. To really appreciate 4D we have to start with our familiar 3d objects, like a cube, a sphere, a tetrahedron. I think that trying to visualize 4D starting with tesseract or 3-sphere is rather hard.

4d objects visualization