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This video and this image show a tesseract, which is a 4d cube:


In what sense is this cube 4 dimensional? Where is time? (commonly called the 4th dimension, although I realize here its probably some sort of direction).

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You are "seeing" a two dimensional model of a three dimensional model of a 4-dimensional figure. In four dimensions, any two edges meet at right angles; it's of course impossible to picture such an object in 3-dimensions, just like it is impossible to draw a cube in 2 dimensions in such a way that any two edges meet at right angles. You are seeing a projection of a projction. – Arturo Magidin Aug 3 '11 at 21:19
Cube? Just looks like a bunch of 2D squares to me... ;) – anon Aug 3 '11 at 21:20
In addition: mathematically, "time" is not "the fourth dimension"; "space time" is a particular physical model, but in mathematics, 4-dimensional Euclidean space is simply the set of all ordered 4-tuples $(a,b,c,d)$ with $a,b,c,d$ real numbers, satisfying certain axioms. Time doesn't enter into it. – Arturo Magidin Aug 3 '11 at 21:20
If you looked at a still holographic projection of a 1x1x1 in^3 cube for 1 second, and identified 1 in = 1 sec, then you'd be seeing a tesseract using time as a "dimension." Although only one 3D cross-section of it at any given moment... – anon Aug 3 '11 at 21:30
Nice animation! Who made it? – TonyK Aug 3 '11 at 21:30

The tesseract is the four dimensional analog of the cube. It lives in $\mathbb{R}^4$, four dimensional euclidean space. This four dimensional space has all dimensions equivalent, with none of them being special like time. The space is the set of points $(x,y,z,w)$ where the coordinates range over the reals. One of the tesseracts has $16$ vertices, with all combinations of $\pm 1$.

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You picture displays a 3-D model of a 4-D object. Time is not one of the dimensions in this case.

Wikipedia has a nice description:

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Or rather, a 2-D picture of a 3-D model of a 4-D object.... – Arturo Magidin Aug 3 '11 at 21:21

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