This is a 4-dimensional analog of a 3-dimensional square--a tesseract--projected from the 4th to the 3rd dimension.

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This is a 4-dimensional analog of a 3-dimensional square--a tesseract--projected from the 4th to the 3rd dimension.

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this is the same thing a dimension higher. if you knew 4d as normal, those edges would all be fixed length and its just a rotating shape.

but its interesting because of the two "cubes" with connected edges and the whole multiple axees of rotation thing...

Don`t worry, that just means you`re normal. ;)

The real headspin comes when you appreciate how differently a 4-dimensional being would be able to see those same 3D objects that surround us, as well as a 4D object like the tesseract in this video. They`d see in true 3D, and would consequently be able to see all six sides of the 3D cubes that make up each `surface` of the tesseract simultaneously. They`d also be able to see inside each constituent cube at the same time. This is analogous to how we can easily see all four sides of a 2D square and its interior too, because we`re looking at that 2D object in its entirety from the enhanced perspective of a third dimension. A 2D being would by contrast be able to see at most two sides of a square at a single time, with the interior of it hidden from them completely.

We see in 2D, because we can only ever see one two-dimensional `window` of the 3D objects that surround us at any one time. We just extrapolate from the fact that we can see different sides of objects with consistently-predictable features that we`re seeing a single two-dimensional aspect of each three-dimensional object at a time as we move through the spacetime we and those 3D objects share.

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You`re right of course. This video, as with everything seen by the human eye, is really two-dimensional (it`s projected onto a flat computer screen). It has 3-dimensional cues that help us to understand implied depth, but in reality it`s completely flat. However, the thing to realise is, the world as we perceive it is *always* 2-dimensional, even though we intuitively understand what we see around us to be 3-dimensional space, and even though binocular eyes have certain amount of `depth perception` out to about 30 feet.

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This video explains more about how considering how 2D relates to 3D, helps understand how 3D relates to 4D.

Kind of. Not a plane, though, but a 3-dimensional object: every `surface` of a tesseract is a 3D cube.

A 3D cube is comprised of six 2D squares, all of which can be fitted into a single square in 2D space (look down on a cube directly from above, and you`ll see a square: four of the edges will be side-on to you in 3D, and two - the top and the bottom faces - will be one in front of the other). Similarly, a 4D tesseract is comprised of eight 3D cubes superimposed onto the same 3-dimensional space, taking up the same position in 3D, but shifted along a perpendicular axis into a fourth dimension we can`t observe directly. 3D objects can be turned "inside out" by shifting them through a fourth dimension, and pass through one another in 3D space.

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If you rotate the cube in the picture, the 2D projection changes, if you rotate the tesseract, the 3D projection changes.

It`s nice to be able to relate lower dimensions to higher ones.

It`s the fourth *spatial* dimension that the Tesseract is rotating in, not a temporal (time) dimension. Although time is sometimes referred to as "the fourth dimension", that`s not what Mathematicians are referring to when they talk of higher spatial dimensions. There`s an excellent Arthur C Clarke short story entitled "Technical Error" that talks about the difference between the two concepts.

and fun in my drink!

Technically, a hypercube is a cube extended into any number of dimensions higher than three, such as this demonstration of extensions up to the sixth dimension shows. A Tesseract is specifically a 3D cube (or 2D square) extended into the fourth spatial dimension by extending every edge of the 3D cube out at 90 degrees to the up/down, left/right and forwards/backwards dimensions we`re familiar with. Just like when you project a cube onto a 2D surface like a piece of paper and you see two squares connected by lines:

So too can you project a 4D Tesseract onto 3 dimensional space. When you do, you see a series of cubes that appear to warp and turn inside out as the object rotates in 4D space. This is what the video shows.

A four-dimensional representation of a cube (such as this is) is caleed a Tesseract. A tesseract is to the cube as the cube is to the square.

A tesseract is always a hypercube, but a hypercube is not always a tesseract. It could also be a penteract (5 dimensional) or a hexeract (6 dimensional) or on up to infinity.

Before the 80`s, all I had was some words on paper, describing a hypercube. It was hard to imagine it.

I don`t like looking at it.

It makes me feel like one of the apes from `2001:A Space Odyssey` encountering an obelisk

http://youtu.be/XjsgoXvnStY

- This is a 4-dimensional analog of a 3-dimensional square--a tesseract--projected from the 4th to the 3rd dimension.