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Hypercubed?

Posted Friday, April 01, 2005 4/01/2005 07:20:00 PM

Some of you (is anyone actually there?) may be wondering what it is to be Hypercubed. Well Hypercubed is the inflected verb meaning to form into a Hypercube. According to Merriam-Webster a Hypercube is "a geometric figure... in Euclidean space of n dimensions that is analogous to a cube in three dimensions" (see reference). Notice it is a geometric figure in space of

What is a hypercube (tesseract) look like? A line is a one (1) dimensional object. Now imagine if you were to make a duplicate of this line and then connect these two lines by more lines of the same length. This would be a two (2) dimensional square. Now take this square and connect it to a duplicate square using more duplicate squares. You now have two squares connected by four squares, six faces in all. This is a three (3) dimensional cube (look here). Now the hard part; imagine two cubes where each face of the first cube is connected to one face of the other cube by a duplicate cube. This is difficult to imagine because we live and think in three dimensions. But mathematically there is no difference between space with three dimensions and space with four dimensions. Well obviously, due to limitations of 3-d space, we cannot see a four dimensional object. However, using the same techniques that one uses when drawing a 3 dimensional cube on a two dimensional piece of paper we can project an image of a four dimensional hypercube (or the wire frame of one) into three dimensions. In 2000 I posted an article on my website explaining how to use POV-RAY to display four dimensional objects here. The image below is a projection of a hypercube made with POV-RAY.

There are other ways to view a hypercube. We all know how to unfold a 3D cube; it is also possible to unfold a hypercube. Check out Salvador Dali's painting "Crucifixion" (aka "Corpus Hypercubus").

**n**dimensions. Therefore Hypercubed means to be formed into a n-dimensional equivalent of a cube. However, when people mention a hypercube they most commonly are referring to the tesseract, the four-dimensional version of a cube. Doesn't Hypercubed sounds better then tesseracted?What is a hypercube (tesseract) look like? A line is a one (1) dimensional object. Now imagine if you were to make a duplicate of this line and then connect these two lines by more lines of the same length. This would be a two (2) dimensional square. Now take this square and connect it to a duplicate square using more duplicate squares. You now have two squares connected by four squares, six faces in all. This is a three (3) dimensional cube (look here). Now the hard part; imagine two cubes where each face of the first cube is connected to one face of the other cube by a duplicate cube. This is difficult to imagine because we live and think in three dimensions. But mathematically there is no difference between space with three dimensions and space with four dimensions. Well obviously, due to limitations of 3-d space, we cannot see a four dimensional object. However, using the same techniques that one uses when drawing a 3 dimensional cube on a two dimensional piece of paper we can project an image of a four dimensional hypercube (or the wire frame of one) into three dimensions. In 2000 I posted an article on my website explaining how to use POV-RAY to display four dimensional objects here. The image below is a projection of a hypercube made with POV-RAY.

There are other ways to view a hypercube. We all know how to unfold a 3D cube; it is also possible to unfold a hypercube. Check out Salvador Dali's painting "Crucifixion" (aka "Corpus Hypercubus").

**Related Links**

- Hypercube - Visualization
- Hypercube's Hompage - What are Hypercubes?
- Hypercubed - Wolfram Research
- Four-Space Visualization of 4D Objects - Masters Degree Thesis by Steven Richard Hollasch

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