Every now and again I come across a mathematical image so powerful that it quite stops me in my tracks. So it was with this magnificent animation:
which shows a “tesseract” (a four dimensional hypercube), being rotated. I wrote to the creator of this astonishing image, Jason Hise, asking how he created it. He was kind enough to write back, and this is how it was done:
- First, all the vertices of the tesseract were placed at the coordinates
.
- Next, the vertices were rotated according to the rotation matrix
- After each rotation, the vertices were translated by
to make the
value positive.
- The vertices were projected into
using
.
For display, the vertices, edges, and “squares” were rendered using Maya and its scripting language MEL. (I know nothing about these.)
For me, this animation is a superb use of high powered computing for mathematical visualization.
Another one of Jason’s beautiful animations, this one with two simultaneous rotations, is
And here is another rotating tesseract, rendered differently:
There is a nice discussion and explanation about tesseract rotation, with very elegant diagrams, at http://traipse.com/hypercube/index.html.
Filed under: Visualization
Just seen your question about the 4×4 matrix problem on the WordPress.com Forums at http://en.forums.wordpress.com/topic.php?id=28520&replies=12#post-207589 and answered it there.
All it needed was a space to make the formula parse but I also suggested you use the pmatrix environment instead.
Most people find it difficult to visualize objects in three dimensions (architects and sculptors excepted) without something concrete. However, this rendition of a four dimensional unit “cube” is a good illustration only after one first understands how to visualize it! I like to think this way. If you imagine a unit cube (3D) and visualize it moving along a unit interval (from 0 to 1 along a line) then it traces the tesseract. (The cube is traced if a square is moved the same way.) Just like in a 2D rendition of the cube, some square faces becomes distorted by perspective, some “faces” (which are cubes) of the tesseract are distorted in a 2D/3D rendition of the tesseract. Also, for the cube, if you hold one dimension constant (take a “cross-section”, or equivalently, delete that dimension by reducing it to a point) you get a square; so in 4D, with the tesseract, if you hold one dimension constant, you get a cube. The tesseract can be visualized as a cube moving along any remaining dimension.
Note, however, in the 2D picture on http://traipse.com/hypercube/index.html, what is pictured looked like a 3D object, not a 4D object; the reason being that the red cube represents all the cubes in between 0 and 1; so it should not be interpreted as stationary, in the middle of the tesseract. In fact, the red cube also represents all the cubes in between 0 and 1 along any dimension! It “becomes” one of the “faces” cubes as it “reaches” 0 or 1 (or front/back, left/right, top/bottom). That is exactly what the animations show. The “rotations” are really viewing the tesseract along a particular dimension with the red cube tracing (recalling) the movement in its formation.
It won’t be difficult to imagine the five-dimension hypercube now. Easy quiz: What is the one-dimension “cube” and how is the two-dimension “cube” obtained from the one-dimension “cube”? Is there a zero-dimension “cube”?
i love animations on computer because i can send them, put them for my desktop, make them as my icon or just have lots of fun with it! Thanks for this Cool Site! ITS GREAT! please if you have animations send me them on sweet.nalie@hotmail.com thanks!
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