Harvard University,FAS
Fall 2003

Mathematics Math21b
Fall 2003

Linear Algebra and Differential Equations

Course Head: Oliver knill
Office: SciCtr 434
Email: knill@math.harvard.edu

The Tesseract

(Visualization of a higher dimensional object)
The tesseract is a four dimensional cube. It has 16 edge points v=(a,b,c,d), with a,b,c,d either equal to +1 or -1. Two points are connected, if their distance is 2. Given a projection P(x,y,z,w)=(x,y,z) from four dimensional space to three dimensional space, we can visualize the cube as an object in familar space. The effect of a linear transformation like a rotation
         |   1  0    0    0      | 
R(t) =   |   0  1    0    0      |
         |   0  0  cos(t) sin(t) |
         |   0  0 -sin(t) cos(t) |
in 4d space can be visualized in 3D by viewing the points v(t) = P R(t) v in R3.


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