The hypercube: Exploring the fourth dimension, and further: "With apologies to those who know all about the hypercube, this is written for those who don't, to indicate the interesting shapes you can get in 3D by extending 3D mathematics into an imaginary fourth dimension. I first learnt of this from Prof. A K Dewdney's 'Computer Recreations' column in the April 1986 issue of Scientific American.
Let us imagine that another Big Bang created a 2D universe and we are 2D critters living in a plane world. We come to appreciate our position in relation to x and y axes at right angles, can perceive points, lines, squares and circles and can do Pythagoras. Our 2D mathematicians (2DMs) can rotate a point about the origin with two equations looking like 'SUB rot' in the QBasic code in Listing 1. As it rotates, the new x and y co-ordinates each depend on the old x and y, and the angle of rotation. We might consider that the simplest square has its corners at the four points (x,y) = (�1, �1)
Figure 1 Hypercube with one cube smaller (Hypertrapezoid?)
Figure 2 Hypertrapezoid after several real and unreal rotations"
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