## Kenny Liu

Chapter 6 demonstrated effectively the limitations of writing a book on the fourth dimension. While words help to convey an idea of what is going on (or should be going on) and pictures try to provide an image for these ideas, there is nothing like the animation and 3-d models that the book hinted at to understand what goes on in 4-d. Hopefully in the future a new edition of Beyond the Third Dimension can be published that would incorporate both of the elements mentioned above, or better yet a combination - a book with animated 3-d models. Ah the future!

There are several ways the the 52nd edition B3D could have helped me out. For example the Schlegel diagrams for regular polyhedra on page 117 would be much more effective if the actual polyhedra were present to offer up their perspective. For some reason the Schlegel diagrams for tetrahedron, hexahedron, and dodecahedron were immediately apparent to me but the octahedron and icosahedron took some working out on scratch paper. The main confusion I had with these diagrams were that for a cube and tetrahedron you can easily look at it and see one face. For the other three it is necessary to greatly distort perspective such that one face "envelopes" the rest of it. Normally if you had one of the greater faced regular polyhedra lying in your hand you could not position it such that you could only see one face at a time - which has to do with their obtuse angles. However the pentagons which made up the dodecahedron seemed to simplify the structure such that I didn't have a problem "pulling" it out in my mind into 3 dimensions. For some reason triangles are harder for me.

In the same way, the Shlegel polyhedra of the 5, 16, and 24 cell were all confusing to me (damn triangles!). However, the central projection of the hypercube is much more comprehensible, especially when animated so that it moves inside out.

Another way a holographic B3D would be helpful would be to show how the 4 by 4 grid on page 123 results in a polyhedral torus in the hypercube. I couldn't visualize that too well.

Finally, a question. If the polyhedral torus in the hypercube is equivalent to the Clifford torus in the hypersphere could a hypersphere simply be constructed by capping off the Clifford torus with a hemisphere on either end of the outer circumference and the inner circumference?

Prof. Banchoff's Response