Art Meets Mathematics in the Fourth Dimension by Stephen Leon Lipscomb

Author:Stephen Leon Lipscomb [Lipscomb, Stephen Leon]
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham
Published: 2015-10-10T19:25:35+00:00

or, upon substitution,

A rearrangement of terms then yields

We now let , and make a substitution to obtain

Continuing, since each term is squared we also have

Upon squaring each expression and then recognizing the dot product

for example, we arrive at a classical 2nd-degree equation with variable t:

where A = m ⋅ m, B = −2m ⋅ b, and C = b ⋅ b − r 2.

It is well known that such equations have at most two solutions, i.e., there are at most two values of t that provide a solution. So the edge [p, q] contains at most two points a t that are also points of our 3-sphere.

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