Reaction to Chapter 5 of *Beyond
the Third Dimension*

Andy Miller

`Comments/Questions`

- The socio-historical aspects of the race to discover the existing four-dimensional regular polytopes in the 1880s was of most interest to me in this chapter. The question seemed to challenge mathematicians' in a competitive way. Instead of working in the name of mathematics, they worked to prove themselves superior, and, in the case of the one reputable mathematician who published incorrect results, did so hastily and slopily. Do certain subjects become particularly fashionable in contemporary mathematics in a similar way? Are some mathematicians motivated by the search for glory just as much as others are motivated to assist pure mathematical progress? I thought it comically ironic that the answer to the questions had been answered earlier and overlooked by other mathematicians. There seemed to exist little of a cooperative work ethic, including a reliance on previous research in a given subject.
- What was Abbott's mathematical background? Was he involved in or motivated by the contemporary interest in regular higher dimensional polytopes?
- Dali's use of the unfolded hypercube
intrigued me in his
*Corpus Hypercubicus*. His work brings to mind the association between higher dimensions and religion alluded to in*Flatland*. What is the symbolism of the woman below the "Cross" and the four rectangular boxes in front of the Christ figure?

`Exercise`s

- Using two geometric two-dimensional representations (i.e. pictures), show how the existence of a dodecahedron composed of pentagons could be possible, but that of a regular polyhedron composed of hexagons is impossible.
- Following the Greek method of regular polygon construction, using only a compass and a straightedge, construct a regular triangle, hexagon, or dodecahedron.

`Links`

- An autobiography of Dali's life and work
- Interesting math-based artwork
- Download a GIF file of one of H.S.M. Coxeter's regular polytopes
- Mail the author of this page