Reaction to Chapter 5 of Beyond
the Third Dimension
- 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?
- 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.
Prof. Banchoff's Response