Reaction to Chapter 8 of Beyond
the Third Dimension
- I am awed to see how often beautiful mathematical formulas
describe natural phenomena, as is the case with the appearance of the golden
ratio in many organic growth patterns (p. 169) and the feature of quaternions
that describe pendulum movement (p. 176). Are there any general explations
or theories for these interesting correlation? I might guess that the formulas
describe the physical path of least resistance, or some maximization of
evolutionary development. I'm very curious.
- I have heard of a commercial computer graphic rendering
program that produces three-dimensional shapes by an input of formulas,
rather than a customary, user-friendly, point-and-click program that modifies
preprogrammed shapes, such as 3D StudioTM.
To what extent can or do mathematicians use purely symbolic formulas on
computers to generate desired shapes, such as a distorted sphere for example,
for artistic, industrial, or commercial purposes?
- I'm surpised that pre-17th century mathematicians sparingly
used symbolic algebra (p. 157-8). How did they record theorems and explicit
rules in a synthetic manner? Although algebraic forms may have removed
some of beauty of geometric representations, synthetic forms seem more
cumbersome, imprecise, and inefficient.
- Is there some known evolutionary advantage for plants
and animals to employ the golden
ratio in their physical structure and growth? (p. 169)
Prof. Banchoff's Response.