Reaction to Chapter 8 of *Beyond
the Third Dimension*

A`ndrew
`M`iller`

**Comments**:

- 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 Studio
^{TM}. 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?

**Questions**:

- 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)

**Link**:

Prof. Banchoff's Response.