From: dtd@world.std.com
Subject: Re: Ram horn spiral
Date: Tue, 11 May 1999 15:24:12 GMT
Newsgroups: sci.math

Hop wrote in message <37337E08.5DDC82F1@tabletoptelephone.com>...

> I have made a cardboard model that I call a ram's horn. ...
> http://www.tabletoptelephone.com/~hopspage/crdmdls/ramshorn.html

you did a beautiful job; it's very nice handiwork.
i wish your page had printable patterns for the
three pieces, so that my daughter could cut it out
and assemble it herself.  i know your page describes
how you did it, but i don't have the time to recon-
struct the pattern from your description.

fwiw, the way a real ram's horn grows, is that the
head's growth patches are roughly triangular, and the
three vertices grow horn at three different rates.
the growth rate varies more-or-less smoothly across
the triangular patch.  you can think of the resulting
horn as being either:

  * three nestled, similar spirals, of three different
    arc-lengths, but sharing an endpoint.

  or,

  * a stack of thin triangular plates of horn, all
    exactly similar in shape, but tapering linearly
    in size.  each plate's thickness varies, so that
    its three corners have three different heights.

note that a real horn's growth patches and growth-
plates are bumpy, not flat, but this doesn't prevent
the "flat patch" model from accurately reproducing
the real horn's shape.

if the the three corners' growth rates are nearly
the same, then the horn grows as a long tight spiral,
like certain antelopes' horns.  otoh, i think the
chamois' horns' hooked shape result when the corners'
rates are very different in the animal's youth, but
become more similar at maturity.

d'arcy wentworth-thompson's book, "on growth & form"
has an interesting discussion of the ram's horn's
growth.  the book also discusses many other math-
ematical features of biological growth.  the book is
available in a nice abridged edition from cambridge u.
press, and in a monumental unabridged dover paperback
of maybe 800 pages.

                                        - don davis
                                          boston