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Topic: Names of polygons
Replies: 26   Last Post: Apr 29, 2008 12:08 PM

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Michael Keyton

Posts: 138
Registered: 12/3/04
Re: 11-gon
Posted: Oct 27, 1994 7:42 PM
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On Wed, 26 Oct 1994, David Masunaga wrote:

> 1. Doesn't the Susan B. Anthony $1 coin have an "inscribed" regular
> hendecagon on it and is not itself a regular hendecagon? Regular
> hendecagon coins include the Canadian "loonie" $1 coin as well as the
> Canadian penny.

Yes, inscribed regular hendecagon on the SBA coin. Are the Canadian coins
polygonal shaped or inscribed polygons in a circular coin? Does anyone
know why? Is it political, geographical, or just quirky? Not nr. of
provinces (10) plus territories (1). And how about Susan B. Anthony, what
is the significance of 11 for her? Or is it just someone being cute?
> 2. Maybe I'm just a naive high school geometry teacher, but I think that
> the non-straightedge/compass constructible regular polygons are very
> interesting. My recollection is that Gauss proved (at age 18?) that a
> regular n-gon can be straightedge/compass constructible if n is a prime
> Fermat number or the product of different prime Fermat numbers. I would
> think that this would then put the non-constructibles into an interesting
> class of numbers. This also leads to some amazing attempts to construct
> the regular 17-gon (see Sci Am., and Rouse-Ball/Coxeter "Mathematical
> Recreations and Essays"). Maybe my lack of background/sophistication in
> pure math research may account for such continual awe and wonder of this
> result linking number theory and geometry, but then again, as I recall,
> this result was also worthy of inclusion in his 1801 Disquisitiones
> Arithmeticae. This is a story which I relate with excitement to my high
> school geometry students. However, with the tenor of the e-mail postings
> regarding constructions, maybe it's not the big deal that I thought it
> was...

Gauss: Constructible iff N is a Fermat prime, a product of distinct
Fermat primes, or a power of two times one of the former. This is what
makes the regular 17-gon so important historically. The first
construction discovered after the Greeks. And what makes so many of the
others lacking in mathematics. Is there any property know about the
regular 11-gon, 13-gon, 19-gon, or any of the other non-constrctible
regular polygons other than it is not constructible with compass and
straghtedge? Froma density point of view, since there are only 5 known
Fermat primes and all indications are that there are no more; (but this is
not solved yet), there are really very few interesting regular polygons.
Some guy named Hermes of Lingen devoted 10 years of his life to trying to
construct the 65537-gon (the largest known Fermat prime) The 17-gon is
inscribed on the side of the monument in Braunschweig to Gauss, there was
a picture of it in Mathematical Intelligencer several years ago.

My argument is that if the only point of the exercise it to name the
objects (regular polygons), then it is not a mathematical problem but one
only of taxonomy or in this case philology. One should not exclude
necessarily these from a mathematics course, but I question treating them
as mathematical questions.

Michael Keyton

> > In provincial wonderment, > Dave Masunaga > > Iolani School
> 563 Kamoku St.
> Honolulu HI 96826
> (808)949-5355
> On Sun, 23 Oct 1994, Michael Keyton wrote:

> > An elderly acquaintance of mine who recently celebrated his 100th
> > birthday used the phrase "lexicographical heteromorphs" for these
> > linguistic abominations that use mixed roots. Undecagon and duodecagon
> > are such in that they unite Latin prefixes with Greek suffices.
> > A subsequent reply correctly gives hendecagon for 11 sides and dodecagon
> > for 12. Also the Susan B. Anthony $1 coin for the U.S. was in the shape
> > of a regular hendecagon. These still exist, but are rarely seen in
> > circulation.
> > Since the 13-gon, 14-gon, 18-gon, 19-gon are not constructable with
> > compass and straight edge, other than for linguistic curiosity, why would
> > one want to name them? However, what name does one give to the 24-gon,
> > the 48-gon, and the 96-gon; those polygons which were useful in the early
> > approximations of PI?
> >
> > Michael Keyton
> > St. Mark's School of Texas
> >
> >


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