Newsgroups: sci.math
From: heyes@troy.win-uk.net (Lindsay HEYES)
Date: Wed, 23 Oct 1996 23:13:35 GMT
Subject: PARADOXICAL CONSTRUCTION OF A HEPTAGON ?
Wantzel proved (from Gauss) that a regular heptagon could
not be constructed through Euclidean Geometry because the
roots of x^n-1 = 0 could only be solved if
n = 2^a.F(0)^b(0).F(1)^b(1).F(2)^b(2).F(3)^b(3).F(4)^b(4)
where 0 < a < oo and b = 0 or b = 1, and where F(x) is a
number of the type 2^2^x + 1 (i.e. a Fermat Prime).
T.R Dawson demonstrated that every point that can be
constructed with a ruler and compasses, and no other
points, can be constructed with identical matchsticks
(i.e. with identical, movable straight-line segments).
Ref: T R DAWSON Match-stick Geometry, Mathematical Gazette,
No 254, 1939.
However:
It is possible to construct an angle of pi/7 by using 7
toothpicks (or matchsticks), after which the regular
heptagon is easily constructed.
Ref: C JOHNSON, A Construction for a Regular Heptagon,
Mathematical Gazette, No. 407, 1975.
How so, and can anyone send me copies of the original
references ?
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% Lindsay Heyes %
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