Constructing a Regular Heptagon Using Plemelj's Method
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This Demonstration shows Plemelj's method for constructing a regular heptagon, using the following steps:
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Contributed by: Izidor Hafner (September 2017)
Open content licensed under CC BY-NC-SA
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This method for constructing a regular heptagon using angle trisection was found by Plemelj in 1892 and published in 1912. The construction is taken from [1, pp. 183–184]. The approximation for was known to Abûl-Wefâ and Heron of Alexandria [1, p. 184].
Start with the trigonometric identity
.
Since
,
replace by and by in the trigonometric identity to get the cubic
.
Then
.
Since and , take .
Substitute to get
.
Substitute to get the Vieta form of the equation,
.
Set
to get the positive solutions
,
,
.
So
,
where
.
Then
.
Reference
[1] G. E. Martin, Geometric Constructions, New York: Springer, 1998.
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