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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1. Draw a circle with center and radius .

2. Draw an equilateral triangle with on .

3. Let be the midpoint of .

4. Construct a point on so that .

5. Construct a point on so that .

6. Successively measure out points on at distance starting with .

Verification

Suppose . Then

.

Since ,

.

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Contributed by: Izidor Hafner (September 2017)
Open content licensed under CC BY-NC-SA


Snapshots


Details

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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