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Constructing a Regular Heptagon Using Lill's Method
This Demonstration shows how to construct a regular heptagon using Lill's method for solving cubic equations.
The points of a regular heptagon with vertices on a circle of radius 1 are given by
. Since
is a solution, if we divide the polynomial
by
, we get
.
If
, then
.
Substituting
leads to the cubic equation
.
It has solutions
,
,
.
This follows from the trigonometric identity
.
Set
and
to get
,
which factors as
.
There are solutions when the points
and
coincide.
Contributed by:
Izidor Hafner
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Reference
[1] G. E. Martin,
Geometric Constructions
, New York: Springer, 1998 p. 45.
RELATED LINKS
Heptagon
(
Wolfram
MathWorld
)
Theobald's Heptagon-to-Square Dissection
(
Wolfram Demonstrations Project
)
Mechanism for Constructing Regular Polygons
(
Wolfram Demonstrations Project
)
Lill's Method for Calculating the Value of a Cubic Polynomial
(
Wolfram Demonstrations Project
)
PERMANENT CITATION
Izidor Hafner
"
Constructing a Regular Heptagon Using Lill's Method
"
http://demonstrations.wolfram.com/ConstructingARegularHeptagonUsingLillsMethod/
Wolfram Demonstrations Project
Published: September 8, 2017
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