Equilateral pentagon

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Equilateral pentagon" – news · newspapers · books · scholar · JSTOR (August 2014) (Learn how and when to remove this message)

In

Four intersecting equal circles arranged in a closed chain are sufficient to determine a convex equilateral pentagon. Each circle's center is one of four vertices of the pentagon. The remaining vertex is determined by one of the intersection points of the first and the last circle of the chain.

Examples

Simple				Collinear edges	Complex polygon
Convex		Concave
internal angles )	Adjacent right angles (60° 150° 90° 90° 150°)	Reflexed regular pentagon (36° 252° 36° 108° 108°)	Dodecagonal versatile[1] (30° 210° 60° 90° 150°)	Degenerate into trapezoid (120° 120° 60° 180° 60°)	Regular star pentagram (36°)	Intersecting (36° 108° −36° −36° 108°)
				Degenerate into triangle (≈28.07° 180° ≈75.96° ≈75.96° 180°)	Self-intersecting	Degenerate (edge-vertex overlap)

Internal angles of a convex equilateral pentagon

When a convex equilateral pentagon is dissected into triangles, two of them appear as

${\displaystyle \alpha }$

${\displaystyle \beta }$

According to the law of sines the length of the line dividing the green and blue triangles is:

${\displaystyle a=2\sin \left({\frac {\beta }{2}}\right).}$

The square of the length of the line dividing the orange and green triangles is:

${\displaystyle {\begin{aligned}b^{2}&=1+a^{2}-2(1)(a)\cos \left(\alpha -{\frac {\pi }{2}}+{\frac {\beta }{2}}\right)\\&=1+4\sin ^{2}\left({\frac {\beta }{2}}\right)-4\sin \left({\frac {\beta }{2}}\right)\sin \left(\alpha +{\frac {\beta }{2}}\right).\\\end{aligned}}}$

According to the

${\displaystyle \cos(\delta )={\frac {1^{2}+1^{2}-b^{2}}{2(1)(1)}}\ .}$

Simplifying, δ is obtained as function of α and β:

${\displaystyle \delta =\arccos \left[\cos(\alpha )+\cos(\beta )-\cos(\alpha +\beta )-{\frac {1}{2}}\right].}$

The remaining angles of the pentagon can be found geometrically: The remaining angles of the orange and blue triangles are readily found by noting that two angles of an isosceles triangle are equal while all three angles sum to 180°. Then ${\displaystyle \epsilon ,\gamma ,}$ and the two remaining angles of the green triangle can be found from four equations stating that the sum of the angles of the pentagon is 540°, the sum of the angles of the green triangle is 180°, the angle ${\displaystyle \gamma }$ is the sum of its three components, and the angle ${\displaystyle \epsilon }$ is the sum of its two components.

A

There are two infinite families of equilateral convex

Equilateral pentagons can intersect themselves either not at all, once, twice, or five times. The ones that don't intersect themselves are called simple, and they can be classified as either convex or concave. We here use the term "stellated" to refer to the ones that intersect themselves either twice or five times. We rule out, in this section, the equilateral pentagons that intersect themselves precisely once.

Given that we rule out the pentagons that intersect themselves once, we can plot the rest as a function of two variables in the two-dimensional

The periodicity of the values of α and β and the condition α ≥ β ≥ δ permit the size of the mapping to be limited. In the plane with coordinate axes α and β, the equation α = β is a line dividing the plane in two parts (south border shown in orange in the drawing). The equation δ = β as a curve divides the plane into different sections (north border shown in blue).

Both borders enclose a continuous region of the plane whose points map to unique equilateral pentagons. Points outside the region just map to repeated pentagons—that is, pentagons that when

Inside the region of unique mappings there are three types of pentagons: stellated, concave and convex, separated by new borders.

Stellated

The

The concave pentagons are non-stellated pentagons having at least one angle greater than 180°. The first angle which opens wider than 180° is γ, so the equation γ = 180° (border shown in green at right) is a curve which is the border of the regions of concave pentagons and others, called convex. Pentagons which map exactly to this border have at least two consecutive sides appearing as a double length side, which resembles a pentagon degenerated to a quadrilateral.

Convex

The

References

Wikimedia Commons has media related to Equilateral pentagons.