In geometry an equilateral pentagon is a polygon with five sides of equal length. Its five internal angles, in turn, can take several values, thus permitting to form a family of pentagons. In contrast, the regular pentagon is unique, because is equilateral but at the same time its five angles are equal.
Four intersecting equal circles disposed in a closed chain, are sufficient to describe an equilateral pentagon. Every center of the circles corresponds to one of four pentagon vertexes. The remain vertex is determined by the intersection of the first and the last circle of the chain.
Is possible to describe any equilateral pentagon with only two angles (to say α and β) following two simple conditions. First condition is that α ≥ β must be hold. Second condition is that fourth angle (δ) must be the lowest of the rest of angles. By doing so, the general equilateral pentagon can be regarded as a two-variables function f(α, β) where the rest of the angles can be obtained by using trigonometric relations. The equilateral pentagon described in this manner will be unique regardless its rotation in the plane.
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Internal angles
By dissecting the equilateral pentagon in triangles two of them appear as isosceles (triangles in orange and blue) while the other one is more general (triangle in green).
According to the law of sines the length of the line dividing the green and blue triangles is:
The length (squared) of the line dividing the orange and green triangles is:
According to the law of cosines, cosine of δ can be seen from the figure:
Simplifying, δ is obtained in function of α and β so far:
![\delta = \arccos\left[\cos(\alpha) + \cos(\beta) - \cos(\alpha + \beta) - \frac{1}{2} \right]](http://wpcontent.answers.com/math/c/2/a/c2a8c5bf1daf90a9eb2ef36e3f83c4c7.png)
Two dimensional mapping
The equilateral pentagon as a function of two variables can be plotted in the two-dimensional plane. Each pair of values (α, β) maps to a single point of the plane and also maps to a single pentagon.
The periodicity of the values of α and β and the condition α ≥ β ≥ δ permit to limit the size of the mapping. In the plane with coordinated axes α and β, α = β is a line dividing the plane in two parts (south border shown in orange in the drawing). δ = β as a curve divides the plane in different sections (north border shown in blue).
Both borders enclose a continuous region of the plane whose points maps to unique equilateral pentagons. Points outside the region just maps to repeated pentagons, that is, pentagons that when rotated or reflected can match others already described. Pentagons which maps exactly those borders, has a line of symmetry.
Inside the region of unique mappings there are three types of pentagons: Stellated, concave and convex, separated by new borders.
Stellated
The stellated pentagons have sides intersected by others. A common example of this type of pentagons is the pentagram. A condition for a pentagon to be stellated, or self-intersecting, is to have 2α + β ≤ 180°. So, in the mapping, the line 2α + β = 180° (shown in orange at the north) is the border between the regions of stellated and non-stellated pentagons. Pentagons which maps exactly this border has a vertex touching another side.
Concave
The concave pentagons are non-stellated pentagons having at least an angle greater than 180°. The first angle which opens wider than 180° is γ, so 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 maps exactly this border, has at least, two consecutive sides appearing as a double length side, which resembles a pentagon degenerated as a quadrilateral.
Convex
The convex pentagons have their five angles smaller than 180° and no sides intersecting others. A common example of this type of pentagons is the regular pentagon.
External links
- Equilateral Pentagons With Java animations.
- Equilateral Pentagons Tilings Equilateral Pentagons that tile the plane.
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