of an equilateral pentagon that can tile the plane but does not belong to either of those two families; its angles are roughly 89°16', 144°32.5', 70°55', 135°22', and 99°54.5', no two supplementary. A two-dimensional mapping 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 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 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 rotated or reflected can match others already described. Pentagons that map exactly onto those borders have 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 pentagon 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 map exactly to this border have a vertex touching another side. Concave 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