The Five Pointed Star

 Exploring the Geometry of the Pentagram The Golden Star

In the elegant domain of star polygons, the be pentagram—denoted by the Schläfli symbol {5/2}—stands as a pinnacle of geometric harmony. This five-pointed star emerges from connecting every second vertex of a regular pentagon, or equivalently, five equally spaced points on a circumscribed circle. Unlike a simple pentagon {5/1}, which forms a convex hull, the {5/2} introduces self-intersection, creating a star with a density of 2, where lines cross to form a central pentagonal core.

It is come to represent several different forms including the commonly seen 5 elements pentagram, alchemical, chaos, the encircled pentacle, the Tetragrammaton, as well the inverted version that represents a second degree witch and the  3rd degree and Elder broom versions. The inverted pentagram also represents the Baphomet and Satanism in some circles.

Construction and Basic Form

To construct the {5/2} pentagram:

1.  Inscribe five points equidistantly on a circle (separated by 72° angles, as 360°/5 = 72°).

2.  Connect each point to the one two steps away (skipping one vertex). This yields a single, continuous line that traverses all points before closing, manifesting as a sharp, interlaced star. The result is chiral—its mirror image is {5/3}, which is equivalent to {5/2} traced in the opposite direction, as 3 ≡ -2 mod 5.

Geometrically, the pentagram divides the circle into arcs and creates internal intersections. Each side intersects two others, forming five intersection points that themselves outline a smaller inverted pentagon. This self-similar property allows for infinite recursion: within the core pentagon lies another pentagram, scaled by the golden ratio, ad infinitum.

Symmetry and Properties

•  Symmetry Group: The pentagram exhibits dihedral symmetry Dih₅ (or D₅), with 10 elements: 5 rotations (multiples of 72°) and 5 reflections across axes through vertices and midpoints.

•  Area and Metrics: For a pentagram with circumradius R, the area is (5/2) R² (√(25 + 10√5) - 5)/2, derived from decomposing into triangles. Its vertex angle is 36°, with internal angles at intersections of 36° as well.

•  Dual and Compounds: The pentagram is self-dual; its line arrangement is topologically identical to itself. It forms part of the pentagonal compound {5/2} ∪ {5/3}, known as the pentagrammic bipyramid in higher dimensions.

•  Topological Density: As a star polygon, its Euler characteristic is V - E + F = 5 - 10 + 1 = -4 (considering the star as a single face with windings), highlighting its non-convex nature.

Beyond pure geometry, the {5/2} pentagram whispers of cosmic order, symbolizing the five elements in ancient traditions or the human form in esoteric lore. Yet in its lines and ratios, it reveals mathematics’ timeless dance—a star where finite form echoes the infinite.