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Polygram — Definition, Formula & Examples

A polygram is a star-shaped figure formed by connecting non-adjacent vertices of a regular polygon, or equivalently, by extending the sides of a polygon until they intersect. Common examples include the hexagram (Star of David) and the pentagram (five-pointed star).

A polygram is a regular star figure denoted {n/k}\{n/k\} in Schläfli notation, constructed by connecting every kk-th vertex of a regular nn-gon, where nn and kk are integers with 2k<n2k < n and k≥2k \geq 2. When gcd⁡(n,k)=1\gcd(n, k) = 1, the result is a single continuous star polygon; otherwise, it is a compound of multiple regular polygons.

How It Works

To construct a polygram, start with nn equally spaced points on a circle. Instead of connecting each point to its immediate neighbor (which would give a regular polygon), connect each point to the vertex kk positions away. For instance, a pentagram {5/2}\{5/2\} connects every 2nd vertex of a regular pentagon, producing a five-pointed star. A hexagram {6/2}\{6/2\} connects every 2nd vertex of a regular hexagon, which yields two overlapping equilateral triangles rather than a single continuous path.

Worked Example

Problem: Identify the polygram formed by placing 8 equally spaced points on a circle and connecting every 3rd point.
Step 1: Determine the Schläfli symbol. With 8 vertices and a step of 3, the symbol is
{8/3}\{8/3\}
Step 2: Check whether it forms a single star or a compound by computing the GCD of 8 and 3.
gcd⁡(8,3)=1\gcd(8, 3) = 1
Step 3: Since the GCD is 1, every vertex is visited in a single continuous path before returning to the start. This produces a single star polygon — an octagram — with 8 points.
Answer: The figure is an octagram {8/3}\{8/3\}, a single continuous eight-pointed star polygon.

Why It Matters

Polygrams appear in art, architecture, and tiling patterns across many cultures — from Islamic geometric art to national flags. Understanding their construction also builds fluency with modular arithmetic and symmetry groups, topics that recur in advanced geometry and abstract algebra.

Common Mistakes

Mistake: Confusing a polygram with a regular polygon.
Correction: A regular polygon connects each vertex to its nearest neighbor (k=1k = 1). A polygram requires k≥2k \geq 2, skipping vertices to create a star shape.