Heptadecagon

In geometry, a heptadecagon, septadecagon or 17-gon is a seventeen-sided polygon. It is a special shape that mathematicians have studied for many years.

One interesting fact about a heptadecagon is that it can be made using only a compass and straightedge. This was proven by the mathematician Carl Friedrich Gauss in 1796. His discovery showed that some regular polygons with a prime number of sides can be drawn perfectly with these simple tools.

Heptadecagons appear in different areas, from buildings to art. Even though a seventeen-sided figure may seem unusual, it shows how numbers and shapes can come together in beautiful ways. Learning about shapes like the heptadecagon helps us understand more about the rules of geometry and the patterns in our world.

Regular heptadecagon

Publication by C. F. Gauss in Intelligenzblatt der allgemeinen Literatur-Zeitung

A regular heptadecagon is a shape with seventeen sides that are all the same length and all the same angles.

It can be drawn using just a compass and straightedge because the number 17 is special in math. This was shown by Carl Friedrich Gauss in 1796. It was a big discovery because shapes like this could not be drawn that way for a very long time.

The way to draw it uses simple math ideas, like square roots. This lets people draw the shape perfectly with basic tools. Many smart people have found fun ways to make this shape using clever steps.

Symmetry

Symmetries of a regular heptadecagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.

The regular heptadecagon has Dih₁₇ symmetry. This means it has special patterns when you flip and turn the shape. Because 17 is a prime number, it has one subgroup with dihedral symmetry and two cyclic group symmetries: Z₁₇ and Z₁. These symmetries can be seen in four different ways on the heptadecagon.

Related polygons

A heptadecagram is a 17-sided star polygon. There are seven regular forms, shown by special symbols called Schläfli symbols: {17/2}, {17/3}, {17/4}, {17/5}, {17/6}, {17/7}, and {17/8}. Because 17 is a prime number, these make regular star shapes.

The regular heptadecagon also acts as a Petrie polygon for a higher-dimensional shape, shown in a special kind of flat view called an orthogonal projection.