Decagon

A decagon is a polygon characterized by having ten sides and ten angles. The name is derived from the Greek words "deka," meaning ten, and "gonia," meaning angle.

In geometry, a simple decagon has an interior angle sum of 1440 degrees. While decagons can be irregular, a regular decagon is defined by total equality. Every side in a regular decagon is the same length. Additionally, every internal angle measures exactly 144 degrees. This specific shape can be represented by the Schläfli symbol {10}.

Constructing a regular decagon is a classic geometric challenge. It is possible to build one using only a compass and a straightedge. This feasibility exists because the number ten is a power of two multiplied by a Fermat prime.

One method involves the use of a regular pentagon. You can construct a pentagon inside a circle first. Then, you draw lines from each vertex through the center of the circle. The points where these lines intersect the circle become the new vertices. By joining these new points with the original pentagon vertices, you form a decagon. Another way to view it is as a truncated pentagon, denoted as t{5}.

Mathematical construction often involves the golden ratio. This ratio is a fundamental constant that helps divide line segments through exterior division. In a decagon constructed within a given circumcircle, a circular arc produces a segment that corresponds to this ratio.

Similarly, when constructing a decagon from a given side length, the golden ratio appears in the division of segments. This relationship between the decagon and the golden ratio is a deep connection in geometry. It highlights how simple shapes are often tied to universal mathematical constants.

Symmetry describes how a shape can be moved and still look identical. A regular decagon possesses Dih10 symmetry, which has an order of 20.

This includes ten distinct symmetries involving reflections and rotations. The reflection lines, or mirrors, can pass through the vertices or through the edges. Mathematicians like John Conway use specific labels to describe these patterns. For example, reflections through vertices are labeled "d" for diagonal, while reflections through edges are labeled "p" for perpendiculars. The shape also features cyclic group symmetries, such as Z10, Z5, Z2, and Z1.

Even irregular decagons can exhibit high levels of symmetry. An isogonal decagon, labeled d10, is constructed by five mirrors. This shape can alternate between long and short edges. Conversely, an isotoxal decagon, labeled p10, has equal edge lengths but alternating internal angles. These two types of irregular decagons are considered duals of each other. They each possess exactly half the symmetry order of a regular decagon. This demonstrates how varying one property can create new, organized structures.

Decagons can also be broken down into smaller, simpler shapes through a process called dissection.

A regular decagon is a type of zonogon. A zonogon is a polygon where opposite sides are parallel and equal in length. Because it is a zonogon with an even number of sides, it can be dissected into rhombi. For a decagon, this means it can be divided into exactly 10 rhombi.

This specific decomposition is related to the projection of a 5-cube. It also relates to the faces of a rhombic triacontahedron.

Finally, decagons do not always have to lie flat on a single plane. A skew decagon is a polygon with ten sides that zig-zags through three-dimensional space.

These shapes are vertex-transitive and have equal edge lengths. They can be seen in the edges of a pentagonal antiprism or a pentagrammic antiprism. In higher-dimensional mathematics, these are known as Petrie polygons. They appear as the perimeter of projections of complex shapes like the dodecahedron or the icosahedron.

This shows that a ten-sided shape connects simple geometry to the study of complex, high-dimensional polytopes.