Nonagon

In geometry, a nonagon (/ˈnɒnəɡɒn/) or enneagon (/ˈɛniəɡɒn/) is a nine-sided polygon or 9-gon. It is a special shape with nine straight sides and nine angles. Nonagons appear in many designs, like buildings and art, and are often used in math problems and puzzles.

The name nonagon is a prefix hybrid formation. It comes from Latin (nonus, meaning "ninth") mixed with gonon. People have used this word since the 1500s in French (nonogone) and since the 1600s in English. Another name for this shape is enneagon. This name comes from the Greek word enneagonon (εννεα, meaning "nine," plus γωνον from γωνία, meaning "corner"). While enneagon might be more correct, nonagon is the name most people use today.

Regular nonagon

A regular nonagon is a shape with nine sides that are all the same length. All of its angles are also the same, and each one measures 140 degrees. You can find the area of a regular nonagon if you know the length of one side, by using special math rules. There are also ways to find the size of the circle that fits perfectly inside or outside the nonagon.

Construction

Nonagon, an animation from a neusis construction based on the angle trisection 120° by means of the Tomahawk, at the end 10 s break

A regular nonagon has nine sides. It cannot be made perfectly using just a compass and straightedge. This is because the number 9 is special and does not follow the rules for this kind of drawing.

People have found smart ways to make it look almost perfect for a long time.

One way to create a nonagon is by using a tool called neusis. Another way is to use a device that can split angles into three equal parts, called an angle trisector. These methods help make the shape very close to perfect.

Symmetry

Symmetries of a regular enneagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center.

The regular enneagon, a nine-sided polygon, has a special kind of balance called Dih₉ symmetry, which has an order of 18. This means it can be flipped and turned in 18 different ways while still looking the same.

There are also smaller patterns of symmetry within the enneagon. These include three cyclic symmetries and two subgroup dihedral symmetries. Each of these symmetries can be seen in different ways on the shape, either through its corners or its sides.

Tilings

A regular enneagon, a nine-sided shape, can fit together to cover a flat surface. But small spaces will be left in between. These spaces can be filled with regular hexagons, which have six sides, and isosceles triangles, triangles with two sides of equal length. This special pattern is called H(*;3;*;[2]) in geometry. It shows a kind of balance and order in shapes on a flat surface.

Graphs

Sound Catcher in Lithuania

The K₉ complete graph can be shown as a regular enneagon with all 36 edges connected. This graph also shows an orthographic projection of the 9 vertices and 36 edges of the 8-simplex.

Architecture

Some famous buildings have nine-sided shapes. Temples of the Baháʼí Faith, called Baháʼí Houses of Worship, must have nine sides. The U.S. Steel Tower is an example of a building with nine sides. In Lithuania, the Sound Catcher is a wooden structure with nine sides. The city of Palmanova in Italy is planned in the shape of a nine-sided figure.

In popular culture

The band They Might Be Giants has a song called "Nonagon" on their children's album Here Come the 123s. The song is about a party with many-sided shapes and a dance they do.

Slipknot's logo looks like a nonagon, a nine-pointed star made from three triangles. This shows the nine members of the band. King Gizzard & the Lizard Wizard also made an album called Nonagon Infinity. The album cover has a nonagon shape, and it has nine songs that play in a repeating cycle.