Nonagon

In geometry, a nonagon (/ˈnɒnəɡɒn/) or enneagon (/ˈɛniəɡɒn/) is a nine-sided polygon or 9-gon. It is a special shape that has nine straight sides and nine angles. Nonagons can be found in various designs, from architecture to art, and they are often used in mathematical problems and puzzles.

The name nonagon is a prefix hybrid formation, coming from Latin (nonus, meaning "ninth") combined with gonon. This word has been used since the 16th century in French (nonogone) and in English from the 17th century. Another name for this shape is enneagon, which comes from the Greek word enneagonon (εννεα, meaning "nine," plus γωνον from γωνία, meaning "corner"). While enneagon might be more correct, nonagon is more commonly used today.

Regular nonagon

A regular nonagon is a nine-sided polygon with all sides and angles the same. Each internal angle in a regular nonagon measures 140 degrees. The area of a regular nonagon can be calculated if you know the length of one side, using special math formulas. There are also ways to find the size of the circle that fits perfectly inside or outside the nonagon.

Construction

A regular nonagon, which has nine sides, cannot be perfectly made using just a compass and straightedge. This is because the number 9 is related to a special kind of number called a Fermat prime, and it doesn’t fit the rules needed for this type of construction. However, people have found clever ways to make it look almost perfect for a very long time.

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

Symmetry

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, which is a nine-sided shape, can fit together in a special way to cover a flat surface, but there will be small spaces left in between. These spaces can be filled using regular hexagons (six-sided shapes) and isosceles triangles (triangles with two sides of equal length). This special pattern is known in geometry by the name H(*;3;*;[2]) and it shows a kind of balance and order in shapes on a flat surface.

Graphs

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

Architecture

Some famous buildings have nonagonal (nine-sided) shapes. Temples of the Baháʼí Faith, called Baháʼí Houses of Worship, must be nonagonal. The U.S. Steel Tower is an example of an irregular nonagon. 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 nonagon.

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 talks about a party where everyone is a many-sided polygon and includes a dance the polygons perform.

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