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"Apeirogram" is a term that can be used to describe two different types of star polygon.

Regular apeirograms

A regular apeirogram is an infinitely-edged regular compound polygon that can be constructed by stellating an apeirogon.

These apeirograms are compounds of some number of multiple apeirogons. These have the Schläfli symbol $ \{\frac{\infty}{n}\} $ where $ n $ is the number of apeirogons the apeirogram is a compound of. Like the apeirogon, these are degenerate polytopes in the Euclidean plane, are circumscribed by the Euclidean line when embedded in a Euclidean plane and are circumscribed by horocycles when embedded in the hyperbolic plane.

Failed star polygons

Another kind of shape that can go by the name "apeirogram" (more descriptively, circular apeirogram or circular apeirogon) is a polygon with infinitely many edges that is circumscribed by a circle. These are called failed star polygons as they have an infinite winding number. Like regular apeirograms, they are also distinct objects from an apeirogon that is circumscribed by a line. Similar to how the Schläfli symbol of a pseudogon is a pure imaginary number that encodes properties of the shape (the distance between consecutive lines of symmetry), the Schläfli symbol of a failed star polygon is an irrational number that tells one the interior angle measure of the shape in a Euclidean plane. Each interior angle of an $ x $-gon embedded in a plane for real $ x $ in radians is $ 2 \arccos (|\sin (\frac{\pi}{x})|) $. For example, a $ \pi $-gon has an interior angle of $ 2 \sin (1) $ radians.

See Also

$ \{1\} $ $ \{2\} $ $ \{3\} $ $ \{4\} $ $ \{5\} $ $ \{\frac{5}{2}\} $ $ \{6\} $ $ \{7\} $ $ \{\frac{7}{2}\} $ $ \{\frac{7}{3}\} $ $ \{8\} $ $ \{\frac{8}{3}\} $ $ \{9\} $ $ \{\frac{9}{2}\} $ $ \{\frac{9}{4}\} $ $ \{10\} $ $ \{\frac{10}{3}\} $ $ \{11\} $ $ \{\frac{11}{2}\} $ $ \{\frac{11}{3}\} $ $ \{\frac{11}{4}\} $ $ \{\frac{11}{5}\} $ $ \{12\} $ $ \{\frac{12}{5}\} $ $ \{13\} $ $ \{\frac{13}{2}\} $ $ \{\frac{13}{3}\} $ $ \{\frac{13}{4}\} $ $ \{\frac{13}{5}\} $ $ \{\frac{13}{6}\} $ $ \{14\} $ $ \{\frac{14}{3}\} $ $ \{\frac{14}{5}\} $ $ \{15\} $ $ \{\frac{15}{2}\} $ $ \{\frac{15}{4}\} $ $ \{\frac{15}{7}\} $ $ \{16\} $ $ \{\frac{16}{3}\} $ $ \{\frac{16}{5}\} $ $ \{\frac{16}{7}\} $ ... $ \{\infty\} $ $ \{x\} $ $ \{\frac{\pi i}{\lambda}\} $
Monogon Digon Triangle Square Pentagon Pentagram Hexagon Heptagon Heptagram Great heptagram Octagon Octagram Enneagon Enneagram Great enneagram Decagon Decagram Hendecagon Small hendecagram Hendecagram Great hendecagram Grand hendecagram Dodecagon Dodecagram Tridecagon Small tridecagram Tridecagram Medial tridecagram Great tridecagram Grand tridecagram Tetradecagon Tetradecagram Great tetradecagram Pentadecagon Small pentadecagram Pentadecagram Great pentadecagram Hexadecagon Small hexadecagram Hexadecagram Great hexadecagram ... Apeirogon Failed star polygon ($ x $-gon) Pseudogon ($ \frac{\pi i}{\lambda} $-gon)
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