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An apeirogon is a 2-dimensional polygon with a countably infinite number of edges. It can be thought of as a tiling of a one-dimensional line by line segments. In the hyperbolic plane, regular apeirogons are circumscribed by a horocycle, their lines of symmetry converging to a point at infinity. More general apeirogons include the apeirograms ($ \{\frac{\infty}{n}\} $ for natural $ n $), failed star polygons ($ \{x\} $ for irrational real $ x $), and pseudogons ($ \{z\} $ for nonzero imaginary $ z $).

It's Bowers acronym is aze.

Structure and Sections

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Apeirogon in an order-3 apeirogonal tiling and circumscribed horocycle.

Hypervolumes

  • vertex count = $ \aleph_0 $
  • edge length = $ \infty l $
  • surface area = $ \infty {l}^{2} $

Sub-facets

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)
Regular
$ t_0 \{\infty\} $
Rectified
$ t_1 \{\infty\} $
Truncated
$ t_{0,1} \{\infty\} $
Apeirogon Apeirogon Apeirogon
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