Apeirogram

Apeirogram
An '"`UNIQ--postMath-00000006-QINU`"' inside an apeirogon in an order-3 apeirogonal tiling
Dimensionality	2
Schläfli Symbol	'"`UNIQ--postMath-00000001-QINU`"' for irrational '"`UNIQ--postMath-00000002-QINU`"' (failed star polygons) '"`UNIQ--postMath-00000003-QINU`"' for natural '"`UNIQ--postMath-00000004-QINU`"' (regular apeirograms)
Symmetry	'"`UNIQ--postMath-00000005-QINU`"'-gonal dihedral symmetry (failed star polygons) Apeirogonal dihedral symmetry (regular apeirograms)
Subfacets
Vertices	ℵ0 points
Edges	ℵ0 line segments
Faces	1 apeirogram

"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

$\{0\}$	$\{1\}$	$\{2\}$	$\{3\}$	$\{4\}$	$\{5\}$	$\{\frac{5}{2}\}$	$\{6\}$	$\{7\}$	$\{\frac{7}{2}\}$	$\{\frac{7}{3}\}$	$\{8\}$	$\{\frac{8}{3}\}$	$\{9\}$	$\{\frac{9}{2}\}$	$\{\frac{9}{4}\}$	$\{10\}$	$\{\frac{10}{3}\}$	$\{11\}$	$\{\frac{11}{2}\}$	$\{\frac{11}{3}\}$	$\{\frac{11}{4}\}$	$\{\frac{11}{5}\}$	$\{12\}$	$\{\frac{12}{5}\}$	$\{13\}$	$\{\frac{13}{2}\}$	$\{\frac{13}{3}\}$	$\{\frac{13}{4}\}$	$\{\frac{13}{5}\}$	$\{\frac{13}{6}\}$	$\{14\}$	$\{\frac{14}{3}\}$	$\{\frac{14}{5}\}$	$\{15\}$	$\{\frac{15}{2}\}$	$\{\frac{15}{4}\}$	$\{\frac{15}{7}\}$	$\{16\}$	$\{\frac{16}{3}\}$	$\{\frac{16}{5}\}$	$\{\frac{16}{7}\}$	$\{17\}$	$\{\frac{17}{2}\}$	$\{\frac{17}{3}\}$	$\{\frac{17}{4}\}$	$\{\frac{17}{5}\}$	$\{\frac{17}{6}\}$	$\{\frac{17}{7}\}$	$\{\frac{17}{8}\}$	...	$\{\infty\}$	$\{x\}$	$\{\frac{\pi i}{\lambda}\}$
Zerogon	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	Heptadecagon	Tiny heptadecagram	Small heptadecagram	Heptadecagram	Medial heptadecagram	Great heptadecagram	Giant heptadecagram	Grand heptadecagram	...	Apeirogon	Failed star polygon ($x$-gon)	Pseudogon ($\frac{\pi i}{\lambda}$-gon)