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# Gigagon

Gigagon
Edges and vertices1,000,000,000
Schläfli symbol${\displaystyle \{10,9\}}$ (using Bowers' arrays)
Area${\displaystyle {\frac {\left\langle 10,9\right\rangle }{4\tan {\frac {\pi }{\left\langle 10,9\right\rangle }}}}l^{2}}$
Internal angle (degrees)179.99999964°

A gigagon or 1,000,000,000-gon is a polygon with 1 billion sides (giga-, from the Greek γίγας gígas, meaning "giant").[1][2] A gigagon with a radius of 1 ly (approximately the size of 100 solar systems) would have its edge length differ from a circle by 9.78 centimetres (3.85 in). It has the Schläfli symbol ${\displaystyle \{10,9\}}$ (using Bowers' arrays).[3][4]

## Regular gigagon

A regular gigagon is represented by the Schläfli symbol {1,000,000,000} and can be constructed as a truncated 500,000,000-gon, t{500,000,000}, a twice-truncated 250,000,000-gon, tt{250,000,000}, a thrice-truncated 125,000,000-gon, ttt{125,000,000}, or a four-fold-truncated 62,500,000-gon, tttt{62,500,000}, a five-fold-truncated 31,250,000-gon, ttttt{31,250,000}, or a six-fold-truncated 15,625,000-gon, tttttt{15,625,000}.

A regular gigagon has an interior angle of 179.99999964°.[5] The area of a regular gigagon with sides of length a is given by

${\displaystyle {\frac {\left\langle 10,9\right\rangle }{4\tan {\frac {\pi }{\left\langle 10,9\right\rangle }}}}l^{2}.}$

Because 1,000,000,000 = 29 × 59, the number of sides is not a product of distinct Fermat primes and a power of two. Thus the regular gigagon is not a constructible polygon. It is not even constructible with the use of neusis or an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a product of powers of two and three.

## References

1. Gibilisco, Stan (2003-06-27). Geometry Demystified. McGraw Hill Professional. ISBN 978-0-07-141650-4. Search this book on
2. "Geometric Basics for Raytracing - Geometry with POV-Ray - Regular Polygon". www.f-lohmueller.de. Retrieved 2020-12-05.
3. Looijen, Prof Dr Ir Maarten. Over getallen gesproken - Talking about numbers: Een wiskundige ontdekkingsreis. Van Haren. ISBN 978-94-018-0469-1. Search this book on
4. "Bowers' Array Notation - Allam's Numbers". sites.google.com. Retrieved 2020-12-05.
5. "BDMNQR Essays: Math & Science". www.erictb.info. Retrieved 2020-12-05.

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