The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A231985 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A231985

Decimal expansion of the side length (in degrees) of the spherical square whose solid angle is exactly one deg^2.
(history; published version)

#16 by Harvey P. Dale at Wed Jun 09 13:22:06 EDT 2021

STATUS

editing

approved

#15 by Harvey P. Dale at Wed Jun 09 13:22:02 EDT 2021

MATHEMATICA

RealDigits[(360/Pi)*ArcSin[Sqrt[Sin[(Pi/360)^2]]], 10, 120][[1]] (* Harvey P. Dale, Jun 09 2021 *)

STATUS

approved

editing

#14 by T. D. Noe at Wed Jan 29 14:17:27 EST 2014

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reviewed

approved

#13 by Joerg Arndt at Wed Jan 29 14:04:55 EST 2014

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proposed

reviewed

#12 by Rick L. Shepherd at Wed Jan 29 12:49:22 EST 2014

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editing

proposed

#11 by Rick L. Shepherd at Wed Jan 29 12:48:26 EST 2014

COMMENTS

This answers the inverse problem of A231984 (not to be confused with its inverse value): what is the side arc-length of a spherical square required to subtend exactly 1 deg^2. Since the solid angle of a spherical square with side s (in rads) is Omega = 4*arcsin(sin(s/2)^2)) (in sr), we have s = 2*arcsin(sqrt(Omega/4)). Converting Omega = 1 deg^2 into steradians (A231982), applying the formula, and converting the result from radians to degrees (A072097), one obtains the result.

FORMULA

(360/Pi)*arcsin(sqrt(sin((Pi/360)^2))).

PROG

(PARI)

default(realprecision, 120);

(360/Pi)*asin(sqrt(sin((Pi/360)^2))) \\ or

(180/Pi)*solve(x = 0, 1, 4*asin(sin(x/2)^2) - (Pi/180)^2) \\ Rick L. Shepherd, Jan 29 2014

STATUS

approved

editing

#10 by N. J. A. Sloane at Sun Nov 17 10:48:22 EST 2013

STATUS

editing

approved

#9 by N. J. A. Sloane at Sun Nov 17 10:48:19 EST 2013

REFERENCES

G. V. Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton University Press, 2012, ISBN 978-0691148922.

STATUS

proposed

editing

#8 by Stanislav Sykora at Sun Nov 17 10:39:06 EST 2013

STATUS

editing

proposed

#7 by Stanislav Sykora at Sun Nov 17 10:32:43 EST 2013

LINKS

Wikipedia, <a href="http://en.wikipedia.org/wiki/Solid_angle#Pyramid">Solid angle</a>, Section 3.3 (Pyramid)

Wikipedia, <a href="http://en.wikipedia.org/wiki/Square_degree">Square degree</a>

Wikipedia, <a href="http://en.wikipedia.org/wiki/Steradian">Steradian</a>