Plouffe asked if the above processes could be "inverted." He considered

(19)

(20)

giving

(21)

(22)

and

(23)

(24)

giving

(25)

(26)

and

(27)

(28)

giving

(29)

(30)

(31)

(OEIS A086203), where the identity was conjectured
by Plouffe and proved by Borwein and Girgensohn (1995).

is sometimes known as Plouffe's constant (Plouffe 1997), although this angle had
arisen in the geometry of the icosahedron dating
back at least to the Ancient Greeks (Smith 2003).

Plouffe's constant is transcendental, as are all numbers of the form for rational and (Smith 2003, Margolius).

The positions of the 1s in the binary expansion of this
constant are 3, 6, 8, 9, 10, 13, 21, 23, ... (OEIS A004715).

Note that the essential idea behind such binary expansions was already known as the "CORDIC" algorithms for computing inverse trigonometric functions (Volder 1959), arguably known to Archimedes, and has been the subject of numerous papers (Walther 1971) and implemented inside numerous commercial electronic calculators such as the HP-35 (Smith 2003).

Borwein and Girgensohn (1995) extended Plouffe's to arbitraryreal , showing that if

(32)

then

(33)

Borwein and Girgensohn (1995) also give much more general recurrences and formulas.

Borwein, J. M. and Girgensohn, R. "Addition Theorems and Binary Expansions." Canad. J. Math.47, 262-273, 1995.Chowdhury,
M. "A Formula for 0.4756260767...." Unpublished note, 2001a.Chowdhury,
M. "On Iterates of the Chaotic Logistic Function ." Unpublished note, 2001b.Finch,
S. R. "Plouffe's Constant." §6.5 in Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 430-433,
2003.Finch, S. R. "Errata and Addenda to Mathematical Constants."
Aug. 11, 2005. http://algo.inria.fr/csolve/erradd.pdf.Margolius,
B. H. "Plouffe's Constant Is Transcendental." http://www.lacim.uqam.ca/~plouffe/articles/plouffe.pdf.Plouffe,
S. "The Computation of Certain Numbers Using a Ruler and Compass." J.
Integer Sequences1, No. 98.1.3, 1998. http://www.math.uwaterloo.ca/JIS/VOL1/compass.Sloane,
N. J. A. Sequences A004715, A049541,
A086201, A086202,
A086203, and A111953
in "The On-Line Encyclopedia of Integer Sequences."Smith,
W. D. "Pythagorean Triples, Rational Angles, and Space-Filling Simplices."
2003. http://math.temple.edu/~wds/homepage/diophant.pdf.Volder,
J. "The CORDIC Trigonometric Computing Technique." IRE Trans. Elec.
Comput.EC-8, 330-334, 1959.Walther, J. S. "A Unified
Algorithm for Elementary Functions." In Spring Joint Computer Conference.
pp. 379-385, 1971.