A294968 Decimal expansion of sqrt(7 + 4*sqrt(2))/2.

1, 7, 7, 8, 8, 2, 3, 6, 4, 5, 6, 6, 3, 9, 2, 4, 4, 5, 0, 8, 5, 8, 3, 3, 4, 8, 2, 0, 4, 1, 5, 0, 2, 6, 7, 6, 0, 7, 6, 5, 0, 1, 7, 3, 7, 2, 9, 5, 2, 5, 7, 8, 5, 4, 4, 0, 7, 9, 2, 2, 8, 5, 1, 0, 5, 0, 8, 1, 8, 3, 5, 3, 5, 4, 5, 4, 7, 6, 7, 2, 3, 1, 0, 6, 4, 7, 0, 1, 9, 7, 1, 1, 0, 7, 9, 9, 9, 5

Offset: 1

Author

Wolfdieter Lang, Nov 16 2017

Keywords

• nonn
• cons

Comments

A construction of this length using a circle of radius of 1 (length unit) and the circumscribed and inscribed square as well as a square in the middle between them is given in figure 15 of the footnote 13 on p. 26 of M. Gardner's book. Point A is on the middle of one side, say the left, of the middle square and point B is at the intersection of the prolonged right side of the inscribed square with the middle square. Then the length AB is sqrt(AC^2 + CB^2) with point C on the middle of the right side of the inscribed square. AC = (2 - sqrt(2))/4 and CB = 1/sqrt(2) + (2 - sqrt(2))/2 = (2 + sqrt(2))/4. Therefore, AB = sqrt(7 + 4*sqrt(2))/2. See the link.

This is a not a good approximation to the squaring of the circle problem: (AB)^2 is not Pi, or AB = 1.778... is not sqrt(Pi) = A002161 = 1.772... Gardner writes that he was told "of an extremely good approximation".

An elegant construction for the reflexible Archimedean solids was devised by Alicia Boole Stott. In the process called expansion, certain sets of elements (i.e., edges or faces) are moved directly away from the center, retaining their size and orientation, until the consequent interstices can be filled with new regular faces. The reverse process is called contraction. The final edge length is the same as that of the starting solid. By contracting the truncated cube according to its triangles, the cuboctahedron is obtained. (Cf. W. W. Rouse Ball, H. S. M. Coxeter, Mathematical recreations and essays, pp. 139-140.) The factor of this contraction is 1/{a(n)} = (2/17)*sqrt(119-68*sqrt(2)) = 0.56216927542964050970... - Martin Renner, Dec 31 2019

Examples

			1.778823645663924450858334820415026760765017372952578...
		

References

• Martin Gardner, Logic Machines and Diagrams, Second Ed., 1982, The Harvester Press, p. 26, Figure 15.
• W. W. Rouse Ball, H. S. M. Coxeter, Mathematical recreations and essays, New York, Dover, 13th ed., 1987, pp. 139-140 (Mrs. Stott's Construction), fig. 3.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Alicia Boole Stott, Geometrical deduction of semiregular from regular polytopes and space fillings. In: Verhandelingen der Koninklijke Akademie van Wetenschappen, section 1, part 11, nr. 1. Amsterdam, Müller 1910, pp. 3-24.
• Wolfdieter Lang, Some poor approximation of sqrt(Pi).

Programs

Magma

SetDefaultRealField(RealField(100)); Sqrt(7+4*Sqrt(2))/2; // G. C. Greubel, Sep 30 2018

Mathematica

RealDigits[Sqrt[7 + 4*Sqrt[2]]/2, 10, 100][[1]] (* G. C. Greubel, Sep 30 2018 *)

PARI

sqrt(7+4*sqrt(2))/2 \\ Felix Fröhlich, Nov 16 2017