A240969 Decimal expansion of the breadth of the "caliper", the broadest worm of unit length.

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

Offset: 0

Author

Jean-François Alcover, Sep 04 2014

Keywords

• nonn
• cons

Comments

A caliper consists of 2 circular arcs with 4 tangent segments, specifically configured (see link to Figure 8.3 from the book by Steven Finch).

Examples

			phi = 0.29004634452825946320905124629823276955932638591519522257237...
psi = 0.480931237564380337681715512959999015584157793267187574483...
beta = 0.43892536925946645674088526115852377421914938651438872683...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.4 Moser's Worm Constant, p. 493.

Links

• Jean-François Alcover, Figure 8.3 A caliper.
• Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 60.

Crossrefs

Cf. A227472.

Programs

Mathematica

phi = ArcSin[1/6 + (4/3)*Sin[(1/3 )*ArcSin[17/64]]];
psi = ArcTan[(1/2)*Sec[phi]];
beta = (1/2)*(Pi/2 - phi - 2*psi + Tan[phi] + Tan[psi])^(-1);
RealDigits[beta, 10, 100] // First

Formula

See trig. formulas in Mathematica code.

Sec(phi), an algebraic number, is the positive root of 3x^6 + 36x^4 + 16x^2 - 64.