A341859 Decimal expansion of 4 - (8/5)*sqrt(5).

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

Offset: 0

Author

Gleb Koloskov, Mar 07 2021

Keywords

• nonn
• cons

Comments

In a triangle inscribed in a unit circle this is the maximal value of its inradius, such that a minimal closed Steiner chain of circles (10 circles) can be sandwiched between the incircle and circumcircle of the triangle.

It can be found as follows.

The squared distance between the centers of the two chain-defining circles is known to be d^2 = (R-r)^2 - 4*r*R*tan(Pi/n)^2.

On the other hand, the squared distance between the circumcenter and the incenter of triangle is known to be d^2 = R*(R-2*r).

Thus, in order to make a valid closed chain of circles, the inradius of triangle inscribed in the unit circle must be equal to 4*tan(Pi/n)^2.

Given that the maximum of such inradius is 0.5, the minimal number of chained circles is n=10, which gives the maximal value r = 4*tan(Pi/10)^2 = 0.42... < 0.5.

Examples

			0.4222912360003364857453221300299580232950106246215588411665644073...
		

References

• Liang-Shin Hahn. Complex Numbers and Geometry (Mathematical Association of America Textbooks). The Mathematical Association of America, 1994, 140-141.

Links

• Wikipedia, Steiner Chain.

Crossrefs

Cf. A019916, A001622.

Programs

Mathematica

RealDigits[4*Tan[18 Degree]^2, 10, 120][[1]]

PARI

4-8/5*sqrt(5)

Formula

Equals 4*A019916^2 = 4*tan(Pi/10)^2 = 4 - (8/5)*sqrt(5) = (4/5)*(7 - 4*phi) = (4/5)*(7 - 4*A001622), where phi is the golden ratio from A001622.