The decimal representation of 𝜋 is thought to have random digits that are uniformly distributed. Since there is a 0.9 probability for any digit in a random decimal sequence to be non-zero, it’s fascinating to investigate when a 0 or a sequence of 0s will appear in this sequence.
Remarkably, a string of eight zeroes (00000000 !) emerges at position 172,330,850 in the decimal representation of 𝜋, counting from the digit immediately following the decimal point.
Below is the string, along with the digits surrounding it:
A “Kepler triangle” is a right triangle having edge lengths in a geometric progression, in which the common ratio is √ϕ, where ϕ represents the golden ratio.
Well, let’s construct a square with side length √ϕ that inscribes a Kepler triangle, that is, a right triangle with edges 1 : √ϕ : ϕ (or approximately 1 : 1.272 : 1.618), as shown in the picture. Draw then the circumcircle of the Kepler triangle (highlighted in orange in the picture) whose diameter is the hypotenuse of the triangle.
Then, the perimeters of the square (4√ϕ≈5.0884) and the circle (πϕ≈5.083) coincide up to an error less than 0.1%. From this, we can get the approximation coincidence π≈4/√ϕ