accidental similarity number

The accidental similarity number is a kind of overlap between numbers. I came up with this concept after creating typographical art about the 4ness of π.

To construct this number for π, φ and e we first write the numbers on top of each other and then identify positions for which the numbers have the same digit.

3.1415926535897932 … 21170679821 … 10270193852 … 
1.6180339887498948 … 93911374847 … 08659593958 … 
2.7182818284590452 … 51664274274 … 32862794349 … 

These digits are then used to create the accidental similarity number. In thise case,

0.979 …

By definition, the decimal is held in place.

accidental similarity art

The poster shows the accidental similarity number for π, φ and e created from the first 1,000,000 digits of each number. There are 9,997 positions in which these numbers have the same digit, but only 9,996 are shown because the distance between positions is used to color the digit and I was limited by input files with 1M digits.

▲ The accidental similarity number for π, φ and e created from the first 1,000,000 digits of each number. (PNG, PDF)

The distribution of distances follows a Poisson distribution with an average of 100, with about 1-1/e values being smaller than 100.

The font is Neutraface Slab Display Medium.

▲ Segments of π, φ and e are connected by thin links if the same digit is shared between two numbers, and thick links if among all three. Shown for the first 10,000 digits. (PNG)

properties of the accidental similarity number

Any properties are accidental, but curiously ASN(π, φ, e) ≈ 1.

If you find other curiously accidental properties, let me know.

data files

Download the first 9,997 digits of the accidental similarity number. This file provides the ASN digit index, the digit and the position from which it is sampled.

other number art

I invented the Accidental Similarity Number immediately after creating this poster of the overlap between π, φ and e.

▲ The overlap between the three interesting numbers π, φ and e (nixie theme). (PNG, PDF)

This thought stream started with the 4ness of π.

▲ The 4ness of π — a measure of how similar each 4 is to its neighbours. (read more)