The First Pixel

Inspiration

In the StarTalk video, “Why can’t you divide by zero?”, 3Blue1Brown’s Grant Sanderson‬ discusses non-periodic tiling, known as Penrose Tiling, in which a pattern fills all of space without repeating. Grant compares this to the digits of Pi, which never repeat. This created the inspiration of finding repeat finite-sized patterns in the non-repeating infinity — both in the geometric tiling in Penrose, and in the digits of Pi.

Penrose Tiling

Penrose tiling never repeats, but it creates a tiling pattern forever. This means you can continually add onto the tiling pattern, and produce large finite sections, as in this Veratasium video, “The Infinite Pattern that Never Repeats”:

But whatever creation you lay out, it will never repeat. However, if the tiling is infinite, wouldn’t any finite pattern be found elsewhere, just with different surroundings? Wouldn’t it be neat to travel along this infinite, not-repeating pattern and find the exact same conglomeration you’ve created — way out there in infinite “Space Harrier“-esque tile-land?

Pi Digits Repeating

I had the same thought about the digits within Pi. Pi is known to many digits, “nearly infinite” as some would say, but finite numbers even in the trillions, is oh so much smaller than infinity.

The digits of Pi never repeat. If they did, it would be a rational number (a number that can be defined as a ratio — rational — of two integers). The thought that any number could be irrational seemed irrational before the square root of 2 was proved irrational, which is where the adjective “irrational” took on the meaning of someone or something being illogical or unreasonable.

Couldn’t you find some interesting patterns in the digits of Pi, such as the Feynman Point (six nines in a row: …999999… at the 762 decimal digit), and then also find it elsewhere?

How about the digits of Pi itself?

Acquire Digits of Pi

You can get the digits of Pi from various sources.

• Download: Internet Archive: “Various Mathematical Constants to 1 Billion Digits“
• Compute: Y-Cruncher

I chose to compute them myself, up to 10 billion digits, using Y-Cruncher:

• y-cruncher.exe custom pi -dec:10b
• y-cruncher.exe custom pi -hex:10b

First 100 Digits of Pi

• Decimal (base-10):
  3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706 …

• Binary (base-2):
  11.0010010000111111011010101000100010000101101000110000100011010011000100110001100110001010001011100 …

I don’t have a direct source data for binary, but I extract it in real-time from hexadecimal digits.

Searching for Pi inside Pi

Digit Index

Note: for binary, there are two digits before the radix point (“decimal point” in base-10) unlike decimal and hexadecimal:

• (base-10) Decimal: 3.141592653 …
• (base-16) Hexadecimal: 3.243f6a888 …
• (base-2) Binary: 11.00100100 …

In all cases, we count the first digit as index 0, and the second digit as index 1. Therefore the first fractional part (“decimal digits” in base-10) in binary starts at digit index 2. In decimal and hexadecimal, it starts at index 1. The index is actually the count of decimal (fractional) digits for decimal and hexadecimal (1st decimal digit = index 1, 2nd decimal digit = index 2, etc.), but not for binary (1st fractional digit = index 2, 2nd fractional digit = index 3, etc.)

Decimal (base-10):

First 10 billion digits:

Integer Sequence:

• 9, 137, 2120, 3496, 88008, 176451, 25198140, 50366472, 1660042751, 7902183159
• The On-Line Encyclopedia of Integer Sequences: OEIS: A081876

Results:

• Found 3 at decimal digit 9
  Found 31 at decimal digit 137
  Found 314 at decimal digit 2120
  Found 3141 at decimal digit 3496
  Found 31415 at decimal digit 88008
  Found 314159 at decimal digit 176451
  Found 3141592 at decimal digit 25198140
  Found 31415926 at decimal digit 50366472
  Found 314159265 at decimal digit 1660042751
  Found 3141592653 at decimal digit 7902183159

Binary (base-2):

First 40 billion digits (from 10 billion hexadecimal digits):

Integer Sequence:

• 1, 12, 16, 48, 77, 246, 246, 418, 418, 513, 513, 513, 513, 44458, 109628, 201504, 201504, 260229, 260229, 260229, 260229, 5195536, 5195536, 5195536, 16400799, 71861116, 71861116, 71861116, 88885576, 88885576, 465466168, 465466168, 2612839361, 2612839361, 5728737753, 5728737753
• The On-Line Encyclopedia of Integer Sequences: OEIS: A385937

Results:

• Found 1 digits = 1 at digit 1
  Found 2 digits = 11 at digit 12
  Found 3 digits = 110 at digit 16
  Found 4 digits = 1100 at digit 48
  Found 5 digits = 11001 at digit 77
  Found 6 digits = 110010 at digit 246
  Found 7 digits = 1100100 at digit 246
  Found 8 digits = 11001001 at digit 418
  Found 9 digits = 110010010 at digit 418
  Found 10 digits = 1100100100 at digit 513
  Found 11 digits = 11001001000 at digit 513
  Found 12 digits = 110010010000 at digit 513
  Found 13 digits = 1100100100001 at digit 513
  Found 14 digits = 11001001000011 at digit 44458
  Found 15 digits = 110010010000111 at digit 109628
  Found 16 digits = 1100100100001111 at digit 201504
  Found 17 digits = 11001001000011111 at digit 201504
  Found 18 digits = 110010010000111111 at digit 260229
  Found 19 digits = 1100100100001111110 at digit 260229
  Found 20 digits = 11001001000011111101 at digit 260229
  Found 21 digits = 110010010000111111011 at digit 260229
  Found 22 digits = 1100100100001111110110 at digit 5195536
  Found 23 digits = 11001001000011111101101 at digit 5195536
  Found 24 digits = 110010010000111111011010 at digit 5195536
  Found 25 digits = 1100100100001111110110101 at digit 16400799
  Found 26 digits = 11001001000011111101101010 at digit 71861116
  Found 27 digits = 110010010000111111011010101 at digit 71861116
  Found 28 digits = 1100100100001111110110101010 at digit 71861116
  Found 29 digits = 11001001000011111101101010100 at digit 88885576
  Found 30 digits = 110010010000111111011010101000 at digit 88885576
  Found 31 digits = 1100100100001111110110101010001 at digit 465466168
  Found 32 digits = 11001001000011111101101010100010 at digit 465466168
  Found 33 digits = 110010010000111111011010101000100 at digit 2612839361
  Found 34 digits = 1100100100001111110110101010001000 at digit 2612839361
  Found 35 digits = 11001001000011111101101010100010001 at digit 5728737753
  Found 36 digits = 110010010000111111011010101000100010 at digit 5728737753

What’s Next?

A few ideas:

• Try more digits. The world record is 30,000x more than 10 billion that I tested:
  • Current World Record: 300 trillion digits = 300,000,000,000,000 = 3×10^14.
  • Date: April 2, 2025
  • Who: Linus Media Group, Kioxia
  • Software: y-cruncher v0.8.5.
  • Computation: 2x AMD EPYC 9684X 3D V-Cache @ 2.55GHz (192 cores, 3.0 TiB DDR5 RAM)
  • Storage: 2.2 PB (80x 15.36TB + 32x 30.72TB)
  • OS: Ubuntu 24.04 (x64)
  • Time: 226 days
• Search for more things
  • How about your favourite number?
  • The rarest / hardest to find numbers?
• 3D Visualization
  • Use binary digits for movement in X, Y, and Z in a random walk.