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March 14th, 2015, 01:03 AM   #1
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Smile Happy Pi Day! New fun fact

Happy Super Pi Day!

On this special day I'd like to share a new fun fact on Pi.
Although no repeating pattern for Pi exists it has not stopped many to look for it anyway. The Feynman point is a well known example.

The Ulam spiral is a simple method of visualizing the prime numbers. I wondered if representing Pi digits this way would reveal something interesting.
Some artist already did this in 2006 but only for the first 144 digits.
In this spreadsheet (mirror) I generated formulas that show the first 2025 digits of Pi in a spiral. By entering a specific digit in cell C2 you can visualize the results.
For example: entering a 9 clearly shows the Feynman point (cell range AF40:AK40).
As you may have guessed: the results only reveal static...
...except for this:
A sequence of seven 1s turns up at the following decimal places of Pi: 850, 971, 1100, 1237, 1382, 1535, 1696. That's one more than the Feynman point!

This particular 'pattern' emerges because the delta between the positions of these points increases by 8.
If it's all the same to you I'd like to call the sequence the Nijhuis points .

Since this is a math forum; the probability of the Feynman point occurring that early in the decimal representation is about 0.0762%.
What is the probability of a sequence of 7 identical digits occuring that early (including diagonals)?

By the way: in the spreadsheet you can apply offset to investigate other parts of Pi. Try for instance: digit=1, offset=6553, 7959, 29533 or 44766.
Let me know if you discover something else!


Last edited by enijhuis; March 14th, 2015 at 01:48 AM.
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March 14th, 2015, 06:12 AM   #2
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Originally Posted by enijhuis View Post
the probability of the Feynman point occurring that early in the decimal representation is about 0.0762%.
I would dispute that. It's a figure that's only true if you assume that the decimal expansion of $\pi$ is random, but it's not.

It might be true to say that 0.0762% of all numbers have the Feynman point that early, but that's a different claim. Specifically, it's not surprising, because for most properties of the decimal expansion of numbers, there is one that is at the limit of the 0.0762th percentile.
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March 15th, 2015, 03:07 PM   #3
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I did a little more research and found that the occurence of other sequences of 7 identical digits in Pi represented as a spiral is rare.
In a spiral containing the first million digits of Pi only four such sequences exist:
1. 0000000: starting at decimal place 892847
2. 1111111: starting at decimal place 850
3. 3333333: starting at decimal place 710100 (7 consecutive digits)
4. 4444444: starting at decimal place 210173

Surprisingly I found no diagonal sequences. I would expect the chance of them occuring to be the same as horizontal or vertical sequences.

Send me a PM if you're interested in the spreadsheet containing the 1000x1000 spiral.

Last edited by enijhuis; March 15th, 2015 at 03:10 PM. Reason: typo
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