Pi might look random but it's full of hidden patterns

March 14, 2016 by Steve Humble, Newcastle University, The Conversation

After thousands of years of trying, mathematicians are still working out the number known as pi or "π". We typically think of pi as approximately 3.14 but the most successful attempt to calculate it more precisely worked out its value to over 13 trillion digits after the decimal point. We have known since the 18th century that we will never be able to calculate all the digits of pi because it is an irrational number, one that continues forever without any repeating pattern.

In 1888, the logician John Venn, who also invented the Venn diagram, attempted to visually show that the of pi were random by drawing a graph showing the first 707 . He assigned a compass point to the digits 0 to 7 and then drew lines to show the path indicated by each digit.

Venn did this work using pen and paper but this is still used today with modern technology to create even more detailed and beautiful patterns.

But, despite the endless string of unpredictable digits that make up pi, it's not what we call a truly random number. And it actually contains all sorts of surprising patterns.

Normal not random

The reason we can't call pi random is because the digits it comprises are precisely determined and fixed. For example, the second decimal place in pi is always 4. So you can't ask what the probability would be of a different number taking this position. It isn't randomly positioned.

But we can ask the related question: "Is pi a normal number?" A decimal number is said to be normal when every sequence of possible digits is equally likely to appear in it, making the numbers look random even if they technically aren't. By looking at the digits of pi and applying statistical tests you can try to determine if it is normal. From the tests performed so far, it is still an open question whether pi is normal or not.

For example in 2003, Yasumasa Kanada published the distribution of the of times different digits appear in the first trillion digits of pi:

Digit—Occurrences

0—99,999,485,134
1—99,999,945,664
2—100,000,480,057
3—99,999,787,805
4—100,000,357,857
5—99,999,671,008
6—99,999,807,503
7—99,999,818,723
8—100,000,791,469
9—99,999,854,780
Total—1,000,000,000,000

His results imply that these digits seem to be fairly evenly distributed, but it is not enough to prove that all of pi would be normal.

Every sequence

We need to remember the surprising fact that if pi was normal then any finite sequence of digits you could name could be found in it. For example, at position 768 in the pi digits there are six 9s in succession. The chance of this happening if pi is normal and every sequence of n digits is equally likely to occur, is 0.08%.

This block of nines is famously called the "Feynman Point" after the Nobel Prize-winner Richard Feynman. He once jokingly claimed that if he had to recite pi digits he would name them up to this point and then say "and so on".

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Other interesting sequences of digits have also been found. At position 17,387,594,880 you find the sequence 0123456789, and surprisingly earlier at position 60 you find these ten digits in a scrambled order.

Pi-hunters search for dates of birth and other significant personal numbers in pi asking the question: "Where do I occur in the pi digits?" If you want to test to see where your own special numbers are in , then you can do so by using the free online software called Pi birthdays.

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16 comments

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julianpenrod
1.7 / 5 (6) Mar 14, 2016
How many have noticed that, this year, "pi day" is 3/14/16, the approximation for pi out to four decimal places?
There is a discovery I made that I have tried to place in various science related journals, but I've been turned down by each.
If pi is the ratio of the circumference of a circle to its diameter and p(i) is the i'th prime number, where I is a positive integer, then the scattershot graph of the points (p(i)/pi) – [p(i)/pi], where [] is the greatest integer function, take on the form of straight, parallel lines, ten to a bunch, each bunch separated by a space large enough for another line. This characteristic is not displayed by any other irrational number, including e = 2.716261626459045, which is close to pi, or sqrt(10) = 3.16227766..., which is very close.
RealScience
5 / 5 (6) Mar 14, 2016
@JP: Check your value for e - it looks like you typed a 6 for each 8.

Did your submission on pi versus prime suggest any meaning to the pattern pi's scattershot graph?
kochevnik
5 / 5 (4) Mar 15, 2016
@julianpenrod Your approximation of PI with sqrt(10) may be why man prefers base ten digits in his numbers, as it approximates "squaring the circle"
antialias_physorg
5 / 5 (6) Mar 15, 2016
may be why man prefers base ten digits in his numbers,

...or just because we have 10 fingers.

In the end the digit hunting is a bit weird, as the digits look totally different if PI is written down in other bases. (The normality analysis stays the same, though)

e once jokingly claimed that if he had to recite pi digits he would name them up to this point and then say "and so on".

In that case I would recite PI in binary - as the repetition of six same digits occurs much earlier (starting from the 11th to 16th decimal place is all 1's)
compose
Mar 15, 2016
This comment has been removed by a moderator.
antialias_physorg
5 / 5 (5) Mar 15, 2016
but it's full of hidden patterns

If PI contains all possible sequences of digits then it should, by definition, also contain ranges which contain any subset of all possible patterns.
compose
Mar 15, 2016
This comment has been removed by a moderator.
Whydening Gyre
5 / 5 (3) Mar 15, 2016
All I know is -
I brought up Pi day a couple of weeks ago. And I didn't sell a SINGLE one of my "Pi" fork pendants...:-(
compose
Mar 15, 2016
This comment has been removed by a moderator.
Whydening Gyre
5 / 5 (3) Mar 15, 2016
may be why man prefers base ten digits in his numbers,

...or just because we have 10 fingers.

(unless you've had a nasty incident with a table saw...)
What if - we have that number of digits because of the way the universe "counts" (arranges) numbers?
Captain Stumpy
3.7 / 5 (3) Mar 15, 2016
@AA_P
or just because we have 10 fingers
Wait... what? crap... i think i dropped one...LOL
we used to get in trouble calling the thumb a "finger" in school... we could call it a digit or one of the phalanges, but not a "finger"...

.

What if - we have that number of digits because of the way the universe "counts" (arranges) numbers?
@whyde
were you being silly or philosophical?
i can't always tell with artists...LOL

what if the reciprocal were the case? what if the universe was designed around our digits (or phalanges, or fingers... whatever)
[jk]
obama_socks
1 / 5 (2) Mar 16, 2016
may be why man prefers base ten digits in his numbers,

...or just because we have 10 fingers.

(unless you've had a nasty incident with a table saw...)
What if - we have that number of digits because of the way the universe "counts" (arranges) numbers?
- WhydG
I doubt that it has anything to do with the number of human or animal fingers and toes. Some humans are even born with 6 fingers on each hand and/or 6 toes on each foot. While 6 isn't the norm, it also has to be considered.
But the fact that the 10 number is the sum of the amount of fingers on BOTH hands must be reckoned with also. In that case, it is perhaps the number 5 as an odd number, even 6 as an even number, that is the determining factor.

On the other hand, (pun intended) it MIGHT be said that the number 10 (5 of the right hand; 5 of the left hand) is a reflection of the Yin-Yang that exists in the Universe. It might be construed as a Philosophical explanation...but quite plausible.
obama_socks
1 / 5 (2) Mar 16, 2016
All I know is -
I brought up Pi day a couple of weeks ago. And I didn't sell a SINGLE one of my "Pi" fork pendants...:-(
- WhydG
Not everyone is a math whiz, so don't take it to heart. You might want to advertise in science and other magazines like "Popular Mechanics" and "Popular Science". You should also think of making designs in plastics, which are easier to work with than sterling...or a combination thereof, if possible. e.g. silver encased in plastic.
obama_socks
not rated yet Mar 16, 2016
torbjorn_b_g_larsson
5 / 5 (1) Mar 19, 2016
Humble makes a silly slight-of-hand in order to publish a trivial article.

If the transcendent number pi has random digits (which it likely has), it has an infinite number of them. Then you _expect_ an infinite number of patterns therein, as in any other random noise description.

ADDED: Reading the thread now, I am ninja'ed by aap.
baudrunner
not rated yet Mar 19, 2016
21.99113 ÷ 7 = 3.14159

I wonder if there is a pattern in dividends that use 7 as the divisor to produce pi numbers with finite numbers of decimal places.

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