This image has landed on my facebook feed (original source). The lines connect neighboring digits in the decimal representation of Pi for 10000 digits. A lot of people has commented how the Pi is special and beautiful. They are right but the special feature of Pi in this case is a bit hidden. Any sequence of independent random variables with uniform distribution would produce similar result.
There exists a special term for numbers that have this interesting property. These are called normal numbers and wikipedia says this about them:
In mathematics, a normal number is a real number whose infinite sequence of digits in every base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b^2 pairs of digits are equally likely with density b^−2, all b^3 triplets of digits equally likely with density b^−3
The thing is that it is not yet proven if Pi is normal number or not. However it is tested for a vast amount of digits so the first 10000 digits really have the uniform distribution.
Because I couldn’t quickly find the transition probabilities on the internet, I have made a simple statistics myself, you can look at it here:
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I have tested the transition probabilities for consecutive digits of Pi. The output can be understood this way: The rows represent 'from' and the cols represent'to'. Indexing starts from 0. So for example if you wanted to know the probability that 5 would follow after 2. You would look into the third row and the sixth column. Digits tested: 56109 Absolute quantities: 568 577 530 553 539 575 560 570 561 583 587 593 542 552 580 565 558 542 601 585 532 592 532 563 515 572 583 555 530 533 543 570 550 525 577 598 507 542 546 612 565 562 556 563 536 563 591 588 537 552 575 531 588 577 597 579 564 599 563 539 589 542 573 567 577 573 529 540 522 561 538 610 539 577 532 528 605 573 578 536 546 550 568 532 571 600 543 567 571 553 573 577 530 561 589 559 533 540 592 541 Probability matrix: 0,1011 0,1027 0,0944 0,0985 0,0960 0,1024 0,0997 0,1015 0,0999 0,1038 0,1029 0,1039 0,0950 0,0968 0,1017 0,0990 0,0978 0,0950 0,1053 0,1025 0,0966 0,1075 0,0966 0,1022 0,0935 0,1039 0,1059 0,1008 0,0962 0,0968 0,0975 0,1023 0,0987 0,0943 0,1036 0,1074 0,0910 0,0973 0,0980 0,1099 0,1007 0,1001 0,0991 0,1003 0,0955 0,1003 0,1053 0,1048 0,0957 0,0983 0,1007 0,0930 0,1029 0,1010 0,1045 0,1014 0,0987 0,1049 0,0986 0,0944 0,1057 0,0973 0,1028 0,1017 0,1035 0,1028 0,0949 0,0969 0,0937 0,1007 0,0958 0,1086 0,0960 0,1027 0,0947 0,0940 0,1077 0,1020 0,1029 0,0954 0,0975 0,0982 0,1014 0,0950 0,1019 0,1071 0,0969 0,1012 0,1019 0,0987 0,1024 0,1031 0,0947 0,1003 0,1053 0,0999 0,0953 0,0965 0,1058 0,0967 |