Rescaled range analysis and the fractal dimension of pi

P. Vanouplines

University Library, Free University Brussels,
Pleinlaan 2, 1050 Brussels, Belgium

Historical facts about pi

The Indiana House of Representatives tried to fix by law the value of pi to 4 or 64/25 in Bill 246 of 1897. The motion was accepted, but did not pass the senate due to a lucky intervention by Professor C.A. Waldo of the Purdue University (Borwein and Borwein, 1987, p 342).

The knowledge of the number of digits of pi has increased over the years. This evolution was certainly not linear or exponential, as is shown in Table 1. The number of known digits has dramatically increased since the advent of computers. But it is remarkable that more than 500 digits where obtained by manual (and mental) computations. It would be most interesting to add a next column in Table 1, showing the total time of computation for each 'record'. The data in Table 1 are compiled from: Almkvist and Berndt (1988); Bailey (1988); Bailey, Borwein and Plouffe (1995); Borwein and Borwein (1986); Borwein, Borwein and Bailey (1989); Burton (1984); Chudnovsky and Chudnovsky (1989); Kanada (1995a); Kanada (1995b); Knuth (1971).

Table 1 Knowledge of the decimals of pi in dates
Year Author Decimals
+/-2000 BC Egyptian Rhind (or Ahmes) Papyrus 1
+/-1000 BC Egyptians and Babylonians 2
+/-500 BC Bible (1 Kings 7:23) 0
+/-250 BC Archimede 1
+/-150 Ptolemaeus 3
+/-490 Tsu Chhung-Chih 6
+/-500 Aryabhata 3
+/-1430 Al-Khashï 16
+/-1600 Ludolph van Ceulen 34
1666 Newton 15
+/-1700 Sharp (E. Halley) 71
1706 Machin 100
1844 Johann Dase 205
1853 William Shanks 526
1945 D.F. Ferguson 530
1947 D.F. Ferguson 808
1949 Metropolis, Reitweiser and Von Neumann 2,037
1957 Felton 7,480
1958 Genuys 10,000
1962 D. Shanks and Wrench 100,000
1967 J. Guillod 500,000
1973 Guillard and Bouyer 1,000,000
1981 Miyoshi and Kanada 2,000,000
1981 J. Guillod 2,000,000
1982 Y. Tamura and Y. Kanada 4,196,393
1982 Y. Tamura and Y. Kanada 8,000,000
1983 Kanada, Tamura, Yoshino and Ushiro 10,013,395
1983 Kanada, Tamura, Yoshino and Ushiro 16,000,000
1985 W. Gosper 17,000,000
Jan.1986 D.H. Bailey 29,360,000
Sep.1986 Kanada 33,554,000
Jan.1987 Kanada 134,217,728
Jan.1988 Kanada 201,326,000
end 1988 D.V. Chudnovsky and G.V. Chudnovsky 480,000,000
Jul.1989 D.V. Chudnovsky and G.V. Chudnovsky 1,011,196,691
Sep.1989 D.V. Chudnovsky and G.V. Chudnovsky 1,130,000,000
Nov.1989 Kanada 1,073,741,799
Jun.1995 Kanada 3,221,220,000
Aug.1995 Kanada 4,294,960,000
Oct.1995 Borwein (hexadecimal expansion) 10,000,000,000

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Last updated on Sunday 19 November 1995.
©Patrick Vanouplines.