“On the shoulders of giant numbers”
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Σ Reference
bigΨ · ref 4
updated: 2013126
Σ.4. Library of Links
The mathematical literature and webpages we've consulted
and recommend for further reading.
To show more library items, select the page extension option

or ++
in the control form on the top left.
The items are marked as:
 Introductory information, suitable for an interested amateur
 Professional books and articles, comprehensive or ground breaking, dry
 Online sources – our earlier webpages are meant for inspiration only
 Entertainment and fiction, religious and philosophical sources
σ.4.1. Big number news, art & fun
 Yee & Kondo's improved New World Record  10 Trillion Digits of Pi 17102011
 Ai Weiwei's 100 million Sunflower seeds at Tate Modern Guardian 11102010
 Carl Bialik The Numbers Guy The World's Best Agers 2372010
 "The 2,000,000,000,000,000th digit is 0" Pi record smashed... BBC News 1692010
 Austin Sendak's Make Hella Official links to CNN, New York Times, etc. 1432010
 "$1,784 billion trillion" Bank claim that is out of this world BBC News 23102009
 Tallest structure on Earth is the Dubai 828 meter Khalifa Tower on 1102009
 Steve's The Googillion Journal Google vs. Googillion 1122007
 Mary & John Gribbin "Big Numbers" 2003
 Kokogiak media – The MegaPenny Project
 Math cats – Can you say really big numbers?

Eames Office film for IBM – APOD YouTube –
Powers of Ten
1968
after the book: Cosmic View by educator Kees Boeke 1957
σ.4.2. Home grown Big numbers
 F.G. Lelieveld "Beatrix matrix system sets number record!" 3042013
 Novaloka "A binary variation with brackets" in functions and matrix 2712011
 Novaloka "Construction of Big and Infinite Numbers" 942009
 Novaloka "Constructie van Grote getallen" [Dutch] 2912009
 Novaloka Binary Big Number Systems the signs and the story 18102008
 Novaloka blog "Spreading fingers across the universe" [Dutch] 2008922
 Novaloka blog "Operator generation as a motor for an operator function" 2582008
 Novaloka & Mathmuis "Plaatsen een Record Getal" [Dutch] 200888
 Novaloka blog "Number distances" Difficult numbers + BigOpera 10000 Jul.2008
 Novaloka blog "Neuropol 0" goes far beyond counting in dimensions 252008
 Novaloka blog "Numbers from 0 to many more..." milk & jazz numbers 2312008
 Novaloka blog series "On the number horizon" Big information theory Oct.2007
 Novaloka essay "One Last Thing" Buddha in recursive progression 2292007
 Novaloka blog "I love big numbers" tetration up to 7^^3 in Java 2462006
σ.4.3. Big numbers on the web
 Christopher M. Bird "Super Huge Numbers" – Bird's array notations, 442006

Jonathan Bowers –
Big Number Central –
Exploding Array Function
Richard Morris tidied up all Bowers' names for large numbers from BEAF.  G.A. Edgar "Transseries: Composition, Recursion, and Convergence" arXiv 792009
 Ioannis Galidakis – Mathematics Lambert's W Function, Infinite Exponents, etc.
 Ioannis N. Galidakis "On Extending hyper4 and Knuth's Uparrow..." 2005
 Peter Hurford – Large numbers part 1, 2, 3, 4, essays on Moser & Conway
 Susan Stepney Factoids ..Steinhaus polygons, Moser polygons.. 2008
 Heiner Marxen Busy Beaver 2009
 Alistair Cockburn's CCWiki with Stephan Houben Really Big Numbers 2007
 Robert Munafo – Large Numbers with references 2009

Andrew Robbins
"Solving the Analytic Extension of Tetration..."
2005
Henryk Trappman, Andrew Robbins "Tetration Forum FAQ" [drafts] 2008  Rubtsov, Romerio "Ackermann's function and new arithmetical operations" 2004
 Sam Hughes on Bowers' linear, planar and multidimensional array notation, 2007
 Sbiis Saibian's blog "1977 IS LONG OVER, AND SO IS G" 2008, and new website

The
Grzegorczyk hierarchy,
extended to the
fastgrowing LöbWainer hierarchy,
the BachmannHoward ordinal from Ordinal collapsing function in Wikipedia.  Wolfram MathWorld – Large Numbers index in online maths encyclopedia
σ.4.4. Big numbers on paper

Wilhelm Ackermann
"On Hilbert's construction of the real numbers"
1928
in Jean van Heijenoort's source book "From Frege to Gödel" 1967 
Archimedes
"The Sand Reckoner"
annotated by James R. Newman, 1956
from: "The works of Archimedes" 3d century BC, by Th. Heath 1897
also in: Hutchins "Great Books of the Western World, vol.11" 1952  Brian Clegg "A brief history of Infinity" 2003
 John H. Conway, Richard K. Guy "The Book of Numbers" 1996, pp.5962
 R. Crandell, C. Pomerance "Prime Numbers, a computational perspective" 2005
 Philip J. Davis "The lore of large numbers" 1961
 Martin Gardner "Mathematical Games" 1977, on Graham's number
 R.L. Graham, B.L. Rothschild "Ramsey's Theorem for nParameter Sets" 1971
 Bhikshu Jin Yong "How Large is One Asamkhyeya?" Nov. 2008

Edward Kasner, James R. Newman "New Names for Old" 1940
in James R. Newman "The world of mathematics, vol.3" 1956, p.2009 
Donald Knuth
"Mathematics and Computer Science: Coping with Finiteness" 1976
in Campbell, Higgins "Mathematics people problems results, vol.2" 1984, pp.210211  David Wells "The Penguin Dictionary of Curious and Interesting Numbers" 1997
σ.4.5. Recursion theory
 C.J. Ash, J.F. Knight "Computable Structures and the Hyperarithmetical Hierarchy" 2000
 Gregory Chaitin "Meta Maths, the quest for omega" 2005
 P. Clote, J.B. Remmel, ed. "Feasible Mathematics II" 1995

Ershov, Goncharov, Nerode, Remmel
"Handbook of Recursive Mathematics"
vol.1 "Recursive model theory" 1998
vol.2 "Recursive algebra, analysis and combinatorics" 1998  M.C. Fitting "Fundamentals of Generalized Recursion Theory" 1981
 R.L. Goodstein "Recursive Number Theory" 1957
 Edward R. Griffor, ed. "Handbook of Computability Theory" 1999
 Andrzej Grzegorczyk "Some classes of recursive functions" 1953
 Peter G. Hinman "RecursionTheoretic Hierarchies" 1978
 A.I. Mal'cev "Algorithms and recursive functions" 1970
 Dag Normann "Recursion on the countable functionals" 1980

Piergiorgio Odifreddi "Classical Recursion Theory" 1989
and his "Classical Recursion Theory Volume II" 1999  Rózsa Péter "Recursive Functions" 1967, 3d ed. (1st ed. German 1951)
 H. Rogers "Theory of recursive functions and effective computability" 1967
 H.E. Rose "Subrecursion, Functions and hierarchies" 1984
 Luis E. Sanchis "Recursive Functionals" 1992
 Andrea Sorbi, ed. "Complexity, logic and recursion theory" 1997
 R.M. Smullyan "Recursion Theory for Metamathematics" 1993
σ.4.6. Foundations
 J.L. Bell, M. Machover "A Course in Mathematical Logic" 1993
 R.D. Carmichael "Groups of Finite Order" 1937
 C.C. Chang, H. Jerome Keisler "Model Theory" 1977
 Dirk van Dalen "Logic and Structure" 4th ed. 2004
 Paul R. Halmos "Naive Set Theory" 1960

David Hilbert "On the infinite" 1925
in: Heijenoort "From Frege to Gödel" 1967  Wilfred Hodges "Building Models by Games" 1985
 Thomas Jech "Set theory" 2002
 Akihiro Kanamori "The Higher Infinite" 1997
 Stephen C. Kleene "Introduction to Metamathematics" 1952
 Ulrich Kohlenbach "Applied Proof Theory: proof interpretations..." 2008
 S. Mac Lane "Categories for the working mathematician" 1997
 S. Mac Lane, I. Moerdijk "Sheaves in Geometry and Logic" 1992
 Jerome Malitz "Introduction to Mathematical Logic" 1979
 Yu. I. Manin "A Course in Mathematical Logic (for Mathematicians)" 2010
 Elliott Mendelson "Introduction to Mathematical Logic" 1964
 J. Donald Monk "Mathematical Logic" 1976
 Wolfram Pohlers "Proof Theory, the first step into impredicativity" 2009
 S. Shelah "Classification Theory" 1990
 Edwin H. Spanier "Algebraic Topology" 1966
 PaulHermann Zieschang "Theory of Association Schemes" 2005
σ.4.7. Mathematics
Gregori Perelman:
"I don't need anything!"
 Stephen Abbott "Understanding Analysis" 2001
 P. Abramenko, K.S. Brown "Buildings: theory and applications" 2008
 Martin Aigner "A Course in Enumeration" 2007
 Jörg Arndt, Christoph Haenel "Piunleashed" 2001
 Avner Ash, Robert Gross "Fearless Symmetry" 2006

Jerome Barkley
Improved lower bound on a Euclidean Ramsey problem
Graham's hypergraph has minimal dimension 13, arXiv 2008  John L. Casti "Five Golden Rules" 1996
 P.M. Cohn "Further Algebra and Applications" 2002
 Mariana Cook "Mathematicians, an outer view..." 2009

Dedekind
"Essays on the Theory of Numbers" 2nd ed. 1893
in Stephen Hawking "God created the Integers" 2007
online: Project Gutenberg ebook, Internet Archive 1901 ed.  Ebbinghaus, Hermes, et al. "Numbers" 1991
 George Gheverghese Joseph "The Crest of the Peacock" 1991
 Gowers, ed. "The Princeton Companion to Mathematics" 2008
 Graham, Rothschild, Spencer "Ramsey Theory" 1990
 Peter D. Grünwald "The Minimum Description Length Principle" 2008
 Julian Havil "Gamma" 2003
 Heikkilä, Lakshmikantham "Monotone iterative techniques..." 1994
 Georges Ifrah "... A universal history of numbers" 1985
 Nathan Jacobson "Basic Algebra I" 1974
 Khalil Jaouiche India's contribution to Arab mathematics 1997
 Eli Maor "e: the Story of a Number" 1998
 JunIti Nagata "Modern Dimension Theory" 1965
 Marcus du Sautoy "Finding Moonshine" / "Symmetry" 2008
 H.A. Thurston "The Number System" 1956
 Eric W. Weinstein "CRC Concise Encyclopedia of Mathematics" 1999
σ.4.8. Science and programming
 Abiteboul, Hull, Vianu "Foundations of Databases" 1995

Darren Aronofsky "
Π
faith in chaos" 1999, film (+book fiction)  Roel Bloo "Preservation of Termination for Explicit Substitution" 1997
 Peter J. de Bruin "Inductive Types in Constructive Languages" 1995
 Doets, van Eijck "The Haskell road to logic, maths and programming" 2004
 David Flanagan "JavaScript, the definitive guide" O'Reilly 2006
 Franklin, Powell, EmamiNaeini "Feedback Control of Dynamical Systems" 2002
 Jeffrey Friedl "Mastering Regular Expressions" O'Reilly 2006
 Douglas R. Hofstadter "Gödel, Escher, Bach" 1979
 Jan van Leeuwen "Rulelabeled programs" 1972
 Seth Lloyd "Computational capacity of the universe" arXiv 24102001
 Nielsen, Chuang "Quantum Computation and Quantum Information" 2000
 Jennifer Niederst Robbins "Web Design in a Nutshell" O'Reilly 2006
 Sullivan, Goerzen, Stewart "Real World Haskell" O'Reilly 2008
 Hermann Weyl "The Theory of Groups and Quantum Mechanics" 1931
 Ludwig Wittgenstein "Tractatus logicophilosophicus" 6.4, 1918
σ.4.9. Religious literature

John Daido Loori "The True Dharma Eye" 2005,
case 300 (Linji d. 867)
comments on: Zen master Dogen's 300 koans, collection 1227 
John Stevens "The Marathon Monks of Mount Hiei" 1988
Cites on kaihôgyô (founder Sôô 859) "The Story of Wandering Saints" 1387  Fred "Philémon – Simbabbad de Batbad" 1974 (Dutch)

Yram "Practical Astral Projection" ch.8,
translation of "Le Médicin de l'Ame" 1925
Yram is the mystic Marcel Fourhan (18841927), electrical engineer in Shanghai 
B. Malinowski
"Baloma; the Spirits of the Dead in the Trobriand Islands" 1916
Journal of the Royal Anthropological Institute of Great Britain and Ireland, vol.46
in his: "Magic, Science and Religion" 1954 (courtesy P.C. Willis), p.161 
Basho "Journal of a weathered skeleton" 1684, haiku 1,9,15
see: Nobuyuki Yuasa "The Narrow Road to the Deep North" 
Sri Caitanya "Šrî Brahmasamhitâ"
16th century, vs.40
from: Joan Wilder "The Hare Krishna Music Book" 1994, p.48 
Keizan "Denkoroku" 14th century, case 30 (Huike 6th century)
translation: Thomas Cleary "Transmission of Light"
in: Classics of Buddhism and Zen vol.4, Shambhala 2001 
Nichiren Shonin "Letter to Niike" 1280
in: "The major writings of Nichiren Daishonin" vol.1 1991, p.260
quotes: "Mahâparinirvâna sûtra" 2nd? century 
Wumen Huikai (Mumon Ekai) "Wumenguan" (Mumonkan) 1228, case 3
translation: Thomas Cleary "Unlocking the Zen Koan" 1993 
Snorri Sturluson "Gylfaginning" (in Prose Edda) 1220, p.XXXVIII
Quoted poem: "Edda" Grímnismál 20 (Codex Regius 13th century) 
Wansong "Ts'ung jung lu" (Shoyo Roku) 12th century compilation, case 3
in: Thomas Cleary "Book of Serenity" One hundred Zen dialogues 1988 
Xuedou Chongxiang (Setchô) "Piyen lu" (Hekigan Roku)
11th century, case 3
after: R.D.M. Shaw "The Blue Cliff Records" 1961
and: Katsuki Sekida "Two Zen Classics" 1977 
The "Great Vows" of the Bodhisattva, as chanted in Zen buddhism
own rendering 2009, based on: D.T. Suzuki "Manual of Zen Buddhism" 1950 &
Maha Karuna & Kanzeon Sangha "Shiguseigan" & San Francisco Conference 1997 
"The Sutra of Complete Enlightenment" 6th? century
in: Lu K'uan Yü "Ch'an and Zen Teaching" vol.3, 1962 
"Prajñâpâramitâ Hrdaya sûtra"
(Heart Sutra) 5th? century
in: Thich Nhat Hanh "The Heart of Understanding" 1988 
"Vajrachchedikaprajñâpâramitâ sûtra"
4th? century
compare: "The Diamond Sutra and the Sutra of Huineng" 1990, ch.11 
"Buddhâvatamsakamahâvaipulya sûtra"
3d? century
translation: Thomas Cleary "The Flower Ornament Scripture" 19841993
with: chapter 30 "The Incalculable" annotated by Novaloka 2010 
"Vimalakîrtinirdeša sûtra"
(Wei mo chieh so shuo ching) 2nd? century
from: Charles Luk "The Vimalakirti Nirdesa Sutra" 1972, ch.4 
"Lalitavistara sûtra" 1st? century,
as "The Voice of the Buddha" 1983, ch.12
with: "The mathematical test" annotated by Asamkhyeya dasa 2011  "Lañkâvatâra sûtra" 1st? century, translation: D.T. Suzuki 1932, ch.2:17
 "The Gospel of Mary" 2nd? century, in: Barnstone, Meyer "The Gnostic Bible" 2003

King James version of the Holy Bible,
St. Mark "The Gospel" 1st century, ch.8:19
in Novak from: The Holy Bible NRSV1989, "Gospel of Matthew" ch.19:2326
from Aramaic via Dutch back to English – Mark 15:34, Matthew 27:46
St. John the Divine "Book of Revelation" 2nd? century, ch.13:18 (number 666) 
Apollodorus "The Library" 1st/2nd century,
modelled on a lost 5th century BC mythography by Pherecydes of Leros?
in: Atchity "The Classical Greek Reader" 1996, p.332  "Anuyogadvâra sûtra" (Mûlagrantha 4, Švetâmbara canon) Jaina 1th? century

"Bhagavad Gîtâ" 0th? century, ch.11.13
A.C. Bhaktivedanta Swami Prabhupâda "Bhagavad Gita As It Is" 1968 
"Saddharmapundarîka sûtra"
(Lotus Sutra) 1st? century BC
translation Bunno Kato, et al: "The Threefold Lotus Sutra" 1975, ch.25
online: Digital Sanskrit Buddhist Canon, in article: "Drops of ink and dust..." 
Chuang Tsu "Nuahua chen ching" 4th? century BC
translation: GiaFu Feng "Chuang Tsu, inner chapters" 1974, ch.1 
Confucius "Li chi" 5th? century BC
from: Philip Novak "The World's Wisdom" 1994, ch3.2b
Li Ji  Li Yun (Book of Rites  Evolution of Rites) 
Lao Tsu "Tao te ching" 6th? century BC, ch.1
free after Novak, Blok (NL) and a copy (DE) in: "I, WILL McBRIDE" 1997, p.196 
The Buddha "Udana" 8.3, 6th? century BC
after: Philip Novak "The World's Wisdom" 1994, ch2.34e 
"Atharva Veda" 11th? century BC, hymn 8, vs.710
online: Ralph T.H. Griffith – the Atharva Veda 1895 
"Rg Veda" 11th? century BC, book 10, hymn 90, vs.12
in: Sacred Writings vol. 5 "The Rig Veda", translation: Ralph T.H. Griffith 1896?
SENNICHI KAIHÔGYÔ The 1000day Mountain Marathon of Hiei (& Imuro Valley) 


year  days  track  features 
source: John Stevens, The Marathon Monks of Mount Hiei  
1  100 
30 (40) km each day, 255 stations of worship. 1day Kyoto run, 54 km 
Clock: 1:30 ~ 8:30 "Freshman" No socks yet, hat carried. 
2  100  
3  100  
4  100 
Gyôja on the road clothed in the colour of death wearing straw sandals. 30 (40) km daily, 1 day Kyoto run 
Socks and hat permitted. Upon completion: "WhiteBelted Ascetic" 
100  
5  100 
Wooden staff permitted. On 700th day dôiri: 7½ days without food, water or rest. "Master of Severe Practice" 

100  
6  100  60 km each day  Sekisan Marathon (steep) 
7  100  84 km each day  Great Marathon (0:30 ~ 17:00) 
100  30 (40) km, 1 Kyoto run  "Saint of the Highest Practice"  
1000 days  38,632 (46,572) km  
follows: Great 100,000Prayer FireCeremony  6½ day fast for Fudô Myôô 