|Part of a series of articles on the|
|mathematical constant π|
Piphilology comprises the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant π. The word is a play on the word "pi" itself and of the linguistic field of philology.
There are many ways to memorize π, including the use of piems (a portmanteau, formed by combining pi and poem), which are poems that represent π in a way such that the length of each word (in letters) represents a digit. Here is an example of a piem: "How I need a drink, alcoholic in nature after the heavy lectures involving quantum mechanics." Notice how the first word has three letters, the second word has one, the third has four, the fourth has one, the fifth has five, and so on. In longer examples, 10-letter words are used to represent the digit zero, and this rule is extended to handle repeated digits in so-called Pilish writing. The Cadaeic Cadenza records the first 3834 digits of π in this manner, and a 10,000-word novel, Not A Wake, has now been written accordingly.
However, piems prove to be inefficient for large memorizations of pi. Other methods include remembering patterns in the numbers (for instance, the year 1971 appears in the first fifty digits of pi) and the method of loci (which has been used to memorize π to 67,890 digits).
- 1 History
- 2 Examples in English
- 3 Examples in other languages
- 4 Memorization record holders
- 5 See also
- 6 Notes and references
- 7 External links
Until the 20th century, the number of digits of pi which mathematicians have had the stamina to calculate by hand remained in the hundreds, so that memorization of all known digits at the time was possible. In 1949 a computer was used to calculate π to 2000 places, presenting one of the earliest opportunities for a more difficult challenge.
Later computers calculated pi to extraordinary numbers of digits (2.7 trillion as of August 2010) , and people began memorizing more and more of the output. The world record for the number of digits memorized has exploded since the mid-1990s, and it stood at 100,000 as of October 2006. The previous record (83,431) was set by the same person (Akira Haraguchi) on July 2, 2005, and the record previous to that (42,195) was held by Hiroyuki Goto. An institution from Germany provides the details of the “Pi World Ranking”; see the website at http://www.pi-world-ranking-list.com.
Examples in English
The most common mnemonic technique is to memorize a so-called "piem" (a wordplay on "pi" and "poem") in which the number of letters in each word is equal to the corresponding digit of π. This famous example has several variations, including:
- How I need a drink, alcoholic in nature, after the tough chapters involving quantum mechanics!
Short mnemonics such as these, of course, do not take one very far down π's infinite road. Instead, they are intended more as amusing doggerel. If even less accuracy suffices, the following examples can be used:
- How I wish I could recollect pi easily today!
- May I have a large container of coffee, cream and sugar?
This second one gives the value of π as 3.1415926535, while the first only brings it to the second five. Indeed, many published poems use truncation instead of one of the several roundings, thereby producing a less-accurate result when the first omitted digit is greater than or equal to five. It is advantageous to use truncation in memorizing if the individual intends to study more places later on, otherwise one will be remembering erroneous digits.
Another mnemonic is:
- The point I said a blind Bulgarian in France would know
In this mnemonic the word "point" represents the decimal point itself.
Yet another example is:
- How I wish I could recollect, of circle round, the exact relation Arkimedes (or Archimede) learned
In this example, the spelling of Archimedes is normalised to nine. (Although 'Archimedes' is, today, a more correct spelling of the ancient Greek mathematician's name in English, Archimede is also often seen when this mnemonic is given, since Archimède is the more correct spelling in many languages, such a French.)
Longer mnemonics employ the same concept. This example created by Peter M. Brigham incorporates twenty decimal digits:
- How I wish I could enumerate pi easily, since all these bullshit mnemonics prevent recalling any of pi's sequence more simply.
In the children's book, 'Somewhen' (David Saul and Danielle Mathieson, ISBN 9780473218584) a poem is presented as a riddle. Here, the words describe the ratio and as laid out, the riddle forms a circle. To side-step the zero at decimal position 32, the word 'nothing' is used.
- It's a fact
- A ratio immutable
- Of circle round and width
- Produces geometry's deepest conundrum
- For as the numerals stay random
- No repeat lets out its presence
- Yet it forever stretches forth
- Nothing to eternity.
- For as the numerals stay random
Some mnemonics, such as this poem which gives the three and the first 20 decimal digits, use the separation of the poem's title and main body to represent the decimal point:
- I wish I could determine pi
- Eureka, cried the great inventor
- Christmas pudding, Christmas pie
- Is the problem's very center.
Another, more poetic version is:
- Sir, I have a rhyme excelling,
- In mystic power and magic spelling,
- Celestial spirits elucidate,
- For my own problems can't relate.
Extensions to 30 or 31 decimals of the same proceed as follows:
There are minor variations on the above rhyme, which still allow pi to be worked out correctly. However, one variation replaces the word "lexicon's" with "lesson's" and in doing so, incorrectly indicates that the 18th digit is seven.
- Now, a moon, a lover refulgent in flight,
- Sails the black silence's loneliest ellipse.
- Computers use pi, the constant, when polite,
- Or gentle data for sad tracking aid at eclipse.
- Now I defy a tenet gallantly
- Of circle canon law: these integers
- Importing circles' quotients are, we see,
- Unwieldy long series of cockle burs
- Put all together, get no clarity;
- Mnemonics shan't describeth so reformed
- Creating, with a grammercy plainly,
- A sonnet liberated yet conformed.
- Strangely, the queer'st rules I manipulate
- Being followéd, do facilitate
- Whimsical musings from geometric bard.
- This poesy, unabashed as it's distressed,
- Evolvéd coherent - a simple test,
- Discov'ring poetry no numerals jarred.
Note that in this example, 10-letter words are used to represent the digit zero.
- dreams number us like pi. runes shift. nights rewind
- daytime pleasure-piles. dream-looms create our id.
- moods shift. words deviate. needs brew. pleasures rise.
- time slows. too late? wait! foreign minds live in
- us! quick-minds, free-minds, minds-we-never-mind,
- unknown, gyrate! neuro-rhymes measure our
- minds, for our minds rhyme. crude ego-emanations
- distort nodes. id, (whose basic neuro-spacetime rhymes),
- plays its tune. space drones before fate unites
- dreams’ lore to unsung measures. whole dimensions
- gyrate. new number-games donate quick minds &
- weave through fate’s loom. fears, hopes, digits, or devils
- collide here—labor stored in gold-mines, lives, lightcone-
- piles. fate loops through dreams & pleasure-looms….
Note that "dreams number us like pi" corresponds to "314159," and so on. Sound-based mnemonic techniques, unlike pilish, do not require that the letters in each word be counted in order to recall the digits of pi. However, where sound-based mnemonics use assonance, extra care must be taken to distinguish "nine" and "five," which contain the same vowel sound. In this example, the author assumes the convention that zero is often called "O."
The piku follows the rules of conventional haiku (three lines of 5, 7 and 5 syllables), but with the added mnemonic trick that each word contains the same number of letters as the numerals of pi, i.e.
- How I love a verse
- Contrived to unhusk dryly
- One image nutshell
- Man, I can’t - I shan’t! - formulate an anthem where the words comprise mnemonics, dreaded mnemonics for pi.
In 2013, Huang extended the song to include the first 100 digits of pi, and changed the title to "Pi Mnemonic Song".
There are piphilologists who have written texts that encode hundreds or thousands of digits. This is an example of constrained writing, known as "Pilish". For example, Poe, E.: Near a Raven represents 740 digits, Cadaeic Cadenza encodes 3835, and Not A Wake extends to 10,000 digits.
It is also possible to use the rhythm and sound of the spoken digits themselves as a memorization device. The mathematician John Horton Conway composed the following arrangement for the first 100 digits,
_ _ _ 3 point 1415 9265 35 ^ ^ _ _ _ _ _ _ __ 8979 3238 4626 4338 3279 ** **^^ ^^ **** . _ _ __ _ _ _ . _ . 502 884 197 169 399 375 105 820 974 944 ^ ^ ^ ^ 59230 78164 _ _ _ _ 0628 6208 998 6280 ^^ ^^ ^^ .. _ .._ 34825 34211 70679, ^ ^
where the accents indicate various kinds of repetition.
Another mnemonic system used commonly in the memorization of pi is the Mnemonic major system, where single numbers are translated into basic sounds. A combination of these sounds creates a word, which can then be translated back into numbers.  When combined with the Method of loci, this becomes a very powerful memorization tool.
Examples in other languages
|This section does not cite any references or sources. (September 2010)|
- خرد و دانش و آگاهی دانشمندان
- .ره سرمنزل مقصود بما آموزد
- The wisdom, science, and knowledge of scholars,
- will show us the way to destination.
Note: By counting the number of each letter in each word you are able to memorize pi to the 10th decimal point. ex: خرد (kherad) = 3, و (va) = 1, دانش (daanesh) = 4, و (va) = 1, آگاهی (aagaahi) = 5, ...
- Nem a régi s durva közelítés,
- Mi szótól szóig így kijön
- Betűiket számlálva.
- Ludolph eredménye már,
- Ha itt végezzük húsz jegyen.
- De rendre kijő még tíz pontosan,
- Azt is bízvást ígérhetem.
- It is not the old and rough approximation,
- What comes out word by word
- Counting their letters.
- Ludolph's result is already here,
- If we do it, on twenty digits.
- But come out ten more precisely
- I also promise definitely.
An interesting (not math themed) alternative:
- Bír-e, érez-e ember nyugalmat,
- Ha lelkét nehéz bús emlék zaklatja.
- Szüntelen felhőbe burkolózó idő az,
- Ami változni ámha akarna se tudhat,
- Mert azt nem írhattya már le halandó kívánsága.
- Whether has, whether feels a man a peace of mind
- If his soul is harassed by heavy, sad memories.
- The continuously clouded time is,
- What cannot change although it want,
- Because it cannot be written by a mortal's desire.
- Íme a szám: a görög periféria pi betűje.
- Euler meg Viète végtelen összeggel közelít értékéhez.
- Lám, őt már Egyiptom, Kína, Európa is akarta, hogy
- „ama kör kerülete úgy ki lehetne számlálva”.
- Kur i hyej, e kryej, sigurisht, po përtoj andaj nuk fitoj — "If I start dealing with it, I will do it, but I am lazy therefore I do not win."
This statement yields π to twenty-two decimal places:
- Wie, o dies π macht ernstlich so vielen viele Müh. Lernet immerhin, Kinderlein, leichte Verselein, wie zum Beispiel dieses dürfte zu merken sein.
English translation that doesn't encode pi:
- How, oh this π seriously makes so many struggles to so many. Learn at least, little children, simple little verses, just such as this one should be memorizable.
Looser English translation that encodes pi:
- Woe! O this π makes seriously so muchly many's woe.
- Que j’aime à faire apprendre un nombre utile aux sages !
- Immortel Archimède, artiste ingénieur,
- Qui de ton jugement peut priser la valeur ?
- Pour moi, ton problème eut de pareils avantages.
- Jadis, mystérieux, un problème bloquait
- Tout l’admirable procédé, l’œuvre grandiose
- Que Pythagore découvrit aux anciens Grecs.
- Ô quadrature ! vieux tourment du philosophe !
- Insoluble rondeur, trop longtemps vous avez
- Défié Pythagore et ses imitateurs.
- Comment intégrer l’espace plan circulaire ?
- Former un triangle auquel il équivaudra ?
- Nouvelle invention : Archimède inscrira
- Dedans un hexagone ; appréciera son aire,
- Fonction du rayon. Pas trop ne s’y tiendra :
- Dédoublera chaque élément antérieur ;
- Toujours de l’orbe calculée approchera ;
- Définira limite ; enfin, l’arc, le limiteur
- De cet inquiétant cercle, ennemi trop rebelle !
- Professeur, enseignez son problème avec zèle !
- How I like to teach this number useful to the wise.
- Immortal Archimedes, artist, engineer,
- in your opinion who could estimate its value?
- For me, your problem had equal advantages.
- Long ago, mysterious, a problem blocked
- All the honorable process, the great work
- That Pythagoras revealed to the Ancient Greeks.
- Oh quadrature! Old philosopher's torment
- Unsolvable roundness, for too long you have
- Defied Pythagoras and his imitators.
- How to integrate the plain circular space?
- Form a triangle to which it is equivalent?
- New invention: Archimedes will inscribe
- Inside a hexagon; will appreciate its area
- Function of a ray. Not too much to hold onto there:
- Will split each previous element;
- Always the calculated orb will approach
- Will define the limit; finally, the arc, the limiter
- Of this disturbing circle, an enemy too rebellious
- Teacher, teach its problem with zeal
An alternative beginning:
- Que j’aime à faire apprendre un nombre utile aux sages !
- Glorieux Archimède, artiste ingénieur,
- Toi de qui Syracuse aime encore la gloire,
- Soit ton nom conservé par de savants grimoires !
Katharevousa (archaizing) Greek
Yielding π to 22 decimal places:
- Ἀεὶ ὁ Θεὸς ὀ Μέγας γεωμετρεῖ,
- τὸ κύκλου μῆκος ἵνα ὁρίσῃ διαμέτρῳ,
- παρήγαγεν ἀριθμὸν ἀπέραντον,
- καὶ ὅν, φεῦ, οὐδέποτε ὅλον θνητοὶ θὰ εὕρωσι
- The Great God applies geometry forever;
- To define the length of the circle using its diameter,
- He produced an infinite number,
- Which, alas, mortals will never find in its entirety.
- Fue y cayó. Y queda solamente la inútil cifra con pocos destinos poderosos, tristes devenires sin el más sencillo bien. Idiota, re idiota, sabe que sus encantos son ya latosos decimales. Pobre...
- It went and it fell. And only the useless figure remains, with little powerful destinies, sad future without the simplest goodness. Idiotic, very idiotic, it knows that its charms are now boring decimals. Poor...
Another. This piem gives π (correctly rounded) to 10 decimal places. (If you prefer to not round π, then replace "cosmos" with "cielo".)
- Sol y luna y mundo proclaman al eterno Autor del Cosmos.
- Sun and moon and world proclaim the eternal Author of the Cosmos. (Or "heaven", not Cosmos, if using "cielo".)
Níl i mata, a shaoi, eolaíocht nó feidhm. (7 decimal places) — "Wise one, mathematics has neither science nor use."
One of the Romanian versions of Pi poems is:
- Dar o ştim, e număr important ce trebuie iubit
- But we know, it's an important number which should be loved
- Din toate numerele însemnate diamant neasemuit,
- Of all the significant numbers it is a peerless diamond
- Cei ce vor temeinic asta preţui
- Those who will sincerely appreciate it
- Ei veşnic bine vor trăi.
- Will live happy for ever.
There is another phrase known in Romanian that will help to memorize the number by eight decimal places: Aşa e bine a scrie renumitul şi utilul număr. — "This is the way to write the renowned and useful number."
In the Russian language, there is a well-known phrase in the reform of 1917 orthography of old tradition: "Кто и шутя, и скоро пожелаетъ «Пи» узнать число — ужѣ знаетъ." (The one who would wish to know the number pi easily and quickly already knows it.)
A more modern rhyme is:
- Это я знаю и помню прекрасно,
- I know the following and remember it perfectly,
- Пи многие знаки мне лишни, напрасны.
- Multitudes of the digits of pi are unnecessary and idle for me.
A short approximation is: "Что я знаю о кругах?" (What do I know about circles?)
In addition, there are several nonfolklore verses that simply rhyme the digits of pi "as is"; for examples, see the Russian version of this article.
Under verse of polish mathematician Witold Rybczyński:
- "Daj, o pani, o boska Mnemozyno, pi liczbę, którą też zowią ponętnie ludolfiną, pamięci przekazać tak, by jej dowolnie oraz szybko do pomocy użyć, gdy się problemu nie da inaczej rozwiązać, pauza - to zastąpić liczbami." (35 decimal places) - "Give, the lady, the divine Mnemosyne, pi number, which also is called enticingly ludolfina memory to pass so that it freely and quickly to help you use when the problem can not be otherwise resolved, pause - (dash means nil) is replaced by the numbers."
Under verse of polish mathematician Kazimierz Cwojdziński:
- „Kuć i orać w dzień zawzięcie, bo plonów niema bez trudu. Złocisty szczęścia okręcie kołyszesz … Kuć. My nie czekajmy cudu. Robota to potęga ludu." (23 decimal places) - "Hammer and plow in a day obstinately, because not fruits without trouble. Golden happiness ship rocking ... Hammering. We do not let us wait miracle. Labor is the power of the people.:"
- Był i jest i wieki sławionym ów będzie, który kół obwód średnicą wymierzył. (12 decimal places) — "There was, and there is, and through centuries renowned will be, who circle's circumference measured with its diameter."
- "Kto z woli i myśli zapragnie Pi spisać cyfry, ten zdoła" (10 decimal places) - "Who will and mind wishes Pi write the numbers, this succeed."
Under is occasional verse related to Mundial Argentina and polish football team.
- "Już i Lato i Deyna strzelili do bramki obcej dwa karne. Lubański dostrzegł mistrza Szarmacha, gdy on tak wypuścił cios szacha, że zdobyć musi cel gry krzyknął Gol na Mundial Argentyna." (30 decimal places). - "Already and Lato and Deyna scored two goals to foreign football goal. Lubański saw master Szarmach so when he released the hit of the check, the gain must be goal of the game shouted goal at the World Cup Argentina.
- Cai a neve e novas ferrovias de marfim serão por casas trocadas. (11 decimal places) — "The snow falls and new ivory railroads will be exchanged by houses."
- Com o zero o lente reprovará os alunos. (8 decimal places) — "With zero the university professor will fail the students."
Or in Brazilian Portuguese:
- Sim, é útil e fácil memorizar um número grato aos sábios. — "Yes, it's useful and easy to memorize a number dear to the wisemen."
- Nós e todo o mundo guardamos pi usando letra por número. — "We and all the world memorize pi using letter for number."
- Sou o medo e temor constante do menino vadio. — "I'm the constant fear and dread of the stray boy."
A piem written in a more poetic manner:
- Sou o amor,
- o homem impetuoso da libido
- Homem que ataca mulheres atraentes,
- meninas pecadoras que no céu imiscuem amor, paixão, fé, desejo, tudo!
- Até que idolatro com as sereias pecadoras tanta fé!
- Esbeltas mulheres para o musculado,
- sereias e fêmeas pecadoras
- Até idolatram serpentes com ardente macho.
- O viril desejará as pecadoras iníquas doravante para amar.
- I am the love,
- The impetuous man from the libido
- Man who attacks attractive women,
- sinfull maidens who on heaven intrude love, passion, faith, desire, everything!
- I even idolize with the mermaids so much faith!
- Luscious women for the brawny,
- sinfull mermaids and females
- They even idolize serpents with the burning buck.
- The virile man will wish the sinfull and the iniquitous henceforth to love.
Japanese piphilology has countless mnemonics based on punning words with numbers. This is especially easy in Japanese because there are two or three ways to pronounce each digit, and the language has relatively few phonemes to begin with. For example, to 31 decimal places:
身 一つ 世 一つ 生 く に 無 意 味 い わ く な く 身 ふ み や 読 む 似 ろ よ さん ざん 闇 に な く 3. 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 mi hitotsu yo hitotsu iku ni mu-imi iwakunaku mi fumiya yomu niro yo san zan yami ni naku
This is close to being ungrammatical nonsense, but a loose translation prioritizing word order yields:
- A person is one; the world is one:
- to live this way, it's meaningless, one says, and cries,
- "step on it, will ya!" then reads—be the same!
- Crying uncontrollably in the dark.
Japanese children also use songs built on this principle to memorize the multiplication table.
It is possible to construct piphilogical poems in Chinese by using homophones or near-homophones of the numbers zero through nine, as in the following well known example which covers 22 decimal places of π. In this example the character meaning "mountain" (山 shān) is used to represent the number "three" (三 sān), the character meaning "I" (吾 wú) is used to represent the number "five" (五 wǔ), and the characters meaning "temple" (寺 sì) and "die" (死 sǐ) are used to represent the number "four" (四 sì). Some of the mnemonic characters used in this poem, for example "kill" (殺 shā) for "three" (三 sān), "jug" (壺 hú) for "five" (五 wǔ), "happiness" (樂 lè) for "six" (六 liù) and "eat" (吃 chī) for "seven" (七 qī), are not very close phonetically in Simplified Mandarin Chinese.
山 巔 一 寺 一 壺 酒 shān diān yī sì yī hú jiǔ 3 . 1 4 1 5 9 爾 樂 苦 煞 吾 ěr lè kǔ shā wú 2 6 5 3 5 把 酒 吃 酒 殺 爾 bǎ jiǔ chī jiǔ shā ěr 8 9 7 9 3 2 殺 不 死 樂 爾 樂 shā bù sǐ lè ěr lè 3 8 4 6 2 6
This can be translated as:
- On a mountain top a temple and a jug of wine.
- Your happiness makes me so bitter;
- Take some wine and drink, the wine will kill you;
- If it does not kill you, I will rejoice in your happiness.
Sen, o alan o çevre bölününce ve sonsuz rakam ile çıkan değişken dizilimli sayısın. — "You are the number with infinite digits in changing order, which is found when the circumference and the area is divided."
Sám u sebe v hlavě magického pí číslic deset mám. (nine decimal places) — "I have ten digits of magical pi in my head."
Lín a kapr u hráze prohlídli si rybáře, udici měl novou, jikrnáči neuplovou. (12 decimal places) — "Tench and carp by the dam watched the fisher. He has a new rod, fish will not escape."
Dej, ó Bože, a číslo Ludolfovo já navždy pomnu, pro větší naplnění moudrosti početní. (13 decimal places) — "Oh God, let me to remember the pi forever, for the increase of mathematical skills."
Mám ó bože ó velký pamatovat si takový cifer řad, velký slovutný Archimedes, pomáhej trápenému, dej mu moc, nazpaměť nechť odříká ty slavné sice, ale tak protivné nám, ach, číslice Ludolfovy! (30 decimal places) — "Shall I, God oh almighty, remember such a long string of numbers, great and famous Archimedes, help my careworn being, give me the power, to recite by heart all the digits, which may be famous, but also hated by some of us, the digits of Ludolph van Ceulen."
Čak i Grci i stari Vavilonci su kazali: obime kad deliš krugovim prečnikom dobijaš neophodan nam pi. (16 decimal places) — "Even Greeks and Old Babylonians have told: when dividing circumferences with circle's diameter you obtain the indispensable pi."
Non è dato a tutti ricordare il numero aureo del sommo filosofo Archimede. Certuni sostengon che si può ricordare tale numero, ma questi poi non recitano che un centone insensato. (29 decimal places) — "Not anybody can retain the golden number of the great philosopher Archimedes. There are who claim it is possible to recall this number, but then they just recite a senseless cento"
Memorization record holders
Even before computers calculated π, memorizing a record number of digits became an obsession for some people. The current Guinness-recognized record for remembered digits of π is 67,890 digits, held by Lu Chao, a 24-year-old graduate student from China. It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of π without an error. On October 3, 2006, Akira Haraguchi reportedly recited the first 100,000 decimal places of π. Daniel Tammet of the United Kingdom holds the European record for quoting pi to the most places, quoting pi to 22,514 digits.
Notes and references
- "Not A Wake: A Dream Embodying π's Digits Fully For 10000 Decimals". Retrieved 2011-03-03.
- Raz A, Packard MG, Alexander GM, Buhle JT, Zhu H, Yu S, Peterson BS. (2009). A slice of pi : An exploratory neuroimaging study of digit encoding and retrieval in a superior memorist. Neurocase. 6:1-12. doi:10.1080/13554790902776896 PMID 19585350
- "Pi calculation smashes records". Retrieved 2010-05-12.
- O'Connor, J J; Robertson, E F. "A history of Pi". Retrieved 2007-02-02.
- (Japanese) "(broken link)". Yahoo! News. Retrieved 2007-02-02.[dead link]
- "Japanese breaks pi memory record". BBC News. London. 2005-07-02. Retrieved 2007-02-02.
- Mnemonics for the number pi
- Mnemonic Techniques for Numbers
- Borgmann, Dmitri A. (1965). Language on Vacation: An Olio of Orthographical Oddities. Charles Scribner's Sons.
- Doerfel, Joel. "Sonnet 72 (Ode to Pi)". Retrieved 2012-04-26.
- "Poe, E.: Near a Raven", cadaeic.net
- "Not A Wake", cadaeic.net
- "Math Forum » Discussions » History » math-history-list", Math Forum
- , Major Memory System
- "Poezii pentru numărul Pi" (in Romanian). Secretele lui Lovendal. 2009-01-02. Retrieved 2012-12-19.
- (Japanese) 暗記法 Japanese mnemonics for π (and some other languages as well)
- Otake, Tomoko (2006-12-17). "How can anyone remember 100,000 numbers?". The Japan Times. Retrieved 2007-10-27.
- David Fiore's record is memorialized: http://www.pi-world-ranking-list.com/lists/details/fiore.html