Mike Keith

April 1999

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Martin Gardner's fictional "Doctor Matrix" used to say that,
properly interpreted, the number pi (the ratio of the circumference
of a circle to its diameter, whose decimal expansion begins
3.14159265358979323846...) contains the entire history of mankind.
In this article I give some results of looking at pi in a
relatively new way: as an infinite string of letters derived from
its expansion in base 26 or base 27.
(Side note: Ivars Peterson's MathTrek column for April 2000 reports
on some of these findings in a tongue-in-cheek style very reminiscent
of the aforementioned Dr. Matrix.)
```**Base 26**
Base 26 is one of two fairly natural ways of representing numbers
as text using a 26-letter alphabet. The number of interest is
expressed numerically in base 26, and then the 26 different base-26
digits are identified with letters as 0=A, 1=B, 2=C, ... 25=Z.
Here are the first 100 digits of pi expressed in this way:
D.DRSQLOLYRTRODNLHNQTGKUDQGTUIRXNEQBCKBSZIVQQVGDMELM
UEXROIQIYALVUZVEBMIJPQQXLKPLRNCFWJPBYMGGOHJMMQISMS...
Lo! At the 6th digit we find a two-letter word (LO), and only a
few digits later we find the three-letter ROD embedded in the four-letter
TROD. How many other English words can be found if we continue looking?
First, a few pi facts are in order. The digits of pi (in any base)
not only go on forever but behave statistically like a sequence of
uniform random numbers. (Mathematically proving that this is the
case - the "pi is normal conjecture" - is a deep unsolved problem,
but numerical analysis of several billion digits suggests that it is
true.) Consequently, pi in base 26 emulates the mythical army of
typing monkeys spewing out random letters. Among other things, this
implies that any text, no matter how long, should eventually appear
in the base-26 digits of pi!
We can use the seemingly random nature of pi's digits to estimate how
many words of various lengths we can expect to find in its first
million digits (letters). For example, for 4-letter words: each
group of consecutive 4 letters in pi is equally likely to be one of
the 26^{4} possible combinations. My dictionary has roughly 5600
4-letter words, so on average there should be a valid 4-letter word
about once every (26^{4})/5600 = 81 digits.
Here are the corresponding estimates (of how many digits we should
expect to scan before finding an N-letter word) for N=2 to 10:
N #digits
-- -------
2 4
3 13
4 81
5 1000
6 14800
7 272000
8 5.7 Million
9 140 M
10 3900 M
Dividing these numbers into 1,000,000 gives an estimate of how
many N-letter words should be expected in the first million base-26
digits. For N=7 this gives 1000000/272000 = 3.67, and
indeed we found three 7-letter words: SUBPLOT at digit 115042,
CONJURE at 246556, and DEWFALL at 883265. Counts for the other
lengths were also as expected. No 8-letter or longer words were
found.
The estimates above are for finding any N-letter word; of course, a
specific N-letter word should only occur on average once every 26^{N}
digits. We should expect to need about 2.5 x 10^{18} letters in order
to find the phrase TO BE OR NOT TO BE (without the spaces) once.
We can only get as far as TO BE in the first million.
The very first N-letter word in base-26 pi (for each N) is
notable; remarkably, those words from N=1 to N=8 almost make a
little poem:
O, lo -
Rod trod steel.
(Oxygen subplot.)
These words occur at digits 6, 5, 11, 10, 6570, 11582, and 115042.
The only possible contender for an earlier word that we found is
the OED word (marked "obs.") HELLY ["pertaining to hell"], which
occurs at digit 5458.
That the first 6-letter word is OXYGEN suggests that pi is truly the
very stuff of life! Here are all the 6-letter words we found, in
order of appearance (reading across the rows):
OXYGEN SALIFY MEDICS PANNES CLEDGY VIRIAL REVETE PRINKY LIBYAN THINGY
AMPLER UPSTEP REBUTS POLITY TEENSY HURROO AVOWER CORVES EXARCH FOGDOM
CUPHEA BOGOTA ADHAKA SOPHIC HAVANA RISSOA CLANGS CHINOL BAKUTU UPTUBE
GRANNY SNUDGE DEIFIC ALTERS DESIRE BEGGAR URATIC WORMER MACANA REFLEE
OPTICS URNISM OVIBOS POTGUN AMOUNT DROVER OCTOPI BISLEY ANCONE MURING
SOZZLE DEFIED WARTED WHILST LIVERY MINTER AMBURY ASARON ORGIES STRACK
GEOMYS ZENITH APONIA RETUNE TUNFUL UNFULL EMPERY MUTATE VOICER KUBERA
ALFURO DOOLIE BALDIE BUSHER CAMPER BULLAN SCROFF EXCEED CHEERY SKIERS
We can also look for words that appear as consecutive letters but
running backwards. The first backwards N-letter words we found are:
N Word Position of 1st letter
-- ---- ----------------------
1 O 6
2 OR 12
3 TRY 10
4 FILM 140
5 FILMY 140
6 FLOUTS 6254
7 ALPHORN 458071
and the distribution of these, as expected, is similar to the distribution
of the forward words. For example, we found three backwards 7-letter
words, and no 8-letter ones, just like in the forward case. The other
two 7-letter backwards words are FULLEST (at 408089) and HYLIDAE (at 695340).
Before venturing into two dimensions, we mention one more
recreation involving the linear string of base-26 pi digits, inspired
by noting the first appearance of the word ONE at position 10087.
Where, we ask, does the number N appear in words, for each N? Here are
most of the answers up to N=10:
ZERO 389247
ONE 10087
TWO 13463
THREE ---
FOUR 11324
FIVE 64838
SIX 14295
SEVEN 786958
EIGHT ---
NINE 175372
TEN 15276
No number words larger than TEN appear in the first million base-26
digits, nor does THREE or EIGHT appear (SEVEN appears exactly once.)
Instead of just looking for the first occurrence, we can note a
number word each time it appears. Those which appear in the first
million digits, in order, are:
14261226622521221122121666116612192221261122122666
61162122616261266629221616122066662612201226622156
26226112266611261266222611121266116111666121121722
226522216626221220222611616662262666
(where we have written 1 for ONE, etc). Since SIX only has three
letters it appears a lot, which means the Beast Number 666 also
appears frequently in the string above.

**Late breaking news: **Hans Haverman has extended the search to 5 million
base-26 digits, and has discovered the word THREE at position 1556763
and FIFTY at 2300987 (and also 4896456!). No other new number words
occur in that range. He also found the first eight-letter word:
the Webster's-Third-Unabridged word ARMAGNAC, at position 3095146.
See Hans' page here for more details.
**The Next Dimension**
We can provide another "degree of freedom" by arranging the
the base-26 digits of pi in a two-dimensional array. There are many
ways to do this (a spiral, a diagonal zigzag filling the
quarter-infinite plane, and so on), but for now we just employ one
method, which is to fill an infinite vertical strip S units wide, by
writing the first S digits in one horizontal row, then the next S
digits in the row below that, and so on. We can then select any
portion of the array and look for words that occupy consecutive
letters and run in any of the eight possible directions (like a
word-search puzzle). Perhaps some of the words will even interlock.
Perhaps the words will have something in common. Perhaps we will
unlock the Pi code!
Of course, this is the same thing that was done to "discover" the
infamous "Bible code". Since we can choose the letter distance
between rows (S), this gives us many (in our case, a million or so)
different ways of looking at the letter string under study, so the
possibility of finding "interesting" arrangements of words is
considerably increased, compared to a one-dimensional search.
For instance, here's a grid we found (with parameters as shown
at the bottom, where "Pos" denotes the digit position of the
upper left corner):
u r r n d a c i v r g c w n p e u b
w f p r z c v k m m p u d p w g l y
v y u q s v b m u y s n v m r k l i
z a k x u g v s e d h m p l l x l d
g n j d u v m x w e s y i e g q i z
f q o p q t k u s j s r z o k v v d
k e n z k a y j n d c f t s r n b r
s m i z d h p s i r d u f u w f o i
p u f r e c n l f f z f o q l h b j
c h n y e a h O M E G A r d r x k p
w a e a i p l d f l o a H p i u s q
o n i q e n z b r i n d t P v k z h
e q g p l s c r v a s g j s L j l v
p i e x s z t t z y v j k p u A l s
y g n q q e j l e l v k v w o o h m
D r q f n g x k b r p i v e x m f f
O y e q h z a x v b q r t p k r s c
G a k z d h s j j o q x f m b h e i
Pos = 148655 S = 14061
This contains the words ALPHA (shown in capitals going diagonally
starting in the lower right) connected with OMEGA (in the center,
horizontally), with GOD (lower left corner) nearby! On the other
side of the coin, consider:
a c j w c t h v g r o f d k h c
l c o h i t n r c z t y a r k d
u l n j t j c w h w z z e k q p
i b d q u h k e h b d e e d w p
w f j j k c x u c z S n h c g a
c x m t c m m m i h A r l c q j
z w o x r w x z h m T r r e w q
g k t t k a y a c m A o k d v q
z j n a u a D E M O N a k i s n
v z s v y h d i f f b w x b a s
t a n k g h e h j e j h j y u u
j x v y i m d h u t q v g j x u
f y c d q z o u y l k d v j q t
p i j c m j w z u t w g m t t e
c c s b v n g a j h q c w x w j
a h y r D E A R j k x z r u c v
Pos = 255717 S = 13771
which has DEMON and SATAN interlocked, with DEAR on the bottom row.
In this case no diagonals are used, which is even more remarkable.
Many other words are present in both of these arrays; we merely noted
the ones that seemed to have a common theme.
Words don't even have to be in straight lines, if that fits
our purpose. Consider the S=2736 array in the vicinity of the
word CONJURE (one of the three 7-letter words in linear pi):
e a e b y t n q t d
v t h h p j H q a t
o w f z z P O b b d
o s c i b x C v p l
h l C O N J U R E u
b t s j g r S n z v
w w g r e h j s b u
Connected with CONJURE is HOCUS (going vertically) and POCUS
(in an L-shape) - two quite appropriate words.
Both truth and crackpottery can be found within pi.
Remember the bizarre theories of Alfred Lawson? One of the fundamental
principles of Lawsonomy was "zig-zag-and-swirl"; turning our attention
to position 49287, with S=49076, we find the lovely little arrangement:
d v j l k a
r u m n t z
e m z Z A G
h S W I R L
d e n G v h
i q i t p o
We can explore the arts as well as the sciences. At pos=505070 with S=3999
the following array appears:
j g s o t w q q c c d r h k e k
g v a x y f l f j l k u a f e l
o z t d l g b l M p w i h s D M
j n z g c t j D M z s r q d c n
d o e q l j d o r m r w l u z g
i l u i z n s q x l s y g p a q
x g t z s o k i b z l v b l r i
t D M v s d D I G M y c p o i q
h p n l j u B E B O P d t h g m
j e t c r y q d w D f e n i l y
n n e u z i c v e A l j q b l n
u x p l z v j l d L i p f v i o
j y y t f c y b q g x p d h e p
In the center we are exhorted to DIG MODAL BEBOP, a popular form of
jazz from the 1950's. If we do we'll certainly feel GLAD (start at the
'g' below MODAL and read upwards). One of the giants of modal jazz
was Miles Davis, whose initials appear no less than four times in
the grid.
Some grids are rich enough to contain entire sentences or poems.
This array is at pos=554766 with S=1058
T L Y P T S W W I B B B D M O T
N B T S U N I L L S X A Q H F J
U L K R G X K F C D K R O U Q U
Y C K D Y Z K U A M A H S I T Z
H N Y O E M C H D F E E M K P C
V L X I T Y B M Q P M R R Z B R
V S B A C Z W U B P D O Z M S S
Z R D Z B E J I V Z C N P Q S H
Q P I N M M W A T E R X P H W Y
V D R P X I T V V F T X Z L N G
O R G D A V P X F S T U V N V X
V O I C D L N V J J H C K I T L
I F E S H W W S F U C A U Z G C
V Y B L I M H I C T Q A B C M I
P M G O K J J G R Q B O U Z W K
E R Y K Z O I K V G W H P G V L
and is fruitful enough that we can write a complete 5-7-5 haiku
using only words found in the grid:
Sun, elk in water;
Oho! For her I'll try to
Be a hero yet.
Another interesting type of grid is illustrated by this one:
f s z u y x h t p p
d n u e a q o p c i
u e q o m a x x g v
a b w D A W N r a w
i p e M E A L n z m
m f y L E R P r g v
c t c A R I L g j a
l e i L I C K q t c
Pos = 65340 S = 103986
Note the five four-letter horizontal words grouped into
a 4x5 rectangle. This is the largest such rectangle we found for
any values of Pos and S less than a million, and it's even more
remarkable because the five words have a similar theme (since ARIL
is a seed covering and LERP is an edible insect deposit on a plant).
Thus we could say, "For my meal at dawn, I will lick lerp from an aril."
The next step is to look for an NxN word square with words both
horizontally and vertically. By choosing (Pos,S) it is easy to find
3x3 word squares, so we attempted the more difficult feat of
finding a 4x4 square. Alas, the best we found is the following
near-miss that has 7 of the required 8 words:
o h e h r a l w p o
r z p W I S T k i x
x d r O V E R d m g
f t q R A T E f q y
h z i D U S T y c a
Pos = 173387 S = 199449
This square contains WIST, OVER, RATE, and DUST horizontally, plus
WORD, SETS, and TRET ("an allowance made for damage in goods
during transit" [OED]) vertically. A perfect 4x4 square does not appear to
exist in the first million letters of pi (regardless of the value of S),
but since it all depends on the completeness of one's dictionary it is
hard to be sure.
The 6x6 square below is, perhaps, an indication that it's time to
stop this discussion and move on to something else:
M I K Eu r
z K i n h l
u E q c b m
j I u p r f
s T b y j f
m H h b o h
x k c i s k
Pos=278978 S=18909
After all, it's obvious to whom the square speaks, and it clearly
spells out the message "U R" (see upper right corner) "SICK" (bottom
row, backwards).
**Base 27**
Another way to look at the digits of pi is to express it in base 27,
with the extra digit assigned to a space, so that we get a series
of words (strings of letters surrounded by spaces), not just letters.
In the May 1993 issue of "Word Ways", Lee Sallows suggested that the
most natural assignment is 0=space, so that all the letters are
assigned non-zero values. (Otherwise, one of the letters (say, A)
will have to have the value zero, which leads to word pairs like
WAKE and AWAKE, which have the same numerical value, even though
it seems more natural for them not to.) Given 0=space, the most
obvious scheme for the letters is A=1, B=2, and so on.
The beginning of pi in this system is:
c.cvezcvbmlyzxmswprpiijzhweemupdrxou jhcfmobyhsijlpjsca
zgxlhqunzwkhdfphtstzoprsnu nhawsjlquvbnqpvzqlwwliytpdauuddkzfgmpcu
fnwsavktwroffceijqrhtlvuqlqnox mjrjmq sqmqscvymhqwjrzkwqdathn
fmwfr fzugxgdjsqpk ckjirtxtiq c
where we have divided the lines at word boundaries (i.e., there is
a space at the end of each line).
It is harder to construct an N-letter word in base 27 than in base 26,
because we have to find an (N+2)-letter string consisting of the
word with a space preceding and following it. If there are W(N)
N-letter words in our dictionary, then in D base-27 digits of pi
we should expect to find W(N)·D/27^{N+2} N-letter words. For
D=1000000 this works out to be (for N=1, 2, ...) approximately
152, 291, 94, 14, 1, 0.07, ...
whereas the actual number of words we found was
137, 244, 83, 10, 0, ...
The 10 four-letter words that appear in the first million letters
of base-27 pi are:
Pos Word
--- ----
27074 AWRY
168376 FULL
186597 WAAR alt. form for WARE [OED]
263682 BUSS
318822 PUPA
554259 BALE
575129 CHIC
695434 KAYO
822868 KISH "a wicker basket" [OED]
943143 RUSE
The first 1-letter word, O, is at digit 6456. The first two-letter
word, the Greek letter NU, appears at digit 10351, followed not much
later by US at 10868.
The first 3-letter word is a bit of a poser, because we find a
great number of obscure specimens before finally hitting on a common
one (WHO) at digit 115288. The earlier possibilities include LIV
(short for Olivia) at 29998, DUP (OED: contraction for "do up") at
41107, AAM (a liquid measure), YAD (OED: obsolete past tense of
"go"), DAR (OED: obsolete form of "dare") at 85782, and GES (OED:
obsolete form of "guess") at 95679. Which of these should be
considered the first three-letter word in pi is left to the reader
to decide.
The first 4-letter word is, as shown above, AWRY. Five-letter
or longer words do not, apparently, show up until after the
first million digits. The word PI occurs twice, at positions
212659 and 979046.
The first backward words of each length are O (6456), TO (696),
PUD (41107 - the reversal of DUP mentioned above), and VETO (10354).
**Pi as Cipher Text**
Another interesting way of looking at base-27 pi is to consider it
as a a text encoded with a substitution cipher. As with the
two-dimensional approach to base-26 pi, this way of looking at the
digits allows us to find a lot more syntactically-correct English
texts. It might seem that this would produce many long strings
of words (after all, there are 26! ways of assigning letters in
a substitution cipher), but as we add more words the letter-pattern
constraints they induce rapidly curtail the number of possible decodings.
Here are a few two and three-word ciphers, with plausible English
translations:
57029: rfsyrcllx eugtyocv = fearfully misproud
(How we should feel as we contemplate the mysteries of pi?)
155865: dlahfwi dswzavznr = Wolfram weanlings
(An indication that pi was invented by Stephen Wolfram? Or maybe
several exceptionally young members of his company?)
76615: iaig fdbizsrqz lvfrixma = eyes numerator hintedly
(What the math student does on seeing the fraction 355/113.)
592835: eupplcycw ch = snookered 'em
(What pi did to everyone who tried to plumb its mysteries.)
Some of pi's short ciphertexts only have one solution using
Unabridged Mirriam-Webster words. Two such are:
edemymksb u rqoqhibut = Anacyclus, I redeposit.
(The customer addresses the bank teller's plant.)
vtm rrpgegtmt = Psi oogenesis.
(The psychic farmer says he can increase egg production via brain waves.)
The longest solvable ciphertext we found has five words, but none
of its solutions are grammatically interesting. The longest single
word with a valid English counterpart is
814790: wpbjngstikmnuydo = VENTRICULOGRAPHY
and this is the only 16-letter specimen we found. The 15-letter
ones we discovered are:
lmrqvdbzyoianjp
bgcnmqpjruylkhx
kiaptmhgzfyvxwo
gdqaborwyjxkput
ezlfybqgpkasoiu
sypudkxgfmictob = DERMATOGLYPHICS
qmerjilgmaudxuv
sgvyhqpngwcdtcb = POLYDAEMONISTIC
bvtxuhzpoyperwg
vyzmqngwhuwjkde = SULPHOCARBAMIDE
vmbhpewspojgwdt = AMPHIBOLIFEROUS
mqnykhpqwoajkut = HYPERGLYCOSURIA
ksinnpdchrifaeg = UNAPPROXIMATELY
aciawzysqhaxmni = ENCEPHALOMETRIC
vlvqejfjclsumia = MEMBRANACEOUSLY
jqhjtpubsrxkagv = SWASHBUCKLERING
bwnyikdimofctdj = INTERPAROXYSMAL
fobmxdpzsaukujb = PHENYLCARBIMIDE
ojsxfwrzbtknvus = PREDISCOUNTABLE
The reason DERMATOGLYPHICS appears six times is because the
cipher words associated with it are 15-letter isograms; it is well known
that DERMATOGLYPHICS is the longest unabridged Mirriam-Webster isogram.
Two other letter patterns appear twice, as shown in the list above.
In this article we have just scratched the surface in exploring
the digits of pi as text. Many challenges remain, including extending
the search past 1,000,000 letters, searching for text in other languages,
and using non-Roman alphabets.

**Post Script**
Though this does not seem to be a useful way of looking at all the digits
of pi, we mustn't fail to note one last logological property. Write pi as
usual in decimal, and group the digits as follows:
3. 14 15 9 26 5...
and then make the obvious substitution A=1, B=2, etc. You get C.NOIZE, which
is rather fitting, because the random nature of pi's digits means that when
you look at it you SEE NOISE!