131

REFLEXICONS

LEE SALLOWS Nijmegen, The Netherlands

Reflexicon struction

A l exicon is a dictionary or a list of words. Hence my u se o f " re fl e x i ve lexicon " or, more crisply, reflexicon , for a self-descript­ive word list that describes its own letter frequencies:

* trois a ,

trois c, trois d, neuf e , quatre f, de ux h.

neuf i, six n, quatre o . deux p. cinq q. six r, sept s,

huit t. neuf u, cinq x

* Immortal verity sans superfluity . Now that is what 1 cal1 belles lettres ! Below we shall look at some English examples. But first, since the answer is far from obvious, how are reflexicons produced?

Imagine a book with pages inscribed as follows. The text on page 1 might be anything--a colophon, an epithet, a dedication to Ross Eckler . For convenience we assume something short . But page 2 and all subsequent pages each comprise a descriptive list, in words of the number of a's, b's, c's, etc., appearing on the previous page. Thus, if page 1 features a single x , then our volume begins like this:

Page: 1 2 3 4 5 6 7

x one x one e five e's five e's one n five n's three f' s one 0 five o':.s three i's one x one x two n's

two o's three s's three v's

one x

This may not be the recipe for a best seller, but the plot does have its appeal. You can hardly help wondering how it ends! Will list lengths continue to expand? Clearl y, 26 items is the limit. In fact , here none will exceed 16. There will be totals for E,F,G, H,l,L,N , O ,R,S,T,U,V,W,X,Y which are the only letters occurring in English cardinals under ONE HUNDRED, a number much higher than feasible list entries, assuming brevity in the opening text. Our e x ample is therefore a lipogram, a work in which A,B,C,D,],

132

K,M,P,Q,Z will b e absent because missing from page 1, the only page on which they could fi rst occu r. The end of our story can now b e dis cern ed.

Ev e r y new page shows a list of at most 16 totals, none of them large. The possible variations are thus finite. Sooner or later, the numb rs on one page will recur on anoth er , albeit diffe rently ordered . Suppos e the totals on page N a r e the same as those on page M. Then N is an anagram of M; their letter f re que ncies agree. But this means that page N+l will be identical to page M+l, which shows that our book must wind up in a repetitive cycle. And the same will be true wh atever the starting te x t. Call the number of pa g e s occurring in such a cycle its period. If the p e r i od is p, th e n we have a closed loop of p sequentially descriptive lists. If P 2, they will form a mutually descriptive pair. If p = 1, then we have a list whose description is a copy of itself: a reflex-icon.

Let distinct letters stand for distinct lists. The onset of a p e r iod 1 loop, R, then looks like this: ... L,M,N,O,P,Q , R ,R,R ... This shows that the refl exicon R not only describes itself, but it de­scribes list Q as well. So Q must be an anagram of R, most prob­ably a diffe r ent ordering of the same set of totals. Once any of its an a grams turn up, the reflexicon itself follows immediately. No reflex icon is reached except via one of its anagrams.

Question: assuming no ABCD]KMPQZ on page 1, how many differ­ent loops a r e th e r e? Using a computer to extend the above shows that its pages converg e on a loop of period 155. Ex tended trials re veal that , provided we stick to priming texts using the 16 car­dinal letters only (none to occur more than 99 times), there are Just four possible outcomes. One is the loop of period 155, another is of period 14, while the rem ain ing pair are both of period 1, the two basic English refl ex icons:

fifteen e' s, sixteen e's, seven f's, four g's, five f's,three g's, six h's,eigbt i's, six h's,nine i 's, four n's,five o's, five n's, four o's,

six r's,eighteen s's, six r's,eighteen s's, eight t's,four u's, eight t's, three u's, three v's, two w's, three v's, two w's,

three x's. four x's.

The two longer loops are readily recon st ructed by ext rapolating from any of t h e ir constitutent 1ists, such as the following (con­densed into digits):

E F G H L N 0 R S T U V W X Y length

14 7 0 3 6 0 7 6 6 18 8 4 3 3 3 0 155 17 3 1 5 5 0 4 5 5 14 8 1 2 2 2 0 14

Thus, in English, all the multitude of different possible st a rting positions lead inexorably into one of these four whi:dpools or strange attrac tors as mathematician s call them (se e Doug Hofstadter' s a d ­mirably lucid e x p osition in [IJ). This convergence is easy to un­

de r s tand. A tions (= ana wh ich is its and so on. pI ica tions.

Consid e r a book, st

page N, total s in th OUT is now so on. 1 text as inp to the form tha t a refl to subvert One. way to going and still other loops of 10]

we going tc How do we v

My answe I nstead of let the nex letter chose! viou r is loa a total ent already cor In this wa ) until it ev, is that a n ti lost every 1!

ed on one the former ' up in loops to evolve one exists,

Reflexic onog

Skipping can we do tence is r padded out contains ... "

to success i' associated "contains" simplest in mechanism in (a) a

1, the only r story can

lOne of th e m ~r or l ate r, t differently as those on nci e s agree. ~ M+1, which =le. And the

the numb e r le period is iptive lists.

If P = 1, If: a re flex -

of a period R,R ... This

but it de­, most prob­8nce any of immediate ly.

nany differ­above shows ended tria l s the 16 car­, there a re 155, another )f period 1,

g's, " s, o's, s's, u's,

v's,

xtrapolating ::lwing (con­

length

IS:') 14

ble starting s or strange adter's ad­easy t o un­

133

d erstan d. A 16-element list has 16! 20,922,789,890,000 permuta­tions (= anagrams), all of them giving rise to a common descripti on which is itself one among 16' new lists having a common successor, and so on. The resultant funnelling effect carries interesting im­plications.

Consid e r a computer program able to generate p age s for such a book, starting from any text. A basic routine scans TEX TIN

page N, initially page I, counts its letter s and writes their totals in the form of number-words to TEXTOUT = page N+I. TEXT­OUT is now substituted for TEXTIN, the routine reiterated, and so on. like to picture this process as a machine that takes text as input and yields text as output, the latter coupled back to the former via a feedback path. This makes it easier to see that a reflexicon is effectively a virus: a code sequence able to s ub vert the machine so as to get itself prepetually reproduced. One way to hunt for reflexicons is therefore t o set such a machine going and just wait for contagion to set in. However, there are still other viruses that may easily usurp it first. These are the l oops of longer period, all of them similarly infectious. How are we going to immunize the device against the se unwanted invaders? How do we write a book that ends specifica lly in a loop of period I?

My answer is to alter the mechanism so as to neutralize cycles. Instead of updating the totals for every letter on every page, let the next page result from correcting the total for a single letter chosen at random each time. The resulting haphazard beha­viour is loop-free by definition e x c e pt i n on e case: when updating a total entails no change in the subsequent list because it is already correct--because the list is already a self-descriptor' In this way the program is forever free to keep juggling number s until it eventually succumbs to a self-reproducer. The only snag is that a n agrams then pass unheeded, which means 16! chances lost every time. But n ot if we alternate methods: all totals updat­ed on one pass, one random correction the next, and so on. Now the former will catch any anagram, while the latter prevents latch­u p in loops. A few million iterations (mutations) normally suffice to evolve (nat urally select) a viable solution (virus)--assuming one exists. of course, failing which the process grinds on forever.

Reflexlconography

Skipping refinements, so much for the basic machinery. What can we do with it? For a start, note that a self-descriptive sen­tence is really a sugar-coated reflexicon, the essential kernel padded ou t with some palliative dummy text such as "This sentence contains ... ". Thus. on appending these constant ballast letters

to succeSSIve counts, our standard process again issues in an associated self-descriptive list, provided it exists. If not, change "contains" to "employs", say, and try again. Passing over the simplest instances, a few special finds made after adapting the mechanism to suit the purpose deserve notice here. These are seen in (a) a (British) letter-totalling reflexicon, (b) an (American)

!

134

l e tter-totalling s elf- descriptiv e pangram, and (c ) a ( trans - Atlantic) mutuall y - descriptive (pangrammatic) pair . Details of th e program c hange s entailed b y th e se special t y pes wo uld occup y u s undul y ; the basic mechanism rem ai ns the same.

This This sentence pangra_ contains contains tvo

one hundred Bnd hundred nineteen ninety-seven letters: letters: five a's, one b.

four a'srone brthree c'sr tvo c's,£our d's, thirty-one e's five d'srthirty-four e'sr eight f's, three g'sr six h's, seven f'srone g, six h's, fourteen i's, one j, one k, tvelve i's, three l's, tvo l's,two .'s,tventy-six n's

tventy-six n'sr ten o'sr seventeen o's, tvo p's,one q, ten r's,tventy- nine s's, ten r's, tventy-nine s's, nineteen t'srsix u's, twenty-four t's,six u's,

seven v'sr four v's, five v's, nine v's, four x's,five y'sr four x's, five y's,

and one and one z. z.

The The right - hand left.-hand

sentence contains sent.ence contains four B'Srone brthree c'sr four a's,one b,t.hree c's,

three d'srthirty - nine e 's, t.hree d's, thirt.y-five e's, ten f's, one g, eight h's, seven f's,four g's, eleven h's, eight i's, one jr one k, eleven i'sr one j, one k,

four l's,one _rtventy- three n'sr one 1, one _, tventy-six n's fifteen o'sr one Pr one qr fifteen o's, one p, one q. nine r'srtventy- three s's, t.en r's, t.venty-t.hree s's, tventy- one t's,four u's, tvent.y-tvo t's,four u's,

seven v's, six v's, t.hree v's, five v's, tvo x's.five y's, t.vo x's, five y's,

and one and one z. z.

Returning to reflexicons proper. in line with French practice above, the plural S is dispensible . Two instances are then found, one trivial ( F IVE F, FIVE I, F IVE V, FlVE E ), the other (see below ) less dull. This is compact. but logologists like their alpha­bet soup thick . Now, there are 14 words and 12+6+3+7+2+2+5+5+5+6+3+

6+4+4 = 70 letters in the text, an aver­TWELVE E, FIVE R, age of 5 letters per word , or 0.2 wordsS IX F, FIVE S, per le tter . Plura l S has been dropped .THREE H, SIX T , Is there another wa y to increase theSEVEN I, THREE U, seman tic density through d isca rdingTWO L, SI X V, still further r edundant symbols? OneTWO N., FOUR W, ap proac h (see nex t page) leaves aFIVE 0 , FOUR X. ce rtai.n amount up to the viewer; letters

under the

( AB C

Here we a stiffer inferred

Iii

D

fint br

(pa

0 T WN

E 0

0 0 N F

E

N

E

E ERHT

THREE No. involving fe This brings

Twelve stI Yet . N strip!

WO

H R

E E

T

F

V E

R

135

s-At l anticl le program

s unduly;

under the line are

T 0 E N N E

properl y

o T N W E 0

invisible.

S E

S V I E X N

F o T I N W V E 0 E

T H

T R 0 W E N o E E

:l ( ABC D E F G HI] K L M N 0 P Q R STU V W X Y Z ~en Here we find 12 words uswg 41 letters 3 .416 Ipw or 0.29 wpl,

one b, a stiffer broth, although the 12 referent letters are now to be rty-one e'a i n ferred ( partly through word positions), a dubious device, per­~. aix h' s, haps. However, a breakthroughj. one k,

0 T T N W W E 0 0

0 0 N N F I V E

E E ~

T

IT H R E E W 0

T E ~ f--

N E

,---

S f--

I X

S E1VlE N 101

comes in seeing that the same ty-six n's letters can be used twice as In p'a, one q,. a crossword. Writing out TEN ine s'a, E, ONE F, on twelve squaredsix u's, strips of ca rd, at length tria l­e ,," a, and-error shuffling led to my e y'a, first s elf- intersecti ng refle x icon

(at left) ,,-hich solve s the problem of the referent letters.

This was a good start, but OTT, NWW, EOO, etc. struck me as pseudoword blemishes. A differ­d ent la y out wouldn't help, either,tains

three c' s, since ONE H and ONE R alwa y s y - :five e's, remain bonded together with HR in s, el.even h' s. THREE. No. to escape this problem called for a new set of items j, one k, involving fewer intersections per strip so as to win elbow-room.

enty - six n's This brings us to a key insight. e p, one q. Twelve strips bearing 12 excess letters imply 12 intersections. - three B 'a, Yet. N strips can cross at most N-1 times unless li nked to i n c lude ,:four u's, a closed chai n. (Look ative v's, ONE X, SIX N, SEVEN 0 ve y' s, T- and TWO S in the upperw

It diagram. ) Contriving such

S X I 10 a loop is the major con­---I ...­

N straint in devising solution - la y outs. Thus, a new listh practice I-N requiring fewer intersections

other (see hen found,

- than strips makes for aEIT big gain in layout flexibili­leir alpha­ ~I ty (and vice versa), a 1­

an aver- EI IF 5+5+5+6+3+

R -E-E

TI ~ R reI E though two or more fewer will impl y a non-connected

1 dropped. 0.2 words

pattern. To avoid this,IOINIEI the obvious course then:rease the I~ is to seek an N item list

boIs? One discarding

invrlving N-1 excess letters leaves a (= intersections); an ex­

er; letters amp le is given at the left.

i

136

This is more like it: no pseudowords and 3.846 Ipw or 0.26 wpl, which, with the words now spatially interlocking, is virtual-l y alphabet jelly' The trouble is that now one letter, F, is alone [Qin not occupyin g an intersection, a niggling asymmetry. At some loss in semantic density, however, restoring plural S is another way to win room for maneuver, as in the diagram below.

F

0 H R E

F N

V R S E T

E 0 N

E E N N E R

S S X S

Here we are back to 12 strips and 12 intersections (necessitating a loop ) , each occu ied b y one of the 12 letters occurring. On consideration. this is a rema rkable property, more so than first sight suggests. since it depends on finding a list in which the letters outnumber their totals by one exactly, the excess then vanishing in intersects. The list used in the above crossword is thus exceptional. For example. no French or Italian equivalent exists. Unusually. however, English enjoys two such lists, the second compns1ng 13 words, although its internal peculiarities impede the construction of elegant self-intersecting layouts. Some readers may like to try their hand at constructing the crossword; the totals are E:15, F : 8, G:1, H:3, 1:5, L:1, N 4, 0:5, R:5, S:11, T:4, U:3 and V:4.

Of course. in general there is nothing against letters appearing on intersects more or less than once, as with the French and 1tal ­ian reflexicons given at the top of the next page (neither lan­guage calls for plural S). Both of these illustrate a further trick in the reflexiconographer's repertoire: the use of ONE # (here, UN B, U,O A, UNO B) as unobtrusively appended d ummy text. This is a useful stratagem when "pure" solutions cannot otherwise be found, although the arbitrariness of letter used (UN B could equal­ly be UN Z. say) detracts from their logological elegance, a point to bear in mind when assessing the merits of different specimens. Dummy text may take more conventional forms, of course, as in the reflexicon at the bottom of the ne x t page, where intersections outnumber strips, a fact reflected 1n multiple loops. However, the construction of such specialties i s demanding, to say the least.

o

a U A T

U A

R P T

T

:

l

1

eI~

t

I

137 ,w or 0 . 26 is virtua l­

is a l o ne f · At some is another

ecessi t a ting urring . On

than first which the

0

a

*-T

IT A E -R '---'

U N

-0 S I-­E E E U X P

X ..!.-u A T R E

o I S 0

!..'S ~ .!:!.. E X

f-­ I-­

P

T R 0

I-­T

'--­

C I-N

a :-­ r-­

S X I

X----< s

,-­ --'

u N

H U I TI

I S a

r0­

T T R E S

E I-­

U N 0 A

S E I N E 0

f0­ r-I ~ r-­

U 0 U E R to­ ~T N '-­ 0" 0

'-­ - 0­

S C I N a u E U

u T N N

f...­ 0 T T 0 lEI i-­B

I-­

E- f.­

0 ~

'-­

xcess then crossword

equiva lent lists , the

'eculiarities

::e,

r-:r IT H

A E-E -

,FI I V S-

Is E

I

10lNIE I G H T

---< T-S-

RI E E W

fo N E ,--N L

E\ I S N-E-

vlE N 0 S N

--J ---E -WIE N T Y

r--

1.. H

S fF 0 U R H I N E

i-- -X E E i-- - r--

T T E E I Is A L L 0 W E S 0 '- '-- f--N

f--F 0 u

r--S I S

i-- '--S ;:!...

E .........

-R

Els l S-

sl outs . Some crossword ; R:5 , S :l1,

s l appearing and lta l­

i ther lan ­r t her trick RT IFlsl

# (here, text. This

herwise be ould eq ua1­

a point specimens .

rse, as in ntersections

However, the least .

138

Some loops are not what they seem. The reflex icon below exhibits pseudoloops and the two wa y s they arise: via intersection on a blank . v iz . THIRTE EN SS and FIVE FS, and via abutment onto a bl a nk, vi z. T1-I I RTEEN ES and FOUR HS. The single real loop here is formed by FOUR OS, FOUR NS, FI VE FS, and FI VE IS.

Pseudoloops can make for comp a cter layouts, a fact s ee n i n comparing the diagra m below with the one two pages earlier, both of which, be it not e d, use the same entries (the special set of st r ips), wh e re a s the two patterns occupy rect angles of 14x16 a nd 14x18, respective l y .

0­S E[V E N R S

E r-­ -T ~ [5 I X T 5H

~ H

R - FI

T F 0 U R 0 51 - r- ­UT E I T

~ I-­

IT H R E EI V 5 R - r- ­

R N E E - - - N I-­E N

re - IF E F 5E 1 V I-­ S 5 5

IF\O u R HiS rs '-­

5 L..-­

Two natural q ue stions now arise: (a) how man y distinct (fully connect e d) self-in tersecting refl exicons can be form e d from this set of strips ? (b) which of them is the most compa c t? To s eek answe r s. Victor EiJkhout, a mathem a tical friend, wrote a recursive s tr i p -shuffling computer program able to scan for solutions. How­eve r, although several days running on a mainframe computer produced thou s ands of alternative solution layouts, it became clear there was no chance of the job terminating within any feasible ti me-scale. The two questions thus remain unanswered. The reflexi­con g iven above. which was hand-produced, remains the most com­pact s p ecimen known.

Neverth c· less. at my suggestion Eijkhout set his (slightly modi­fied) progcam to work on a new but related se arch which was to bear fantastic fruit. The two reflexicons on the next page em­body two jewels of logolog y (we seem to have rea c hed alphabet ice ). Here are the classic strips again, the loop now realized as the entire s e t holding hands in a single twelve-linked bracelet' The pair shown are among 18 such specimens found b y the prog r m, not counting rotations a nd reflections, but including trivial vari ­ations such as when FI VE lS is switched with FIVE FS.

F

Marve llous probably awa

icon wi th a solution <::ntr (p o sslbly int con (without meantime. on flexicon belo one of the 1 S. It is a

j

,11

Ie

xt

l O

:he

139 ow exhibits ;e c tion on a )utment onto e real loop ~ IS.

xct seen in ~arlier , both ecia 1 set of J 14x16 a n d

F -0-u--R

-0 s

lu lsl ~ -F

S

~

rnHIRI E ~

F Iu X ~ONE U-

Fl H ~ stinct (fu lly

fro m this cP To s e e k

a r (": ursi v e Jtions. How ­

e com puter )ecame clear :lny feas ible S The r e flexi­

most com­

ightl y modi­which was Marvellous as EiJkhout' s finds are, further collector's pieces

page em- probabl y await discovery. For example, could there exist a reflex­d alphabet icon with a s y mme tr ical la y out? A congruent pair showing distinct W realized solution entries? A 3-dimensional brac elet that forms a knot? A

ed brac elet ' (possibly interlacing) co-descriptive pair? A p an grammatic reflexi­program, con (without dumm y text)? The list is easily extended . In the

:rivial vari­ meantime. one classic specimen has passed unmentioned. The re­flexicon below again features 12 intersectio ns , each occ upied by one of the 12 letters occurring, although now there is no plural S . It is a relative of the first self- inter s e c tor examined, where

N

v s

'

: ,

140

the number of e xcess letters also matches the number o f items, Last l y , t but which cannot be s olved without pseudowo rds . As with the li st the precedi used e arlier , a second list with the same property (but minus flex icon the plural S) has been found. (A third trivial case is FOUR [F], FOUR using still [0], FOU R [U], FOUR [R]. ) The analog ous question then a rises : the comput, how many distinct solutions can be formed from the entries 1n problem tha the reflexicon below? de sc ri ptor t

1F1' V E 0 N r-E

r- ­ r-T

T W E L vlE El 0 ~I- ­

H

T H R E ET R -£.

w E , I- ­

'Iv0 F E V I-­

0 .E..I- ­10 N E U W '- ­ '-­

R F - '-­

F , viE T

...... ~ W

s EJV E NI 0 s-l r- ­

~ T wlE L vlEI T sl w w s I-­ rF U1R Fisl r ­

~ 0 T ...... ,......, I- ­T T H

loiN E H UIN D R E D s11 X TL E TITlE R sl R ~ N ~ ~- I-­E E

I-­ ...... );; -E t.I­ - r-­ -I S 0 H

F 0 U RI v s N 11N E TJEJE NI Is s ~ ~ ~ E

l-V

~ t-=

IT W E N T Y OIN E lEIs s

Is 1 X Ris ~

[1] Hofstadt

METAM

Willar ter, 1 abilia lng, Espy '~ a sea ized unifyi its Ie rebus tests) . rzng I

about prints that c. 1903 r A sam j

Punctl.. Dicki

Revise contr

Clemen Edna

Presid much

John ( the s

er of items, with the list

(but minus JR [F], FOU R then a rises :

e entrie s in

141

Lastly, to conclude this brief reVlew, I offer at the bottom of the preceding page a final example of the state-of-the-art, a re­flexicon that incorporates its own letter-total. Can one be found using still fewer than 106 letters? Another tough challenge for the computational logologist I However, the seemingly-insuperable problem that keeps me awake at nights is how to produce a self­descriptor that will tell us what letters it uses where ?

REFERENCE

[1] Hofstadter, D. lvletamagical Themas (Basic Books, 1985 ) , p.364.

METAMORPHOSES OF COMMONPLACES

Willard Espy subtitles The Word's Gotten Out (Clarkson Pot­ter, 1989) "a commonplace book" -- that is, a book of memor­abilia, youthful adventures, passages encountered In read­ing, clippings sent In by readers, and the like. Yet, in Espy's skilled hands, this often-mundane material suffers a sea-change, illuminated by clever turns of phrase, satir­ized by bits of light verse. The English language IS the unifying theme: its logical or orthographic inconsistencies, its long-forgotten words, its playfulness as exemplified by rebus or palindrome (but not by word square, whi ch he de­tests). One can sample some of the book's fla vor by refer­ring to Espy's Kickshaws in the May 1987 Word Wa ys, where about half of the material is recycled in his book; he also prints a set of a trocious literary-name puns by Derek Pell that appeared in February 1983, and a Polish pangram and 1903 rebus cited by Will Shortz in the May 1986 Kickshaws. A sampler of other ma terial:

Punctured poetry: I heard a fly buzz when I died (Emily Dickinson) / Of fruit sprayed with insecticide (Espy)

Revised Etymologies: CONDOM originally came from the Latin contra Domine, which means contrary to the will of God

Clement Wood's expandible palindrome: Di, Al, Togo, Boll, Edna, Todd, Adolf ... Flo, Dad,Dot and El Lobo got laid

Presidential Mini-biographies: GERALD FORD Proved, which much astounded s ome / That he could walk and still chew gum

John Culkin: Recently, I realized that the word dollop reads the same way upside-down. Would you call it an invertogram?