by Ross Eckler
Word Ways, 1980
Place the different letters of a word on a sheet of paper and, without lifting pencil from paper, trace out a path linking one letter to another, spelling out the word. Such a path may have loops emanating from a letter and returning to it (if the word has a doubled letter), or may have more than one link joining a pair of different letters (if a bigram appears int he word, and the same bigram or its reversal appears elsewhere in the word. This pattern of letters and links is called a spelling net; it can take on many different shapes depending on the way the letters and links are arranged on the paper. However, certain properties of the spelling net are invariant--that is, do not depend upon the arrangement of letters and links. One of these is the property of non-planarity: for some words, no matter how the spelling net is drawn, it must contain two links which cross each other. A word (or, more generally, a phrase such as stray satyrs) which has a non-planar spelling net has been christened an eodermdrome by Gary S. Bloom, John W. Kennedy and Peter J. Wexler, in their paper "Ensnaring the Elusive Eodermdrome" in the August 1980 Word Ways. An eodermdrome is itself an eodermdrome.
How can one recognize whether or not a word is an eodermdrome without tedious experimentation with its spelling net? (It is easy to make inept choices of letter placement, leading to non-planar spelling nets, even when a planar net is possible.) If one replaces the letters in a spelling net with points, a spelling net is simply a graph, a network of links joining nodes. There exists a famous mathematical theorem in graph theory, originally formulated by Kuratowski in 1930, which uniquely characterizes non-planar graphs; however, to understand this theorem a bit of preparation is necessary. There are two relatively simple non-planar graphs: K(5), the complete graph on five points (draw a star, and connect its points with a pentagon), and K(3,3), the bipartite graph on six points (place three points in each of two vertical rows, and draw a set of nine links connecting each point in the left row with each point in the right row). Kuratowski's theorem states that a graph is non-planar if and only if it can be reduced to either K(5) or K(3,3) by elimination of superfluous links and by elimination of nodes which are reached by only two links (all nodes in K(5) and K(3,3) have three or more links emanating from them).
Kuratowski's theorem enables one to eliminate most words from consideration merely by look-ing at the number of letters in the word repeated two, three or more times. To contain K(5) a word must have at least four different letters each repeated at least twice, and at least one letter repeated at least three times. To contain K(3,3) a word must have at least six different letters repeated at least twice. Further arguments involving the position of the thrice-repeated letter in K(5) can be invoked--this letter must be sufficiently spread out in the word that it brackets all eight letters of the twice-repeated ones.
In short, the search for eodermdromes first involves determining the distribution of repeated letters in a word. The tediousness aof this preliminary screening is markedly eased by using Jack Levine's pattern word lists which group all likely candidates together through words of length sixteen. For words of seventeen letters or more, a transposal dictionary is helpful, for it groups multiple letters in a word together and enables one to spot potential eodermdromes more quickly; I used one based on Webster's Second Edition supplied me by Tom Kurtz of Dartmouth College. Once a word passes this screening, one identifies by inspection superfluous letters and links, forming the remainder into a spelling net in the format of K(5) or K(3,3) as appropriate.
1. What dictionary eodermdromes contain only K(3,3)?
At first glance, it might appear that a word of ten letters is sufficient to contain the nine links of K(3,3); however, if one remembers that the word must be trraced out without lifting pencil from paper, two more links are needed, making a twelve-letter word the shortest possible. Although a handful of twelve-letter words consisting of six letter repeated twice do exist (these are known as pair isograms) there are none which are eodermdromes. In the shortest dictionary eodermdrome, METASOMATOSES, the ES link is traversed twice. (Place M, T and S in the left row and E, A and O in the right one.)
A list of all known eodermdromes from Webster's Second or Third Editions based on K(3,3) is given below; the letters to be connected by links lie on opposite sides of the slash. This list is almost certainly incomplete, for it does not include plurals, past tenses and the like for words of seventeen or more letters, unless they are explicitly given in boldface in Webster's. Furthermore, words only in Webster's Third of seventeen through twenty-two letters are omitted (word lists of these are not available). For long words it becomes increasingly difficult to identify the six letters forming K(3,3); not all are given.
13 metasomatoses mts/eao
14 orchioscirrhus irs/och, supersaturates aeu/str, electroculture clr/etu, duodenojejunal dnj/eou, satellitesimal stl/aei
15 hyperpharyngeal ypa/her, preconspiracies rsi/pce, enterogastrones eot/nrs, ethnohistorians tni/hso
16 intranscalencies sia/enc, magnetogenerator rgt/aeo, postconvalescent ote/scn, unprosperousness ope/rsn, Sericostomatidae sia/eot
17 transformationist otn/irs, unadulteratedness uae/ndt, tetrachloroethane tao/erh, intraorganization tao/irn, transexperiential tae/rni, transportationist aso/rnt, antispectroscopic ico/stp, historicocultural oiu/crt, overconcentration nro/tce, contradictoriness oit/cnr, orthopsychiatrist hrs/iot, colicystopyelitis cls/oiy
18 proletarianization oai/nrt. hyperbrachycephaly cap/ehy, transrectification ari/cnt, orthopsychiatrical hrc/iot, tetrachloromethane hre/aot, tuberculosectorial cel/oru, intercontradictory ncr/iot, microrefractometer coe/mrt, ultraconscientious inu/ost, pharyngorhinoscopy prn/hoy, Heautontimorumenos eot/num, superconsciousness eos/unc, erythrocytoschisis yho/crt, duodenojejunostomy uoe/jnd, forethoughtfulness feh/uto
19 Helminthocladiaceae eic/ahl, ultradolichocranial ori/acl, hyperarchaeological aec/hor, monochloranthracene aho/ncr, encephalomyelopathy lyp/eho, counterorganization inr/aot, hyperdolichocephaly ehl/coy, cholecystolithiasis hlt/cio, electrohorticulture clr/eut, unconscientiousness ins/ceo
20 pectinatodenticulate tnc/aei, electrometallurgical aer/lct, ophthalmodiastimeter aot/ihm, hydroxyanthraquinone aho/nry, thoracogastroschisis hsr/cot, spinulosodenticulate uot/iln disproportionateness ots/ipn
21 protransubstantiation aot/irn, mandibulosuspensorial lis/aou, duodenopancreatectomy aco/ten, aminoacetophenetidine ant/eoi, microseismometrograph ioe/smr, pseudohermaphroditism eom/drh
23 pancreaticoduodenostomy aco/ent, blepharosphincterectomy eho/prt
24 formaldehydesulphoxylate ahy/ole, tetraiodophenolphthalein eap/ohl
25 tetrabromophenolphthalein eap/ohl
26 hydroxydeoxycorticosterone, cystoureterophelonephritis tpe/yor
One word in the list is worth special comment. DUODENOPANCREATECTOMY has a dictionary transposal, PANCREATODUODENECTOMY, with a planar net, the only transposal known in Webster's in which the planarity of the two words is different.
The 45-letter word PNEUMONOULTRAMICROSCOPICSILICOVOLCANOKONIOSIS has been omitted from the eodermdrome analysis. It is not hard to show that this word is non-planar. Gary Bloom discovered that an embedded K(4,3) graph in the spelling net of a word guarantees the existence of at least two crossings. Such a graph can be found in the spelling net of the lung disease, by connecting CLMN in all possible ways with AIO.
2. What dictionary eodermdromes contain only K(5)?
Eodermdromes containing only K(3,3) are rare, but those containing only K(5) are far rarer, probably because of much more stringent restrictions on the letters. As pointed out in Bloom's article, the shortest possible eodermdrome based on K(5) is eleven letters long. Only four Webster-ian examples have been found:
15 saponaceousness s/e/n/o/a
16 sphincteroscopes s/p/e/o/c
18 hypsibrachycephaly y/h/c/a/p
19 craniorhachischisis c/a/r/i/h
3. What dictionary eodermdromes contain both K(5) and K(3,3)?
These eodermdromes contain at least five letters repeated at least twice, and at least one letter repeated at least three times, for a total of thirteen or more letters. The minimal graph of such an eodermdrome is a K(5) graph with one outer link removed, but with three added links to a sixty point. Only two eodermdromes in Webster's are known:
17 overconscientious sie/nco c/n/o/i/e
26 phenoltetrachlorophthalein hot/rle h/o/t/e/l
4. What is the longest planar word (non-eodermdrome)?
Thie longest known word in any dictionary with a planar spelling net is non-unique; in the Random House Unabridged, the 34-letter words SUPERCALIFRAGILISTICEXPIALIDOCIOUS and DIAMINOPROPYLTETRAMETHYLENEDIAMINE both qualify. Graph theorists have proved that a planar net can always be formed without using curved lines (excluding self-loops for doubled letters). To construct the spelling net of the former word, start with the letter I and surround it with a ring of the letters P, F, G, L, A, C, D, O, T and S which are all linked to it; the remaining links can be easily added.
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