Foldedness factor in letter arrays.

Foldedness factor in letter arrays. An idle diversion with pencil and graph paper involves writing someordered sequence of letters in the cells of a grid and searching thehorizontal, vertical and diagonal rows of the resulting array foraccidentally-formed new words. Since there is usually a multiplicity ofpossible ways to inscribe in��scribe?tr.v. in��scribed, in��scrib��ing, in��scribes1. a. To write, print, carve, or engrave (words or letters) on or in a surface.b. To mark or engrave (a surface) with words or letters. a given letter string into a grid, thequestion arises as to whether some inscription patterns are likely to bemore productive of new words than others. Logic suggests that the answeris yes. Because any new words in the array must form along straightlines, and because no new words can form entirely within the line of theoriginal letter sequence, grid space that is occupied by straight-linesegments of that sequence is partly unavailable for new-word formation.Ergo, those inscription patterns that break or fold the original letterstring into the shortest straight-line segments presumably pre��sum��a��ble?adj.That can be presumed or taken for granted; reasonable as a supposition: presumable causes of the disaster. offer thegreatest opportunity for new words to form. But is this foldedness factor, as it might be called, a potentenough force to have any noticeable effect at the modest dimensions ofordinary letterplay? In an effort to find out, a letter sequence wasinscribed in��scribe?tr.v. in��scribed, in��scrib��ing, in��scribes1. a. To write, print, carve, or engrave (words or letters) on or in a surface.b. To mark or engrave (a surface) with words or letters. into a series of identical small grids with varying degrees offoldedness and the resulting arrays compared for rates of new-wordproductivity. The Alpha-Omega Sequence The letter sequence chosen for the trial was the familiar sequenceof the spelled-out Greek letter alphabet: alpha beta gamma Alpha Beta Gamma is an International Business Honor Society established in 1970 to recognize and encourage scholarship among two-year college students in business curricula. delta epsilonzeta eta theta iota kappa lambda mu nu xi omicron om��i��cronn. Symbol The 15th letter of the Greek alphabet. pi rho sigma tauupsilon up��si��lonor yp��si��lonn. Symbol The 20th letter of the Greek alphabet. phi chi psi omega. In addition to being well-known andalphabet-related, this sequence commends itself by having a favorable(45 to 55) ratio of vowels to consonants and by happening to consist ofexactly 100 letters, which may be neatly arrayed as a 10x10 square. Thisis advantageous because a square is the optimal rectangular shape inwhich to array a letter sequence in order to maximize the opportunitiesfor new-word formation. The Inscription Pattern Diagrams of the fifteen 100-cell inscription patterns selected forcomparisonare shown in the figure on the next page. Note that in every patternbut two (A1 and C5), the letter sequence is a single unbroken string, inpattern AI the sequence is divided into ten segments whose serial orderis from top to bottom, and in pattern C5, a knight's tour, thesequence is completely disjoined dis��join?v. dis��joined, dis��join��ing, dis��joinsv.tr.To undo the joining of; separate.v.intr.To become separated. , with no two consecutive letters in itbeing adjacent in the grid In the Grid is a game show that airs on UK broadcaster Five at 6.30pm week nights. It first aired on Monday 30 October 2006.In the Grid is hosted by Les Dennis and is produced by Initial West, one of the Endemol UK companies. . A way to quantify the foldedness of an inscription pattern is tocount the number of cells in the grid that it fills in which a patternline does not enter and exit the cell on opposite sides or corners. Inpractice, this amounts to simply counting the number of vertices The plural of vertex. See vertex. in apattern, with each bend and each line end counting as one vertex. Thus,pattern A1, with 20 line ends and no bends, has a vertex number of 20,while pattern A2, with 18 bends and two line ends, also has a vertexnumber of 20. Patterns that jump between non-adjacent cells areconsidered to create a vertex in each of those cells, and so pattern C5,which consists of 99 such jumps, has a vertex number of 100. In patternsin which pattern lines cross, the crossings are not counted as vertices;such crossing lines might be thought of as passing over and under eachother without intersecting. Once a pattern's vertex number is known, its foldedness indexmay be found by dividing its vertex number by the number of cells in thegrid that it occupies and multiplying the quotient by 100. For 100-cellpatterns, the vertex number and the foldedness index, will obviously bethe same. The reason for converting vertex numbers to foldedness indexesis to permit degrees of foldedness in patterns of different magnitude tobe compared. For example, a 100-cell pattern with a vertex number of 72has a foldedness index of 72, whereas a 144-cell pattern with the samevertex number has a foldedness index of only 50, reflecting the factthat the larger pattern is proportionately less folded. in the figure, each pattern's foldedness index is shown inparentheses See parenthesis. parentheses - See left parenthesis, right parenthesis. . As may be seen, the 15 patterns have been divided intothree groups of five on the basis of their foldedness indexes. In groupA the pattern foldedness indexes range from 20 to 28, in group B theyrange from 36 to 56, and in group C they range from 86 to 100. If thefoldedness factor does in fact exert an effect at this scale, we wouldexpect to find that group B pattern expressions produce more new wordsthan those of group A, and that group C pattern expressions engendermore new words than those of either of the other groups. This was,incidentally, a blind trial in that the 15 inscription patterns chosenfor it were settled upon before any of their alpha-omega expressions hadbeen generated. Word Search Results The alpha-omega sequence expressions of the trial patterns aredisplayed in the bottom half of the figure. All countable (mathematics) countable - A term describing a set which is isomorphic to a subet of the natural numbers. A countable set has "countably many" elements. If the isomorphism is stated explicitly then the set is called "a counted set" or "an enumeration". new wordsfound are shown in capitals. Counted were complete uncapitalized wordsof four or more letters that are printed in boldface type in the mainsection of Merriam-Webster's New Collegiate Dictionary, TenthEdition, and their inferred plurals. Not counted were words alreadyembedded in the alpha-omega sequence such as ABET, AGATE agate(ăg`ĭt), translucent, cryptocrystalline variety of quartz and a subvariety of chalcedony. Agates are identical in chemical structure to jasper, flint, chert, petrified wood, and tiger's-eye, and are often found in association with opal. , CHIPS, LAMB,LISP, MADE, MICRON, PEAT, SILO, TAKA ta��ka?n.See Table at currency.[Bengali , TEAT and various other. However,these words were deemed countable if their formation in an array was atleast in part the product of chance. Reversals were counted as twowords. Three methods of measuring the amount of new-word formation in aletter array suggest themselves. One is by a simple word count, which isthe total of all countable new words in an array; another is by aweighted word count, which arbitrarily assigns higher point values tolonger words; and the third is by word density, which tallies the numberof letters in an array that contribute to countable new words, in eachcase, dividing the raw count by the number of letters in the arrayyields a ratio that is comparable between arrays of different size. Itis not readily apparent which of these standards provides the fairestmeasure of new-word formation, but word density--i.e., the percentage ofan array that is occupied by countable new words--may be the simplest ofthe three to work with. In the figure, each array's word density isshown in brackets. Twenty-three five-letter and 188 four-letter countable new wordswere found in the trial arrays. No countable new words of six or moreletters were found, although the touring knight does, in a seven-letterphrase, bid us with BRIO to EAT PEAS. The 23 five-letter words arelisted below. Among the 188 four-letter words (not listed) were suchpossibly unfamiliar terms as agon, baal, deet deet(det) diethyltoluamide. DEETdiethyltoluamide. , geta, holp holp?v. ArchaicA past tense of help. , lota, luna,mana mana:see animism; taboo. manaAmong Polynesian and Melanesian peoples, a supernatural force or power that may be ascribed to persons, spirits, or inanimate objects. , mano, mora, nett, puna puna(p`nä), high plateau region, 12,000 to 16,000 ft (3,658–4,877 m) high, between ridges of the Andes in Peru and Bolivia. , sett, sori, taka, tala, tali, tapa, tele,tepa, tipi and toom, the phrase fragments alai, alma, lese, noms andpima and the hyphenated hy��phen��at��ed?adj.1. Having a hyphen: a hyphenated adjective.2. Often Offensive Of or relating to naturalized citizens or their descendants or culture. words nono and non-U. In the following summaryof results, each pattern's foldedness index is shown in parenthesesand its alpha-omega expression's word density is shown in brackets: A1 (20)--SIRUP and 9 fours [36%]; A2 (20)--IMPEL and 8 fours [29%]; A3 (28)--RATES, SHOOT and 14 fours [39%]; A4 (20)--AORTA and 9fours [30%]; A5 (20)--8 fours [25%] B1 (36)--12 fours [36%]; B2 (41)--ADMIT, ANOLE anole:see chameleon. , PIPIT pipit,common name for a group of chiefly Eurasian and African birds that together with the wagtails constitute a subfamily of songbirds related to the Old World warblers and thrushes. Pipits are trim, slender birds with thin, pointed bills. and 10 fours[34%]; B3 (40)--BATTS, EASEL, POLIS and 12 fours [36%]; B4 (56)--TOTES and13 fours [38%]; B5 (36)--TOAST and 14 fours [42%] C1 (91)--LEMMA, TUTOR, UNHIP un��hip?adj. SlangNot aware of or following the latest fashions or developments. and 8 fours [35%]; C2 (92)--20 fours[52%]; C3 (100)--MEDAL and 23 fours [50%]; C4 (86)--STELA and 13 fours[43%]; C5 (100)--GAMES, GLENS, HEMIN hemin/he��min/ (he��min)1. a porphyrin chelate of iron, derived from red blood cells; the chloride of heme. It is used to treat the symptoms of various porphyrias.2. hematin (1). , MILPA, NITRE ni��tre?n. Chiefly BritishVariant of niter.nitreor US niterNounChem same as potassium nitrate [Latin nitrum]Noun 1. and 15 fours [51%] Discernible in these results is a generally consistent correlationbetween the degree of pattern foldedness and the rate of new-wordproduction in the trial arrays. Note that the most new words, (24) arefound in pattern C3, one of two 100-vertex patterns, and that the twonext-highest scorers, (20 new words each) are also group C patterns, C2and C5. At the other end of the scale, the fewest new words (8) arefound in the 20-vertex pattern A5 and the next fewest (9) are in anotherfoldedness index 20 pattern, A2. In the arena of five-letter words, the100-vertexed knight is the clear champion of the lists with a total offive, a muscular two more than his nearest competitors. (It's thepower of peas!" he trumpets, as his Peasco sponsors beam. Once upona more innocent time, he championed peace, not peas; note his storiedname, now somewhat transposed trans��pose?v. trans��posed, trans��pos��ing, trans��pos��esv.tr.1. To reverse or transfer the order or place of; interchange.2. but still infused with pax, leading offhis array.) Analysis Average folded indexes and average new-word counts of the patternsin each of the three inscription pattern groups are collected in thefollowing table. For the weighted word Counts, four-letter words wereassigned a value of one and five-letter words a value of six. Foldedness Simple Weighted Word Index Word Count Word Count Density (%)Group A 21.6 10.6 15.6 31.8Group B 41.8 13.8 21.8 37.2Group C 93,8 17.8 27.8 46.2 Even more strikingly than the individual pattern results, thesegroup results imply a definite connection between the degree ofinscription pattern foldedness and the rate of new-word productivity inthe trial arrays. Where there is an increase in pattern foldedness indexfrom group A to group B, there is a corresponding increase in groupB's new-word productivity over group A'.s, and where there isan increase in pattern foldedness index from group B to group C, thereis a corresponding increase in group C's new-word productivity overgroup B's. These consistent correspondences would appear to leavelittle doubt that the foldedness factor does indeed exert a significantinfluence in arrays of this size. A less-expected finding was that new-word production in the arraysdoes not increase in direct proportion to the increase in patternfoldedness, but rather increases at a decreasing rate as foldednessincreases. In retrospect that relationship might have been anticipated,given that nothing hinders pattern foldedness from rising to 100%,whereas additional new-word formation becomes increasingly unlikely asthe word density of an array approaches 100%JAMES PUDERSaratoga, California