Solid and hypersolid forms.

Solid and hypersolid forms. I have been thinking about extending Forms into three or moredimensions for a few years now and have so far only inflicted mythoughts on the National Puzzlers' League and G4G participants.Sadly, it has become time to end this journal's readership'sbliss and begin to bloviate blo��vi��ate?intr.v. blo��vi��at��ed, blo��vi��at��ing, blo��vi��ates SlangTo discourse at length in a pompous or boastful manner: "the rural Babbitt who bloviates about 'progress' and 'growth'"here.You have my sympathy.The first thing I need to discuss is changing the standardmeaning of the terms "single" and "double" withrespect to forms. As currently used, a single form has one set of wordsthat go across and down, and a double form has two. This works wellenough in two dimensions and extends to three in a fairly obviousmanner, but it breaks down in four. The problem is that a DoubleTesseract (a Tesseract that has two sets of words) is not a well-definedshape: there are two possible ways of creating Double Tesseracts, eitherwith two sets of words each running in two dimensions, or a set of wordsrunning in three and another set that runs in one only. For example(imagine the Squares stacked on top of each other left-to-right to formCubes, and the Cubes stacked on top of each other top-to-bottom to formaTesseract): Double_2,2 Tesseract FOG ACE ATE OBI CAR THE GIG ERAEELACE NAP DNA CAR ALI NAG ERA PIPAGEATE DNA OTT THE NAG TWO EEL AGETONDouble_3,1 TesseractCAT AHI TIC AHI HAD IDA TIC IDACABABA BAN ANA BAN AGO NOR ANA NORARMLOG OWN GNP OWN WEE NEE GNP NEEPEW The left Tesseract has the words Across and Down each Square thesame, and the words that go Through and Hyperthrough the same (forexample, the F in the upper-left begins the word FOG Across and Down,but the word FAA Through and Hyperthrough). The right Tesseract, on theother hand, has the same words in three directions (Across, Down, andThrough) and different words in the Hyperthrough direction (for example,CAT running in three directions starting in the upper left, but CALrunning in the Hyperthrough direction). Note that all these shapes canbe rotated so the words can run in any set of dimensions you want; Ifind these pairings convenient, but the Double_2,2Tesseract could, for example, just as easily have been written as havingone set of words Across and Hyperthrough and the second set Down andThrough. Performing that rotation is left as an exercise for thereader.The careful Reader has no doubt already deduced that I am fixingthis problem by adding a disambiguating subscript to the word"Double" to indicate how many directions the words run in(unless the Reader is a member of the National Puzzlers' League, inwhich case he is most likely pondering the question "How many timesis Jabberwock planning on recycling his one tired idea that nobody caresabout?" The answer, friend, is "Clearly, at least oncemore.")Before I detail the subscripts, I need to redefine"Single" and "Double" from affecting words toreflecting letters. Rather than defining a Single Square as one with oneset of words running across and down, I need it to mean that the twocoordinates that define the position in the Square are grouped such thatall positions with coordinates that permute per��mute?tr.v. per��mut��ed, per��mut��ing, per��mutes1. To change the order of.2. Mathematics To subject to permutation. to each other must have thesame letter. So a Single 5-Square has a structure: 1 2 3 4 5 2 6 7 8 9 3 7 10 11 124 8 11 13 14 5 9 12 14 15 All cells with the same number must have the same letter inthem; thus 7, for example, represents all positions with the coordinates(2,3) in any order. It is clear to see that the same words must runacross and down because of symmetry. (Note that there is no restrictionon the same letter appearing for multiple numbers; the letter E (say)could appear as both 7 and 10).When we move up to Cubes there are three possible types: Single,Double, or Triple. Single and Triple have obvious meaning in mynotation, but Double is more subtle. A Double Cube (which has one set ofwords going in two directions and another set in one) is formed bygrouping the coordinates into two sets, one with two coordinates and onewith one. Any cells in which the groups match, allowing permutationswithin the groups, must have the same letter. Thus, if we group theAcross (y) and Down (x) coordinates together, and leave the Through (z)coordinate separate, we end up with a Cube that looks like: 1 2 3 7 8 9 13 14 15 2 4 5 8 10 11 14 1617 3 5 6 9 11 12 15 17 18 The positions (z, y, x) of (1,2,3) and (1,3,2) must have thesame letter (5 in this example), but position (3,1,2) may have adifferent letter (14 in this example). You can compare this to the mapof a Single 3-Cube: 1 2 3 2 4 5 3 5 6 2 4 5 4 7 8 5 8 9 3 56 5 8 9 6 9 10 in which all three coordinates share the same group, so allcells with coordinates that are permutations of (1,2,3) have the number5.The numbers in the subscripts to the word "Double"above indicates how many dimensions are in each coordinate group. Sincethe order is irrelevant, the numbers are always written in decreasingorder. Thus, a Double_2,2 Tesseract has two groupsof coordinates, each spanning two dimensions. TheDouble_3,1 Tesseract has two groups ofcoordinates, one of which spans three dimensions, the other spans one.Because of symmetry, you get as many sets of words as you have groups ofcoordinates, with each word going in as many directions as its group hasdimensions.It is possible, though unnecessary, to subscript all instances ofSingle, Double, etc., even where not needed for disambiguation dis��am��big��u��ate?tr.v. dis��am��big��u��at��ed, dis��am��big��u��at��ing, dis��am��big��u��atesTo establish a single grammatical or semantic interpretation for. ; thusTesseracts can appear as Single₄,Double_3,1, Double_2,2,Triple_2,1,1, orQuadruple_1,1,1,1. Note that a Single form alwayshas its coordinates partitioned into one group, a Double is alwayspartitioned into two, etc.By considering the individual letters, instead of words, as beinglinked it becomes possible to build solid forms in non-cubical shapesthat are surprisingly elegant. Joseph DeVincentis suggested thecuboctahedron as a likely shape to use for solid forms--it offers a nicepacking of reasonably long words. For example:PARGAREREPREPAREGRATEEREEARGONEOKANEMONEREMODELGOODALLKNELTELLAREREANEMONEREFUSALEMULATEROSALIAENATIONELEANPREPAREREMODELEMULATEPOLEMICADAMANTRETINUEELECTEDGRATEGOODALLROSALIAADAMANTTALAYOTELINORELATTEEREKNELTENATIONRETINUEELINORETOURSNEEEELLELEANELECTEDLATTENEEDThe Through words all appear as Across and Down words as well;note, for example, that the middle P in PREPARE on the top layer beginsthe word POLEMIC going through. This falls out from the symmetryrules--it's really quite elegant I think.Now that we have well-established rules for extending forms tohigher dimensions, the question (first asked of me by Ravi Vakil)"so how high can you go?" leaps to mind. The answer is"higher than is interesting." English has a huge number ofthree and four letter words, acronyms, and abbreviations--it is possibleto construct a form of almost any dimensionality you like, but it willhave short words and not be particularly interesting. It certainly makesfor a lousy puzzle because of all the repetitions. For example, theSingle Cube R A V I A M I R V I N A I R A N A M I R M A B A I B A D R A D AV I N A I B A D N A A M A D M II R A N R A D A A D M I N A I A can be extended into a Single 4-Dekeract (a 10-dimensionalhypercube of order 4) that is composed of the following 220 words andabbreviations:RAVIAMIRVINAIRANMABAIBADRADANAAM Naam(nā`ăm), in the Bible, son of Caleb. ADMINAIAAAALBAAEALEAAAFPDEPRAARPAFFAMPAAIRASAPSEAAASAANILSISANDREIRE Eire:see Ireland; Ireland, Republic of ASEAFDRYPRYSRESH resh?n.The 20th letter of the Hebrew alphabet. See Table at alphabet.[Hebrew r�� PAHAFRAUAYUSASSESHEDEADSAAHSAHIRSSRI SSRIselective serotonin reuptake inhibitor. SSRIn.Selective serotonin reuptake inhibitor; a class of drugs that inhibit the reuptake of serotonin in the central nervous system, used to treat depression and other NICEIREDSIDEDCHEREEDEDDOAEONRHETYETASDAKHOKAANAGAERAUTAHSAHOEKOIDAIMSGMLAAUPHUAISPICIALCRICEICEDCLARECRU ec��ru?n.A grayish to pale yellow or light grayish-yellowish brown.[French ��cru, raw, unbleached, from Old French escru : es-, intensive pref. DEUTEDTA EDTA:see chelating agents. HABUERUCDUCOOTOSNASAEBENTUNEACERKOREASEMGAMAREADANDAHEAFORFE ORFE Operations Research and Financial Engineering (Princeton University)IEEEMMESLASAAAREURDEPEELADAIIEIECLEALAIRCIRREERYDAYSAIONRRNA rRNAribosomal RNA. rRNAn.The RNA that is a permanent structural part of a ribosome. Also called ribosomal RNA.rRNAribosomal RNA. See ribonucleic acid. URALTYLIASIABOBOUNOPCAPEOLEASIAMAAMCEBON eb��on?adj.1. Made of ebony.2. Black in color.n.Ebony.[Middle English eban, ebony wood, from Old French, from Latin hebenus, ebenus, NONI noni,n See morinda. EPICRECREARPMMPI MMPIabbr.Minnesota Multiphasic Personality InventoryMMPIChild psychiatry A personality assessment tool widely used in making psychologic evaluations, which is normally given at age 16 and older. Personality testing ACIDAONEDNETAITSFCSTERTHEPHASIALADLEAAUWRUEREWRYDECAERALLYLYACISIASIELIAAYAH a��yah?n.A native maid or nursemaid in India.[Hindi y IIWI General InformationThe ʻIʻiwi (Vestiaria coccinea) or Scarlet Hawaiian Honeycreeper is a Hawaiian bird of the family Drepanididae, and the only member of the genus Vestiaria. RSISRISPYAPSSHSHOWEDNIDI NIDI Nederlands Interdisciplinair Demografisch Instituut (Dutch)NIDI Nickel Development InstituteNIDI National Internet Diagnostics Infrastructure ASINLPNSISSAAHAUBEDAODASPISTENTOASORMARLCULMODESNASEISESCTSSROSEPREKILKA il��ka? also ilkadj. ScotsEach; every.[Middle English ilk a, each one : ilk (variant of ech, each; see each) + a, one, a DMASNEAPESPNTENGSSGTTSTMHEMAAKALLALOESOP ESOPSee: Employee Stock Ownership PlanESOPSee Employee Stock Ownership Plan (ESOP). ABEYUELEWYESELATRETEYSERCALKATKALEAL LEAL. Loyal; that which belongs to the law. YRLYILEASKAG skag?n. SlangVariant of scag.Noun 1. skag - street names for heroinbig H, hell dust, nose drops, scag, thunder, smack IAGOALOPHYPEWEDEIAEA IAEAInternational Atomic Energy Agency. SGADPODOSPORHERDEDENDENTIATA IATAInternational Air Transport Association, which sets the rules for air transport, including those concerning air transport of animals. NDAKSOKE soke?n.1. In early English law, the right of local jurisdiction, generally one of the feudal rights of lordship.2. The district over which soke jurisdiction was exercised. AREAUDALDEBTANTESTETTATEOKEYREYNLANDMLDREBROSTOLEELYSTYE stye(sti) hordeolum. sty, styesee external hordeolum. SEENEYNEKNEEADENSRNAARNOPOORNLRBGYBE gybe?v. & n.Variant of jibe1.gybeor jibe NautVerb[gybing, gybed] or jibing, TEETMNTNAENDLEDAONANPANEIt is possible to represent a Tesseract on a single sheet ofpaper (Through going across the Squares, Hyperthrough going down theSquares). You can then pile a stack of them to represent a new dimension(HyperHyperthrough the Penteract), and then have more stacks on othertables, in other rooms, on other floors, in other buildings, etc, forother directions. This rather small 10-dimensional form will require4096 sheets of paper to hold all the Tesseracts that comprise it.Because of symmetry, almost every word, every Square, every Cube, etc.,will appear many many times over; I cannot imagine anyone beinginterested enough to actually try to physically construct this (though Ihave made a pdf file available athttp://www.madhouse.us/~ronnie/npl/RaviDekeract.pdf if you would like totry). To give a sense of exactly how much duplication is present in aSingle Dekeract, consider that there are 4[sup.10] (1,048,576) uniquewords and unique letters possible in a Decuple 4-Dekeract (a Dekeractwith 4-letter words that are unique in each direction). A Single4-Dekeract has 220 unique words and 286 unique letters; 99.979% of thewords, and 99.973% of all letters, are duplicates.The number of unique letters and unique words are fairly easilycomputed. The following equations are all in terms of the order R of theform (the length of the words that comprise it):Square Form Number of unique Number of unique (oforder R) letters wordsPoint (0-dimensional) 1 undefinedSingle₁ R 1 LineSingle₂ R(R + 1) / 2 R SquareDouble_1,1 R[sup.2] 2R SquareSingle₃ R(R + 1)(R + 2) / 6 R(R + 1) / 2CubeDouble_2,1 R[sup.2](R + 1) R(R + 3) / 2 Cube / 2Triple R[sup.3] 3R[sup.2]_1,1,1 CubeSingle₄ R(R + 1)(R + 2) R(R + 1)(R +2) / 6 Tesseract (R + 3) / 24Double_3,1 R[sup.2](R + 1) R(R + 1)(2R + 1) / 3 Tesseract (R + 2) / 6Double_2,2 R[sup.2](R + 1) R[sup.2](R + 1) Tesseract [sup.2] / 4Triple R(R - 1)[sup.3] R[sup.2](2R + 1) _2,1,1 / 2 TesseractQuadruple R[sup.4] 4R[sup.3]_1,1,1,1 Tesseract Note that as nice as it would be for the number of unique wordsto just be the first derivative of the number of unique letters (whichdoes hold for some cases), it seems to be just a coincidence; thederivative of R(R + 1) / 2 (the letters in a Single Square), forexample, is R + 1/2, not R.There are four simple formulae from which the number of lettersor words can be computed for any square form. L(x) means the number ofunique letters for grouping x; W(x) is the number of unique words, allin terms of R, the order of the form. The variable n stands for a groupwith only one member (a Single form), any other variable represents agroup of an arbitrary number of (comma-separated) members. * L(n) = L(n - 1)(R + n - 1) / n [equation 1] * W(n)= W(n - 1)(R + n - 2) / (n - 1) [equation 2] * L(x,y) = L(x)L(y)[equation 3] * W(x,y) = W(x)L(y) + L(x)W(y) [equation 4] Note that the last two rules handle an arbitrary number ofgroups elegantly:* L([g_l],[g₂],[g₃], ..., [g_n]) =L([g₁])L([g₂])L([g₃]) ... L([g_n])* W([g_l],[g₂],[g₃], ..., [g_n]) =W([g₁])L([g₂])L([g₃]) ... L([g_n]) +L([g₁])W([g₂])L([g₃]) ... L([g_n]) +L([g₁])L([g₂])W([g₃]) ... L([g_n]) +L([g₁])L([g₂])L([g₃]) ... W([g_n])So we can compute the number of unique words and letters in a[Quintuple_4,3,1,1,1] 5-Dekeract thus: L(4,3,1,1,1)= L(4)L(3)L(1)L(1)L(1) by equation 3] L(0) = 1[given by first line of table] L(1) = L(0)(5 + 1 - 1) / 1 = 5 [byequation 1 and result of L(0)] L(2) = L(1)(5 + 2 - 1) / 2 = 15 [byequation 1 and result of L(1)] L(3) = L(2)(5 + 3 - 1) / 3 = 35 [byequation 1 and result of L(2)] L(4) = L(3)(5 + 4 - 1) /4 = 70 [byequation 1 and result of L(3)] so L(4,3,1,1,1) = 70 * 35 * 5 * * 5 = 306,250 [substituting for L(4), L(3), and L(1)]W(4,3,1,1,1) = W(4)L(3)L(1)L(1)L(1) + L(4)W(3)L(1)L(1)L(1) +L(4)L(3)W(1)L(1)L(1) + L(4 )L(3)L(1)W(1)L(1) + L(4)L(3)L(1)L(1)W(1) [byequation 4]W(1) = 1 [given by second line of table]W(2) = W(1)(5 + 2 - 2) / (2 - 1) = 5 [by equation 2 and result of W(1)]W(3) = W(2)(5 + 3 - 2) / (3 - 1) = 15 [by equation 2 and result of W(2)] W(4) = W(3)(5 + 4- 2) / (4 - 1) = 35 [by equation 2 and result of W(3)] so W(4,3,1,1,1) =35 * 35 * 5 * 5 * 5 + 70 * 15 * 5 * 5 * 5 + 70 * 35 * 1 * 5 * 5 + 70 *35 * 5 * 1 * 5 + 70 * 35 * 5 * 5 * 1 = 468,125 So the [Quintuple_4,3,1,1,1] 5-Dekeract has306,250 unique letters, forming themselves into 468,125 uniquewords.Note that the similarity of the numbers of words and letters isnot a coincidence--all Single Square forms have the same number ofunique words as the Single Square of one smaller dimension had uniqueletters (note the equations in the table for the number of words in aSingle Cube versus the equation for the number of letters in a SingleSquare).RONNIE KONLos Gatos, California