US 3904208 A Abstract Dice games are disclosed in which each die has a three dimensional configuration conceptually representative of a four dimensional system consisting of the usual three spatial dimensions together with a fourth dimension such as time. The time dimension corresponds to movement of a three dimensional object, such as a cube, along a time axis. One or more of these dice may be used in the game, each in the shape of a dodecahedron having 12 rhombic shaped faces and representing the four dimensional system in three dimensions much as a three dimensional cube can be pictorially represented in a two dimensional drawing. Each of the 12 faces of each die are provided with number and color indicia means for distinguishing the various faces and providing the probability determining events characteristic of dice. The face numbering and color coding system is derived from the four dimensional concept used in creating each die. The die may be used individually or together, or as disclosed herein in combination with a game board having betting zones provided thereon by which a player can select a bet based on the numbers or colors or both represented on the dice faces.
Claims available in Description (OCR text may contain errors) Grossman 1451 Sept. 9, 1975 PSEUDO FOUR DIMENSIONAL DICE AND GAME 1 Jack J. Grossman. 328 Seventh St.. Manhattan Beach. Calif. 90266 [22] Filed: May 14.1973 [21] Appl.No.:359.773 [76] Inventor: OTHER PUBLICATIONS v Scientific American; Mathematical Games, Nov. l96o, Vol. 215. N0. 5; pp. 138-143. by Martin Gard- Prinu ry Esfimiinery-Richard C. Pinkham AA'A'I'SHUII Examiner-Joseph R. Taylor Attorney. Agent, or Firm-Miketta. Gienny. Poms & Smith i 57 1 ABSTRACT Dice games are disclosed in which each die has a three dimensional configuration conceptually representative of a four dimensional system consisting of the usual three spatial dimensions together with a fourth dimension such as time. The time dimension corresponds to movement of a three dimensional object, such as a cube, along a time axis. One or more of these dice may be used in the game, each in the shape of a dodecahedron having 12 rhombic shaped faces and representing the four dimensional system in three dimensions much as a three dimensional cube can be pictorially represented in a two dimensional drawing. Each of the 12 faces of each dlie are provided with number and color indicia means'i'o'r distinguishing the various faces and providing the probability determining events characteristic of dice. The face numbering and color coding system is derived from the four dimensional concept used in creating each die. The die may be used individually or together, or as disclosed herein in combination with a game board having betting zones provided thereon by which a player can select a bet based onthe-nurnbers oricolors or both represented on the dice faces. 1 Claim, 12 Drawing Figures PATENTED 1975 SHKET 1 [IF 5 SHEET 2 OF 5 PATENTEU 91975 (+)orz -INDICATE S cuss OR FACE as AT OR() END OF A I )INDICATES TWO EDGES: OF FACE ARE AQALLEL. TO GIVEN AXIS i="AcE: DE$\G. NO. VALJ-JE K Y Z w E M F E a u c Nq VALUE Y Z w cuaa r-uqcs 6:516. INDICATES EXTERNQLyFAEZES q=- PSEuDo ,TESSVERACT O' INDICATES JINrEQNA FACF-i's. or PsEuDo T'EssEpAc-r PATENTED' snmsurs prior games.the probability detennining devices are PSEUDO FOUR DIMENSIONAL DICE AND GAME BACKGROUND in general, the present-invention relates to games and educational toys and in particular to gamesan d toys based on the use of dice. The history of dice games and related games of chance extends back to ancient times. The Latin word for dice "tesseraef is,d erived from the Greek "tes seres" and ionic tessares", meaning four. corresponding to the four edges of the square comprising each side of the standard die. The classic form of the die is a cube with six square faces, 12 edges and eight comers. Since the ancient origin of the dice game, many variations on this basic die have been derived. t 1 f t For example, U.S. PataNos. l.S l 7,l 13; 3,208,754; and 3,399,897 show-severaivariations on thebasie or standard game of dice. in general, a wide variety of regular and irregular polyhedra have been proposedxfor use asthe probability determining element of a dice game. Dice or other three dimensional objects are variously used to define *word games. has ball-games, horseracing, card gamesqncluding poker. etc. In these representative of only two "and three "dimensional shapes orobjects. U SUMMA The object'of this invention is to provide a dice game. in Twhich the die or dice represent a .multi-dimensional concept or system having more than three dimensions. ' Starting with a multi-dimensional system. such asafour 2 for a baseball game in which the various die faces represent typical baseball plays. Furthermore. because this prior art patent does not derive the rhombic dodecahedron shape from a four dimensional system. it does not disclose the face numbering formula characterizing the present invention as more fully disclosed herein. 'Thus, it is an object of the present invention to provide a new game using dice formed by degenerating or reducing a four dimensional concept into a three dimensional multi-faced solid object or die. Also, it is an object of the present invention to provide an educational toy demonstrative of the inter-relatlonship between a multi-dimensional concept and the representation thereof as three dimensional objects. such as one or more die. .7 It is another object of the present invention to derive su'ch 'athree' dimensional muiti-faced die having its faces numbered in accordance with a self-consistent numbering systems based on the four dimensional concept. i A still further object of the present invention is to provide one or more die derived in the foregoing man- J ner todefine a solid object having the shape of a dodecahedron defining l2 rhombic shaped faces. These 12 faces arenumbered by. anumbering system for the eight individual and interrelated cubes composingthe dodecahedromln the resulting dodecahedron die, only t 12 external faces show, however the numbers or values dimensional system. dice may be created bytthe use of formulae disclosed herein to-reduce thefour'dimensions to three dimensional 'soiid representations. reduction to three dimensional form-is much like the pictorial representation of a three dimensional cube a two dimensional picture. in one particular and preferredembodime closed herein; the "four-dimensional system-in accoras probabiiity determining events." dance with the present inventionisreduced to a die having the three dimensional shapefof a dodecahedron? defining 12 individual die faces.jThese -l 2 faces are sys tematicaliy numbered in accordance with a formulade eightcubes comprising the dodecahedron maybe identitied suchas with the numbers lithrough 6 ,"inafman ner similar to the numbering of thejsitt sides of a start da rd cubic; die; .Howevei',- as more .fullyl disclosed 1 herein, the interrelationship of the various faces of the eight cubes result in a systematic numbering scheme for the entire set of cubes and for theda'decahedron composedthereof. One or.v more die formed in thislmanner can beuscd '45 rived by reducing the four dimensional system to a re'pid object, such as the ula forthis h 1 he dodecahedron created 0 I rived from the fourdimensiona l in three dimensionaiiform. as ageneraldice game, as well asforcard games andn other games ln one respect. the prior art US. Patiflo. l.-5l7.l l3"'mentioned above, issimilarto the present invention in that it shows a die structure having a' rhomf bic dodecahedron configuration. i-loweverithisipi'ior teaching proposesthat the dietbeu sed "as a single die p rin and hidden face nt as dis- 4 pendent on bothbr eitherof the number or detailed description of of these] 2 faceshave been determinedby the remainsof the above mentioned eight "cubes; V v 1 it is another object of the invention to provide such a l2.faced die,in which additional indicia means are applied to thedie faces also in accordance with the eight cubes com prisingthe l2faced die. For example, the indicia means may be a color code in which different facesare marked'with' a different color. One or more die formed inthis manner may be employed in a in which both thefaee numbers and set colors are Thisinven'tion'also has-an object.- thei provision of a game 5 board. such as a crap board; in which the 'standardbettingzones of, the crap board are provided withsub-zones to'permit a player to place a bet decolor of the fupj faces of the rolied dice. 1 4 f g 1 'An additional objectef the present invention is to dic'e'gam'ein whichfthe'hS individual cubes I, as the g degenerate four: "dimensional f object v are broken from the ,;roar' dimensional object and "'usiedas agamecomposed ef 8 cubic dice. values'sf suchf sag-card values including a designated suit; inijwhieh the valuesjand suits aredetermined from] the numbering jsystemr characterizing this. in- vention; ln othertwordsathe faeevalues of the 8 dice p are all interrelatedin accordance with the numbering -system-foftiie;present invention-which in tumis del concept represented .E'These and furtherobjects 'andivarious advantages of thepseudo' four dimensional dice and game according to thepresent invention will become apparent to those skilled. in theartfrom'a consideration of the following an exemplary embodiment I BRIEF DESCRIPTION OF THE DRAWINGS Reference will be made to the appended sheets of drawings in which: FIG. I is a perspective view of a pair of degenerate, pseudo four dimensional dice formed in accordance with the present invention. FIG. 2 is an isometric diagrammatic representation of the generation of a pseudo four dimensional die from a pair of intersecting three dimensional cubes. FIG. 3 is another isometric diagrammatic representation f the generation of the dice in FIG. I in which the of the three dimensional cubes shown In FIG. 2 are j ined to form a solid figure which is shown to compri 8 separate cubes shown surrounding the central ure. cubes forming'the'l2 face die and the 12 faces of the pseudo four dimensional die'are numbered in accor- .dance with the present invention. FIG. 5 is a front elevationalview of a 12 sided die in which the faces thereof have been provided with number and color indicia means in accordance with the present invention. FIG. 6 is a top plan view of the die shown in FIG. 5 as viewed from VI-VI therein. FIG. 7 is a rear planiview of the die as seen from VII VII in FIG. 6. FIG. 8 is a bottom view of the die as seen from VIII- -VIII in FIG. .7. FIG; 9 is a front elevation view of a die, similar to FIGS. 5- 8. but showing a different-arrangement of the numbers still in accordance with the self-consisten numbering formula of the present invention. FIG. 10 is a top plan view of the die of FIG. 9taken from X- therein. b FIG. 11 is a plan view of a conventional game board forthe dice game'sometimes called craps, showing the I various betting zones available to the players. F IO; 12 is a view of a game board similar to FIG. 11, but heremodif ed in accordance with the presentinvention to provide betting sub-zones in accordance with the various indicia means provided on the dice shown in FIGS. l and! through 10. ' DESCRIPTI ONi Because thesedice are-derivc'din this manner. they maybe referred to as. pseudo four dimensional dice. FIG. 4 is a table setting forth the self-consistent nu rnand thus the dice may also be called pseudo tesseracts. Although l2 sided die have been known for some time, the manner in which a particular type of i2 sided die, namely a rhomic dodecahedral die, may be employed to represent a four dimensional concept and the selfconsistent numbering formula used in marking and distinguishing the 12 die faces in accordance with the present invention are unique. To understand the derivation of these dice from tesseracts', it is helpful to consider'the following background in geometrical representations. In mathematics at point has no dimensions. A one dimensional line may be generated by translating a point through a length L so that it consists of two points or vertices and a single line or edge. If this line is translated through a distance L in a direction perpendicular to the single edge of the line, the figure generated'is a two dimensional square with four vertices, four edges and one face; From this square a cube may be generated by translating the two dimensional square through a distance L perpendicular to the two dimensional plane to form a three dimensional co-ordinate system. The cube has eight vertices, l2 edges, six faces and one volume. It is thus observed that it is possible to draw-a point, a line and a'square since the page is two dimensional and since these representations are equal to or less than two dimensional objects. The cube can be represented in the fonn of a two dimensional 'picture". However, this is only a representation and its true shape is one of a'three dimensional object. - Extending ,this analysis one further step into a fourth dimension, it is possible :to conceptually construct a four dimensional system or tesseizact hypercube by imagining a cube-at'a time}, and the infinite sequence of cubes ending at a time I In other words, the fourth dimension is realized' by using the dimension of time rather than a spatial dimension. Stated differently, the fourth dimension is obtained by translating the three dimensional solid object along a time axis for a length l.',such that-L c (r, r where c is the velocity of light. a b This four dimensional hypercube can be constructed in pictorial form in three dimensions in the same sense that a three dimensional solid object can be pictorially represented in a twodimensional picture. Thus, the tesseract or four dimensional hypercube is bounded by 16 vertices, 32 edges, 24 faces and eight cubes or volumes. This is compared'with a cube in which the'volume is bounded by eight vertices. I2 edges and-six faces; and 7 .with atwo dimensional square which has four vertices, four edges. one face and-no'volume. The dice in'a'ecordance with the present invention may also be termed degenerate or pseudo ftesser actsl. The term tesseractis the mathematicalname of afoun dimensional hyp ercube dcrivedfrom "the Ionic tessercs' meaning four. plus .aktis meaning ray. The coordi natc system comprising the four dimensional system includes fourvectors (rays). each perpendicularto the other three vectors as will be explained morefully herein.-Thus, the i2 sided die is a three dimensional object representativeof the four dimensional tesseract. By representing the four dimensional hypercube or tesseract in three dimensions and then taking a two dimensional picture of the three dimensional representation, the tesseract or four dimensional hypercube may 1 be depicted as shown in FIGS. 2 and 3. The result may be referred to as a pseudo four dimensional hypercube or pseudo tesseract. l a Y *The foregoing is graphically illustrated in FIG. 2, by a cube 11 shown intersecting with a cube 12. The edges 'of each of cubes II and I2 are drawn along the three spatial dimensions or co-ordinates represented as x. y. z. The fourth dimension is shown as the translation of cube I2 along an axis w which is conceptually perpendiculari tothe two dimensional picture representation of cube II in the two dimensional plane of the drawing. This conceptually represents the four dimensional hypercube or tesseract in which the fourth dimension is a perpendicular translation of the two dimensional pic ture of the three dimensional cube 11. The cube 12 is the result of translating the two dimensional picture of cube 11 along the time or fourth dimensionalaxis w. The representation in FIG. 2 may be developed into a two dimensional picture of a three dimensional tesseract as shown in FIG. 3 by diagonally connecting the vertices or cornersof the intersecting cubes "and 12 as illustrated. By connecting these vertices and then dissecting the resulting picture into its component parts. it is possible to demonstrate the presence of eight separate identifiable cubes or volumes which comprise the tesseract. These eight separate but interrelated cubes areshown as cubes 21 22, 23, 24. 26, 27, i and 28 surrounding the central tesseract. With refercnc eto FIG. 4, each of these eight cubes comprising the te ssei ract and each of "the sixfaces'of each such separate cube may be uniquely identified the indicated positive (4-), negative and parallel (l) notation. Basically. the edges of the cubes define four .25, and this same face is shared with the last face of cube 28 is indicated by the interconnecting line. Ac- cordingly. this face of cube 28 also has the face value I. Similarly, the face value 6 of cube 25 is shared with the last face of cube 24; face value number 2 of cube 25 is shared with the last face of cube number 22; the face having a value f cube 25 is shared with the last face of cube number 26; the face value 3 is shared with the last face of cube number 27; and face value 4 is shared sets of eight parallel lines comprising the tesseractfigure. if the direction to the right along ear. of the tes seract edges is called positive"(+), the direction to'the left along each edge called negative and the set of lines parallel to a given dimension as (i), then each of the sixfaces of each of the .eight cubes 21 through 28 may beuniquely identified as shown in the table of FIG. 4. Thusthecube'2l is comprised of a face paralleljto the y and z axis (denoted as y =1 and z l) and to the right of the .r axis (denoted x and to the left of the w axis (denoted a Similarly the other facesof cube 2! are identified by the plus sign minus sign" and parallel(l)notation under each ofthe x, y s. and waxis. H g To create a self-consistent numberin g'syst'em forthe eight cubes com prising the'tes'seract, the numbering system for a standard or regular six sided cubic die be used as a starting point. Starting with any one ofthe with the last face of cube number 23. It is observed that the shared faces have the same sign notation even though;. they are associated with different numbered "cubes. Thus the face value 1 of cube 25 may be located by finding the same sign notation, which as indicated above re-appears as the last face of cube number 28; {Thus in completing the numbering scheme, there is a limitation on the freedom of choice for the face values. For example, once the facevalues for cube number 25 have been assigned, thisnecessarily determines the val ues of at least six and as many as .l 8 other faces of the 48 faces defined by the 8 cubes. it is noted with respect to FIG. 3,that when the'8 tesseract cubes are consolidated into the three dimensional pseudo tesseract, that only certain of the 48 cube faces appear externally of the figure. That is, the eight cubes comprising the tcsseract of FIG. 3, each of g the six faces of such cubeis designated as a number or indicia l through 6 with the numbers on opposite paralicl faces adding to seven in a manner. similar to the numbering of a regular six sided die. Thus as shown for cube 21. the 2. 3 and 6 value faces are shown. This aur -.1 tomatically fixcs the remaining face values as 5, 4 an 1 respectively located on thejopposite parallel faces csf thisdie. The numbers for the 5, 4 and l faces for die 9' tributes three external faces and three internal faces to the degenerate tesseract figure. Thus in FlG. 3, cube have been deleted for clarity. Having allocatedzvalues for. the six faces'of onetesseract cube. the'six faces' oneac h of the remaining seven cubes can be similarly determined." This dc'termination h'oweverofthc remaining faces must takeinto account the fundamental characteristicsof-the-tesser prised. a These characteristics are basedon l );The intersection'of tesscractcubcs ZI through 28 incommon' faces, i.e., eachface of the *geometrical structure is shared by two of the tcsser'act cubes. Accordingly, this face pseudo. tess'eract is a 12 sided or dodecahedral object in flwhich the'l2 external faces correspond to certain of the facesjdefinedj by the eight six-sided tesseract cubesQln- 4',-the external faces can readily identified from the face notation, andthese are shown in 40- i i FIG. 4by the square around certain face values. Similarly, "the internalor hidden faces can be identified and these are' shown in FlC i.f4 by the circled face valuesfltis' observed that the extemal facevalues are to a certain extent controlled the internal face values of the complete set of eight tesseract cubes. As a result of this self eonsistent numbering system, the 12 external faces of thep eudo tesseract carry .two sets of face val- Furthermore, e'ac'h'or the eight tesseract cubes connumber 25 contributes external faces: 1, 5, and 3; cube number 23contributes exterior faces2, 6, and 3, etc. furtherf'characteristic of this system is that the act and the eight volumes orjc ubes 'of which itis com c'ubes 21 through 28 fo'rrn'mirror image pairs'of one another. Thus cubes 21 and'25'are mirror images of one anothencubes22 and'26 are mirror images; cubes 23 and 27 are mirror images; and] cubes 24 and 28 are mirshared by two cubes rnust havethesam'c value orindicia. unless the l'ace is. internally of {the three dimensional pseudo tesscract. in which cuse anfarbitrary value may be assigned as the face is hiddcn from view. (2) There must be some rule'relating opposite faces of each of the six sidedtcsser act cubes. For example. the rule derived.fromregular dicc that opposite faces of rorimagcsf, if an'attemptis made to number the 8 cubes inaccordance withtraditionalpractice associated with regular die. thenonly certain of thecubes will exhibit the standardcounter-elocltwise numbering convention. in particular; this convention provides that if a regular die is viewedaloneone of its diagonals. that the faces designated oncrtwo and three will be adjacent one another andwill exhibit a'counter-clockwise movement when going from face number I to number 2 to number 3. their face values. Furthermore. the opposite cube pairs may be identitle as a plus or minus cube of an associated pair. depc ing upon which end (+or of the fourth axis, i.e. th 'axis not used in forming the cube, that the desigted cube is found. Thus, the cubes2l and25 are opposite pairs having negative and positivefvalues respectively as indicated in FIG. ssmnmy; the cubes 2 2 and 26, 23 and 27, 24 and 28 are oppositeplus and minus pairs as indicated in 'FIG. 3. 1 It takes six face'assignments on three tesseract cubes of the set of eight here disclosed to consistently number 8 For the pseudo tesseract shown by the central three dimensional solid in FIG. 3. a simplified or short-hand notation of the numbering system can be devised as follows: Rule A: The face up and the face down of the pseudo tesseract will have thesame value. or stated differently. opposite parallel faces are provided with the same indiera. Rule B: The sum of the represented number or value on any given face of the pseudo tesseract plus the number or value on another face connected thereto only by a vertex equals the sum of seven. Rule C: The sum of the value or number on any given face plus the number or value on another face connected theretoby an edge does not equal seven. (Note: Rule C is redundant and follows as a necessary result of the application of Rules A and B if the six faces of each a of the eight tesseract cubes 2ll through 28 are assigned all six values of numbers 1 through 6.) all 48 faces of the tesseract. 'Ihis may be demonstrated as follows. Using the rule thatcipposite'fae ,s'of each individual cubemust'add toseven, and fetlo'wing the" chart in FIG. 4'to indicate common or shared faces of the tesseract. the following faces values may be tracedf Given theface value l'on cube number thisautog matically and necessarily designates the value of the last face of cube 28. Since opposite faces of each cube add to seven, this determines theface value 6 for the second to last face of cube 28 observe sign notation for opposite faces, which in mm establishes the value of the first face of Cube nurnb er-ZS. The opposite face of cube 21 is" thus designated 1,' here the second face of cube 21, which inturn'sets the .valve for the same face which is shared with cube number 24. The opposite face of 'c'ube 24, here the lastiface thereof, is thus 6 in accordance with the addition to seven rulerThe face designated 6 of cube 24 is shared with the second face of cube 25, which istherebyassigned the valueo. This forms acomplete set of eight faces one 4 cubes eonsisiti ing of two pairs of mirroriimage cubes. Accordingly, It is observed that the Rules A through C above are merely the result of the self consistent numbering systern taught in connection with FIGS. 2, 3 and 4 in the abovedisclosure. i I The pseudo four dimensional die or tesseract resulting" from FIGS. land} is the three dimensional representation or picture of the tesseract from some arbitrary'direetion with respect 'to the four axis x, y, z and w. The two cube vertices interior to the solid can have almost any position. constrained only by the distortions which maintain constant length for all cube edges. Although any such solid three dimensional object formed and numbered in this manner may be used as. the pseudo four dimensional die, a particular configuration is preferred. This preferred configuration is the equivalent unique degenerate pseudo four dimensional tesser- I act derived by joining the two interior cube vertices, of cubes 11 and 12 in FIGS. 2 and 3, to forma dodecahedron having rhombie faces. A rhombie dodecahedron or degenerate pseudo four dimensional configuration is preferred because of. itssymmetry, permitting a dieto be formed which will have an equal probability of coming tO II'CS'. on' a'ny one of its 12 faces. Although the given one numben'on a particular cube face,"this auto; matically sets th e numbers on eight faces of four cubes. The numbersl',"2 and 3 can"be assigned any of three faces e f-the firstcubenamely,cubenumber automatically Sets. 1 5 shown "above;- the other threef faces of thesselecte d cube and 18 other faeeson the remaining seven cubes) Examining cube numberj23, vonlyf two faces have assignedvaluesa and'4.. Therefore ithe numbers and f 2lm aylbe assigneidi to twoof theremain-. ing' faces which automaticallysets leimore ras, o'rfa total our faces thusfarljOne remaining pair of raes r cube number 28 can beassigned the remaining number pair-Sand d soithat 48 faces ha ve now beenassig'n ed numbers such that"each cube of the tesseractfhasgall numbers-1 through j6'assigne d. pictureof a true-tese seract together l'vwith' itsiself-consistent numberinsi y r tem as described above may formedintojathfi dimensional picture-likerepresentational solid byconthree'dimensional object formed merely by joining diagonal corners of the cubes 1! and 12 as shown in FIG. 3 has 12 sides like the rhombiic faced dodecahedron, it . does not havethe three-fold and four-fold symmetry axes ofa rhombic-faced dodecahedron and thusthere 5 is an unequal,probability that its various sides will land faeeup after a diceroll. The preferred geometrical configuration foreach dielmay be formedby starting with the' l2 sided figurefas shown in FIG. 3 and, keeping all edges equal in length, distorting its various faces until it'degenerates-into the rhombic-faced dodeeahe- -dron. This leavesthe l2faces of the' dodecahedron with the same numbering system discussed above in connection "with FIGS. 3 and 4. I Theresulting die may be used in a game either by itselforrwith 'one or more additional die having the same 3 or=similar face mar k ings. In FIG. 1 a pair of rhombic necting opposite diagonalcornersof thcl' cubes'l lli'and, 12 to define the outer twelvebbundary surfacesofthe solid. Thefremaining twelve faces are interior togthe three dimensional representationfilhe faces for such a pseudo tesseract or pseudo four dimensional "die will carry the face values as developed in the foregoing self-' consistent numbering system. dodecahedral die are illustrated with their faces appro- .priately numbered in accordance with the foregoing system. With referenceto FIGS. 5 through 10 an enlarged viewof one of die is illustrated. As indicatedi'eachsof the l 2 die faces is divided into two separate sets of six faces each corresponding to the numbering system derived from the true four dimensional hypercube or tesseract die. FIGS. 5 through 8 depict separate plan views of the die as it is rotated in each view. Accordingly, all l2 faces of the die are shown collectively in the FIGS. 5 through 8. Each set of six faces is provided with number indicia means for representing the numbers 1 through 6. Thus. for example, in FIG. 5 the numbers 1, 2, 5, 3 and 4 are represented-by indicia means, such as dimples. pips or other suitable means. on the die faces 31. 32, 33, 34 and-35 respectively. With reference to the remaining faces of the die shown in FIGS.6, 7, and 8, it will be observed that the I2 faces are numbered'in accordance with the above defined rules A, B and C as follows. The number indicia means on opposite parallel faces, such as faces 31 and 36 shown in FIGS. 5 and 7 respectively represent the same number, in this instance the number Secondly, the sum of the represented number on any given face. such as face 31, and the number on another face which is perpendicular to the first chosen face equals the total of seven. Thus, the number 1 on face 31 plusthe value 6 represented onfaee 37 in FIG. 6 equals the total of seven. Similarly, the value I on face 36 plus the value 6 on face 38 in FIG. 8 again total seven. Finally, the sum of the represented number on any giveniface, such asv the number 1 on face 3I in FIG. 5 plus the number on another face. such as any one of faces 32, 33, 34 and.35 does not equalthe total of sevenQAccOrdingiy, the value I on face 31 plus the value 2 on face 32: the'vaIue l on face 31 plus the value 5 on face 33; the value 1' on face 31 plus the value 3 on face 34; and the value 1 on'face 31 plus the value 4 on face35 do not equal seven.'Note that this may be anecessary'result of the application of the first and second rules above in accordance with the discussion herein- 35 above. are provided with different color indicia means. such as the colors red and black. Accordingly, as Opposite faces have the same numerical value. this differentiates between the two faces carrying the same .numcrical value. Thus each I2 sided die in FIGS. through will have a red 2 and a black 2. a red 3 and a black 3. etc. In this instance, in addition to the red and black color indicia means for the two sets of face values 2-6, a third color or indicia means is provided for the faces 31 and 36 carrying the value I as discussed below. By utilizing the color or set indicia means for differentiating between the two sets of six faces each on a die, it is possible to use the die or dice in a variety of games. For example. the colored faces may be used in a game similar to Roulette. in which a number, such as l through 6 may be called out in combination with a color such as black or red with the probability of the event being determined by both the numberindicia Although one or more die formed and numbered in this manner may be used in accordance with the game of the present inventioma preferredembodiment of this invention provides further indicia means on,the various die faces. 7 V Such'additional or further indicia means may be pro vided in the form of differently colored faces.'These different colors may be superimposed or otherwise means and the set orcolor indicia means. If, for exampic, at red-2 is called out, that means the player must roll the die such that the face 32 is up-right at the end of the dice roll. Furthermore, a pair of dice numbered and colored in this manner may be used to advantage in a game which depends both on the combined or total number that is rolled and also on the combined color of the roll. Thus the odds or probability of rolling two black fours will be different and less'probable than rolling a pair of fours, one black and the other red. As a further example. given a total point count, such as nine, it is possible to roll this pointby' any of the following' combinations: black-5 plus black-4, black-5 plus red-4, red-5 plus black-4, red-5 plus red-4; black-6 plus black-3, black-6 plus red-3, red-6 plus black-3, red-6 plus red-3; and black-7 plus black 2, black-7 plus red-2, red-7plus black-2, red-7 pIus'red-2. Thus the possible variations on rolling a particular "point" have .been substantially increased. The combined probability of rolling any of these various numbers and colors can of slightly alterating the odds so as to favor one player bering the die faces,the first rule to be observed is that I shared; faces of the tesseract cubes must be identical both in colorand numerical value. Secondly. no more than three faces of each cube shall have the same color. , If theserules are followed "all exposed faces of "the pseudo tesseract will have identical color: for identical face. number values. If however the odd numbers are provided with one color suchas red, as in rouletteLthen the even numbers'can be provided with a different color,'suchas black. Alternatively, exterior faces 1,2,- and'3 may be designated red and the remaining faces 4, s, and aback} if r Astill further and preferredaltemative is to, relax the rules requiring that shared faces-have the same color Accordingly. in accordance with the presently 'pre ferrcd embodiment. opposite .or parallel faces of the degenerate tesseract die shown in FIGS. 5 through I0 or the'other'asdesired. In the present embodiment. as illustrated in FIGS. 5 through. 8, the faces 31 and 36 car'ryingtheunumber indicia means representing one are colored green or any other third color different from the pairof colors, here red and black. differentiating the two sets of six faces. Thus in'a game depending on the .color that is rolled, the green faces 31 and 36 function as the zero and double iero in Roulette altering theodds in'favor of the house. It is observed that whena one spot. comes up, it pairs with black, red or green. Thus introducing an additional variable in the color combination discussed above and appearing only in the event a value I is rolled on one of the die. A roll resulting ina pair of green die faces corresponds to snake-eyes, i. e..'one and one. The degenerate tesseract or I2 sided die shown in FIGS. 5 through 8 has its number face values arranged according to one of several possible arrangements satisfying the rules above. The'die shown in FIGS. 9 and 10 illustrates another arrangement of the face values. Not all 12 faces of the die are shown in FIGS. 9 and I0 as the values for the faces which are shown determine the 11 i values for the remaining faces. Thus, based on the illustration in FIGS. 9 and I it is possible to add the remaining face values by merely applying rule A which requires that opposite parallel faces carry the same number value. In one embodiment of the dice game in accordance with the present invention. a game board is provided for use in combination with the degenerate tesseract dice. The game board may be of a type having betting zones for allowing players to place bets with or against the occurrence of certain numbers. colors or combinations i both. The numbers and colors provided by the num er and set indicia means serve as the probability det mining events, in a manner similar to the use of ndard dice. The game board may-be based on a standard game board, such as shown in'FIG. II for use in indicated in FIG. 11, may provide point designations for the dice values 4, 5, 6, 8, 9 and 10 as indicated at 46 to provide a place where the players point may be indicated by placing his bet either in front of or behind the respective numbers to indicate a PASS or DON'T PASS bet respectively. Similarly, the box as indicated at 47 over-lying the boxes for the values 4, 5 6, 8, 9 and 10 providebet indicating zones for the and DON'T COME bets. Particular numberbets together with the 'odds therefore may bedesignated in a zonee 48 called inthis instance NUMBER BETS. Similarly. bets for ANY CRAPS for betting on any roll that 2, 3 or 12 will occur, is provided in a zone 49 as shown in FIG. II. In accordance with the present invention this conventional gaming board is' reconfigured to provide for the additional combinations of face colors and face values of the degenerate four dimensional hypercube dice shaped and.marked as described above.-Thus, in FIG. 12, the conventional betting zones of PASS LINE 41 FIELD 42', COME 43', and DONT FASS/DON'T COME 44." are provided with zone indicia means corresponding toboth the numberindicia means and set indicia meanson the die faces. This permitsthe player to place a bet dependent on the occurrence of a particular number or total of numbers on a pair of dice and/or dependent on the color or combinations of colors represented by the set indicia means difi'erentiating between the two sets of values I through 6 on each die. Accordingly, the PASS LINE betting zone 41' is divided into betting sub-zones 51 and 52 representing and corresponding to the different colors, such as black Similarly, each of the FIELD, COME. DON'T PASS- IDONT COME zones 42'. 43' and 44 are marked with color sub-zones corresponding to the color indicia means on the dice. Thus, FIELD betting zone 42' is provided with a black sub-zone 56, a red sub-zone 57, a green sub-zone 58 and a plain sub-zone 59 as shown. COME betting zone 43' has a black sub-zone 6], a red sub-zone 62, a green sub-zone 63 and a plain sub-zone 64. Black, red, green and plain sub-zones 66, 67, 68 and 69 aresimilarly provided for the DON'T PASS- IDONT COME betting zone 44'. Corresponding color sub-zones are formed in the bet indicating boxes 46' and 47' as shown. Preferably, the game board color sub zones are laid out to be adjacent one another so that a player can straddle the various sub-zones with his betting token. This enables the player to select the color individually or in combinations. As two dice are usedin the game, the color possibilities upon any roll are as follows: both black, both red, both green, one black and one red, one black and one green, and one red and one green. It is observed that all of the color sub-zones, such as the PASS LINE sub-zones 51, 52 and 53 are disposed so thateach color borders the other two colors permitting a token to straddle a border between any two color subzones. Thus, for example, a player may place a token bet with the PASS LINE zone 41' so as to straddle the border between black and red color sub-zones 52 and 51. This means the player is betting that he will roll a 7 L or 11 with a color split of red and black. Thus, a roll of Similarly, the bet can be places so as to straddle zones 51 and 53 or zones 52 and 53 requiring a combination of red and green and black ZIIdLgIiI1 respectively. It will be appreciated that the probability of winning on a bet in which the color of the die faces is desig- "ANY CRAPS betting zone 49'. Region 7l is a special betting zonefor placing bets dependent only on the occurrence of the colors of the set indicia means on the dice. In other words, a player can place'a bet based on the colors "only while other players are rolling the dice l for either the numbers or numbers and colors in combiand red in this instance, of the set indicia means on each die. An additional sub-zone 53 representing and corresponding to the additional colorindicia means, in this instance green, may also be provided. To allow a player to bet on the number value of the dice only, a plain subzone 54 may be provided as part of the overall PASS LINE zone 41 as indicated. natiomThus in zone 71 aplurality of color sub-zones gether with the odds or probability of rolling the particular combination. These number bets may include not only the numbers, such assnake-eyes. double twos', double threes, etc.,- but also the color combination of black-black, red-black, green-black. green-green, etc. The following schedule is illustrative of the various die value and color combinations possible together with the random odds. These odds may be used as a basis for token pay-offs. Forexamplc, a pay-off might be 30:] for random odds of 35: l. PROBABILITY TABLE Point Combinations Random Odds BR RR/BB GR/GB GG BR RR/BB GR/GB G 2 4 3$zl 3 4 35zl 4 2 l 4 7l;l l43:l 3$:l s 4 2 4 35.1 71; 35a 6 6 3 4 23:] 47:l 35H 7 8 4 4 I71] 35:] 35:l II II) S 6715 I395 9 g 4 l7:l 35:! l0 6 3 231l 47:l l l 4 2 35H 7l:l l2 2 I 7l:l l43:l 7-H l2 6 4 ll'.l 23:! 35:1 2.3.l2 2 l 4 4 7l:l 1432i 35:] 351i Color Only U 25 20 4 47:25 H9225 3H5 35:l The rhombic faced dodecahedron dice as shown in FIGS. 1 and 5 through may be fabricated from known materials and in accordance with existing techniques. For example. the dice can be either opaque or transparent. To keep the dice honest. transparent material may be used. In a transparent die. the number indicia means may be provided by pips or dimples. as suggested in FIGS. 5 through 10. and the pips may be colored selectively on each face to serve as the color indicia means disclosed herein. A further preferred embodiment of the invention provides for the creation of card dice for playing a card game such as poker. Although poker dice are in general known. the present invention provides a particular and unique form of poker dice derived from the tesseract discussed above in connection with FIGS. 2, 3 and 4. One form of the poker dice in accordance with the present invention involves the use of the eight tesseract cubes 21 through 28 of FIGS. 3 and 4. in which the six faces of each cube are assigned card values including the face value such as Ace. King. Queen. Jack. 10. 9. 8 etc.. and the suit such as Spade. Heart. Diamond, Club. As only 48 faces are available, six faces on each of the 8 tesseract cubes. a 48 card deck is represented. in which the four twos are omitted. The game may be played by taking the eight tesseract cubes. having their respective faces marked with the card values in accordance with the present invention as discussed herein. and rolling the eight cubes as dice such that the Lip-face values represent the players card 'hand. To provide the uniformity of probability that any given card will'show up in a players hand. it is necessary to distribute the 48 card values and suits evenly throughout the 48 available faces of the tesseract cubes. This can be accomplished in accordance with the present invention as follows. Reference is made to the translation table set forth "TRANSLATION TABLE Code Nnminul Translated Code Nominal Translated Nu. Face Face Nu. Face Face Value Value Value Value l l 6 6 6 7 l I2 25 2 l l II 5 it 5 5 2 2 3 III 4 9 4 4 J 3 I I l 5 5 5 K 2 2 2. I l 27 (s 6 6 7 I I2 "TRANSLATION TABLE-continued Code Nominal Translated Code Nominal Translated No. Face Face No. Face Face Value Value Value Value 2 2 5 S 5 8 2 l I 24 3 3 28 4 4 4 9 3 I0 I I2 6 6 6 7 I l I I2 6 6 6 7 l l 26 3 I0 22 4 4 4 9 3 3 2 2 5 8 5 5 2 l I Translated Value determined by applying rule that opposite faces of tesseract IldL tn l3 and adjacent pair number differences are greater than or equal to 2. herein, in which the nominal face values of 1 through 6 for each of cubes 21 through 28 as set forth in FIG. 4 are translated to face values 1 through 12. This translation is effected by applying the rule that opposite faces of the degenerate tesseract in FIG. 3 add to 13 and that adjacent pair number differences are greater than or equal to 2 as a result the individual face values for cubes 21 through 28 previously having only the nominal values I through 6, now have translated face values of I through 12. Having accomplished this translation, it is possible to substitute the translated face values of 1 through I2 with the face value of a deck of cards. such as card values 3 through 10. Jack. Queen. King and Ace. If this is done in the manner disclosed herein. a set of card dice may be developed in which a variety of card games including poker may be played. The card dice game may take several! forms in accordance with the present invention. one of which involves the use of the eight cubes forming the tesseract as a combined card hand in the following manner. In one fonn. the die scheme in the present invention contemplates the use of the eight tesseract cubes of FIG. 3. having their various faces representing the face values and suits of a deck of cards. For example. the 48 faces available with the eight tesseract cubes. may be used to represent a 48 card deck. with the four cards having the face value of two omitted. These eight cubes or dice may be rolled together as a unit. and the resulting up faces. showing the face value in suit of the card. 16 card set of poker dice. consisting of 8 cubes or eight six sided dice representing the indicated card values and suits. To allow the formation of such hands as four of a kind". and "fiushes, it is necessary to arrange the suit *POKER DICE TABLE SUIT 8L VALUE FOR EACH FACE Tesseract-Cubc Translated Poker Face Cube No. Dice Value Value l C H S D 2 C S H D 4 3 D H S C 5 4 S D H C 6 5 D C H S 7 6 S D C H 8 7 H D C S 9 8 D C S H H) 9 H C D S J l() H- C D S Q l l D C S -H- K 12 S D H A S SPADE H HEART D DIAMOND C CLUB A self cunsktent set of face numbering and suit dlalrlhutlfin which depend: on tesseract cubes. tem disclosed above in connection with FIG. 4, and the translation table discussed above. With these tables, a poker dice table may be developed as shown herein, in which the 48 faces of the eight available tesseract cubes are assigned card values and suits. With reference to the poker dice table set forth herein, the top row shows the number of the cube, corresponding to the reference numerals 21 through 28 of FIGS. 3 and 4. The left-hand column shows the translated face value of I through 12 derived from the translation table above. The right-hand column shows the assigned card value of 3, 4, 5, 6, 7, 8., 9, 10 Jack, Queen. King and Ace. These assigned card values eorrespond directly with the translated face values 1 through 12. The row of cube numbers and the column of poker dice values form a matrix in which the circled letter S. H, D and C correspond to assigned suits of Spade, Heart, Diamond and Club. This is accomplished in the following manner. For the card value 3, it is necessary to have four different suits of Spade, Heart, Diamond and Club from the available 48 cube faces. Thus starting with cube 21 from the translation table it is observed that the translated value 1 does not appear on cube 2!, therefore the corresponding poker dice value of three is not assigned to any face on cube 21. Next, observing that cube 22 does have a translated face value of 1, this may be arbitrarily assigned the club suit or C, indicating that cube number 22 carries a face representing the three of clubs. Surprisingly, it has been found that each translated face value and corresponding poker dice value occurs four times among the 8 available cubes, such that each occurrence can be designated by a different suit. In this manner the matrix shown in the poker dice table is completed to form a 48 designations in the illustrated poker dice table such that each suit is separated by at least four values or numbers. Thus, cube number 21, for example, does not carry both the four of spades and the five of spades or any other spade within four cards of the four of spades. Similarly, for cube number 22, the three of clubs is separated by four faces before the club suit reappears, in this instance at the eight of clubs. By designating adjacent card values of the same suit on different cubes, it is therefore possible to roll straight flushes and royal flushes. While only a limited number of embodiments of the present invention have been disclosed herein, it will be readily apparent to persons skilled in the art that numerous changes and modifications may be made thereto without departing from the spirit of the invention. I claim: 1. A game comprising: one or more die each having a three dimensional configuration representing a four dimensional tesseract composed of 16 vertices, l2 edges, 24 faces, and eight volumes or cubes, said eight cubes being interrelated in that each face of said tesseract is shared by a pair of said cubes; and at least one of said die is in the shape of a dodecahedron having 12 external faces of a degenerate tesseract; and I further comprising a set of eight additional dice in the form of the eight cubes which compose the tesseract to show the relationship between the degenerate tesseract configuration and the eight volumes or cubes contained therein. i i i i Patent Citations
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