A. J. R. Allsobrook, M. E. Brown, and 1. Glasser Rhodes University Grahamstown, south Africa
Xtal-Line A b o a r d g a m e in crystallography
The board game described below is primarily an entertainment, hut also serves to illustrate some concepts of space symmetry and provides training in three-dimensional observation. The game has a number of variants; each, in succession, incorporates additional space-group elements and becomes more complex and challenging. The basic game is "noughts and crosses" (English) or " , tlc-tac-toe" (American), where two players, in turn, place one token a t a time on a 3 x 3 square grid; the first to place three tokens in a line (length, width, or diagonal) is the winner. This has long been played in a three-dimensional version, suitable for two to four players,l where the basic unit is a 4 X 4 square grid of transparent material, with four layers of such grids, one placed vertically above the other to form a cube; the first player to place four tokens in a line is the winner. The extension from three to four units in each direction adds interest to the game.
The variations which follow differ with respect to how the expression "four tokens in a line" is interpreted. Standard Game: Translation Only (Fig. Part 1 ) The "line" of tokens may he along the length, width, or diagonal of any horizontal or vertical plane (lines 1-51. This includes the body-diagonal of the cube (line 6). Variation I: Glides (Fig. part 1 ) Translation may he combined with reflection across a glide line or glide plane to yield a new kind of "line" of tokens (line 7); the glide line (or trace of the glide plane) 'e.g., "Checkline," catalog no. 501 of Crestline Manufacturing Co., Santa Ana, Calif. or "Qubic," catalog no. 400 of Parker Bros., Inc., Salem, Mass. The various kinds of "lines" of tokens are depicted here. A is taken as the topmast layer, in sequence down to D, the lowest layer. Arrows indicate the translation utilized; solid arrows indicate a translation within a layer, broken arrows a translation to an adjacent layer. Lines 1-6 depict translation only. Line 4 is a vertical translation, indicated by the small circle. Line 7 is generated by a glide line ($haw" as a broken line). Lines 8-13 depict screw axes. Line 8 is a 41 screw (right-hand rotation), while line 9 is a 43 screw (left-hand rotation). Lines 10 and 1 1 are 21 screws, with the screw axis for line 1 1 lying horizontally between layers B and C. Lines 12 and 13 are 41 and 43 screws. respectively. the screw axes lying between planes A and 8."Line" 14 results from a pure rotation. The two- and four-sided figures are the standard crystallographicsymboisfor 21.4.41.and 43screw axes. Lines 15 and 16 are sloping screw axes. 41 and 43. respectively. The hatched areas and arrows show how the area of operation of the screw move9 from layer to layer. Corners E and F are regarded as identical in the reduced representation of a lattice. Lines 17-20 indicate sloping translations, which are continued on the opposite side of the cube when the cube is regarded as a reduced representation of an infinite lattice. Sloping glides and screws are not shown.
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/ Journal of Chemical Education
has been marked in the figure with the standard crystallographic notation.2 There is no equivalent, in the game, to a glide translation along the body diagonal. The glide plane is appropriate to the crystallographic space groups, the glide line to the plane groups. Since we are here dealing with a single line of tokens in the space of the cube, the glide line is equivalent to the glide plane; the former description is preferred for its simplicity. Variation 11: Screws (Fig. Part 2) Translation may be combined with rotation about an axis, to yield a "line" of tokens aligned about a screw axis.2 The standard crystallographic rotation direction is that of a right-hand screw (line 8). The screw is denoted n, (41 for line 8), where n is the order of the axis and p i n is the "throw" of the screw (Ya of a rotation for each step of translation for a 41 screw axis). Line 9 is a 43 axis, and line 10 is a 21 axis. Line 11 is a Z1 axis, where the screw axis is regarded as lying between the two planes. The 21 screw axis isequivalent to a glide line, for symmetrical objects such as the tokens used in this game, so that lines 7 and 11 may he regarded as generated by either 21 axes or else glide lines. The equivalence is not general, for a reflection alters an object into its enantiomorph, but a rotation does not. Horizontal 4-fold screw axes are more difficult to visualize than the vertical axes of lines 8-10. These are, however, illustrated in lines 12 and 13, which represent 41 and 43 axes, respectively. The axes in these examples are taken to be half-way between the topmost and second layers, A and B. "Line" 14 is the limit of a screw axis, where there is no translation, only rotation (4-fold axis). An intermediate type of screw axis, between the 41 (or 43) and the 4 axis, is the 42 axis. A portion of such an axis is generated if a diagonal pair of tokens in the 14 set is raised into the next layer. This may he included as a variation, or ignored since it requires to be continued on to two further levels for the 42 axis to he completed.3 Variation 111: Inclined Screws (Fig. Part 3) The screw axes so far described are simple, hut can he made more complicated by permitting the screw axis to be inclined (lines 15 and 16; the hatched areas show the region on each layer in which the rotation of the screw is taken to he operating). Variation IV: Reduced Representation (Fig. Part 4 ) The basic 4 x 4 x 4 cuhe may he regarded as the reduced volume of an infinite lattice in which the line of four tokens is to be placed. The lattice is considered as
constructed by repeating the basic cuhe in all three axial directions; thus, for example, the corner at the top of the cube, labelled E, is regarded as equivalent to the corner F, at the bottom (Fig. 3). This has no consequences with respect to lines of tokens along axial directions. For a nonaxial direction, however, the reduction of the infinite lattice to the basic cuhe brings a line of tokens extending heyond the limits of the cuhe (in the infinite lattice) to the opposite side of the cuhe (in the reduced lattice). This is most clearly seen in line 17, in one plane, while line 18 illustrates this in a diagonal direction, and lines 19 and 20 in a body-diagonal direction. The complexity of t h i s r e duced representation version may he extended to include glide lines and screw axes to create a crystallographer's nightmare. The scientific use of a reduced representation of a lattice is not uncommon; the most familiar example' appears in the representation of the energy of electrons in a crystal lattice; the various Brillouin zones are then included in the range of the single reduced representation. The motion of an electron within a given Brillouin zone is exactly reproduced5 by lines such as 17-20. Play The game is played with the standard rules, plus one or more variations, increasing in complexity as the players become mare expert. There are so many lines possible when a number of the variations are allowed that the game is not worthwhile if it is eoneluded as soon as a player forms a line. Instead, completion of a line may be counted as a scare for that particular player (it is also possible to weight the scoring for various types of lines, although this is hardly necessary because the more complex the relationship between tokens, the less likely it is that opponents will spot it). The winner is the player with the highest scare when the boards are filled or, the mare likely event, when the game is abandoned. Note: The authors disclaim all responsibility for arguments among players! 2"International Tables for Crys+410graphy," Kynach Press, Birmingham, 1952, vol. 1; Glasser, L., J. CHEM. EDUC., 44, 502 119fi7\ -- * . ,. 3It should he carefully noted that the lengths of the steps of translation are taken to he equal for the 2- and the 4-fold screw axes in this game, so that the cube is treated as having the unit cell axial length for the 41 and 43 axes, since one rotation of the screw spans it, but the cube has twice the axial length for the 2, and 42 axes, which go through two rotations in spanning it. 'Dekker, A. J., "Solid State Physics," Macmillan, London, 1958, Figure 10-4ih),p. 246. 5Ricic F. 0.. and Teller. E.. "The Structure of Matter." ~eience'~ditionh i . e . , ~ e ~wo r k ,1966 (1961 edition). ~ i g u i e 8.17(1), p. 163. \
Volume 50, Number 10, October 1973 / 689
