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Introduction to geometry second edition - Coxeter H.S.M

Coxeter H.S.M Introduction to geometry second edition - Wiley publishing , 1969. - 243 p.
Download (direct link): coxeterintroductionto1969.pdf
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406
TWO-DIMENSIONAL GEOMETRY
the face is a p-gon of side 21, its inradius is (see 2.91, 2.92); therefore, the density is
/ cot 7r/p and its area is plr
plr
1L L P I
= 7i- COt E = !L
P P P/
tan
This is an increasing function of p, and tends to 1 when p tends to infinity. But since the p-gon is a face of a regular tessellation, the only relevant values of p are 3, 4, 6. Therefore the “best” value of p is 6, and the closest regular packing consists of the incircles of the faces of {6, 3}, the density being
f cot^ = V3 = = 0.9069 . . .
6 6 6 v 2y3
[Hilbert and Cohn-Vossen 1, p. 47].
HONEYCOMBS
407
It can easily be proved that this is still the closest packing when we abandon the requirement of regularity but insist instead that the centers of the circles form a lattice [Hilbert and Cohn-Vossen 1, pp. 33-35]. Actually, even this restriction can be abandoned [Darwin 1, p. 345; Fejes Toth 1, p. 58], as the bees discovered millions of years ago (Plate IV).
An analogous packing of spheres in three-dimensional space may be obtained by taking the inspheres of the cells of a honeycomb of equal poly-hedra. The density is naturally defined as the ratio of the volume of a sphere to the volume of the cell in which it is inscribed. In the case of (4, 3, 4), the honeycomb of cubes of edge 21, this is
UP
(2/)3
0.5236_______
A greater density can be obtained by using the midspheres (§ 10.4) of alternate cells, as we shall soon see.
Figure 22.4c
If we imagine the cells of the cubic honeycomb to be colored alternately black and white, like a three-dimensional chessboard, we may dissect each white cube into six square pyramids (by planes joining pairs of opposite edges) and attach each pyramid to the neighboring black cube. Each black cube is now covered with six white pyramids, one on each face, to form a rhombic dodecahedron (Figure 22.4c), whose twelve rhombic faces have the twelve edges of the black cube for their shorter diagonals [Steinhaus 2, p. 152]. Thus the insphere of the rhombic dodecahedron is the midsphere of the cube, of radius sj2l, and the volume of the rhombic dodecahedron is twice that of the cube, namely, 2(2/)3 = 16/3. In the honeycomb of such larger cells, each insphere is the midsphere of a black cube, and such spheres touch one another at the centers of the rhombic faces, that is, at the midpoints of the edges of the original honeycomb of cubes. Thus each sphere touches twelve others, the points of contact being the midpoints of the twelve edges of a cube. The density of this cubic close packing is evidently
408
THREE-DIMENSIONAL GEOMETRY
i*(V2!)3 _ _JL_ _ 0.74048 ...
16/3 3y2
[Hilbert and Cohn-Vossen 1, p. 47].
The rhombic dodecahedron occurs in nature as a crystal of garnet, and the three-dimensional chessboard occurs as the arrangement of atoms in a crystal of common salt, with a sodium atom in each black cube and a chlorine atom in each white cube (or vice versa). The centers of the black cubes, which are the centers of the spheres in cubic close packing, are easily seen to form the face-centered cubic lattice. It follows from § 18.4 that this is the densest possible packing of spheres whose centers form a lattice.
In old war memorials we often see a pyramidal pile of cannon balls: one at the apex resting on four others which, in turn, rest on nine, and so on. Each interior ball touches 12 others: 4 in its own layer, 4 above, and 4 below. In fact, these cannon balls are arranged in cubic close packing [Kepler 1, pp. 268-269]. The base of the square pyramid consists of (say) n2 balls arranged like the circles in Figure 22.4a. When n is large, the shape of the whole pyramid is essentially the “top” half of a regular octahedron (regarded as a square dipyramid); each sloping face is an equilateral triangle formed by 1 + 2 . . . + n balls.
By turning the pyramid over so that such a sloping face becomes horizontal, we obtain a different aspect of the same packing. In this aspect we begin with a horizontal layer of spheres whose “equators” are the incircles of the hexagons of (6, 3}, as in Figure 22.4b. The next higher layer is just like this but shifted slightly to the right (say), so that each sphere rests on three, its center being vertically above a vertex of {6, 3} from which an edge goes off to the right. Since all the centers form a three-dimensional lattice, the spheres in the third layer (resting on the second) are shifted again to the right, so that each center is vertically above a vertex of {6, 3} from which an edge goes off to the left. The fourth layer is vertically above the first, and thereafter the sequence recurs.
In 1883, the crystallographer Barlow described an equally dense packing in which the centers do not form a lattice. This can be derived by taking the same horizontal layers in a different order. More precisely, we discard the “third layer” just described and substitute a new third layer vertically above the first. Then we add a fourth layer vertically over the second, and so on; the shifting from one layer to the next is alternately to the right and left, like a zigzag. This nonlattice packing is called hexagonal close packing [Ball 1, p. 150; Hilbert and Cohn-Vossen 1, p. 46; Steinhaus 2, p. 170; Fejes Toth 1, pp. 172-173].
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