Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-20T22:41:18.133Z Has data issue: false hasContentIssue false
  • English
  • Français

The densest and the least dense packings of equal spheres

Published online by Cambridge University Press:  14 March 2018

Sidney Melmore
Get access Rights & Permissions [Opens in a new window]

Extract

Of the results which have so far been achieved by study of the packing of equal spheres one of the most remarkable, both by reason of its simplicity and of its fundamental importance, was that announced by Barlow in 1883. He called attention to the fact that equal spheres can be most densely packed in two ways, one possessing cubic symmetry and the other hexagonal (fig. 1).

Already in 1862 Tait had investigated the piling of marbles of equal size and had noticed that ‘there are two obvious ways of constructing the layers, and two of applying layer to layer’: nevertheless his two densest arrangements are in fact identical. He saw that the cubic structure could be begun either upon a square base or a triangular base but failed to perceive the possibility of the hexagonal arrangement.

Type
Research Article
Information
Mineralogical magazine and journal of the Mineralogical Society , Volume 28 , Issue 204 , March 1949 , pp. 479 - 485
Copyright
Copyright © The Mineralogical Society of Great Britain and Ireland 1949

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

page 479 note 1 Barlow, W., Nature, London, 1883, vol. 29, p. 186.CrossRefGoogle Scholar

page 479 note 2 Tait, P. G., Proc. Roy. Soc. Edinburgh, 1862, vol. 4, p. 535.CrossRefGoogle Scholar

page 479 note 3 Greenhill, A. G., Nature, London, 1889, vol. 40, p. 10.CrossRefGoogle Scholar

page 479 note 4 Thomson, W., Proc. Roy. Soc. Edinburgh, 1890, vol. 16, p. 693.CrossRefGoogle Scholar

page 479 note 5 Barlow, W., Proc. Roy. Dublin Soc., 1898, n. ser., vol. 8, p. 527 (preprints dated 1897).Google Scholar

page 480 note 1 Barlow, W. and Pope, W. J., Journ. Chem. Soc. London, 1907, vol. 91, p. 1158.CrossRefGoogle Scholar

page 480 note 2 Melmore, S., Nature, London, 1947, vol. 159, p. 817.CrossRefGoogle Scholar

page 480 note 3 By density is meant the ratio of the total volume of the spheres in a lattice cell to the whole volume of the cell.

page 481 note 1 Reynolds, Osborne, Phil. Mag., 1885, ser. 5, vol. 20, p. 469. The sub-mechanics of the universe, Cambridge, 1903. Rede Lecture, Cambridge, 1902.CrossRefGoogle Scholar

page 481 note 2 Segre, B. and Mahler, K., Amer. Math. Monthly, 1944, vol. 51, p. 261. But I have not seen this paper.CrossRefGoogle Scholar

page 482 note 1 Fairbairn, H. W., Bull. Geol. Soc. Amer., 1943, vol. 54, p. 1305. [M.A. 941.]CrossRefGoogle Scholar

page 482 note 2 Heywood, H., Journ. Imp. Coll. Chem. Eng. Soc., 1946, vol. 2.Google Scholar

page 482 note 3 Heeseh, H. and Laves, F., Zeits. Krist. 1933, vol. 85, p. 443. [M.A. 5–342.]Google Scholar

page 483 note 1 Melmore, S., Nature, London, 1942, vol. 149, p. 669. [M.A. 8–344, 9–139.]CrossRefGoogle Scholar

page 484 note 1 The value is thus given by Heesch and Laves. I have carried the calculation farther, using 7-figure logarithms, with the result 0·055515…

page 484 note 2 Nowacki, W., Homogene Raumteilung und Kristallstruktur. Zürich, 1935, p. 48.Google Scholar

page 484 note 3 Barlow, W., Zeits. Kryst. Min., 1894, vol. 23, p. I; Proc. Roy. Dublin Soc., 1898, n. ser., vol. 8, p. 527.Google Scholar

page 484 note 4 The particular normals which are the digonal axes of the extended structure lie in the surface of the cells as indicated in fig. 7, which represents one of the models exhibited when the paper was read.

page 485 note 1 In order that the spheres shall be in contact, A must bisect the long face diagonal in the ratio 0·758 : 0·875.