Homogeneous packings of circles

Koch, E.; Fischer, W.

doi:10.1107/97809553602060000617

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 9.1, p. 746

Section 9.1.1.2. Homogeneous packings of circles

E. Koch^a and W. Fischer^a

^a Institut für Mineralogie, Petrologie und Kristallographie, Universität Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany

9.1.1.2. Homogeneous packings of circles

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The homogeneous packings of circles in the plane may be classified into 11 types (cf. Niggli, 1927, 1928; Haag, 1929, 1937; Sinogowitz, 1939; Fischer, 1968; Koch & Fischer, 1978). These correspond to the 11 types of planar nets with equivalent vertices derived by Shubnikov (1916). If, in addition, symmetry is used for classification, the number of distinct cases becomes larger (31 cases according to Sinogowitz, 1939).

Table 9.1.1.1 gives a summary of the 11 types. In column 1, the type of circle packing is designated by a modified Schläfli symbol that characterizes the polygons meeting at one vertex of a corresponding Shubnikov net. The contact number k is given in column 2. The next column displays the highest possible symmetry for each type of circle packing. The corresponding parameter values are listed in column 4. The appropriate shortest distances d between circle centres and densities ρ are given in columns 5 and 6, respectively.

Table 9.1.1.1| top | pdf |
Types of circle packings in the plane

Type	k	Symmetry			Parameters	Distance d	Density
3⁶	6	p6mm	1(a)	0, 0		a	0.9069
3²434	5	p4gm	4(c)	$[x, x+{{1}\over{2}}]$	$[x = {{1}\over{4}}\sqrt3 - {{1}\over{4}}]$	$[{{1}\over{2}} ( \sqrt6 - \sqrt2 )a]$	0.8418
3³4²	5	c2mm	4(d)	x, 0	$[x = 1 - {{1}\over{2}}\sqrt3 ; \, b/a = 2 - \sqrt3]$	b	0.8418
3⁴6	5	p6	6(d)	x, y	$[x = {{3}\over{7}}; \, y = {{1}\over{7}}]$	$[{{1}\over{7}} \sqrt7 a]$	0.7773
4⁴	4	p4mm	1(a)	0, 0		a	0.7854
3464	4	p6mm	6(e)	$[x, \bar x ]$	$[x = {{1}\over{2}} - {{1}\over{6}} \sqrt3 ]$	$[{{1}\over{2}} ( \sqrt3 - 1) a]$	0.7290
3636	4	p6mm	3(c)	$[ {{1}\over{2}},0]$		$[ {{1}\over{2}} a]$	0.6802
6³	3	p6mm	2(b)	$[ {{1}\over{3}}, {{2}\over{3}}]$		$[ {{1}\over{3}} \sqrt3 a]$	0.6046
48²	3	p4mm	4(d)	x, 0	$[x = 1 - {{1}\over{2}}\sqrt2 ]$	$[(\sqrt2-1)a]$	0.5390
46.12	3	p6mm	12(f)	x, y	$[x = {{1}\over{6}}\sqrt3 + {{1}\over{6}}; \, y = {{1}\over{6}} \sqrt3 - {{1}\over{6}}]$	$[ ({{1}\over{2}} - {{1}\over{6}} \sqrt3)a]$	0.4860
3.12²	3	p6mm	6(c)	$[x,\bar x ]$	$[x = 1 - {{1}\over{3}}\sqrt3 ]$	$[(2-\sqrt3)a]$	0.3907

With three exceptions (3⁶, 3⁴6, 46.12), all types include circle packings that are not similar in the mathematical sense and that differ, therefore, in their geometrical properties. The highest possible symmetry for a type of homogeneous circle packing corresponds necessarily to the lowest possible density ρ of that type. Therefore, homogeneous circle packings of type 3.12² with symmetry p6mm are the least dense. The highest possible density is achieved by the circle packings with contact number 6 referring to triangular nets with hexagonal symmetry.

All circle packings described in Table 9.1.1.1 are stable in the sense defined above. Only circle packings of types 3.12² and 48² may be unstable.

References

Fischer, W. (1968). Kreispackungsbedingungen in der Ebene. Acta Cryst. A24, 67–81.Google Scholar

Haag, F. (1929). Die Kreispackungen von Niggli. Z. Kristallogr. 70, 353–366.Google Scholar

Haag, F. (1937). Die Polygone der Ebenenteilungen. Z. Kristallogr. 96, 78–80.Google Scholar

Koch, E. & Fischer, W. (1978). Types of sphere packings for crystallographic point groups, rod groups and layer groups. Z. Kristallogr. 148, 107–152.Google Scholar

Niggli, P. (1927). Die topologische Strukturanalyse I. Z. Kristallogr. 65, 391–415.Google Scholar

Niggli, P. (1928). Die topologische Strukturanalyse II. Z. Kristallogr. 68, 404–466.Google Scholar

Shubnikov, A. V. (1916). On the structure of crystals. Izv. Akad. Nauk SSSR Ser. 6, 10, 755–799.Google Scholar

Sinogowitz, U. (1939). Die Kreislagen und Packungen kongruenter Kreise in der Ebene. Z. Kristallogr. 100, 461–508.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 9.1, p. 746