Dichromatic symmetry

Dichromatic triangle illustrating colour symmetry

Dichromatic symmetry,^[1] also referred to as antisymmetry,^[2]^[3] black-and-white symmetry,^[4] magnetic symmetry,^[5] counterchange symmetry^[6] or dichroic symmetry,^[7] is a symmetry operation which reverses an object to its opposite.^[8] A more precise definition is "operations of antisymmetry transform objects possessing two possible values of a given property from one value to the other."^[9] Dichromatic symmetry refers specifically to two-coloured symmetry; this can be extended to three or more colours in which case it is termed polychromatic symmetry.^[10] A general term for dichromatic and polychromatic symmetry is simply colour symmetry. Dichromatic symmetry is used to describe magnetic crystals and in other areas of physics,^[11] such as time reversal,^[12] which require two-valued symmetry operations.

Examples

A simple example is to take a white object, such as a triangle, and apply a colour change resulting in a black triangle. Applying the colour change once more yields the original white triangle.

The colour change, here termed an anti-identity operation (1'), yields the identity operation (1) if performed twice.

Another example is to construct an anti-mirror reflection (m') from a mirror reflection (m) and an anti-identity operation (1') executed in either order.

The m' operation can then be used to construct the antisymmetry point group 3m' of a dichromatic triangle.

There are no mirror reflection (m) operations for the dichromatic triangle, as there would be if all the smaller component triangles were coloured white. However, by introducing the anti-mirror reflection (m') operation the full dihedral D3 symmetry is restored. The six operations making up the dichromatic D3 (3m') point group are:

identity ( $e$ )
rotation by $2 π /3$ ( $r$ )
rotation by $4 π /3$ ( $r 2$ )
anti-mirror reflection ( $m'$ )
combination of $m'$ with $r$ ( $m'r$ )
combination of $m'$ with $r 2$ ( $m'r 2$ ).

Note that the vertex numbers do not form part of the triangle being operated on - they are shown to keep track of where the vertices end up after each operation.

History

In 1930 Heinrich Heesch was the first person to formally postulate an antisymmetry operation in the context of examining the 3D space groups in 4D.^[13] Heesch's work was influenced by Weber's 1929 paper on black-and-white colouring of 2D bands.^[14]

In 1935-1936 H.J. Woods published a series of four papers with the title The geometrical basis of pattern design. The last of these^[15] was devoted to counterchange symmetry and in which was derived for the first time the 46 dichromatic 2D point groups.

The work of Heesch and Woods were not influential at the time, and the subject of dichromatic symmetry did not start to become important until the publication of A.V. Shubnikov's book Symmetry and antisymmetry of finite figures in 1951. Thereafter the subject developed rapidly, initially in Russia but subsequently in many other countries, because of its importance in magnetic structures and other physical fields.

1951 Landau and Lifshitz reinterpret black and white colours to correspond to time reversal symmetry^[16]
1953 Zamorzaev derives the 1651 3D antisymmetric space groups for the first time^[17]^[18]
1956 Tavger and Zaitsev use the concept of vector reversal of magnetic moments to derive point groups for magnetic crystals^[19]
1957 Belov and his colleagues independently derive the 2D and 3D antisymmetric groups^[20]
1957 Zamorzaev and Sokolov begin the generalization of antisymmetry by introducing the concept of more than one kind of two-valued antisymmetry operation^[11]^[21]^[22]^[23]
1957 Mackay publishes the first review of the Russian work in English.^[9] Subsequent reviews were published by Holser (1961),^[24] Koptsik (1968),^[25] Schwarzenberger (1984),^[26] in Grünbaum and Shephard's Tilings and patterns (1987),^[27] and Brückler and Stilinović (2024)^[28]
Late 1950s M.C. Escher's artworks based on dichromatic and polychromatic patterns popularise colour symmetry amongst scientists^[29]^[30]
1961 Clear definition by van der Waerden and Burckhardt of colour symmetry in terms of group theory, regardless of the number of colours or dimensions involved^[31]
1964 First publication of Shubnikov and Belov's Colored Symmetry in English translation^[3]
1965 Wladyslaw Opechowski and Rosalia Guccione provide a complete derivation and enumeration of the dichromatic 3D space groups^[32]
1966 Publication by Koptsik of the complete atlas of dichromatic 3D space groups^[33] (in Russian)
1971 Derivation by Loeb of 2D colour symmetry configurations using rotocenters^[1]
1974 Publication of Symmetry in Science and Art by Shubnikov and Koptsik with extensive coverage of dichromatic symmetry in 1D, 2D and 3D^[34]
1988 Washburn and Crowe apply colour symmetry analysis to cultural patterns and objects^[35]
2008 Conway, Burgiel and Goodman-Strauss publish The Symmetries of Things which describes the colour-preserving symmetries of coloured objects using a new notation based on Orbifolds^[36]

Dimensional counts

The table below gives the number of ordinary and dichromatic groups by dimension. The Bohm^[37] symbol $G_{ol}^{a}$ is used to denote the number of groups where $o$ = overall dimension, $l$ = lattice dimension and $a$ = number of antisymmetry operation types. $a=1$ for dichromatic groups with a single antisymmetry operation .

Overall dimension	Lattice dimension	Ordinary groups				Dichromatic groups
Overall dimension	Lattice dimension	Name	Symbol	Count	Refs	Symbol	Count	Refs
0	0	Zero-dimensional symmetry group	$G_{0}$	1		$G_{0}^{1}$	2
1	0	One-dimensional point groups	$G_{10}$	2		$G_{10}^{1}$	5
1	1	One-dimensional discrete symmetry groups	$G_{1}$	2		$G_{1}^{1}$	7
2	0	Two-dimensional point groups (rosettes)	$G_{20}$	10		$G_{20}^{1}$	31
	1	Frieze (strip) groups	$G_{21}$	7	^[38]	$G_{21}^{1}$	31	^[2]^[34]
	2	Wallpaper (plane) groups	$G_{2}$	17	^[39]^[40]	$G_{2}^{1}$	80	^[14]^[34]^[41]
3	0	Three-dimensional point groups	$G_{30}$	32	^[42]	$G_{30}^{1}$	122	^[2]^[13]
	1	Rod (cylinder) groups	$G_{31}$	75	^[38]	$G_{31}^{1}$	394	^[43]
	2	Layer (sheet) groups	$G_{32}$	80	^[38]	$G_{32}^{1}$	528	^[44]
	3	Three-dimensional space groups	$G_{3}$	230	^[45]	$G_{3}^{1}$	1651	^[17]^[20]
4	0	Four-dimensional point groups	$G_{40}$	271	^[46]	$G_{40}^{1}$	1202	^[47]
	1		$G_{41}$	343	^[48]
	2		$G_{42}$	1091	^[49]
	3		$G_{43}$	1594	^[50]
	4	Four-dimensional discrete symmetry groups	$G_{4}$	4894	^[46]	$G_{4}^{1}$	62227	^[47]

References

^ ^a ^b Loeb, A.L. (1971). Color and Symmetry, Wiley, New York, ISBN 9780471543350
^ ^a ^b ^c Shubnikov, A.V. (1951). Symmetry and antisymmetry of finite figures, Izv. Akad. Nauk SSSR, Moscow
^ ^a ^b Shubnikov, A.V., Belov, N.V. et. al. (1964). Colored symmetry, ed. W.T. Holser, Pergamon, New York
^ Gévay, G. (2000). Black-and-white symmetry, magnetic symmetry, self-duality and antiprismatic symmetry: the common mathematical background, Forma, 15, 57–60
^ Tavger, B.A. (1958). The symmetry of ferromagnetics and antiferromagnetics, Sov. Phys. Cryst., 3, 341-343
^ Woods, H.J. (1935). The geometric basis of pattern design part I: point and line symmetry in simple figures and borders, Journal of the Textile Institute, Transactions, 26, T197-T210
^ Makovicky, E. (2016). Symmetry through the eyes of old masters, de Gruyter, Berlin, ISBN 9783110417050
^ Atoji, A. (1965). Graphical representations of magnetic space groups, American Journal of Physics, 33(3), 212–219, doi:10.1119/1.1971375
^ ^a ^b Mackay, A.L. (1957). Extensions of space-group theory, Acta Crystallogr. 10, 543-548, doi:10.1107/S0365110X57001966
^ Lockwood, E.H. and Macmillan, R.H. (1978). Geometric symmetry , Cambridge University Press, Cambridge, 67-70 & 206-208 ISBN 9780521216852
^ ^a ^b Padmanabhan, H., Munro, J.M., Dabo, I and Gopalan, V. (2020). Antisymmetry: fundamentals and applications, Annual Review of Materials Research, 50, 255-281, doi:10.1146/annurev-matsci-100219-101404
^ Shubnikov, A.V. (1960). Time reversal as an operation of antisymmetry, Sov. Phys. Cryst., 5, 309-314
^ ^a ^b Heesch, H. (1930). Über die vierdimensionalen Gruppen des dreidimensionalen Raumes, Z. Krist., 73, 325-345, doi:10.1524/zkri.1930.73.1.325
^ ^a ^b Weber, L. (1929). Die Symmetrie homogener ebener Punktsysteme, Z. Krist., 70, 309-327, doi:10.1524/zkri.1929.70.1.309
^ Woods, H.J. (1936). The geometric basis of pattern design part IV: counterchange symmetry in plane patterns, Journal of the Textile Institute, Transactions, 27, T305-320
^ Landau, L.D. and Lifshitz E.M. (1951). Course of theoretical physics, vol. 5. Statistical physics, 1st edition, Nauka, Moscow
^ ^a ^b Zamorzaev, A.M. (1953). Generalization of the space groups, Dissertation, Leningrad University
^ Zamorzaev, A.M. (1957). Generalization of Fedorov groups, Sov. Phys. Cryst., 2, 10-15
^ Tavger, B.A. and Zaitsev, V.M. (1956). Magnetic symmetry of crystals, Soviet Physics JETP, 3(3), 430-436
^ ^a ^b Belov, N.V., Neronova, N.N. and Smirnova, T.S. (1957). Shubnikov groups, Sov. Phys. Cryst., 2, 311-322
^ Zamorzaev, A.M. and Sokolov, E.I. (1957). Symmetry and various kinds of antisymmetry of finite bodies, Sov. Phys. Cryst., 2, 5-9
^ Zamorzaev, A.M. and Palistrant, A.F. (1980). Antisymmetry, its generalizations and geometrical applications, Z. Krist., 151, 231-248, doi:10.1524/zkri.1980.151.3-4.231
^ Zamorzaev, A.M. (1988). Generalized antisymmetry, Comput. Math. Applic., 16(5-8), 555-562, doi:10.1016/0898-1221(88)90245-3
^ Holser, W.T. (1961). Classification of symmetry groups, Acta Crystallogr., 14, 1236-1242, doi:10.1107/S0365110X61003612
^ Koptsik, V.A. (1968). A general sketch of the development of the theory of symmetry and its applications in physical crystallography over the last 50 years, Sov. Phys. Cryst., 12(5), 667-683
^ Schwarzenberger, R.L.E. (1984). Colour symmetry, Bull. London Math. Soc., 16, 209-240, doi:10.1112/blms/16.3.209
^ Grünbaum, B. and Shephard, G.V. (1987). Tilings and patterns, W.H. Freeman, New York, ISBN 9780716711933
^ Brückler, F.M. and Stilinović, V. (2024) From friezes to quasicrystals: a history of symmetry groups, 1-42, doi:10.1007/978-3-030-19071-2_132-2, in Sriraman, B. (ed.) Handbook of the history and philosophy of mathematical practice, Springer, pp 3200, ISBN 978-3-030-19071-2
^ MacGillavry, C.H. (1976). Symmetry aspects of M. C. Escher's periodic drawings, International Union of Crystallography, Utrecht, ISBN 9789031301843
^ Schnattschneider, D. (2004). M. C. Escher: Visions of Symmetry, Harry. N. Abrams, New York, ISBN 9780810943087
^ van der Waerden, B.L. and Burkhardt, J.J. (1961). Farbgruppen, Z. Krist, 115, 231-234, doi:10.1524/zkri.1961.115.3-4.231
^ Opechowski, W. and Guccione, R. (1965). Magnetic symmetry in Magnetism, vol. IIA ed. Rado, G.T. and Suhl, H., Academic Press, New York, pp 105-165
^ Koptsik, V.A. (1966). Shubnikov groups: Handbook on the symmetry and physical properties of crystal structures, Moscow University, Moscow
^ ^a ^b ^c Shubnikov, A.V. and Koptsik, V.A. (1974). Symmetry in science and art, Plenum Press, New York, ISBN 9780306307591 (original in Russian published by Nauka, Moscow, 1972.)
^ Washburn, D.K. and Crowe, D.W. (1988). Symmetries of Culture: Theory and Practice of Plane Pattern Analysis, Washington University Press, Seattle, ISBN 9780295970844
^ Conway, J.H., Burgeil, H. and Goodman-Strauss, C. (2008). The symmetries of things, A.K. Peters, Wellesley, MA, ISBN 9781568812205
^ Bohm, J. and Dornberger-Schiff, K. (1966). The nomenclature of crystallographic symmetry groups, Acta Crystallogr., 21, 1000-1007, doi:10.1107/S0365110X66004389
^ ^a ^b ^c Kopský, V. and Litvin, D.B. (eds.) (2010). International Tables for Crystallography Volume E: Subperiodic groups, Second online edition ISBN 978-0-470-68672-0, doi:10.1107/97809553602060000109
^ Fedorov, E.S. (1891). "Симметрія на плоскости" [Simmetriya na ploskosti, Symmetry in the plane]. Записки Императорского С.-Петербургского Минералогического Общества (Zapiski Imperatorskova Sankt-Petersburgskova Mineralogicheskova Obshchestva, Proceedings of the Imperial St. Petersburg Mineralogical Society). 2nd series (in Russian). 28: 345–390.
^ Pólya, G. (1924). Über die Analogie der Kristallsymmetrie in der Ebene (On the analog of crystal symmetry in the plane), Z. Krist., 60, 278–282, doi:10.1524/zkri.1924.60.1.278
^ Alexander, E. and Herrman, K. (1929). Die 80 zweidimensionalen Raumgruppen, Z. Krist. 70, 328-345, doi:10.1524/zkri.1929.70.1.328
^ Frankenheim, M.L. (1826). Crystallonomische Aufsätze, Isis (Jena) 19, 497-515, 542-565
^ Neronova, N.N. and Belov, N.V. (1961). A single scheme for the classical and black-and-white crystallographic symmetry groups, Sov. Phys. Cryst., 6, 3-12
^ Litvin, D.B. (1999). Magnetic subperiodic groups, Acta Cryst. A, 55(5), 963–964, doi:10.1107/S0108767399003487
^ Burckhardt, J.J. (1967). Zur Geschichte der Entdeckung der 230 Raumgruppen [On the history of the discovery of the 230 space groups], Archive for History of Exact Sciences, 4(3), 235-246, doi:10.1007/BF00412962
^ ^a ^b Brown, H., Bulow, R., Neubuser, J. et. al. (1978). Crystallographic groups of four-dimensional space, Wiley, New York, ISBN 9780471030959
^ ^a ^b Souvignier, B. (2006). The four-dimensional magnetic point and space groups, Z. Krist., 221, 77-82, doi:10.1524/zkri.2006.221.1.77
^ Palistrant, A.F. and Zamorzaev, A.M. (1992). Symmetry space groups: on the 100th anniversary of their discovery, ed. Vainshtein, B.K., Nauka, Moscow (in Russian)
^ Zamorzaev, A.M., Karpova, Yu.S., Lungu, A.P. and Palistrant, A.F. (1986). P-symmetry and its further development, Shtiintsa, Chisinau (in Russian)
^ Palistrant, A.F. (2012). Complete scheme of four-dimensional crystallographic symmetry groups, Crystallography Reports, 57(4), 471-477. doi:10.1134/S1063774512040104

External links

Crowe, D.W. (1986). The mosaic patterns of H.J. Woods, Comput. Math. Applic., 12B(1/2), 407-411
Schattschneider, D. (1986). In black and white: how to create perfectly colored symmetric patterns, Comput. Math. Applic., 12B(1/2), 673-695, doi:10.1016/0898-1221(86)90418-9
Senechal, M. (1988). Color symmetry, Comput. Math. Applic., 16(5-8), 545-553, doi:10.1016/0898-1221(88)90244-1
Radovic, L. and Jablan, S. (2001). Antisymmetry and modularity in ornamental art, Proceedings of Bridges: Mathematical Connections in Art, Music, and Science, Kansas, 55–66

[Loeb-1] Loeb, A.L. (1971). Color and Symmetry, Wiley, New York, ISBN 9780471543350

[Shubnikov1951-2] Shubnikov, A.V. (1951). Symmetry and antisymmetry of finite figures, Izv. Akad. Nauk SSSR, Moscow

[Colored_Symmetry-3] Shubnikov, A.V., Belov, N.V. et. al. (1964). Colored symmetry, ed. W.T. Holser, Pergamon, New York

[4] Gévay, G. (2000). Black-and-white symmetry, magnetic symmetry, self-duality and antiprismatic symmetry: the common mathematical background, Forma, 15, 57–60

[5] Tavger, B.A. (1958). The symmetry of ferromagnetics and antiferromagnetics, Sov. Phys. Cryst., 3, 341-343

[6] Woods, H.J. (1935). The geometric basis of pattern design part I: point and line symmetry in simple figures and borders, Journal of the Textile Institute, Transactions, 26, T197-T210

[7] Makovicky, E. (2016). Symmetry through the eyes of old masters, de Gruyter, Berlin, ISBN 9783110417050

[8] Atoji, A. (1965). Graphical representations of magnetic space groups, American Journal of Physics, 33(3), 212–219, doi:10.1119/1.1971375

[Mackay-9] Mackay, A.L. (1957). Extensions of space-group theory, Acta Crystallogr. 10, 543-548, doi:10.1107/S0365110X57001966

[10] Lockwood, E.H. and Macmillan, R.H. (1978). Geometric symmetry , Cambridge University Press, Cambridge, 67-70 & 206-208 ISBN 9780521216852

[Padmanabhan-11] Padmanabhan, H., Munro, J.M., Dabo, I and Gopalan, V. (2020). Antisymmetry: fundamentals and applications, Annual Review of Materials Research, 50, 255-281, doi:10.1146/annurev-matsci-100219-101404

[12] Shubnikov, A.V. (1960). Time reversal as an operation of antisymmetry, Sov. Phys. Cryst., 5, 309-314

[Heesch-13] Heesch, H. (1930). Über die vierdimensionalen Gruppen des dreidimensionalen Raumes, Z. Krist., 73, 325-345, doi:10.1524/zkri.1930.73.1.325

[Weber-14] Weber, L. (1929). Die Symmetrie homogener ebener Punktsysteme, Z. Krist., 70, 309-327, doi:10.1524/zkri.1929.70.1.309

[15] Woods, H.J. (1936). The geometric basis of pattern design part IV: counterchange symmetry in plane patterns, Journal of the Textile Institute, Transactions, 27, T305-320

[16] Landau, L.D. and Lifshitz E.M. (1951). Course of theoretical physics, vol. 5. Statistical physics, 1st edition, Nauka, Moscow

[Zamorzaev-17] Zamorzaev, A.M. (1953). Generalization of the space groups, Dissertation, Leningrad University

[18] Zamorzaev, A.M. (1957). Generalization of Fedorov groups, Sov. Phys. Cryst., 2, 10-15

[19] Tavger, B.A. and Zaitsev, V.M. (1956). Magnetic symmetry of crystals, Soviet Physics JETP, 3(3), 430-436

[Belov-20] Belov, N.V., Neronova, N.N. and Smirnova, T.S. (1957). Shubnikov groups, Sov. Phys. Cryst., 2, 311-322

[21] Zamorzaev, A.M. and Sokolov, E.I. (1957). Symmetry and various kinds of antisymmetry of finite bodies, Sov. Phys. Cryst., 2, 5-9

[22] Zamorzaev, A.M. and Palistrant, A.F. (1980). Antisymmetry, its generalizations and geometrical applications, Z. Krist., 151, 231-248, doi:10.1524/zkri.1980.151.3-4.231

[23] Zamorzaev, A.M. (1988). Generalized antisymmetry, Comput. Math. Applic., 16(5-8), 555-562, doi:10.1016/0898-1221(88)90245-3

[24] Holser, W.T. (1961). Classification of symmetry groups, Acta Crystallogr., 14, 1236-1242, doi:10.1107/S0365110X61003612

[25] Koptsik, V.A. (1968). A general sketch of the development of the theory of symmetry and its applications in physical crystallography over the last 50 years, Sov. Phys. Cryst., 12(5), 667-683

[26] Schwarzenberger, R.L.E. (1984). Colour symmetry, Bull. London Math. Soc., 16, 209-240, doi:10.1112/blms/16.3.209

[27] Grünbaum, B. and Shephard, G.V. (1987). Tilings and patterns, W.H. Freeman, New York, ISBN 9780716711933

[28] Brückler, F.M. and Stilinović, V. (2024) From friezes to quasicrystals: a history of symmetry groups, 1-42, doi:10.1007/978-3-030-19071-2_132-2, in Sriraman, B. (ed.) Handbook of the history and philosophy of mathematical practice, Springer, pp 3200, ISBN 978-3-030-19071-2

[29] MacGillavry, C.H. (1976). Symmetry aspects of M. C. Escher's periodic drawings, International Union of Crystallography, Utrecht, ISBN 9789031301843

[30] Schnattschneider, D. (2004). M. C. Escher: Visions of Symmetry, Harry. N. Abrams, New York, ISBN 9780810943087

[31] van der Waerden, B.L. and Burkhardt, J.J. (1961). Farbgruppen, Z. Krist, 115, 231-234, doi:10.1524/zkri.1961.115.3-4.231

[32] Opechowski, W. and Guccione, R. (1965). Magnetic symmetry in Magnetism, vol. IIA ed. Rado, G.T. and Suhl, H., Academic Press, New York, pp 105-165

[33] Koptsik, V.A. (1966). Shubnikov groups: Handbook on the symmetry and physical properties of crystal structures, Moscow University, Moscow

[Shubnikov-34] Shubnikov, A.V. and Koptsik, V.A. (1974). Symmetry in science and art, Plenum Press, New York, ISBN 9780306307591 (original in Russian published by Nauka, Moscow, 1972.)

[35] Washburn, D.K. and Crowe, D.W. (1988). Symmetries of Culture: Theory and Practice of Plane Pattern Analysis, Washington University Press, Seattle, ISBN 9780295970844

[36] Conway, J.H., Burgeil, H. and Goodman-Strauss, C. (2008). The symmetries of things, A.K. Peters, Wellesley, MA, ISBN 9781568812205

[37] Bohm, J. and Dornberger-Schiff, K. (1966). The nomenclature of crystallographic symmetry groups, Acta Crystallogr., 21, 1000-1007, doi:10.1107/S0365110X66004389

[volE-38] Kopský, V. and Litvin, D.B. (eds.) (2010). International Tables for Crystallography Volume E: Subperiodic groups, Second online edition ISBN 978-0-470-68672-0, doi:10.1107/97809553602060000109

[39] Fedorov, E.S. (1891). "Симметрія на плоскости" [Simmetriya na ploskosti, Symmetry in the plane]. Записки Императорского С.-Петербургского Минералогического Общества (Zapiski Imperatorskova Sankt-Petersburgskova Mineralogicheskova Obshchestva, Proceedings of the Imperial St. Petersburg Mineralogical Society). 2nd series (in Russian). 28: 345–390.

[40] Pólya, G. (1924). Über die Analogie der Kristallsymmetrie in der Ebene (On the analog of crystal symmetry in the plane), Z. Krist., 60, 278–282, doi:10.1524/zkri.1924.60.1.278

[41] Alexander, E. and Herrman, K. (1929). Die 80 zweidimensionalen Raumgruppen, Z. Krist. 70, 328-345, doi:10.1524/zkri.1929.70.1.328

[42] Frankenheim, M.L. (1826). Crystallonomische Aufsätze, Isis (Jena) 19, 497-515, 542-565

[Neronova-43] Neronova, N.N. and Belov, N.V. (1961). A single scheme for the classical and black-and-white crystallographic symmetry groups, Sov. Phys. Cryst., 6, 3-12

[Litvin1999-44] Litvin, D.B. (1999). Magnetic subperiodic groups, Acta Cryst. A, 55(5), 963–964, doi:10.1107/S0108767399003487

[45] Burckhardt, J.J. (1967). Zur Geschichte der Entdeckung der 230 Raumgruppen [On the history of the discovery of the 230 space groups], Archive for History of Exact Sciences, 4(3), 235-246, doi:10.1007/BF00412962

[Brown-46] Brown, H., Bulow, R., Neubuser, J. et. al. (1978). Crystallographic groups of four-dimensional space, Wiley, New York, ISBN 9780471030959

[Souvignier-47] Souvignier, B. (2006). The four-dimensional magnetic point and space groups, Z. Krist., 221, 77-82, doi:10.1524/zkri.2006.221.1.77

[48] Palistrant, A.F. and Zamorzaev, A.M. (1992). Symmetry space groups: on the 100th anniversary of their discovery, ed. Vainshtein, B.K., Nauka, Moscow (in Russian)

[49] Zamorzaev, A.M., Karpova, Yu.S., Lungu, A.P. and Palistrant, A.F. (1986). P-symmetry and its further development, Shtiintsa, Chisinau (in Russian)

[50] Palistrant, A.F. (2012). Complete scheme of four-dimensional crystallographic symmetry groups, Crystallography Reports, 57(4), 471-477. doi:10.1134/S1063774512040104

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Dichromatic symmetry

Examples

History

Dimensional counts

References

External links

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