A matrix approach to symmetry

Himes, V. L.; Mighell, A. D.

doi:10.1107/s0108767387099276

Cited by 11 publications

(19 citation statements)

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“…An important problem is the lack of an appropriate metric for the necessary comparisons. Himes & Mighell (1987) proposed the use of B matrices to determine which symmetry axes are present in a lattice. No metric is described which would indicate how close the test lattice is to a lattice of a particular type.…”

Section: Review Of Older Methodsmentioning

confidence: 99%

Lattices and reduced cells as points in 6-space and selection of Bravais lattice type by projections

Andrews¹,

Bernstein²

1988

Acta Crystallogr A Found Crystallogr

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Section: Review Of Older Methodsmentioning

confidence: 99%

Lattices and reduced cells as points in 6-space and selection of Bravais lattice type by projections

Andrews¹,

Bernstein²

1988

Acta Crystallogr A Found Crystallogr

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“…Using converse-transformation analysis and NIST, LAT-TICE (Karen & Mighell, 1991a, b), the experimentalist can immediately determine symmetry matrices and the highest possible crystal symmetry. With this knowledge, one is then able to determine in a logical and accurate manner the Laue group (Himes & Mighell, 1987) and the space group.…”

Section: Origin and Solution Of The Problemmentioning

confidence: 99%

“…§ Cell 2 = C-centered orthorhombic transformed cell (angles not given, maximum deviation from 90 ° = 0.05 ° for examples 1-3 and 0.73 ° for examples 4-6. ¶ Number of symmetry matrices (Himes & Mighell, 1987). ** Kuriyan et al (1990).…”

Section: Origin and Solution Of The Problemmentioning

confidence: 99%

“…As part of our review of the crystal symmetry in crystalline materials, we have analyzed the metric symmetry* of the proteins in the NIST/CARB Biological Macromolecule Crystallization Database (1990) and in the Brookhaven Protein Data Bank (April 1990 release) (Bernstein et al, 1977). Using converse-transformation theory (Himes & Mighell, 1987;Karen & Mighell, 1991a, b) and reduction techniques (Mighell & Rodgers, 1980), our analyses have revealed that it is * Metric symmetry = the symmetry of a lattice as defined by any arbitrary primitive triplet of noncoplanar translations (a triplet is called primitive when it defines a primitive cell). not uncommon in these macromolecules for the metric symmetry to exceed the reported crystal symmetry.…”

Section: Introductionmentioning

confidence: 99%

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Protein symmetry: metric and crystal (a precautionary note)

Mighell¹,

Rodgers²,

Karen³

1993

J Appl Cryst

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A review of the symmetry of proteins has revealed that in many cases the metric symmetry exceeds the reported crystal symmetry. Regardless of the reason, this observation has important implications for experimental protein crystallography. Standard laboratory procedure should always include a direct determination of the lattice metric symmetry. With full knowledge of the highest possible symmetry, the experimentalist is then able to determine in a logical and accurate manner the Laue group and the space group. For those proteins in which it has been proved that the metric symmetry exceeds the crystal symmetry, the protein crystallographer must proceed with caution (e.g. in relating multiple sets of data, positional parameters etc. on the same or related crystals), because a given lattice will have metrically similar unit cells that are not symmetrically equivalent.

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“…metric and subsequent geometric interpretation of the filtered matrices lead to mathematically stable and rich information on the Bravais-lattice symmetry and deviations from the exact symmetry. The emphasis on the crystallographic features of lattices was obtained by shifting the focus (i) from analysis of a lattice metric to analysis of symmetry matrices [Himes & Mighell (1987). Acta Cryst.…”

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confidence: 99%

Symmetry of semi-reduced lattices

Stróż

2015

Acta Cryst Sect A

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The main result of this work is extension of the famous characterization of Bravais lattices according to their metrical, algebraic and geometric properties onto a wide class of primitive lattices (including Buerger-reduced, nearly Buerger-reduced and a substantial part of Delaunay-reduced) related to low-restricted semi-reduced descriptions (s.r.d.'s). While the `geometric' operations in Bravais lattices map the basis vectors into themselves, the `arithmetic' operators in s.r.d. transform the basis vectors into cell vectors (basis vectors, face or space diagonals) and are represented by matrices from the set {\bb V} of all 960 matrices with the determinant ±1 and elements {0, ±1} of the matrix powers. A lattice is in s.r.d. if the moduli of off-diagonal elements in both the metric tensors M and M(-1) are smaller than corresponding diagonal elements sharing the same column or row. Such lattices are split into 379 s.r.d. types relative to the arithmetic holohedries. Metrical criteria for each type do not need to be explicitly given but may be modelled as linear derivatives {\bb M}(p,q,r), where {\bb M} denotes the set of 39 highest-symmetry metric tensors, and p,q,r describe changes of appropriate interplanar distances. A sole filtering of {\bb V} according to an experimental s.r.d. metric and subsequent geometric interpretation of the filtered matrices lead to mathematically stable and rich information on the Bravais-lattice symmetry and deviations from the exact symmetry. The emphasis on the crystallographic features of lattices was obtained by shifting the focus (i) from analysis of a lattice metric to analysis of symmetry matrices [Himes & Mighell (1987). Acta Cryst. A43, 375-384], (ii) from the isometric approach and invariant subspaces to the orthogonality concept {some ideas in Le Page [J. Appl. Cryst. (1982), 15, 255-259]} and splitting indices [Stróż (2011). Acta Cryst. A67, 421-429] and (iii) from fixed cell transformations to transformations derivable via geometric information (Himes & Mighell, 1987; Le Page, 1982). It is illustrated that corresponding arithmetic and geometric holohedries share space distribution of symmetry elements. Moreover, completeness of the s.r.d. types reveals their combinatorial structure and simplifies the crystallographic description of structural phase transitions, especially those observed with the use of powder diffraction. The research proves that there are excellent theoretical and practical reasons for looking at crystal lattice symmetry from an entirely new and surprising point of view - the combinatorial set {\bb V} of matrices, their semi-reduced lattice context and their geometric properties.

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A matrix approach to symmetry

Cited by 11 publications

References 1 publication

Lattices and reduced cells as points in 6-space and selection of Bravais lattice type by projections

Lattices and reduced cells as points in 6-space and selection of Bravais lattice type by projections

Protein symmetry: metric and crystal (a precautionary note)

Symmetry of semi-reduced lattices

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