Crystallography & Symmetry
patterns derived from their groups
Concept

Cell transformation — where it appears

A stated change from one cell of a lattice to another, recorded as the matrix carrying one basis to the other. It is what makes a claim that two descriptions are the same lattice checkable rather than asserted.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The fourteen Bravais lattices. All fourteen lattices: triclinic P, with 2 symmetries; monoclinic P, with 4 symmetries; monoclinic C, with 4 symmetries; orthorhombic P, with 8 symmetries; orthorhombic C, with 8 symmetries; orthorhombic I, with 8 symmetries; orthorhombic F, with 8 symmetries; tetragonal P, with 16 symmetries; tetragonal I, with 16 symmetries; rhombohedral P, with 12 symmetries; hexagonal P, with 24 symmetries; cubic P, with 48 symmetries; cubic I, with 48 symmetries; cubic F, with 48 symmetries. The corner points are the conventional cell; the points in the second colour are the centring translations, drawn at every position inside the cell rather than one per face. The cell shapes are the picture's, chosen so no two systems look alike; only the angles a system is defined by mean anything.

Twenty-five cells, and fourteen lattices

The usual picture of the fourteen Bravais lattices is a plate of fourteen boxes, which is the answer with the argument removed. The argument is one question asked of twenty-five candidates, and the question has a computable answer.

lattices · The fourteen Bravais lattices 3 lattices. 3 lattices: hexagonal P, with 24 symmetries; rhombohedral P, with 12 symmetries; hexagonal R, with 12 symmetries. The corner points are the conventional cell; the points in the second colour are the centring translations, drawn at every position inside the cell rather than one per face. The cell shapes are the picture's, chosen so no two systems look alike; only the angles a system is defined by mean anything.

A lattice described on somebody else's axes

R-centring a hexagonal cell does lower its symmetry, from twenty-four to twelve — and the lattice that results is the fourteenth, the rhombohedral one, which already appears on the list under its own axes. It is the only row in the enumeration where losing symmetry and being a duplicate are the same verdict.

lattices · The fourteen Bravais lattices

Named alongside it

The objects these essays reach for when they reach for this one.

CentringBravais latticeCrystal systemHexagonal axesHolohedryLattice automorphismMetric tensorObverse reverseRhombohedralSettingTrigonal

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