Crystallography & Symmetry
patterns derived from their groups
Concept

Closed form — where it appears

A crystal form whose faces enclose a volume, so a crystal could be bounded by that form alone. Whether a form closes is decided in integers, by asking whether any half-space contains every one of its face poles, and thirty of the forty-eight distinct shapes are of this kind.

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

{111} in class m3̅m. The form {111} of crystal class m3̅m: 8 faces, being the orbit of one face under the 48 operations of the class, with a stabiliser of order 6. 4 poles lie in the upper hemisphere or in the plane of the page and are drawn filled; the other 4 lie below and are drawn open at the same positions, which is the stereographic convention and the reason only the upper ones carry their indices. The form is closed: the faces enclose a volume, so a crystal can be bounded by this form alone.

A form is an orbit, and whether it closes is an integer question

Name one face of a crystal and its class names the rest. That set is a form, it is an orbit in exactly the sense this site has used since its first essay, and whether it encloses a volume — whether a crystal could be bounded by it alone — is decided without any lengths or angles entering the calculation anywhere.

applied · Forms A hundred and thirteen orbit types, merged into shapes. The three counts, and what stands between them. 113 is the number of kinds of form this site publishes: one for every stabiliser a face can have, in every class. Allowing a stratum to change shape along its own family raises it to 164. Merging entries that are the same solid with the same symmetry, wherever they occur, brings it down to 48 — 30 that enclose a volume and 18 that do not, which is the count every mineralogy text prints, with the dome and the sphenoid kept apart rather than merged. The last line is the warning: throwing away the symmetry of the solid and keeping only its combinatorial type leaves 35, because a rhombic dipyramid, a tetragonal dipyramid and an octahedron are one and the same arrangement of eight triangles.

A hundred and thirteen orbits, and forty-eight shapes

This collection reports 113 kinds of crystal form and every mineralogy text reports 47. That difference was explained here in a paragraph and never computed, which means nobody had checked it. Computing it needs a definition of *shape* a program can decide, and the definition turns out to be the interesting part.

applied · Forms

Named alongside it

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

Crystal formColour refinementCombinatorial typeFace poleGeneric cellMiller indicesMultiplicityOpen formOrbitShape groupSpecial positionStabiliser

All concepts