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Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group. These are the Bravais lattices in three dimensions: P primitive; I body centered (from the German Innenzentriert) F face centered (from the German Flächenzentriert) A centered on A faces only; B centered on ...
The face-centered cubic lattice (cF) has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving a total of four lattice points per unit cell (1 ⁄ 8 × 8 from the corners plus 1 ⁄ 2 × 6 from the faces).
(This 4-dimensional body centered cubic lattice is actually the union of two tesseractic honeycombs, in dual positions.) This is the same densest known regular 3-sphere packing, with kissing number 24, that is also seen in the other two regular tessellations of 4-space, the 16-cell honeycomb and the 24-cell-honeycomb .
It is the Brillouin zone of body-centered cubic (bcc) crystals. Some minerals such as garnet form a rhombic dodecahedral crystal habit . As Johannes Kepler noted in his 1611 book on snowflakes ( Strena seu de Nive Sexangula ), honey bees use the geometry of rhombic dodecahedra to form honeycombs from a tessellation of cells each of which is a ...
The Bravais lattice of the space group is determined by the lattice system together with the initial letter of its name, which for the non-rhombohedral groups is P, I, F, A or C, standing for the principal, body centered, face centered, A-face centered or C-face centered lattices. There are seven rhombohedral space groups, with initial letter R.
The vertex arrangement of the 16-cell honeycomb is called the D 4 lattice or F 4 lattice. [2] The vertices of this lattice are the centers of the 3-spheres in the densest known packing of equal spheres in 4-space; [3] its kissing number is 24, which is also the same as the kissing number in R 4, as proved by Oleg Musin in 2003.
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The Wigner–Seitz cell of the face-centered cubic lattice is a rhombic dodecahedron. [9] In mathematics, it is known as the rhombic dodecahedral honeycomb . The Wigner–Seitz cell of the body-centered tetragonal lattice that has lattice constants with c / a > 2 {\displaystyle c/a>{\sqrt {2}}} is the elongated dodecahedron .
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