the hexagonal closest packing (hcp)
The hcp can be described as L union (L+(0,3,3)), where L is a lattice
with basis
( (3,0,3),(3,3,0),(-4,4,4) ).
Then there are 4 basic symmetric mappings:
-
point symmetry at 1/2 * (0,3,3)
-
rotational symmetry around the axis (-4,4,4) with a rotation of 120 degrees
-
plane symmetry at the plane through 0 with normal vector (-4,4,4)
-
plane symmetry at the plane through 0 with normal vector (2,1,1)
These 4 basic symmetries can be combined to 24 symmetries containing the
identity.
The symmetric mappings of the hcp-configuration:
/ 2/3 1/3 -2/3 \ / 1 \
x = | -2/3 2/3 -1/3 | x + |-1 |
\ 1/3 2/3 2/3 / \-4 /
Densest packings with small paramter use to have dense facets. The densest
facets have tho following normal vectors. The image should explain where the
normal vector lies.
| 1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
| (-3,3,3) |
(3,-3,-3) |
(-7,7,-5) |
(7,-7,5) |
(-7,-5,7) |
(7,5,-7) |
(5,7,7) |
(-5,-7,-7) |
(11,1,1) |
(-11,-1,-1) |
(-1,1,-11) |
(1,-1,11) |
(-1,-11,1) |
(1,11,-1) |
| D |
B |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
The pictures have to be interpreted as follows: You are looking from direction
(-4,4,4). Plane 3,5,7 build a simplex without floor. If you cut this simplex
with a plane orthogonal to the direction (-4,4,4), you get a triangle like in
the picture as floor of a complete simplex...
Peter Scholl
(Peter.Scholl@unix-ag.org)
(last modified: 11.05.2001)