Generalized Waterman Polyhedron from polyhedral packs

single lattice series
the general series are defined as -
The convex hull of the intersection of a closed ball with the lattice(s) of a space filler grouping swept from symmetrically biased locations; polyhedral center, edge mid-point, vertex or face center.
A Generalized Waterman polyhedron, includes any related lattice structuring, wherein pertinent all space fillers become employed to establish a common/combined lattice point set. This Generalized approach is currently able to be manifested and viewed by Antiprism...written by Adrian Rossiter

[ The following displays may become obsolete...should a new applet related to space groups become a reality...stay tuned ( 7 May 2008 ) ]

Meanwhile, there currently is depicted a total there are well over 2000 depictions that can be seen seen from below. This single lattice groups finds 63 differents unlimited series of Waterman polyhedron. Of special interest are those designed as symmetry groups Oh and those labelled as twist. There are 14 related Oh series, 7 related twist series amoung these 63. The remaining 42 series shown... all possess this feature; the ability to slice the polyhedron into equal halves, uniquely, at least twice.

Symmetries are shown; as the count of possible events through a single point, that cut the object into two equal halves. A second point may be required to account for all possible symmetrical conditions. For example, a cube 's face center could have 4 events while an additional unigue halving may also be attributed to an equatorial halving. The cube would accordingly by given a symmetrical halving value of 4 1.


click left hand side below to see related Waterman polyhedron; 1-32, 36 and 48
click image and drag to rotate image


bcc Oh Oh C2v C2v
Oh C4v C3v
cuboctahedron
diamond
fcc Oh Oh C2v Oh
C4v C2v C4v
hcp Oh Oh C2v Oh
C4v C2v C4v
truncated octahedron Oh C4v C2v C3v
Oh C4v Oh C4v
C4v Oh
rhombic dodecahedron Oh Oh C3v C4v
C4v C4v C2v
sc and sc offset combined Oh Oh Oh Oh
Oh
truncated tet tet C2v C3v C4v C4v
C2v C4v C4v C4v
C2v
truncated tet truncated octahedron cuboctahedron C2v C3v C4v C4v



39coordinate points used for sweep analysis

red points extend from 0,0,0 to 2,2,2 - each block is therefore a length of 2



39 tested Origins

Antiprism itself is a collection of programs that facilitates numerous techniques, like weaves and conversion and merging abilities as well as a viewer. It is also free as well as being extremely versatile.
Here are the downloadable programs.
Here is a additional MASTER page for SPHERE packed ccp applets - links - software related - examples - write ups - etc

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