Waterman polyhedra

Waterman Polyhedron

the basic series are defined as -
The convex hull of the intersection of a closed ball with the lattice of sphere- center points of a cubic close packing

A Generalized Waterman polyhedron, would include any related ccp structuring, wherein pertinent all space fillers become employed to establish a common/combined lattice point set. This Generalized approach is currently on the agenda, and as of yet, not finished the applicable programming via computer software.

These basic Waterman Polys were made in Early November 2005. The three rows represent Waterman polyhedron from root 1 through root 10 for two of the series below, the 1wn and the 6wn. Also shown, is a third series, based upon consecutive radial sweeps from a cube.

left row is "cubic" - with center of cube series from Generalized approach
center row is "1wn" - from center of one sphere - the BASIC series of WP
right row is "6wn" - with center of octahedral void series from Generalized approach

Photos taken by Bob Malvasio.

The gray polyhedron are all-space-fillers. Faces have been colored as; red - along the 6 x,y,z axis, blue as per the 8 vertex of a cube and yellow as per the 12 mid-points of the edges of a cube. "Gaps" between those three have been appropriately colored as orange, purple and green.

Click on the phrase above the icons below to go to these other sites.

applet by Mark Newbold [1 to 3000] all 7 series	Great Stella by Rob Webb [1 to unlimited] 2 of the 7 series	generator by Paul Bourke [1 to 1000] 4 of the 7 series	write up by Paul Bourke
ccp matrix schemes depictions by Maurice Starck ( slow loading...great graphics )	write up by Maurice Starck	MuPad on-line article [ inter-reactive ] by Mirek Majewski	MathWorld write up by Eric W. Weisstein
programs by Adrian Rossiter	Adrian Rossiter's site	write-up by Rene K. Muller 1w1 - 1w100	write-up by Rene K. Muller 1w101 - 1w250 removed temporarily due to space concerns
Wikipedia article	missing numbers	edge lengths maximum and minimum	next release 4d Waterman in Great Stella
SpringDance by Alan Ferguson [ 1-50 eig ]	early work by Kirby Urner	LynnClaire Dennis poly	Steinmetz-Waterman solids
dice combinations by Dan Thomasson	ShapeScape by Matt Storz	CCP hex prisms and w10	CCP inter-locking grids

the first few polyhedron from the basic sphere center series

no spheres at root 14

no spheres at root 30

no spheres at root 46

no spheres at root 56
[rt 14 x 4]

no spheres at root 62

no spheres at root 78

no spheres at root 94

102

no spheres at root 110

110

118

Integer sequences related to Waterman polyhedra by Neil J. Sloane.

w1,n series    ORIGIN is center of sphere
A119869 size        A119870 vertex    A119871 faces
A119872 edges     A119873 volumes
A055039 missing numbers

w6,n series    ORIGIN is at center of 6 sphere cluster...the octahedral void
A119874 size        A119875 vertex    A119876 faces
A119877 edges     A119878 volumes

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