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{{more footnotes|date=November 2013}}
{{more citations|date=April 2023}}

In [[geometry]], the '''Waterman polyhedra''' are a family of [[polyhedron|polyhedra]] discovered around 1990 by the mathematician [[Steve Waterman (mathematician)|Steve Waterman]]. A Waterman polyhedron is created by [[Sphere packing|packing spheres]] according to the [[cubic close packing|cubic close(st) packing]] (CCP), also known as the face-centered cubic (fcc) packing, then sweeping away the spheres that are farther from the center than a defined radius,<ref name=Popko2012>{{cite book|last=Popko|first=Edward S.|title=Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere|year=2012|publisher=CRC Press|isbn=9781466504295|pages=174–177}}</ref> then creating the [[convex hull]] of the sphere centers.

<gallery>
Image:Waterman_Packed_Spheres_0024.1.png|Cubic Close(st) Packed spheres with radius {{sqrt|24}}
Image:Waterman_0024.1.png|Corresponding Waterman polyhedron W24 Origin 1
</gallery>
[[File:W5 sphere cluster.jpg|thumb|Waterman/fcc sphere cluster W5]]
[[File:W5 polyhedron.jpg|thumb|Waterman/fcc polyhedra interpretation of sphere cluster w5]]
Waterman polyhedra form a vast family of polyhedra. Some of them have a number of nice properties such as multiple symmetries, or interesting and regular shapes. Others are just a collection of faces formed from irregular [[convex polygon]]s.

The most popular Waterman polyhedra are those with centers at the point (0,0,0) and built out of hundreds of polygons. Such polyhedra resemble spheres. In fact, the more faces a Waterman polyhedron has, the more it resembles its [[circumscribed sphere]] in volume and total area.

With each point of 3D space we can associate a family of Waterman polyhedra with different values of radii of the circumscribed spheres. Therefore, from a mathematical point of view we can consider Waterman polyhedra as 4D spaces W(x, y, z, r), where x, y, z are coordinates of a point in 3D, and r is a positive number greater than 1.<ref>[http://www.mupad.com/mathpad/2006/majewski/index.php Visualizing Waterman Polyhedra with MuPAD] by M. Majewski</ref>

==Seven origins of cubic close(st) packing (CCP)==

There can be seven origins defined in CCP,<ref>[http://dogfeathers.com/java/ccppoly.html 7 Origins of CCP Waterman polyhedra] by Mark Newbold</ref> where n = {1, 2, 3, …}:
* Origin 1: offset 0,0,0, radius <math>\sqrt{2n}</math>
* Origin 2: offset {{sfrac|1|2}},{{sfrac|1|2}},0, radius <math>\tfrac12\sqrt{2+4n}</math>
* Origin 3: offset {{sfrac|1|3}},{{sfrac|1|3}},{{sfrac|2|3}}, radius <math>\tfrac13\sqrt{6(n+1)}</math>
* Origin 3*: offset {{sfrac|1|3}},{{sfrac|1|3}},{{sfrac|1|3}}, radius <math>\tfrac13\sqrt{3+6n}</math>
* Origin 4: offset {{sfrac|1|2}},{{sfrac|1|2}},{{sfrac|1|2}}, radius <math>\tfrac12\sqrt{3+8(n-1)}</math>
* Origin 5: offset 0,0,{{sfrac|1|2}}, radius <math>\tfrac12\sqrt{1+4n}</math>
* Origin 6: offset 1,0,0, radius <math>\sqrt{1+2(n-1)}</math>

Depending on the origin of the sweeping, a different shape and resulting polyhedron are obtained.

[[Image:Waterman Overview.png]]

==Relation to Platonic and Archimedean solids==
{{Unsourced section|date=April 2023}}

Some Waterman polyhedra create [[Platonic solid]]s and [[Archimedean solid]]s. For this comparison of Waterman polyhedra they are normalized, e.g. {{nowrap|W2 O1}} has a different size or volume than {{nowrap|W1 O6,}} but has the same form as an octahedron.{{citation needed|date=April 2023}}

===Platonic solids===
*Tetrahedron: W1 O3*, W2 O3*, W1 O3, W1 O4
*Octahedron: W2 O1, W1 O6
*Cube: W2 O6
*Icosahedron and dodecahedron have no representation as Waterman polyhedra.{{citation needed|date=April 2023}}

===Archimedean solids===
*[[Cuboctahedron]]: W1 O1, W4 O1
*[[Truncated octahedron]]: W10 O1
*[[Truncated tetrahedron]]: W4 O3, W2 O4
*The other Archimedean solids have no representation as Waterman polyhedra.{{citation needed|date=April 2023}}

The W7 O1 might be mistaken for a [[truncated cuboctahedron]], as well W3 O1 = W12 O1 mistaken for a [[rhombicuboctahedron]], but those Waterman polyhedra have two edge lengths and therefore do not qualify as Archimedean solids.{{citation needed|date=April 2023}}

==Generalized Waterman polyhedra==
{{Unsourced section|date=April 2023}}
Generalized Waterman polyhedra are defined as the convex hull derived from the point set of any spherical extraction from a regular lattice.{{citation needed|date=April 2023}}

Included is a detailed analysis of the following 10 lattices – bcc, cuboctahedron, diamond, fcc, hcp, truncated octahedron, [[rhombic dodecahedron]], simple cubic, truncated tet tet, truncated tet truncated octahedron cuboctahedron.{{citation needed|date=April 2023}}

Each of the 10 lattices were examined to isolate those particular origin points that manifested a unique polyhedron, as well as possessing some minimal symmetry requirement.{{citation needed|date=April 2023}} From a viable origin point, within a lattice, there exists an unlimited series of polyhedra.{{citation needed|date=April 2023}} Given its proper sweep interval, then there is a one-to-one correspondence between each [[Integer-valued polynomial|integer value]] and a generalized Waterman polyhedron.{{citation needed|date=April 2023}}

==Notes==
<references/>

==External links==
*[http://watermanpolyhedron.com Steve Waterman's Homepage]
*[http://dogfeathers.com/java/ccppoly.html Waterman Polyhedra Java applet by Mark Newbold]
*[http://www.ac-noumea.nc/maths/polyhedr/Waterman_.htm Maurice Starck's write-up]
*[http://www.software3d.com/UsersModels.php   hand-made models by Magnus Wenninger]
*[http://paulbourke.net/geometry/waterman/  write-up by Paul Bourke]
*[http://paulbourke.net/geometry/waterman/gen/  on-line generator by Paul Bourke]
*[http://www.antiprism.com/programs/waterman.html  program to make Waterman polyhedron by Adrian Rossiter in Antiprism]
*[http://www.progonos.com/furuti/MapProj/Normal/ProjPoly/projPoly2.html Waterman projection and write up by Carlos Furiti]
*[http://www.watermanpolyhedron.com/CART5.html rotating globe by Izidor Hafner]
*[http://earth.nullschool.net/#current/wind/surface/level/waterman real time winds and temperature on Waterman projection by Cameron Beccario]
*[http://bl.ocks.org/mbostock/8428323 Solar Termination (Waterman)] by [[Mike Bostock]]
*[http://www.jasondavies.com/maps/waterman-butterfly/ interactive Waterman butterfly map by Jason Davies]
*[http://www.ac-noumea.nc/maths/polyhedr/Waterman_.htm write-up by Maurice Starck]
*[https://www.youtube.com/watch?v=fJz5cFPi0Qw first 1000 Waterman polyhedra and sphere clusters by Nemo Thorx]
*{{OEIS el|1=A119870|2=Number of vertices of the root-n Waterman polyhedron}}
*[http://www.watermanpolyhedron.com/watermanpolyhedra1.html Steve Waterman's Waterman polyhedron (WP)]
*[https://www.youtube.com/watch?v=ajwR6ku5WGk  Generalized Waterman polyhedron by Ed Pegg jr of Wolfram]
*[https://www.youtube.com/watch?v=06-b-Mjr1eY  various Waterman sphere clusters by Ed Pegg jr of Wolfram]
*[http://www.software3d.com/StellaManual.php?prod=stella4D  app to make 4d waterman polyhedron in Great Stella by Rob Webb]
*[http://www.mathworks.com/help/symbolic/mupad_ref/plot-waterman.html   Waterman polyhedron app in Matlab needs a workaround as shown on the following reference page]
*[http://www.mathworks.com/matlabcentral/answers/93067   Waterman polyhedron in Mupad]

[[Category:Polyhedra]]