geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the Waterman polyhedra are a family of

polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...

discovered around 1990 by the mathematician Steve Waterman. A Waterman polyhedron is created by packing spheres according to the 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, then creating the

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

of the sphere centers. Image:Waterman_Packed_Spheres_0024.1.png, Cubic Close(st) Packed spheres with radius Image:Waterman_0024.1.png, Corresponding Waterman polyhedron W24 Origin 1 W5 sphere cluster

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 In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...

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 geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle' ...

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.

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

There can be seven origins defined in CCP,7 Origins of CCP Waterman polyhedra
by Mark Newbold where n = : * Origin 1: offset 0,0,0, radius

\sqrt

* Origin 2: offset ,,0, radius

\tfrac12\sqrt

* Origin 3: offset ,,, radius

\tfrac13\sqrt

* Origin 3*: offset ,,, radius

\tfrac13\sqrt

* Origin 4: offset ,,, radius

\tfrac12\sqrt

* Origin 5: offset 0,0,, radius

\tfrac12\sqrt

* Origin 6: offset 1,0,0, radius

\sqrt

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

Relation to Platonic and Archimedean solids

Some Waterman polyhedra create

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...

s and

Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...

s. For this comparison of Waterman polyhedra they are normalized, e.g. has a different size or volume than but has the same form as an octahedron.

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.

Archimedean solids

Cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...

: W1 O1, W4 O1 *

Truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...

: W10 O1 *

Truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedro ...

: W4 O3, W2 O4 *The other Archimedean solids have no representation as Waterman polyhedra. The W7 O1 might be mistaken for a

truncated cuboctahedron In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its fac ...

, as well W3 O1 = W12 O1 mistaken for a

rhombicuboctahedron In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at ea ...

, but those Waterman polyhedra have two edge lengths and therefore do not qualify as Archimedean solids.

Generalized Waterman polyhedra

Generalized Waterman polyhedra are defined as the convex hull derived from the point set of any spherical extraction from a regular lattice. Included is a detailed analysis of the following 10 lattices – bcc, cuboctahedron, diamond, fcc, hcp, truncated octahedron,

rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahedro ...

, simple cubic, truncated tet tet, truncated tet truncated octahedron cuboctahedron. 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. From a viable origin point, within a lattice, there exists an unlimited series of polyhedra. Given its proper sweep interval, then there is a one-to-one correspondence between each integer value and a generalized Waterman polyhedron.

Notes

External links

Steve Waterman's Homepage hand-made models by Magnus Wenninger write-up by Paul Bourke on-line generator by Paul Bourke
* ttp://www.progonos.com/furuti/MapProj/Normal/ProjPoly/projPoly2.html Waterman projection and write up by Carlos Furitibr>rotating globe by Izidor Hafnerreal time winds and temperature on Waterman projection by Cameron BeccarioSolar Termination (Waterman)
by

Mike Bostock Michael Bostock is an American computer scientist and data-visualisation specialist. He is one of the co-creators of ''Observable'' and noted as one of the key developers of D3.js, a JavaScript library used for producing dynamic, interactive, o ...

interactive Waterman butterfly map by Jason Daviesfirst 1000 Waterman polyhedra and sphere clusters by Nemo Thorx
*{{OEIS el, 1=A119870, 2=Number of vertices of the root-n Waterman polyhedron
Generalized Waterman polyhedron by Ed Pegg jr of Wolfram various Waterman sphere clusters by Ed Pegg jr of Wolfram app to make 4d waterman polyhedron in Great Stella by Rob Webb
* ttp://www.mathworks.com/matlabcentral/answers/93067 Waterman polyhedron in Mupad Polyhedra