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Introduction

Waterman polyhedra were invented, around 1990, by Steve Waterman---a Canadian enthusiast of mathematics, physics and cartography. In fact, the idea of his polyhedra came up while solving a cartography problem.

Waterman polyhedra form a vast family of polyhedra. Some of them have a number of nice properties like multiple symmetries, or very interesting and regular shapes. Some other are just a bunch of faces formed out of irregular convex polygons. The most popular Waterman polyhedra are those with centers in the point (0,0,0) and build out of hundreds of polygons. Such polyhedra resemble big spheres in 3D. In fact, the more faces has a Waterman polyhedron, the more it shape resembles the sphere circumscribed on it. Its volume and total area are close to those parameters of the circumscribed on it sphere.

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 mathematical point of view we can consider Waterman polyhedra as a 4D space W(x,y,z,r), where x,y,z are coordinates of a point in 3D, and r is a positive number, and r>1.

Due to its complexity, and sometimes huge number of faces, Waterman polyhedra are a very interesting and challenging computational problem. In last 10 years there were a few attempts to produce Waterman polyhedra using Java programming language (Mark Newbold, [2]), Python (Kirby Urner, [3]), C++ and POV-Ray (Paul Bourke, [1]). Most of these implementations produce a family of Waterman polyhedra with center in (0,0,0). Mark Newbold developed a Java applet that is able to calculate and display seven families of Waterman polyhedra associated with some selected points in 3D.

In this paper we will show how Waterman polyhedra can be visualized in MuPAD using its programming language and its powerful visualization tool---Virtual Camera. Finally, Waterman polyhedra developed in MuPAD can be exported to JavaView and displayed online as JavaView virtual models.

2. The concept of Waterman polyhedra and Waterman Polyhedra Space

We will start by examining the concept of Waterman polyhedra. Imagine that we place on a plane a bunch of spheres, all having the same radius, say and organize them in such a way that the empty space between them is minimal, see figure 1.

Figure 1 - Single layer of close packing spheres

In the figure 1 we used only a finite number of spheres. However, in theory we should consider an infinite, in each horizontal direction, layer of spheres. Now, on top of this layer we can put another layer of spheres in such a way that the new spheres will fall into holes between spheres of the first layer. Of course, we still assume that all spheres have the same radius. If we continue this process by adding more layers up and down of the first layer we will create a regular arrangement, often called the lattice arrangement, that is known in crystallography as Cubic Close Packing (CCP) or Face Centered Cubic arrangement (FCC). A finite example of CCP arrangement is shown in the figure 2.

Figure 2 - Cubic Close Packing arrangement of spheres

Now, let us concentrate on centers of spheres of the CCP arrangement (figure 3). In this paper we will call them CCP points. These are located on a grid of horizontal and vertical lines, that in crystallography is used to represent sodium chloride crystals. In this representation, chloride ions shown as big dots in the figure 3 are arranged in a cubic close packing, while sodium ions fill the octahedral gaps between them. Each chloride ion has six nearest neighbors with octahedral geometry.

Figure 3 - Centers of spheres in CCP arrangement

In the next step let us take a point P(x,y,z) in 3D and develop a large sphere with radius R. In such case some of our CCP points will fall inside of the large sphere and some not. If we remove all CCP points that are outside of the big sphere we will obtain a finite number of points that, in this paper, we will call Waterman points associated with the point P and radius R (see figure 4).

Figure 4 - A large sphere with Waterman points

In the final step we will take all Waterman points, i.e. points that lay inside of the big sphere, or on its surface, and use them to build a convex hull. This will be the Waterman polyhedron associated with the point P(x,y,z) and radius R. Figures 5a, 5b and 5c show Waterman polyhedra associated with the point P(0,0,0) and radii equal to 2, 5.5, and 10, respectively. Figure 5d shows a Waterman polyhedron with center (0.2, 0.3, 0.4) and radius equal to 5. We can easily notice that first three polyhedra have a number of symmetries while the last one seems to be completely irregular. Waterman polyhedron with the center in P(x,y,z) and radius R we will denote as W((x,y,z),R).

 

Figure 5a - W((0,0,0),2)

Figure 5b - W((0,0,0), 5.5)

Figure 5c - W((0,0,0), 10)

Figure 5d - W((0.2, 0.3, 0.4), 5)

In our examples, for the CCP arrangement we used spheres with radius equal to sqrt(2)/2. Therefore, we may easily notice that coordinates of centers of spheres forming the CCP lattice are integer numbers m, n and p, such that m+n+p is an even number. A simple consequence of the CCP lattice structure, and how Waterman polyhedra are created, is that

(*) W((x+2k,y,z),R)=W((x,y+2k,z),R)=W((x,y,z+2k),R)=W((x,y,z),R)

where k is an integer number. Therefore, we can restrict investigations of Waterman polyhedra to points (x,y,z) in the cube [0,2)x[0,2)x[0,2) and any positive radius R.

On his web pages Mark Newbold investigated seven sequences of Waterman polyhedra (see [2]). Each sequence was created by starting from a fixed point in 3D as a center of Waterman polyhedra and changing radius of the big sphere. Although Mark Newbold used slightly different points as centers of his Waterman polyhedra sequences than those shown in the figure 6, according to our observation (*) we can use points that are equivalent to Newbold's points and are located in the cube [0,2)x[0,2)x[0,2), or even in [0,1)x[0,1)x[0,1). We show them in the figure 6.

 

Figure 6 - Seven origins of Waterman polyhedra sequences by Mark Newbold

Table 1 presents detailed information about these seven points.

 

Origin	Origin location	Description
1	(0, 0, 0)	Center of a sphere
2	(1/2,1/2,0)	Point where two spheres touch, e.g ((0,0,0)+(1,1,0))/2
3	(1/3,1/3,2/3)	Center of the space between three spheres, e.g. ((0,0,0)+(0,1,1)+(1,0,1))/3=(1/3,1/3,2/3)
3*	(1/3,1/3,1/3)	Center of the space between three empty nodes of the CCP lattice, e.g. ((1,0,0)+(0,1,0)+(0,0,1))/3=(1/3,1/3,1/3)
4	(1/2,1/2,1/2)	Center of the space between four spheres or for empty nodes of the CPP, e.g. ((0,0,0)+(1,0,1)+(0,1,1)+(1,1,0))/4=(1/2,1/2,1/2)
5	(0, 0, 1/2)	Point half way between sphere center and empty node of the CCP lattice, e.g. ((0,0,0) +(0,0,1))/2=(1/2,1/2,1/2)
6	(1, 0, 0)	Empty node of the CPP lattice

Table 1 -- Origins of Waterman polyhedra sequences by Mark Newbold

 

Finally, in order to create his sequences of Waterman polyhedra Mark Newbold used specific radii. The main reason for this was that, quite often, even a very small change of the radius creates a new polyhedron. Table 2 presents the radii formula for each sequence of Waterman polyhedra by Mark Newbold. Note, the radii formulae in the table 2 were modified in such a way that for each sequence the parameter n in calculations of Waterman polyhedra starts from 1 and is exactly the same as the number of the polyhedron in the sequence. In Newbold's original work the parameter n in calculations is not always the same as the number of the polyhedron in its sequence. Note also, that the name of the origin is not always the same as the number of the sequence .

 

Sequence #	Origin name	Radius formula
1	Origin 1
2	Origin 6
3	Origin 4
4	Origin 2
5	Origin 5
6	Origin 3
7	Origin 3*

Table 2 -- Radii formulae for Mark Newbold's sequences of Waterman polyhedra

In further pages of this paper we will use notation w_k,n to represent the n-th Waterman polyhedron from the sequence k, where k=1..7, for example w_3,10 is the tenth Waterman polyhedron from the sequence 3.