I. Introduction Consider a source uniformly distributed over a large ball B in R k, centered at the origin. Suppose that this source is quantized usin

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1 Accepted for publication in IEEE Trans. Inform. Thy., Mar. 2, 200 On Quantization with the Weaire-Phelan Partition * Navin Kashyap and David L. Neuho Department of Electrical Engineering and Computer Science University of Michigan Ann Arbor, MI 4809 fnkashyap,neuhog@eecs.umich.edu Abstract Until recently, the solution to the Kelvin problem of nding a partition of R into equalvolume cells with the least surface area was believed to be tessellation by the truncated octahedron. In 994, D. Weaire and R. Phelan described a partition that outperformed the truncated octahedron partition in this respect. This raises the question of whether the Weaire-Phelan (WP) partition can outperform the truncated octahedron partition in terms of normalized moment of inertia (NMI), thus providing a counterexample to Gersho's conjecture that the truncated octahedron partition has the least NMI among all partitions of R. In this paper, we show that the eective NMI of the WP partition is larger than that of the truncated octahedron partition. We also show that if the WP partition is used as the partition of a -dimensional vector quantizer, with the corresponding codebook consisting of the centroids of the cells, then the resulting quantization error is white. We then show that the eective NMI of the WP partition cannot be reduced by passing it through an invertible linear transformation. Another contribution of this paper is a proof of the fact that the quantization error corresponding to an optimal periodic partition is white, which generalizes a result of amir and Feder. Keywords Gersho's conjecture, normalized moment of inertia, periodic partition, quantization error This work was supported by NSF Grant CCR Portions of this work will be presented at the IEEE International Symposium on Information Theory, Washington D.C., June 200

2 I. Introduction Consider a source uniformly distributed over a large ball B in R k, centered at the origin. Suppose that this source is quantized using a k-dimensional vector quantizer (VQ) specied by a partition S = fs ; S 2 ; : : : ; S N g of B, and corresponding codebook C = fc ; c 2 ; : : : ; c N g of points in R k. Let V i denote the volume of the cell, so that vol(b) = P N i= V i, and let M( ; c i ) = R jjx? c i jj 2 dx denote the moment of inertia (MI) of the cell about the point c i. The mean squared error (MSE) or distortion, per dimension, of this quantizer is then given by D = k = k = k NX jjx? c i jj 2 vol(b) dx i= P Ni= M( ; c i ) P Ni= V i P Ni= N M( ; c i ) N P Ni= V i +2=k vol(b) N 2=k vol(b) 2=k = m(s; C) () N The quantity m(s; C) is called the eective normalized moment of inertia of the partition S about the points in C. This quantity is invariant to any scaling of the partition and the points by an arbitrary real number, and also to translations and rotations in R k. This denition of eective normalized moment of inertia generalizes the usual denition of normalized moment of inertia (NMI) of a cell S about a point c in R k : m(s; c) = k R S jjx? cjj2 dx vol(s) +2=k = k M(S; c) vol(s) +2=k (2) One special case of interest is when we have a partition of R k that is periodic in the sense that it is composed of translates of a fundamental unit consisting of L cells S ; S 2 ; : : : ; S L, and a codebook C that is the set of centroids of the cells forming the partition. By suitably scaling the partition so as to be able to (barely) t N points of the codebook into the ball B, we obtain a partition S of B. It is clear from () that the eective NMI of this partition, in the limit as N!, is given by m(s; C) = k P Li= L M( ; c i ) +2=k () LP Li= vol( ) where c i is the centroid of for each i. We shall take this to be the denition of the eective NMI of a periodic partition about the centroids of its cells.

3 2 It is well-known (cf. [], p. 50) that for a xed codebook C, the MSE in () is minimized by choosing the partition S to be such that each 2 s the Voronoi cell of c i 2 C. Thus, for a xed codebook C, the partition S that minimizes m(s; C) is the Voronoi partition with respect to C, which we shall denote by Vor(C). As is also well-known, for asymptotically large N, the least distortion of any k-dimensional VQ applied to a source that is uniformly distributed over B has the form m k (vol(b)=n)2=k, where m k is ador's constant [2]. In our formulation, m k = lim inf N! fcr k :jcj=ng m(vor(c); C) (4) ador also generalized this result by showing that the least distortion of any k-dimensional VQ applied to a source whose probability density satises certain mild tail conditions, is asymptotically of the form m k kn?2=k, where k is a term that depends on the probability density [2] (see also [4]). It should be noted that since m(vor(c); C) is invariant to any scaling or translation of C, m k is independent of the actual choice of the ball B. So far, the value of m k is known only for k = (m = =2 = 0:08 : : :), and for k = 2 (m 2 = 5=(6 p ) = 0:0808 : : :) []. It has been conjectured [4] that m equals the eective NMI of the Voronoi partition of the BCC lattice, 4 [ 4 + (2; 2; 2), about the centroids of the cells, which are the lattice points themselves. Due to the regularity of the lattice, this partition is a periodic partition with its fundamental unit being a truncated octahedron, which is the Voronoi cell centered at the origin. Hence, by (), the eective NMI of this partition is equal to 9=(92 p 2), which is the NMI of the truncated octahedron about its center. Thus, m is conjectured to be 9=(92 p 2). Until recently, tessellation by the truncated octahedron was also believed to be the solution to the problem, originally proposed in 887 by Lord Kelvin, of nding a partition of R into cells of equal volume with the least surface area (normalized so as to be invariant to scaling) [5]. In 994, Weaire and Phelan [6] described a partition of R that was better in this respect than tessellation by the truncated octahedron. Given the similarities between the two problems presented here (the sphere minimizes both surface area and NMI among all -dimensional bodies of a given volume), it is natural to ask whether the Weaire-Phelan partition can outperform the truncated octahedron in terms of NMI. In this paper, we calculate the eective NMI of the Weaire-Phelan partition and show that this is not better than the NMI of the truncated octahedron. In addition, we show that if the cells comprising the Weaire-Phelan partition are taken to be the cells of a quantizer whose codebook

4 Fig.. The Weaire-Phelan equal-volume partition consists of the centroids of these cells, then the resulting quantization error is white, in the sense that its autocorrelation matrix is a scalar multiple of the identity matrix. Consequently, a generalization of the arguments in [7] shows that the eective NMI of the Weaire-Phelan partition cannot be improved by passing the partition through a linear transformation. II. The Weaire-Phelan Partition Weaire and Phelan's partition of R into equal-volume cells (shown in Figure ) is, in fact, one of a class of partitions based on the Voronoi partition of the non-lattice packing consisting of the points of the BCC lattice and half of its Voronoi corners (a detailed description can be found in [8]). In other words, the packing consists of the cubic lattice 4 and its translates by the points (2; 2; 2) and cyc(; 0; 2) (i.e., the points (; 0; 2) and their cyclic shifts). Each member of the Weaire-Phelan (WP) class of partitions is parametrized by a real number 0 < < =2, and is a periodic partition composed of translates by 4 of a fundamental unit comprising two pentagonal dodecahedra, which are centered at the BCC lattice points (0,0,0) and (2,2,2), and six 4-hedra, which are centered at the non-bcc points cyc(2,,0) and cyc(2,,0).

5 4 These eight points are the points of the packing that lie within the fundamental parallelepiped of the 4 lattice. The dodecahedron, C 2;0, at the origin has vertices (2=; 2=; 2=) and cyc(0; =2; ). The dodecahedron at (2,2,2) is congruent to C 2;0, i.e., it can be transformed to C 2;0 by a rigid motion that consists of a translation by (?2;?2;?2) followed by an orthogonal linear transformation. On the other hand, each 4-hedron in the fundamental unit is congruent to a 4-hedron, C 4;0, centered at the origin, with vertices (? 2=; (2? 2=); 2=), (2=?; 2=; (2?2=)), (?; (2?=2); 0), (?; 0; (2?=2)), (; (2?); =2), (?; =2; (2? )), (?; ; 0) and (; 0; ). Thus, the 4-hedron centered at c, where c is one of the points cyc(2,,0) or cyc(2,,0), can be transformed to C 4;0 by a rigid motion consisting of a translation by?c followed by a suitable orthogonal linear transformation. The WP partition with parameter is a true Voronoi partition of the packing for = 5=4, as shown later in this section; for other values of, this is a \weighted" Voronoi partition, with weights at BCC points dierent from those at other points. A weighted Voronoi partition simply means that each face of a cell passes through a weighted average of the two relevant packing points, while still being perpendicular to the line joining those points. Note that the vertices of C 2;0 lie symmetrically about the planes x = 0, y = 0 and z = 0 (symmetry about the plane x = 0, for example, means that (x; y; z) is a vertex if and only if (?x; y; z) is one). Since the cell is the convex hull of its vertices, these symmetries hold for the cell itself, from which it follows that the centroid of C 2;0 is the origin. By the same reasoning, C 4;0 is symmetric about the planes y = 0 and z = 0, as well as about the line x = 0; y = z (symmetry about this line means that (x; y; z) belongs to the 4-hedron if and only if (?x; z; y) does). It follows from these symmetries that the centroid of C 4;0 also lies at the origin. These facts show that the centroid of each cell in the WP partition is the point of the packing that lies within the cell. We now describe the cells of the partition in more detail. The 2 faces of the dodecahedron C 2;0 are all congruent pentagons, each with four sides of length p 2=6, one side of length, and area p 5 2 =2. Each face lies at a distance of 2= p 5 from the origin. The volume of the dodecahedron is thus 2 (= p 5 2 =2 2= p 5) = 4. A point to be noted is that by choosing < =2, it is ensured that no two dodecahedra in the partition actually touch each other, and so each dodecahedron shares all its faces with the 4-hedra that surround it. A point to be noted is that C 4;0 is not a cell belonging to the Weaire-Phelan partition.

6 5 The faces of the 4-hedron C 4;0 are of three types:. Two congruent hexagonal faces with two parallel sides of length each and four other sides of length (? =2) p 5 each. Each of these faces has area 4? 2, and lies at a distance of from the origin. 2. Four pentagonal faces that are congruent to the faces of C 2;0. These are the only faces that the 4-hedron shares with dodecahedra. Each such face lies at a distance of (5? 2)= p 5 from the origin.. Eight other congruent pentagonal faces each of which has two sides of length (? =2) p 5 each, two sides of length p 2=6 each, and one side of length (2? 4=) p. Each of these faces has area p 6(? 2 )=, and lies at a distance p 6=2 from the origin. Using this data, the volume of the 4-hedron is calculated to be 4(8? )=. Equating the volume of the dodecahedron with that of the 4-hedron, we see that the partition consists of equal-volume cells when = p 2. To determine the value of that gives the Voronoi partition, we rst note that the hexagonal faces of the 4-hedra always lie exactly half-way between the points of the packing that are distance 2 apart (for example, (,0,2) and (?; 0; 2)). Similarly, the pentagonal faces that are shared by two 4-hedra always lie exactly half-way between the points of the packing that are distance p 6 apart (for example, (,0,2) and (0,2,)). Finally, the boundaries between neighboring BCC and non-bcc points, which are separated by a distance of p 5 (for example, (0,0,0) and (,0,2)), are formed by the pentagonal faces shared by a dodecahedron and a 4-hedron. Comparing the distance between the center of a dodecahedron and such a face to the corresponding distance for a 4-hedron, we nd that for = 5=4, the two distances are equal, thus generating a Voronoi partition. From the Voronoi partition for this packing, we can immediately calculate the packing radius for this packing to be p 5=2, and hence the packing density is 5 p 5=48 = 0:7755 : : : (see [] for the denitions of these terms). Similarly, the covering radius for the packing is calculated to be 5 p =6, from which we obtain the covering density (or thickness) to be 25 p =42 = : : : :. III. NMI Calculations In this section, we calculate the eective NMI of an arbitrary member of the WP class of partitions about the centroids of its constituent cells. Let M 2 be the (unnormalized) moment

7 6 of inertia of a dodecahedral cell of the partition about its centroid, and let M 4 be the same for a 4-hedral cell. Similarly, let V 2 and V 4 be the volumes of the respective cells. Since the WP partition is a periodic partition by a fundamental unit consisting of two congruent dodecahedra and six congruent 4-hedra, by (), its eective NMI is given by M M 4 2 V V 5= (5) 4 2 Having already determined the volumes of the cells in the previous section, it remains to calculate their moments of inertia about their respective centroids. For this purpose, we use the following result. Theorem : ([], Chapter 2, Theorem ) Let P be a polyhedron in R containing the origin 0, with faces F ; F 2 ; : : : ; F n. Let a i be the foot of the perpendicular drawn from 0 to the plane containing F i, and let h i = jja i jj. The moment of inertia of P about 0 is then given by M(P; 0) = nx i= We rst apply this result to the dodecahedron C 2;0 h i 5 (M(F i; a i ) + h i2 area(f i )) (6) described in Section II. It is easy to verify that calculating the MI of each face of the dodecahedron about the corresponding point a i is equivalent to calculating the MI, about the origin, of the pentagon in R 2 with vertices (0; =(2 p 5)), (2=; 2=( p 5)), (=2;?= p 5), (?=2;?= p 5) and (?2=; 2=( p 5)). Some simple, but tedious, calculations show that this MI works out to be 67 4 =(44 p 5). Putting this into (6), along with the areas and distances from the previous section, we get M 2 = 7 5 =0. Turning our attention to the 4-hedron C 4;0, it is easily seen that the MI calculations for the four faces shared with dodecahedra are the same as those for the faces of C 2;0. Thus, for each such face, M(F i ; a i ) = 67 4 =(44 p 5). Evaluating M(F i ; a i ) for each of the eight other pentagonal faces is equivalent to evaluating the MI, about the origin, of the pentagon in R 2 with vertices (0; = p 2), ((? =2) p ; (? )= p 2), ((? 2=) p ;?= p 2), ((2=? ) p ;?= p 2) and ((=2? ) p ; (? )= p 2). The MI of this pentagon turns out to be (44? ? 4 4 )=(6 p 6). Finally, for the hexagonal faces, it suces to determine the MI, about the origin, of the hexagon in R 2 with vertices (0; ), (2? ; =2), (2? ;?=2), (0;?), (? 2;?=2) and (? 2; =2). This MI is calculated to be (80? ? 4 )=24. Combining all of these using (6), we get M 4 = (200? ? 7 5 )=90. Putting the expressions for M 2, M 4, V 2 and V 4 into (5), we nd that the eective NMI of

8 7 the WP partition with parameter is given by m W P () = 96 (4? 5 + 0) (7) It is easily veried that the value of this expression is a minimum at = 5=4, which is to be expected since this corresponds to the Voronoi WP partition. The eective NMI of the Voronoi partition is m W P (5=4) = 645=892 = 0:07875 : : :, which is greater than the NMI of the truncated octahedron. When = p 2, we get the equal-volume WP partition, for which the eective NMI is 2?= = 0: : : :. Remarkably, this is exactly the same as the NMI of the rhombic dodecahedron, which is the Voronoi cell for the face-centered cubic (FCC) lattice []. IV. Minimizing NMI via Linear Transforms Although the Voronoi WP partition is not better than the truncated octahedron in terms of NMI, we can use it as the starting point of a search for a partition with lower eective NMI. In this section, we focus our search on partitions that are the images of WP partitions under invertible linear transformations. Let S be any periodic partition of R k, composed of translates of a fundamental unit consisting of L cells S ; S 2 ; : : : ; S L, at least one of which is non-degenerate (i.e., has positive volume). We take the codebook C to be the set of centroids of the cells forming the partition. Given T 2 GL k (R), the group of real, invertible, k k matrices, let T S = ft (S) : S 2 Sg and T C = ft c : c 2 Cg. Clearly, T s a periodic partition of R k consisting of translates of the fundamental unit made up of the cells T (S ); T (S 2 ); : : : ; T (S L ), and T C is the set of centroids of the cells in T S. Thus, by (), we have m(t S; T C) = k P Li= L M(T ( ); T c i ) +2=k (8) LP Li= jdet(t )jvol( ) Here, we have used the fact that vol(t ( )) = jdet(t )jvol( ). Our goal now is to nd a T 2 GL k (R) for which m(t S; T C) is a minimum. To do this, we need an alternative formulation of the denition of the eective NMI of a periodic partition, which we provide next. Consider a random vector X uniformly distributed over the fundamental translational unit, F = [ L = i, of the partition S. Let Q : F! C be the vector quantizer dened by Q(x) = c i, if x 2, for i 2 f; 2; : : : ; Lg. Let E = X? Q(X) be the quantization error, and let R E = E[EE t ] be the corresponding autocorrelation matrix. Note that since at least one of the 's has positive

9 8 volume, R E is non-singular (and hence positive denite). This is because if we assume that R E is singular, then the components of the random vector E would have to be linearly dependent, which can happen only if (with probability one) E lies entirely within some (k? )-dimensional hyperplane. But, this is impossible precisely because X is distributed uniformly over [ L i=, and some is non-degenerate. Noting that the trace, tr(r E ), of the matrix R E is E[jjEjj 2 ] = P Li= M( ; c i )= PLi= vol( ), we have from (), m(s; C) = k tr(r E) L LX i= vol( )!?2=k (9) Now, suppose X 0 is a random vector uniformly distributed over T (F ) = [ L i= T (). Q 0 : T (F )! T C be the quantizer dened by Q 0 (x) = T Q(T? x), and let E 0 = X 0? Q 0 (X 0 ) be the corresponding quantization error. Note that X 0 and T X are identically distributed and hence, E 0 has the same probability distribution as T X? Q 0 (T X) = T X? T Q(X) = T E. Therefore, the autocorrelation matrix of E 0 is given by T R E T t. Thus, by (9), we have m(t S; T C) = k tr(t R ET t ) = L LX i= jdet(t )jvol( ) k tr(t R ET t ) det(t R E T t ) =k det(r E) =k L where we have used the fact that det(t R E T t ) = jdet(t )j 2 det(r E ). LX i=!?2=k Let vol( )!?2=k (0) Following along the lines of the proof of Theorem in [7] (which applies to partitions generated by lattices, rather than periodic partitions in general), we show that T minimizes m(t S; T C) if and only if the corresponding quantization error E 0 = T E is white, i.e., T R E T t = I where I is the k k identity matrix and is some positive real number. We note that since the trace and the determinant of a matrix are, respectively, the sum and the product of its eigenvalues, by the arithmetic-geometric means inequality (cf. [9], p. 7), we have (=k)tr(t R E T t ) (det(t R E T t )) =k. We have equality here if and only if all the eigenvalues of T R E T t are equal, which can happen if and only if the quantization error E 0 = T E is white. This follows from the fact that the autocorrelation matrix is real symmetric and hence orthogonally diagonalizable (cf. [0], p. 7). We summarize these results in the following theorem, which somewhat generalizes the corresponding result in [7].

10 9 Theorem 2: For any T 2 GL k (R), m(t S; T C) det(r E ) =k L LX i= vol( )!?2=k = m (S; C) () with equality if and only if the quantization error E 0 = T E is white. In particular, m(s; C) m (S; C) with equality if and only if E is white. Note that the quantization error E 0 = T E can always be whitened by taking T = R?=2 E, the (unique) real, symmetric, positive denite square root of R E?. Therefore, it follows immediately from Theorem 2 that a necessary condition for a periodic partition to be optimal, in the sense of having the least eective NMI among all partitions, is that the corresponding quantization error should be white. Theorem 2 also provides an alternative way of looking at (9). Observe that (9) may be rewritten as m(s; C) = (S; C)m (S; C) (2) where (S; C) = k tr(r E)=det(R E ) =k is a term that measures how far the quantization error corresponding to the partition s from being white. Equivalently, (S; C) measures how far the NMI of the partition s from the NMI, m (S; C), of the \whitened" version of the partition. We now show that, if we take S to be any WP partition, then the corresponding quantization error E is in fact white. This, by the above theorem, would imply that m(s; C) = m (S; C), and so the eective NMI of a WP partition cannot be improved by any invertible linear transformation. Let S be the WP partition with parameter. Let S and S 2 be the dodecahedra centered at the points (0,0,0) and (2,2,2), respectively. Similarly, let S, S 4, S 5, S 6, S 7 and S 8 be the 4-hedra centered at the points (,0,2), (0,2,), (2,,0), (,0,2), (0,2,) and (2,,0) respectively. Since the random vector X = (X; Y; ) t is uniformly distributed over the fundamental translational unit of S, each element of the correlation matrix R E is of the form 6V 4 + 2V 2 8X i= (x? c i; ) p (y? c i;2 ) q (z? c i; ) r dx () for some p; q; r 2 f0; ; 2g such that p + q + r = 2, where c i = (c i; ; c i;2 ; c i; ) is the centroid of. As noted in Section II, for i = ; 2, T i (? c i ) = C 2;0, and for i = ; : : : ; 8, T i (? c i ) = C 4;0, for suitable orthogonal linear transforms T i. It may easily be veried that the matrices

11 0 corresponding to these transforms are T 5 = T = I; T 2 = 7 5 ; T 6 = ? ; T = I; T 4 = 7 5 ; T 7 = ? ; 5 ; T 8 = ? We use these transforms in conjunction with some simple changes of variables to evaluate the integral terms of (), thus determining the form of the autocorrelation matrix R E. Note that since S = C 2;0 is symmetric about the plane x = 0, it is invariant under the transform T 6. Hence, using the change of variable x 0 = T 6 x, we have R S xy dx = R T 6 (S )?x0 y 0 dx 0 =? R S xy dx. But, this implies that R S xy dx = 0. Similarly, it follows from the symmetry of S about the plane z = 0 that R S xz dx = R S yz dx = 0. Next, observe that if (x; y; z) is a vertex of C 2;0, then so are its cyclic shifts (z; x; y) and (y; z; x). Therefore, it follows that C 2;0 is invariant under the linear transforms T 4 and T 5 which map a point into one of its cyclic shifts. Hence, using the change of variable x 0 = T 4 x, we get R S x 2 dx = R T 4 (S ) (y0 ) 2 dx 0 = R S y 2 dx. Similar reasoning shows that R S y 2 dx = R S z 2 dx. Thus, S x 2 dx = S y 2 dx = S z 2 dx = jjxjj 2 dx = M 2 C 2;0 Turning our attention to S 2, we observe that the change of variable x 0 = T 2 (x? c 2 ) yields R S 2 (x? 2) p (y? 2) q (z? 2) r dx = R C 2;0 (z 0 ) p (y 0 ) q (x 0 ) r dx Combining this with the results for the integrals over S, we see that R S 2 (x? 2) 2 dx = R S 2 (y? 2) 2 dx = R S 2 (z? 2) 2 dx = M 2 =, and R S 2 (x? 2)(y? 2) dx = R S 2 (y? 2)(z? 2) dx = R S 2 (x? 2)(z? 2) dx = 0. Moving on to, i = ; : : : ; 8, the change of variable x 0 = T i (x? c i ) gives us (x? c i; ) p (y? c i;2 ) q (z? c i; ) r dx = C 4;0 (^x) p (^y) q (^z) r dx 0 (4) where ^x = (^x; ^y; ^z) t = T i? x 0. The symmetry of C 4;0 about the planes y = 0 and z = 0 shows that R C 4;0 xy dx = R C 4;0 yz dx = R C 4;0 xz dx = 0. By (4), this implies that for i = ; : : : ; 8, R (x? c i; )(y? c i;2 ) dx = R (y? c i;2 )(z? c i; ) dx = R (x? c i; )(z? c i; ) dx = 0. It also follows

12 from (4) that and 5X i= 8X i=6 (x? c i; ) 2 dx = (x? c i; ) 2 dx = 5X i= 8X i=6 (y? c i;2 ) 2 dx = (y? c i;2 ) 2 dx = 5X i= i=6 (z? c i; ) 2 dx = jjxjj 2 dx = M 4 C 4;0 8X (z? c i; ) 2 dx = jjxjj 2 dx = M 4 C 4;0 Putting together all the above results, we nd that the correlation matrix R E is given by R E = 6M 4 + 2M 2 6V 4 + 2V 2 I (5) where I is the identity matrix, thus showing that E is white. In eect, the above analysis shows that if, for i = ; 2; : : : ; 8, we dene a random vector X i uniformly distributed over, then the corresponding \error" vector E i = X i? c i is white for i = ; 2, and uncorrelated for i = ; : : : ; 8. In other words, the corresponding autocorrelation matrices R Ei are all diagonal, with the diagonal entries being all equal for i = ; 2, but not for i = ; : : : ; 8. The triples formed by the diagonal elements of the autocorrelation matrices for E, E 4 and E 5 are all cyclic permutations of one another, i.e., if R E = diag( ; 2 ; ), then R E4 = diag( 2 ; ; ) and R E5 = diag( ; ; 2 ). Hence, the sum of these three matrices is a scalar multiple of the identity matrix. The same is the case for the autocorrelation matrices corresponding to E 6, E 7 and E 8. Thus, the whiteness of the quantization error over the fundamental unit of a WP partition may be viewed as a consequence of the whiteness over each dodecahedron, and over each of the two sets of three 4-hedra, namely, fs ; S 4 ; S 5 g and fs 6 ; S 7 ; S 8 g. By Theorem 2, the whiteness of the quantization error E implies that the eective NMI of any WP partition cannot be improved by means of invertible linear transforms. Thus, in conclusion, any image of a WP partition under an invertible linear transform has larger eective NMI than that of the truncated octahedron. Acknowledgment The authors wish to express their gratitude to Thomas Hales for sparking their interest in the Weaire-Phelan partition, and to Dennis Hui for pointing out that it is instructive to reformulate (9) as (2).

13 2 References [] A. Gersho and R.M. Gray, Vector Quantization and Signal Compression, Boston, MA: Kluwer, 992. [2] P.L. ador, \Development and evaluation of procedures for quantizing multivariate distributions," Ph.D. dissertation, Stanford Univ., 96. [] J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, rd ed., New York, NY: Springer- Verlag, 998. [4] A. Gersho, \Asymptotically Optimal Block Quantization," IEEE Trans. Inform. Theory, vol. IT-25, no. 4, pp. 7{80, July 979. [5] Denis Weaire (ed.), The Kelvin Problem, London, England: Taylor & Francis, 996. [6] D. Weaire and R. Phelan, \A Counter-Example to Kelvin's Conjecture on Minimal Surfaces," Phil. Mag. Lett., vol. 69, no. 2, pp. 07{0, 994 (reproduced in [5], pp. 47{5). [7] R. amir and M. Feder, \On Lattice Quantization Noise," IEEE Trans. Inform. Theory, vol. 42, no. 4, pp. 52{59, July 996. [8] R. Kusner and J.M. Sullivan, \Comparing the Weaire-Phelan Equal-Volume Foam to Kelvin's Foam," in [5], pp. 7{80. [9] G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities, 2nd ed., Cambridge, UK: Cambridge Univ. Press, 952. [0] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge, UK: Cambridge Univ. Press, 985.