Discrete sets with minimal moment of inertiaâ
Discrete sets with minimal moment of inertia ⋆
arXiv:0710.0911v1 [math.CO] 4 Oct 2007
S. Brlek, G. Labelle, A. Lacasse ∗ LaCIM, Universit´e du Qu´ebec a ` Montr´eal, C. P. 8888 Succursale “Centre-Ville”, Montr´eal (QC), CANADA H3C 3P8
Abstract We analyze the moment of inertia I(S), relative to the center of gravity, of finite plane lattice sets S. We classify these sets according to their roundness: a set S is rounder than a set T if I(S) < I(T ). We show that roundest sets of a given size are strongly convex in the discrete sense. Moreover, we introduce the notion of quasi-discs and show that roundest sets are quasi-discs. We use weakly unimodal partitions and an inequality for the radius to make a table of roundest discrete sets up to size 40. Surprisingly, it turns out that the radius of the smallest disc containing a roundest discrete set S is not necessarily the radius of S as a quasi-disc. Key words: Lattice paths, polyominoes, moment of inertia, discrete set
1
Introduction
In this paper we consider plane sets up to translations. By a discrete set we mean a finite set of lattice points or a finite union of lattice closed unit squares (pixels) (Figure 1 (a)). In particular, the word polyomino means a finite union of pixels in the plane whose boundary consists of a disjoint union of simple closed polygonal paths using 4-connectedness (Figure 1(b)). These sets are well-known combinatorial objects in discrete geometry. The dual of a polyomino, usually called animal, consisting of the set of centers of its pixels, is also considered (Figure 1(c)). Using a ( 12 , 21 ) shift, we can always assume that an animal is a subset of the discrete plane Z × Z. ⋆ with the support of NSERC (Canada) ∗ Corresponding author. Email addresses: [email protected] (S. Brlek), [email protected] (G. Labelle), [email protected] (A. Lacasse).
Preprint submitted to Elsevier Science
16 April 2008
(a )
(b )
(c )
Fig. 1. (a) Discrete set (b) a typical polyomino and (c) its corresponding animal.
Moreover, a polyomino is called v-convex (resp. h-convex) if all its columns (resp. rows) are connected (see Figure 2(a),(b)). We say that a polyomino is hvconvex (see Figure 2(c)) if all its columns and rows are connected and stronglyconvex (see Figure 2(d)) if given any two points u and v in its corresponding animal, the lattice points w in the segment [u, v] are all in the animal.
(a)
(b)
(c)
(d)
Fig. 2. A polyomino (a) v-convex (b) h-convex (c) hv-convex (d) strongly-convex.
The goal of this paper is to study the roundest discrete sets S of N pixels (or N points) in the sense of having minimal moment of inertia I(S), relative to the center of gravity. This problem was raised in a previous paper [2,?] in the context of the study of incremental algorithms based on discrete Green theorem. The present notion of roundness is distinct from the one given in [1] where they consider minimizing the site perimeter of lattice sets, that is the number of points with Manhattan distance 1 from the sets. For a given N, minimizing I(S) is equivalent to minimizing I(A), where A is the associated set of centers of the pixels of S (equation (2)). To simplify notations and computations, in this paper the plane is identified with the complex plane C and Z × Z is identified with Z + iZ. In Section 2, we recall some basic notions about moment of inertia of discrete sets. Section 3 is devoted to properties of roundest discrete sets. More precisely, we establish a useful lemma concerning the moment of inertia of a union of discrete sets. We also introduce the notions of strong convexity and discrete quasi-disc and apply the above lemma to prove that roundest discrete sets are strongly convex and quasi-discs. Then a method is developed for computing the roundest discrete sets according to size (≤ 40) and some parameters associated to them. Finally, we show in Section 4, how to extend our results to other kinds of lattices and to higher dimensions. 2
2
Continuous and discrete moments of inertia
We recall definitions of the basic geometric parameters: Definition 1 Let S be a measurable subset of the complex plane such that Z Z
S
|z|2 dx dy < ∞.
(1)
The center of gravity g and the moment of inertia I(S), relative to the center of gravity are defined by the following equations: 1 g = g(S) = Area(S)
Z Z
S
zdx dy
and I(S) =
Z Z
S
2
|z − g| dx dy =
Z Z 2 1 |z| dx dy − zdx dy , Area(S) S S
Z Z
where
2
Area(S) =
Z Z
S
dx dy.
Note that, in particular, if S = P1 ∪ P2 ∪ · · · ∪ PN is a union of N distinct RR 2 pixels, the condition P1 ∪P2 ∪···∪PN |z| dx dy < ∞ is obviously satisfied and g(S) and I(S) are well-defined. Note also that the moment of inertia of any single pixel P is I(P ) = center of gravity corresponds to its geometrical center.
1 6
and its
Definition 2 Let T = {a1 , a2 , · · · , aN } ⊆ C be a set of N distinct points in the complex plane where the point ak has a mass mk for k = 1, · · · , N. The center of gravity g and the moment of inertia I(T ), relative to the center of gravity are defined by g = g(T ) =
N X 1 mk ak , m1 + · · · + mN k=1
and N X
N X
2
N X 1 I(T ) = mk |ak − g| = mk |ak | − mk ak m1 + · · · + mN k=1 k=1 k=1 X 1 = mk ml |ak − al |2 , m1 + · · · + mN k · · · . At each step, filling such a hole, decreases the moment of inertia by at least N1 . After a finite number of steps, this process must terminate and the resulting set S (k) must be a strongly convex set which is also an animal since it is, in particular, hv-convex. 2
7
3.2 Roundest discrete sets are discrete quasi-discs
Much more can be said. We now show that roundest polyominoes are nearly discs in the following sense: Definition 8 Let c ∈ C, and S ⊆ Z + iZ be a finite set of lattice points. Then S is called a (i) (discrete) disc centered at c of radius r if S = {z : |z − c| ≤ r} ∩ (Z + iZ), (ii) (discrete) quasi-disc centered at c of radius r if {z : |z − c| < r} ∩ (Z + iZ) ⊆ S ⊆ {z : |z − c| ≤ r} ∩ (Z + iZ), where r = maxs∈S |s − c|. A disc and a quasi-disc of radius r = 5 are shown in Figure 5 (a) and (b) respectively. Note that every lattice point on the circumference must belong to a disc while at least only one is necessary in the case of quasi-disc. In both cases, every lattice point lying within the circumference must belong to the disc and quasi-disc.
(a)
(b)
Fig. 5. (a) A (discrete) disc
(b) a (discrete) quasi-disc
Theorem 9 Let S be a polyomino having N pixels with minimal moment of inertia, that is a roundest polyomino. Let A be its associated animal and g = g(A) be its center of gravity. Then A is a quasi-disc centered at g with radius r = maxa∈A |g − a|. Proof. Let A be a minimal animal of size N. Let us prove first that for every point a ∈ A, the lattice points in the interior of the disc Γa = {z ∈ C : |z − ga | ≤ |a − ga | }, 8
ga = g(A \ {a})
are points of A. This is a consequence of Lemma 5 with N = 2: I(S1 ∪ S2 ) = I(S1 ) + I(S2 ) +
m1 m2 |g1 − g2 |2 . m1 + m2
This is seen as follows. Take S1 = A \ {a}, g1 = ga , m1 = N − 1, S2 = {a}, g2 = a, m2 = 1. Then, I(A) = I(A \ {a}) + I({a}) + = I(A \ {a}) +
N −1 |ga − a|2 N
N −1 |ga − a|2 N
since I({a}) = 0. Now suppose that Γa contains in its interior, a lattice point b 6∈ A, that is |ga − b| < |ga − a|. Replace a by b and consider the set B = ((A \ {a}) ∪ {b}). Then, I(B) = I((A \ {a}) ∪ {b}) = I(A \ {a}) +
N −1 |ga − b|2 < I(A), N
which contradicts the minimality of the moment of inertia of A. Now, take a0 ∈ A such that r = |g − a0 | = maxa∈A |g − a| and consider the closed disc Ca0 = {z ∈ C : |z − g| ≤ r = |g − a0 | }. Then, by definition of Ca0 , we obviously have, A ⊆ Ca0 ∩ (Z + iZ).
(5)
Furthermore, it is easy to check that g − a0 =
N −1 (ga0 − a0 ). N
This implies that g belongs to the segment [ga0 , a0 ]. Hence, Ca0 ⊆ Γa0 as we can see in the following figure: Γa 0
a0
g ga 0 Ca 0
9
But we have seen that every lattice point in the interior of Γa0 are in A. In particular all those in the interior of Ca0 must be also in A: (int Ca0 ) ∩ (Z + iZ) ⊆ A. We conclude, using (5). 2 Note that the strong convexity established constructively in Section 2.2 is also consequence of Theorem 9: Corollary 10 A roundest animal with N points is a strongly convex set. Proof. Let A be such a roundest animal. Given any two distinct points u, v ∈ A, any lattice point w ∈ [u, v], w 6= u, w 6= v, is necessarily in the interior of the disc Ca0 . Hence, w ∈ A by Theorem 2. 2 Figure 6 (a) illustrates Theorem 2 with N = 5, as the reader can check. By contraposition, the 7×7 lattice set A is not minimal since the disc Ca0 contains lattice points not in A (see Figure 6 (b)). Note that the converse of Theorem 2 is false since, for N = 3, the quasi-disc of Figure 6 (c) is not minimal (with I = 2). The minimal one for N = 3 (with I = 34 < 2) is shown in Figure 6(d).
(a)
(b)
(c)
(d)
Fig. 6. Illustration of Theorem 2 and of the falsity of its converse.
To pursue our study of roundest discrete sets we need a finer analysis. In particular, given N, the following result gives an upper bound for the radius r of the disc Ca0 . Lemma 11 Let A be a roundest animal having N points. The radius r = |a0 − g| = maxa∈A |a − g| of the disc Ca0 centered at g = g(A) satisfies 1 r≤√ + 2
s
N . π
Proof. Consider the polyomino P associated to a roundest animal A. This polyomino is made of N unit pixels whose centers are the elements of A. We 10
will show that the open disc B ◦ (g, r − satisfies
√1 ), 2
of radius r −
√1 , 2
centered at g
!
1 B g, r − √ ⊆P 2 and the result will follows since (6) implies that ◦
1 π r−√ 2
!2
(6)
≤ area (P ) = N.
To established (6), consider an arbitrary complex number z ∈ B ◦ (g, r − √12 ). We must show that there exists ν ∈ A such that z ∈ pixν , where pixν is the pixel centered at ν. So, let z = x + iy be such that 1 |z − g| < r − √ . 2 Then there exist integers ν1 , ν2 such that, x = ν1 + f1 ,
y = ν2 + f2
where |f1 | ≤ 12 , |f2 | ≤ 21 . Let ν = ν1 + iν2 and f = f1 + if2 . We have z = ν + f ∈ pixν . There remains to show that ν ∈ A. We have, by the triangular inequality 1 |ν − g| − |f | ≤ |ν − g + f | = |z − g| < r − √ . 2 Hence,
1 1 1 |ν − g| < r − √ + |f | ≤ r − √ + √ = r 2 2 2 q
r 2
2
1 + ≤ + 21 = √12 . By Theorem 2, we conclude that since |f | = 2 ν ∈ A since every lattice point in the open disc B ◦ (g, r) necessarily belongs to A. 2
f12
f22
3.3 Computation of the roundest discrete sets according to size In order to generate all the roundest animals of a given size N, we classify animals according to their vertical projections. Let A be a roundest animal of size N with vertical projections (n1 , n2 , · · · , ns ) with N = n1 + n2 + · · · + ns . Then, because of the convexity property, the sequence n1 , n2 , · · · , ns must satisfy 0 < n1 ≤ n2 ≤ · · · ≤ nk < nk+1 = · · · = nl−1 > nl ≥ · · · ≥ ns−1 ≥ ns > 0. 11
Such sequences are called weakly unimodal partitions of N (or stack or planar partitions of N), see Stanley [6], Section 2.5, p. 76. Surprisingly, it turns out that any such sequence of projections corresponds to 0, 1 or 2 roundest animal of size N. More precisely, we have the following result. Lemma 12 Let (n1 , n2 , · · · , ns ) be a weakly unimodal sequence with N = n1 + · · · + ns . Then among all animals having this sequence of vertical projections, (i) there is a unique animal A, with minimal moment of inertia, if n1 , n2 , · · · , ns have the same parity; (ii) otherwise, there are exactly two animals A, A′ , having these projections, with minimal moment of inertia. Moreover, the moment of inertia of A (and A′ ) is given by the formula s s X 1 X 1 1 I(A) = n3k − N + k 2 nk − 12 k=1 12 N k=1
1 + 4N
X
nk even
s X
knk
k=1
!
nk
!2
X
nk odd
(7)
nk .
Proof. Let S be an animal with projections n1 , n2 , · · · , ns having columns C1 , C2 , · · · , Cs . More precisely, Ck is the column of points in S over the point (k, 0), k = 1, · · · , s (see Figure 7). C1 C 2 C3 C4 5 4 3 2 1 0
1
2
3
4
Fig. 7. Ck = {(k, y) | (k, y) ∈ S}, k = 1, · · · , s.
Let gk be the center of gravity of Ck , 1 ≤ k ≤ s. Note that there exists νk ∈ Z such that, for 1 ≤ k ≤ s, gk = In other words,
(k (k
+ iνk )
if nk is odd,
+ i(νk + 21 )) if nk is even.
1 gk = k + i νk + χeven (nk ) 2 12
k = 1, · · · , s
where χeven (n) = 1, if n is even and 0 otherwise. Then, by Lemma 5 and formula (3), we have I(S) = I(C1 ∪ · · · ∪ Cs ) = I(C1 ) + · · · + I(Cs ) + I({g1 , · · · , gs }) s X n3k − nk 1 X = + nk nl |(k − l) 12 N k
arXiv:0710.0911v1 [math.CO] 4 Oct 2007
S. Brlek, G. Labelle, A. Lacasse ∗ LaCIM, Universit´e du Qu´ebec a ` Montr´eal, C. P. 8888 Succursale “Centre-Ville”, Montr´eal (QC), CANADA H3C 3P8
Abstract We analyze the moment of inertia I(S), relative to the center of gravity, of finite plane lattice sets S. We classify these sets according to their roundness: a set S is rounder than a set T if I(S) < I(T ). We show that roundest sets of a given size are strongly convex in the discrete sense. Moreover, we introduce the notion of quasi-discs and show that roundest sets are quasi-discs. We use weakly unimodal partitions and an inequality for the radius to make a table of roundest discrete sets up to size 40. Surprisingly, it turns out that the radius of the smallest disc containing a roundest discrete set S is not necessarily the radius of S as a quasi-disc. Key words: Lattice paths, polyominoes, moment of inertia, discrete set
1
Introduction
In this paper we consider plane sets up to translations. By a discrete set we mean a finite set of lattice points or a finite union of lattice closed unit squares (pixels) (Figure 1 (a)). In particular, the word polyomino means a finite union of pixels in the plane whose boundary consists of a disjoint union of simple closed polygonal paths using 4-connectedness (Figure 1(b)). These sets are well-known combinatorial objects in discrete geometry. The dual of a polyomino, usually called animal, consisting of the set of centers of its pixels, is also considered (Figure 1(c)). Using a ( 12 , 21 ) shift, we can always assume that an animal is a subset of the discrete plane Z × Z. ⋆ with the support of NSERC (Canada) ∗ Corresponding author. Email addresses: [email protected] (S. Brlek), [email protected] (G. Labelle), [email protected] (A. Lacasse).
Preprint submitted to Elsevier Science
16 April 2008
(a )
(b )
(c )
Fig. 1. (a) Discrete set (b) a typical polyomino and (c) its corresponding animal.
Moreover, a polyomino is called v-convex (resp. h-convex) if all its columns (resp. rows) are connected (see Figure 2(a),(b)). We say that a polyomino is hvconvex (see Figure 2(c)) if all its columns and rows are connected and stronglyconvex (see Figure 2(d)) if given any two points u and v in its corresponding animal, the lattice points w in the segment [u, v] are all in the animal.
(a)
(b)
(c)
(d)
Fig. 2. A polyomino (a) v-convex (b) h-convex (c) hv-convex (d) strongly-convex.
The goal of this paper is to study the roundest discrete sets S of N pixels (or N points) in the sense of having minimal moment of inertia I(S), relative to the center of gravity. This problem was raised in a previous paper [2,?] in the context of the study of incremental algorithms based on discrete Green theorem. The present notion of roundness is distinct from the one given in [1] where they consider minimizing the site perimeter of lattice sets, that is the number of points with Manhattan distance 1 from the sets. For a given N, minimizing I(S) is equivalent to minimizing I(A), where A is the associated set of centers of the pixels of S (equation (2)). To simplify notations and computations, in this paper the plane is identified with the complex plane C and Z × Z is identified with Z + iZ. In Section 2, we recall some basic notions about moment of inertia of discrete sets. Section 3 is devoted to properties of roundest discrete sets. More precisely, we establish a useful lemma concerning the moment of inertia of a union of discrete sets. We also introduce the notions of strong convexity and discrete quasi-disc and apply the above lemma to prove that roundest discrete sets are strongly convex and quasi-discs. Then a method is developed for computing the roundest discrete sets according to size (≤ 40) and some parameters associated to them. Finally, we show in Section 4, how to extend our results to other kinds of lattices and to higher dimensions. 2
2
Continuous and discrete moments of inertia
We recall definitions of the basic geometric parameters: Definition 1 Let S be a measurable subset of the complex plane such that Z Z
S
|z|2 dx dy < ∞.
(1)
The center of gravity g and the moment of inertia I(S), relative to the center of gravity are defined by the following equations: 1 g = g(S) = Area(S)
Z Z
S
zdx dy
and I(S) =
Z Z
S
2
|z − g| dx dy =
Z Z 2 1 |z| dx dy − zdx dy , Area(S) S S
Z Z
where
2
Area(S) =
Z Z
S
dx dy.
Note that, in particular, if S = P1 ∪ P2 ∪ · · · ∪ PN is a union of N distinct RR 2 pixels, the condition P1 ∪P2 ∪···∪PN |z| dx dy < ∞ is obviously satisfied and g(S) and I(S) are well-defined. Note also that the moment of inertia of any single pixel P is I(P ) = center of gravity corresponds to its geometrical center.
1 6
and its
Definition 2 Let T = {a1 , a2 , · · · , aN } ⊆ C be a set of N distinct points in the complex plane where the point ak has a mass mk for k = 1, · · · , N. The center of gravity g and the moment of inertia I(T ), relative to the center of gravity are defined by g = g(T ) =
N X 1 mk ak , m1 + · · · + mN k=1
and N X
N X
2
N X 1 I(T ) = mk |ak − g| = mk |ak | − mk ak m1 + · · · + mN k=1 k=1 k=1 X 1 = mk ml |ak − al |2 , m1 + · · · + mN k · · · . At each step, filling such a hole, decreases the moment of inertia by at least N1 . After a finite number of steps, this process must terminate and the resulting set S (k) must be a strongly convex set which is also an animal since it is, in particular, hv-convex. 2
7
3.2 Roundest discrete sets are discrete quasi-discs
Much more can be said. We now show that roundest polyominoes are nearly discs in the following sense: Definition 8 Let c ∈ C, and S ⊆ Z + iZ be a finite set of lattice points. Then S is called a (i) (discrete) disc centered at c of radius r if S = {z : |z − c| ≤ r} ∩ (Z + iZ), (ii) (discrete) quasi-disc centered at c of radius r if {z : |z − c| < r} ∩ (Z + iZ) ⊆ S ⊆ {z : |z − c| ≤ r} ∩ (Z + iZ), where r = maxs∈S |s − c|. A disc and a quasi-disc of radius r = 5 are shown in Figure 5 (a) and (b) respectively. Note that every lattice point on the circumference must belong to a disc while at least only one is necessary in the case of quasi-disc. In both cases, every lattice point lying within the circumference must belong to the disc and quasi-disc.
(a)
(b)
Fig. 5. (a) A (discrete) disc
(b) a (discrete) quasi-disc
Theorem 9 Let S be a polyomino having N pixels with minimal moment of inertia, that is a roundest polyomino. Let A be its associated animal and g = g(A) be its center of gravity. Then A is a quasi-disc centered at g with radius r = maxa∈A |g − a|. Proof. Let A be a minimal animal of size N. Let us prove first that for every point a ∈ A, the lattice points in the interior of the disc Γa = {z ∈ C : |z − ga | ≤ |a − ga | }, 8
ga = g(A \ {a})
are points of A. This is a consequence of Lemma 5 with N = 2: I(S1 ∪ S2 ) = I(S1 ) + I(S2 ) +
m1 m2 |g1 − g2 |2 . m1 + m2
This is seen as follows. Take S1 = A \ {a}, g1 = ga , m1 = N − 1, S2 = {a}, g2 = a, m2 = 1. Then, I(A) = I(A \ {a}) + I({a}) + = I(A \ {a}) +
N −1 |ga − a|2 N
N −1 |ga − a|2 N
since I({a}) = 0. Now suppose that Γa contains in its interior, a lattice point b 6∈ A, that is |ga − b| < |ga − a|. Replace a by b and consider the set B = ((A \ {a}) ∪ {b}). Then, I(B) = I((A \ {a}) ∪ {b}) = I(A \ {a}) +
N −1 |ga − b|2 < I(A), N
which contradicts the minimality of the moment of inertia of A. Now, take a0 ∈ A such that r = |g − a0 | = maxa∈A |g − a| and consider the closed disc Ca0 = {z ∈ C : |z − g| ≤ r = |g − a0 | }. Then, by definition of Ca0 , we obviously have, A ⊆ Ca0 ∩ (Z + iZ).
(5)
Furthermore, it is easy to check that g − a0 =
N −1 (ga0 − a0 ). N
This implies that g belongs to the segment [ga0 , a0 ]. Hence, Ca0 ⊆ Γa0 as we can see in the following figure: Γa 0
a0
g ga 0 Ca 0
9
But we have seen that every lattice point in the interior of Γa0 are in A. In particular all those in the interior of Ca0 must be also in A: (int Ca0 ) ∩ (Z + iZ) ⊆ A. We conclude, using (5). 2 Note that the strong convexity established constructively in Section 2.2 is also consequence of Theorem 9: Corollary 10 A roundest animal with N points is a strongly convex set. Proof. Let A be such a roundest animal. Given any two distinct points u, v ∈ A, any lattice point w ∈ [u, v], w 6= u, w 6= v, is necessarily in the interior of the disc Ca0 . Hence, w ∈ A by Theorem 2. 2 Figure 6 (a) illustrates Theorem 2 with N = 5, as the reader can check. By contraposition, the 7×7 lattice set A is not minimal since the disc Ca0 contains lattice points not in A (see Figure 6 (b)). Note that the converse of Theorem 2 is false since, for N = 3, the quasi-disc of Figure 6 (c) is not minimal (with I = 2). The minimal one for N = 3 (with I = 34 < 2) is shown in Figure 6(d).
(a)
(b)
(c)
(d)
Fig. 6. Illustration of Theorem 2 and of the falsity of its converse.
To pursue our study of roundest discrete sets we need a finer analysis. In particular, given N, the following result gives an upper bound for the radius r of the disc Ca0 . Lemma 11 Let A be a roundest animal having N points. The radius r = |a0 − g| = maxa∈A |a − g| of the disc Ca0 centered at g = g(A) satisfies 1 r≤√ + 2
s
N . π
Proof. Consider the polyomino P associated to a roundest animal A. This polyomino is made of N unit pixels whose centers are the elements of A. We 10
will show that the open disc B ◦ (g, r − satisfies
√1 ), 2
of radius r −
√1 , 2
centered at g
!
1 B g, r − √ ⊆P 2 and the result will follows since (6) implies that ◦
1 π r−√ 2
!2
(6)
≤ area (P ) = N.
To established (6), consider an arbitrary complex number z ∈ B ◦ (g, r − √12 ). We must show that there exists ν ∈ A such that z ∈ pixν , where pixν is the pixel centered at ν. So, let z = x + iy be such that 1 |z − g| < r − √ . 2 Then there exist integers ν1 , ν2 such that, x = ν1 + f1 ,
y = ν2 + f2
where |f1 | ≤ 12 , |f2 | ≤ 21 . Let ν = ν1 + iν2 and f = f1 + if2 . We have z = ν + f ∈ pixν . There remains to show that ν ∈ A. We have, by the triangular inequality 1 |ν − g| − |f | ≤ |ν − g + f | = |z − g| < r − √ . 2 Hence,
1 1 1 |ν − g| < r − √ + |f | ≤ r − √ + √ = r 2 2 2 q
r 2
2
1 + ≤ + 21 = √12 . By Theorem 2, we conclude that since |f | = 2 ν ∈ A since every lattice point in the open disc B ◦ (g, r) necessarily belongs to A. 2
f12
f22
3.3 Computation of the roundest discrete sets according to size In order to generate all the roundest animals of a given size N, we classify animals according to their vertical projections. Let A be a roundest animal of size N with vertical projections (n1 , n2 , · · · , ns ) with N = n1 + n2 + · · · + ns . Then, because of the convexity property, the sequence n1 , n2 , · · · , ns must satisfy 0 < n1 ≤ n2 ≤ · · · ≤ nk < nk+1 = · · · = nl−1 > nl ≥ · · · ≥ ns−1 ≥ ns > 0. 11
Such sequences are called weakly unimodal partitions of N (or stack or planar partitions of N), see Stanley [6], Section 2.5, p. 76. Surprisingly, it turns out that any such sequence of projections corresponds to 0, 1 or 2 roundest animal of size N. More precisely, we have the following result. Lemma 12 Let (n1 , n2 , · · · , ns ) be a weakly unimodal sequence with N = n1 + · · · + ns . Then among all animals having this sequence of vertical projections, (i) there is a unique animal A, with minimal moment of inertia, if n1 , n2 , · · · , ns have the same parity; (ii) otherwise, there are exactly two animals A, A′ , having these projections, with minimal moment of inertia. Moreover, the moment of inertia of A (and A′ ) is given by the formula s s X 1 X 1 1 I(A) = n3k − N + k 2 nk − 12 k=1 12 N k=1
1 + 4N
X
nk even
s X
knk
k=1
!
nk
!2
X
nk odd
(7)
nk .
Proof. Let S be an animal with projections n1 , n2 , · · · , ns having columns C1 , C2 , · · · , Cs . More precisely, Ck is the column of points in S over the point (k, 0), k = 1, · · · , s (see Figure 7). C1 C 2 C3 C4 5 4 3 2 1 0
1
2
3
4
Fig. 7. Ck = {(k, y) | (k, y) ∈ S}, k = 1, · · · , s.
Let gk be the center of gravity of Ck , 1 ≤ k ≤ s. Note that there exists νk ∈ Z such that, for 1 ≤ k ≤ s, gk = In other words,
(k (k
+ iνk )
if nk is odd,
+ i(νk + 21 )) if nk is even.
1 gk = k + i νk + χeven (nk ) 2 12
k = 1, · · · , s
where χeven (n) = 1, if n is even and 0 otherwise. Then, by Lemma 5 and formula (3), we have I(S) = I(C1 ∪ · · · ∪ Cs ) = I(C1 ) + · · · + I(Cs ) + I({g1 , · · · , gs }) s X n3k − nk 1 X = + nk nl |(k − l) 12 N k
