CRYSTALLINE PATTERNS AND DIOPHANTINE EQUATIONS 1. 2D

CRYSTALLINE PATTERNS AND DIOPHANTINE EQUATIONS TOSHIKAZU SUNADA

Abstract. This short note illustrates an interesting relationship among seemingly irrelevant subjects, say 2-dimensional crystalline patterns, Legendre’s theorem of three squares, and rational points on a certain complex quadric.

1. 2D crystalline pattern obtained from the cubic lattice It is my pastime to make various models of crystals by juggling a kit which I bought at a downtown stationer’s shop. Though it is not always possible to make what I want because of the limited usage of the kit, I can still enjoy playing with it. For instance, my kit allows me to produce the model of the diamond crystal whose beauty, caused by its big symmetry, has intrigued me for some time, and motivated to find out another crystal structure with the same symmetric property as the diamond which thus deserves to be called the diamond twin1.

Figure 1. Projected images of the cubic lattice Among all possible models, the simplest one is the cubic lattice (the junglegym-like figure in plain language). As a matter of fact, the cubic lattice is not much interesting as a crystal model2, but from some “view”, this lends itself to another recreation, and gives rise to an interesting mathematical issue which may be immediately generalized to a broad class of crystal models including the diamond and its twin. 1 This is what I call the K4 crystal ([6], [8]), whose structure was for the first time described by Fritz Laves in 1933. 2 Sodium chloride (NaCl) crystallizes in a cubic lattice.

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Let us look at the cubic lattice from enough remote distance. What we find out when we turn it around is that there are some specific directions toward which we may see 2D crystalline patterns (ignoring the effect of perspective). For instance, one can see the square lattice and regular triangular lattice as such crystalline patterns. Mathematically, this means that we are looking at the image in the plane R2 of the cubic lattice placed in R3 by the orthogonal projection P : (x, y, z) 7→ (x, y) (see Fig.1). Here the cubic lattice is supposed to be generated by a basis e1 , e2 , e3 of R3 satisfying hei , ej i = αδij

(α > 0).

Thus the set of vertices in the cubic lattice is {k1 e1 + k2 e2 + k3 e3 | k1 , k2 , k3 ∈ Z}. We shall call {ei }3i=1 an orthogonal basis. We now put vi = (ui , vi ) = P (ei ). Because hx, e1 ie1 + hx, e2 ie2 + hx, e3 ie3 = αx (x ∈ R3 ), we have hx, v1 iv1 + hx, v2 iv2 + hx, v3 iv3 = αx (x ∈ R2 ), or equivalently u1 2 + u2 2 + u3 2 = v1 2 + v2 2 + v3 2 (= α),

(1.1)

u1 v1 + u2 v2 + u3 v3 = 0. (1.2)

Projected images of orthogonal basis are characterized completely by this property. Namely we have Lemma 1.1. If three vectors u1 , u2 , u3 in R2 satisfy hx, u1 iu1 + hx, u2 iu2 + hx, u3 iu3 = αx

(x ∈ R2 ),

then there exists a basis f1 , f2 , f3 of R3 satisfying hfi , fj i = αδij and P (fi ) = ui . Proof For ui = (ai , bi ), we have a1 2 + a2 2 + a3 2 = b1 2 + b2 2 + b3 2 = α, a1 b1 + a2 b2 + a3 b3 = 0, so one can find a vector c = (c1 , c2 , c3 ) such that c1 2 + c2 2 + c3 2 = a1 2 + a2 2 + a3 2 ,

a1 c1 + a2 c2 + a3 c3 = b1 c1 + b2 c2 + b3 c3 = 0,

(indeed, c is a scalar multiple of the vector product (a1 , a2 , a3 ) × (b1 , b2 , b3 )). Having these equalities in mind, we consider the matrix   a1 b1 c1 A = a2 b2 c2  . a3 b3 c3 Then tAA = αI3 where tA stands for the transpose of A, and I3 is the identity matrix ; namely A is a scalar multiple of an orthogonal matrix, and hence we have AtA = αI3 . Putting f1 = (a1 , b1 , c1 ), f2 = (a2 , b2 , c2 ), f3 = (a3 , b3 , c3 ), we find that hfi , fj i = αδij .

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and P (fi ) = (ai , bi ) = ui . We go back to our situation. The projected image of vertices in the cubic lattice is given by {k1 v1 + k2 v2 + k3 v3 | k1 , k2 , k3 ∈ Z}. What we need to notice is that the projected image of the cubic lattice does not always give a crystalline pattern. Actually in almost all cases, the image of vertices in the cubic lattice is not discrete. To have a crystalline pattern, it is necessary (and sufficient) that three vectors v1 , v2 , v3 generate a lattice in R2 , or equivalently there exist a triple of integers (n1 , n2 , n3 ) 6= (0, 0, 0) such that n1 v1 + n2 v2 + n3 v3 = 0, (1.3) where one may assume without loss of generality that the greatest common divisor of n1 , n2 , n3 is 1. Remark The cubic lattice as a graph is the Cayley graph X(A) associated with the free abbelian group A generated by a Z-basis a1 , a2 , a3 . If we put H = {n1 a1 + n2 a2 + n3 a3 | n1 , n2 , n3 ∈ Z}, then the quotient graph X(A)/H by the H-action on X(A) is the Cayley graph associated with the factor group A/H generated by {ai + H}3i=1 . The projected image of the cubic lattice is a realization of X(A)/H (possibly having degenerate edges and/or colliding vertices). The map of X(A) onto X(A)/H associated with the canonical homomorphism A −→ A/H is a covering map, and is compatible with the projection P of the cubic lattice onto the projected image. X(A)   y

−−−−→ R3   yP

X(A)/H −−−−→ R2 2. Rational points on a quadric We √ now identify R2 with the complex plain C as usual, and put zi = ui + vi −1. Then it is readily checked that the equalities (1.2) is equivalent to the equation z1 2 + z2 2 + z3 2 = 0. (2.1) Condition (1.3) is also written as n1 z1 + n2 z2 + n3 z3 = 0.

(2.2)

Assume that n3 6= 0. Eliminating z3 by using (1.3) to obtain the quadratic equation (n1 2 + n3 2 )z1 2 + 2n1 n2 z1 z2 + (n2 2 + n3 2 )z2 2 = 0,

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and solving this, we get p ¡ ¢ 1 2 + n 2 + n 2) z , z2 = 2 − n n ± −(n 1 2 1 2 3 1 n2 + n3 2 p ¡ ¢ 1 2 + n 2 + n 2) z . z3 = 2 − n n ∓ −(n 1 3 1 2 3 1 n2 + n3 2 Thus we obtain a unique solution (z1 , z2 , z3 ) up to non-zero complex factors and the complex conjugate. Note that (z1 , z2 , z3 ) 7→ (αz1 , αz2 , αz3 ) and (z1 , z2 , z3 ) 7→ (αz1 , αz2 , αz3 ) (α ∈ C\{0}) are similar transformations in the plane3. Since we have interest in similarity class of crystalline patterns, our argument gives a satisfactory answer. For example, the square lattice (a), regular triangular lattice (b) and the lattice (c) in Fig. 1 correspond to (n0 , n1 , n2 ) = (1, 0, 0), (n0 , n1 , n2 ) = (1, 1, 1), (n0 , n1 , n2 ) = (1, 1, 2), respectively. What we observed so far is explained in terms of the complex projective space P 2 (C) = C3 /(C\{0}). Introduce the quadric Qcub = {[z1 , z2 , z3 ] ∈ P 2 (C)| z1 2 + z2 2 + z3 2 = 0} and the (projective) line L = {[z1 , z2 , z3 ] ∈ P 2 (C)| n1 z1 + n2 z2 + n3 z3 = 0}, where [z1 , z2 , z3 ] denotes the homogeneous coordinate. The points in P 2 (C) corresponding to the projected image√of the cubic lattice are the intersection of Qcub and L, and belong to P 2 (Q( −D)), where D is the square free part of n1 2 + n2 2 + n3 2 . Here we recall the general definition of rational points. Let K be an algebraic number field, and let V be a projective algebraic variety, defined in some projective space P n−1 (C) by homogeneous polynomials f1 , . . . , fm with coefficients in K. A K-rational point of V is a point [z1 , . . . , zn ] in P n−1 (K)(⊂ P n−1 (C)) that is a common solution √ of all the equations fj = 0. Thus our 2D crystalline patterns yield Q( −D)-rational points on the quadric Qcub . What about the converse?; that is, we ask whether any √ Q( −D)-rational point on Qcub is derived from a 2D crystalline pattern coming from the cubic lattice. The answer √ is “yes”. To see this, we first note −D) over Q is 2, so three complex that the dimension of the vector space Q( √ numbers z1 , z2 , z3 ∈ Q( −D) are linearly dependent over Q, and hence there exists a triple of integers (n1 , n2 , n3 ) 6= (0, 0, 0) such that n1 z1 +n2 z2 +n3 z3 = 0. Next we should note that three points z1 , z2 , z3 with [z1 , z2 , z3 ] ∈ Qcub do not lie on any line of the form Rz (z 6= 0), from which it follows that z1 , z2 , z3 3Reflections are included in similar transformations.

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with [z1 , z2 , z3 ] ∈ Qcub ∩ L generate a lattice in C. As discussed before, from the equality z1 2 +z2 2 +z3 2 = 0 we deduce that vi = (Re zi , Im zi ) (i = 1, 2, 3) are the projected image of an orthogonal basis. This completes the proof of the claim. It is interesting to point out that the quadric defined by z1 2 +z2 2 −z3 2 = 0 has Q-rational points, which, as is well known, are closely related to the socalled Pythagorean numbers, while Qcub has no Q-rational points. 3. Legendre’s theorem of three squares A new √ question arises: For which square free D, does the quadric Qcub have a Q( −D)-rational point? The answer is given in the following. √ Proposition 3.1. The quadric Qcub has a Q( −D)-rational point if and only if D is not of the form 8k + 7. This is, as easily conceived and proved below, a consequence of the theorem of three squares due to Legendre4 (1798) which says that a positive integer n can be expressed as the sum of three squares if and only if n is not of the form 4` (8k + 7). To give the proof of the above proposition, we write √ zi = ai + bi −D (ai , bi ∈ Q). Then the equation z1 2 + z2 2 + z3 2 = 0 is equivalent to the equation a1 2 + a2 2 + b3 2 = D(b1 2 + b2 2 + b3 2 ),

a1 b1 + a2 b2 + a3 b3 = 0.

If we put a = (a1 , a2 , a3 ), b = (b1 , b2 , b3 ) and n = (n1 , n2 , n3 ) = a × b, m = kbk2 , then n1 , n2 , n3 , m is a rational solution of n1 2 + n2 2 + n3 2 = Dm2 . (3.1) √ Thus if the quadric Qcub has a Q( −D)-rational point, the Equation (3.1) has a (non-trivial) rational solution. Conversely if n1 , n2 , n3 , m are a solution of (3.1), then it is easily checked that √ √ £ 2 ¤ n2 + n3 2 , −n1 n2 ± n3 m −D, −n1 n3 ∓ n2 m −D √ is a Q( −D)-rational point of Qcub . If Equation (3.1) has a non-trivial rational solution, then it has a nontrivial integral solution. Therefore to prove Proposition 3.1, it suffices to establish Lemma 3.1. Equation (3.1) has a non-trivial integral solution if and only if D is not of the form 8k + 7. Proof We first show that if D is not of the form 8k + 7, then n1 2 + n2 2 + n3 2 = D has an integral solution n1 , n2 , n3 . To this end, suppose that n1 2 + n2 2 + n3 2 = D has no integral solution. Then D = 4` (8k + 7) for some 4Legendre’ proof is not complete. It is Gauss who gave a complete proof.

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` and k by Legendre’s theorem. Since D is square free, D must be of the form 8k + 7. Next suppose that D is of the form 8k + 7. If n1 2 + n2 2 + n3 2 = Dm2 has a non-trivial integral solution n1 , n2 , n3 , m, then writing m = 2` (2s + 1), we have ¡ ¢ Dm2 = 4` (8k + 7)(2s + 1)2 = 4` 8(8kt + 7t + k) + 7 , where t = s(s + 1)/2. This contradicts to Legendre’s theorem.

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The Diophantine equation (3.1) includes a few classical equations as special cases. For instance, when D = 1, we have the 3D pythagorean equation x2 + y 2 + z 2 = w2 whose general solutions are given by x = 2(mq + np), y = 2(nq − mp), z = m2 + n2 − p2 − q 2 , w = m2 + n2 + p2 + q 2 . A solution of the negative Pell’s equation x2 − Dy 2 = −1 yields a solution (n1 , n2 , n3 , m) = (x, 1, 0, y) of (3.1). 4. The case of diamond and a generalization So far we have treated projected images of the cubic lattice. What happens when handle other crystal models? Are their projected images still related to rational points on certain quadrics? The answer is affirmative provided that a given crystal model sits “canonically” in the space5. Roughly speaking, such a model is characterized by a certain minimal principle, and has “maximal” symmetry among all possible deformations. And any model can be deformed to a canonically sitting model. Here instead of explaining the exact meaning of “canonically”, we illustrate several examples in Fig. 26. The interested reader is directed to [2], [4], [8]. Theorem 4.1. ([9]) Given a crystal model X, one can associate a complex quadric QX defined over Q. If this model sits canonically in the space, then the√2D crystalline pattern obtained as its projected image corresponds to a Q( −D)-rational points on QX . We consider the case of the diamond crystal (Fig. 3) without any detailed interpretation. This being the case, the √ 2D crystalline patterns obtained as its projected images are related to Q( −D)-rational points on the quadric defined by Qdia = {[z1 , z2 , z3 , z4 ] ∈ P 3 (C)| z1 2 +z2 2 +z3 2 +z4 2 = 0, z1 +z2 +z3 +z4 = 0}, which is also described as the quadric in P 2 (C) defined by {[z1 , z2 , z3 ]| z1 2 + z2 2 + z3 2 + z1 z2 + z2 z3 + z3 z1 = 0}.} 5In [2], the term “standard realization” is used for “canonically sitting modle”. 6Source of Fig. 2 (a) and Fig. 3: WebElements [http://www.webelements.com/]. The

CD image of the diamond twin was created by Kayo Sunada.

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Figure 2. (a) Lonsdaleite, (b) Diamond twin (K4 crystal), (c) ThSi2 structure, (d) 3D kagome lattice, (e) Net associated with the face-centered cubic lattice (fcu), (f) Net associated with the body-centered cubic lattice (bcu)

Figure 3. Diamond The square lattice is a projected image of the diamond crystal which corresponds to the intersection of Qdia and the line L = {[z1 , z2 , z3 , z4 ]| z1 + z2 + z3 + z4 = 0, z1 + z3 = z2 + z4 }; more explicitly √ √ Qdia ∩ L = [1, ± −1, −1, ∓ −1]. If we take up the line L = {[z1 , z2 , z3 , z4 ]| z1 +z2 +z3 +z4 = 0, 2z1 = z2 +z3 }, then √ √ Qdia ∩ L = [1, 1 ± −3, 1 ∓ −3, −3], which corresponds to the crystalline pattern depicted in Fig. 4. Remark Since Qdia is biregular over Q to the quadric {[w1 , w2 , w3 ]| w1 2 + w2 2 + w3 2 = 0},

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Figure 4. A projected image of the diamond crystal √ Qdia has Q( −D)-rational points if and only if D is not of the form 8k + 7. References [1] J-G. Eon, Euclidean embeddings of periodic nets: definition of a topologically induced complete set of geometric descriptors for crystal structures, Acta Cryst. A67 (2011), 68–86. [2] M. Kotani and T. Sunada, Standard realizations of crystal lattices via harmonic maps, Trans. Amer. Math. Soc., 353 (2000), 1–20. [3] M. Kotani and T. Sunada, Jacobian tori associated with a finite graph and its abelian covering graphs, Advances in Apply. Math., 24 (2000), 89–110. 633–670. [4] M. Kotani and T. Sunada, Spectral geometry of crystal lattices, Contemporary Math., 338 (2003), 271–305. [5] T. Ono, Gauss sums and Poincar´e’s sum, in Japanese, Nihon Hyoronsha, 2008. [6] T. Sunada, Crystals that nature might miss creating, Notices Amer. Math. Soc., 55 (2008), 208–215. [7] T. Sunada, Lecture on topological crystallography, Japan. J. Math. 7 (2012), 1–39. [8] T. Sunada, Topological crystallography —With a View Towards Discrete Geometric Analysis—, Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 6, Springer, 2012. [9] T. Sunada, Standard 2D crystalline patterns and rational points in complex quadrics, arXiv:submit/0620196 [math.CO] 23 Dec 2012. School of Interdisciplinary Mathematical Sciences, Meiji University, Nakano 4-21-1, Nakano-ku, Tokyo, 164-8525 Japan E-mail address: [email protected]