Crystallographic groups, strictly tessellating polytopes, and analytic eigenfunctions
Julie Rowlett, Max Blom, Henrik Nordell, Oliver Thim, Jack Vahnberg
CCRYSTALLOGRAPHIC GROUPS, STRICTLY TESSELLATINGPOLYTOPES, AND ANALYTIC EIGENFUNCTIONS
JULIE ROWLETT, MAX BLOM, HENRIK NORDELL,OLIVER THIM, AND JACK VAHNBERG
Abstract.
The mathematics of crystalline structures connects analysis, ge-ometry, algebra, and number theory. The planar crystallographic groups wereclassified in the late 19th century. One hundred years later, B´erard provedthat the fundamental domains of all such groups satisfy a very special analyticproperty: the Dirichlet eigenfunctions for the Laplace eigenvalue equation areall trigonometric functions. In 2008, McCartin proved that in two dimensions,this special analytic property has both an equivalent algebraic formulation, aswell as an equivalent geometric formulation. Here we generalize the resultsof B´erard and McCartin to all dimensions. We prove that the following areequivalent: the first Dirichlet eigenfunction for the Laplace eigenvalue equa-tion on a polytope is real analytic, the polytope strictly tessellates space, andthe polytope is the fundamental domain of a crystallographic Coxeter group.Moreover, we prove that under any of these equivalent conditions, all of theeigenfunctions are trigonometric functions. In conclusion, we connect thesetopics to the Fuglede and Goldbach conjectures and give a purely geometricformulation of Goldbach’s conjecture. Introduction In The Grammar of Ornament, published in 1856, Owen Jones wrote [15]:Whenever any style of ornament commands universal admiration,it will always be found to be in accordance with the laws whichregulate the distribution of forms in nature.In the case of crystals, the laws which regulate their shape are dictated by thecrystallographic groups.1.1.
Crystallographic groups.
A crystal or crystalline solid is a solid materialwhose constituents, such as atoms, molecules, or ions, are arranged in a highlyordered microscopic structure; for a two dimensional example see Figure 1. Math-ematically we can identify the locations of the constituents with the points of alattice, also known as a crystal lattice.
Every two dimensional crystal has a sym-metry group which is a plane crystallographic group. Such a group consists ofisometries of the plane. There are three basic types of isometries: translations,rotations and reflections. These form a group under composition. The patterns inFigure 2 have symmetry groups which are plane crystallographic groups.
These aresubgroups of the group of isometries of the plane which are topologically discreteand contain two linearly independent translations. Equivalently, a plane crystallo-graphic group is a co-compact subgroup of the group of isometries of the plane.A subgroup in this context is called co-compact if the quotient space R / Γ by the These are also known as wallpaper groups. a r X i v : . [ m a t h . SP ] D ec ROWLETT, BLOM, NORDELL, THIM, & VAHNBERG
Figure 1.
Graphene is an allotrope of carbon in the form of a two-dimensional, atomic-scale hexagonal lattice such that each pointin the lattice corresponds to an atom. This image is licensedunder the Creative Commons Attribution-Share Alike 3.0 Un-ported license at https://commons.wikimedia.org/wiki/File:Graphen.jpg .subgroup, Γ, is compact. The classification of these groups, up to equivalence, wasachieved at the end of the 19th century by E. S. Fedorov [10] and A. Schoenflies [26].Two groups are equivalent if they are equal up to a translation. In two dimensions,up to this notion of equivalence, there are seventeen crystallographic groups.One can also consider crystals in three dimensions, and mathematically we maygeneralize all of these notions to R n . An n -dimensional crystallographic groupis a discrete group of isometries of R n which is co-compact. Fedorov [10] andSchoenflies [26] proved that there are, up to equivalence, 219 crystallographicgroups in R . In 1910, Bieberbach proved that for any n , there are only finitelymany n -dimensional crystallographic groups up to equivalence [4,5], thereby solvingHilbert’s 18th problem.1.2. Strictly tessellating polytopes and our main result.
Crystal latticescreate a perfectly regular pattern, and the fundamental domain of the associatedfull-rank lattice tessellates space by translation. Another way to create a perfectlyregular pattern is by ‘strict tessellation.’ This is a notion specific to polytopes.
Definition 1.
The set of all one dimensional polytopes is the set of all boundedopen intervals, ℘ := { ( a, b ) : −∞ < a < b < ∞} . Inductively we define the set of polytopes in R n for n ≥ to be the set of connectedbounded domains in R n such that Ω ∈ ℘ n if and only if ∂ Ω = m (cid:91) j =1 P j , P j ∼ = Q j ∈ ℘ n − . RYSTALS, POLYTOPES, AND EIGENFUNCTIONS 3
Figure 2.
Two Egyptian patterns whose symmetry groups areplanar crystallographic groups. These patterns were documentedby Owen Jones in 1856 [15, Egyptian No 7 (plate 10), images 8and 13].
Above, the boundary of Ω consists of the closures of n − dimensional polytopes, P j . Each P j is contained in an n − dimensional hyperplane, which is a set of theform { x ∈ R n : M · x = b } , for some fixed M ∈ R n and b ∈ R . Such a hyperplane is canonically identified with R n − so that with this identification P j is canonically identified with Q j ∈ ℘ n − ,which is the meaning of P j ∼ = Q j . Next we introduce the notion of strict tessellation. We are not aware of the term‘strict tessellation’ in the literature, but it may be known under a different name.An example of a strict tessellation of the plane is given in Figure 3. For an exampleof a tessellation of the plane which is not strict, see Figure 4.
Figure 3.
Equilateral triangles are shown here to strictly tessel-late the plane.
Definition 2.
A polytope Ω ∈ ℘ n strictly tessellates R n if R n = (cid:91) j ∈ Z Ω j . ROWLETT, BLOM, NORDELL, THIM, & VAHNBERG
Above, each Ω j is isometric to Ω , and they are obtained by reflecting Ω across itsboundary faces. Furthermore, the hyperplanes which contain the boundary faces ofeach Ω j have empty intersection with (the interior of ) Ω k , for all j and k . One dimensional polytopes always strictly tessellate because for any real numbers a < b , R = (cid:91) j ∈ Z [ j ( b − a ) + a, j ( b − a ) + b ] . Remark.
Any polytope covers R n by reflection. If there were to be a gap, that is aregion of R n not covered by reflecting the polytope across its boundary faces, such agap will border a face of a reflected copy of the polytope. Then we simply reflect thepolytope again and do this for all bordering faces until we have filled the gap. Thusone can always cover R n by repeatedly reflecting any polytope across its boundaryfaces, but there may be significant overlap. If the (interior of the) reflected copiesof the polytope have empty intersections, then we obtain a tessellation. Figure 4.
Although it is well known that regular hexagons tessel-late the plane by reflection, the tessellation is not strict, becausethe lines which contain the edges of the hexagon cut through theinterior of the reflected copies.In 2008, McCartin proved a remarkable classification theorem [22], connectinggeometry and analysis. Recall the Laplacian on R n is the partial differential oper-ator ∆ := − n (cid:88) k =1 ∂ ∂x k . The Laplace eigenvalue problem for a domain Ω ⊂ R n with the Dirichlet boundarycondition is to find all functions u which are not identically zero and satisfy∆ u = λu, u | ∂ Ω = 0 , for some constant λ. This is a difficult problem, because in general it is impossible to compute thesenumbers λ . However, using the tools of functional analysis [7] one can prove thatthese eigenvalues are discrete and positive and therefore can be ordered0 < λ ≤ λ ≤ . . . ↑ ∞ . In this way we may speak of the first eigenfunction which has eigenvalue λ . In onedimension, by definition 1, a polytope is a bounded open interval, ( a, b ) for some RYSTALS, POLYTOPES, AND EIGENFUNCTIONS 5 real numbers a < b . The Laplace eigenvalue equation with the Dirichlet boundarycondition on such a polytope is to find all functions u such that there exists λ ∈ C with − u (cid:48)(cid:48) ( x ) = λu ( x ) , a < x < b, u ( a ) = u ( b ) = 0 . Solutions to this equation are u k ( x ) = sin (cid:18) x − ab − a kπ (cid:19) , λ k = k π ( b − a ) , k ∈ N . Moreover, Fourier analysis [11, Chapter 4] can be used to show that these are all the solutions to the equation. These eigenfunctions are all trigonometric functions.We can also define trigonometric functions on R n . Definition 3.
An eigenfunction u for the Laplacian is trigonometric if it can beexpressed as a finite sum of trigonometric functions, u ( x ) = m (cid:88) j =1 a j sin( L j · x ) + b j cos( M j · x ) . Above, a j , b j , ∈ C and L j , M j ∈ R n satisfy || L j || = || M j || = λ for all j =1 , . . . , m . Remark.
Since cos( t ) = sin( t + π/ , ∀ t ∈ R , it is equivalent to define a trigonometric eigenfunction to be a function of the form u ( x ) = m (cid:88) j =1 a j sin( L j · x + φ j ) . Above, a j ∈ C , L j ∈ R n , φ j ∈ R , and || L j || are the same for all j = 1 , . . . , m .In general, it is impossible to compute the eigenfunctions of an arbitrary polyg-onal domain. Nonetheless, McCartin proved the following classification theoremwhich shows the equivalence of the analytic property, having trigonometric eigen-functions, with the geometric property, strictly tessellating. Theorem 1 (McCartin [22]) . The only polygonal domains in the plane which havea complete set of trigonometric eigenfunctions for the Laplacian with the Dirichletboundary condition are those which strictly tessellate the plane. There are preciselyfour types: rectangles, isosceles right triangles, equilateral triangles, and hemiequi-lateral triangles as shown in Figure 5.
Remark.
The Laplace eigenfunctions for a rectangular domain with vertices atthe points (0 , a, , b ) and ( a, b ) with the Dirichlet boundary condition canbe computed using separation of variables which reduces the problem to two one-dimensional problems. The resulting eigenfunctions are indexed by m, n ∈ N . ForCartesian coordinates x = ( x, y ) ∈ R , the eigenfunctions are u m,n ( x, y ) = sin (cid:16) mπxa (cid:17) sin (cid:16) nπyb (cid:17) . Using trigonometric identities, u m,n ( x, y ) = 12 (cid:20) cos (cid:18)(cid:20) mπa − nπb (cid:21) · x (cid:19) − cos (cid:18)(cid:20) mπanπb (cid:21) · x (cid:19)(cid:21) . Consequently, these are trigonometric eigenfunctions.
ROWLETT, BLOM, NORDELL, THIM, & VAHNBERG
Figure 5.
From left to right above we have an equilateral triangle,rectangle, hemi-equilateral triangle, and an isosceles right triangle.These are the four types of polygons which strictly tessellate theplane.Our main result is a generalization to all dimensions.
Theorem 2.
Assume that Ω is a polytope in R n . Then the following are equivalent: (1) The first eigenfunction for the Laplace eigenvalue equation with the Dirich-let boundary condition extends to a real analytic function on R n . (2) Ω strictly tessellates R n . (3) Ω is congruent to a fundamental domain of a crystallographic Coxeter groupas defined in Bourbaki [6, VI.25 Proposition 9 p. 180], and is also knownas an alcove [3, p. 179]; see also § The three equivalent statements in Theorem 2 are respectively analytic, geo-metric, and algebraic. Our work therefore reveals an intimate connection betweenanalysis, geometry, and algebra. Moreover, combining our theorem with B´erard’sProposition [3, Proposition 9, p. 181], we obtain the following rather remarkableresult.
Corollary 1.
Assume that Ω is a polytope in R n . If the first eigenfunction for theLaplace eigenvalue equation with the Dirichlet boundary condition extends to a realanalytic function on R n , then it is a trigonometric eigenfunction. Moreover, all theeigenfunctions of Ω are trigonometric. Remark.
Every trigonometric eigenfunction satisfies the first condition of Theorem2. However, there are many functions which satisfy this condition but are not trigonometric. Examples include the eigenfunctions for a disk in R which areproducts of Bessel functions and trigonometric functions. There is no contradictionwith the above corollary because a disk is not a polygonal domain.1.3. Organization. In §
2, we prove that if the first eigenfunction of a polytopesatisfies the hypotheses of Theorem 2, then the polytope strictly tessellates R n .We prove this by generalizing classical results by Lam´e [19]. In §
3, we introducethe notions of root systems and alcoves and prove that all polytopes which strictlytessellate R n are alcoves. We then recall the result of B´erard [3]: all alcoves havea complete set of trigonometric eigenfunctions for the Laplace eigenvalue equationwith the Dirichlet boundary condition. These results together complete the proofsof Theorem 2 and Corollary 1. In § RYSTALS, POLYTOPES, AND EIGENFUNCTIONS 7
Ω is Strictlytessellating Ω is an Alcove First eigenfunc-tion analytic. ⇐ = Proven here P r o v e nh e r e = ⇒ B ´ e r a r d [ ] = ⇒ Figure 6.
This diagram shows the three statements of Theorem2 and how they were proven.2.
The first eigenfunction and strict tessellation
There is no known method to explicitly compute the eigenfunctions and eigen-values for an arbitrary polytope. However, using the tools of functional analysis,one can prove general facts about them. We summarize briefly here. For the Dirich-let boundary condition for the Laplace eigenvalue equation on a bounded domain,Ω ⊂ R n , the eigenvalues form a discrete positive set which accumulates only atinfinity. We can therefore order the eigenvalues as they increase0 < λ ≤ λ . . . ↑ ∞ . We may correspondingly order the eigenfunctions. In this way, we may speak ofthe “first” eigenfunction, which is the eigenfunction whose eigenvalue is equal to λ . The eigenfunctions form an orthogonal basis of the Hilbert space L (Ω). Weshall require the following well-known fact about the first eigenfunction. The proofof this theorem can be found in the classical PDE textbook of Evans [8, § Theorem 3.
Let Ω be a bounded domain in R n . Then the first eigenfunction for theLaplace eigenvalue equation with the Dirichlet boundary condition does not vanishin Ω . The following result is originally due to Lam´e [19] in two dimensions and specifiedto trigonometric eigenfunctions. Here we generalize the result to R n for all n aswell as to all real analytic functions. We include a short proof for completeness. Lemma 1 (Vanishing planes) . Let u be a real analytic function on R n . Assumethat u vanishes on an open, non-empty, subset of a hyperplane, P := { x ∈ R n : M · x = b } . Then u vanishes on all of P .Proof. Let y be any point of P . Let x be a point in the interior of the openneighborhood on which u vanishes. We parametrize a line segment to join thepoints x and y , defining l ( t ) := t x + (1 − t ) y , ≤ t ≤ . We note that M · l ( t ) = t M · x + (1 − t ) M · y ≡ b, ∀ t ∈ [0 , . ROWLETT, BLOM, NORDELL, THIM, & VAHNBERG
Consequently this line segment is contained entirely in P . Now we consider thefunction u ( t ) := u ( l ( t )) . Since u is a real analytic function, u ( t ) is also a real analytic function of t . Since x isin the interior of the open neighborhood on which u vanishes, and this neighborhoodis contained in P , there is ε > l ( t ) is contained in this neighborhood forall t ∈ [1 − ε, u ( t ) vanishes identically on [1 − ε, t , and in particular u ( l (0)) = u ( y ) = 0 . Since y ∈ P was arbitrary, we obtain that u vanishes at every point of P . (cid:3) We will also generalize Lam´e’s Fundamental Theorem, which was originallyproven in two dimensions and for trigonometric functions, to n dimensions andreal analytic eigenfunctions. Theorem 4 (Lam´e’s Fundamental Theorem) . Assume that u is a real analyticfunction on R n which satisfies the Laplace eigenvalue equation with the Dirichletboundary condition on a polytope Ω ∈ ℘ n . Then u is anti-symmetric with respectto all ( n − dimensional hyperplanes on which u vanishes.Proof. Let λ be the eigenvalue corresponding to u , so that on Ω we have∆ u ( x ) = λu ( x ) ∀ x ∈ Ω . Then, since u is real analytic, ∆ u is also real analytic on Ω. The function∆ u − λu is real analytic and vanishes on Ω which is an open subset of R n . Consequently, itvanishes on all of R n , and therefore u satisfies the same Laplace eigenvalue equationon all of R n .Now, let H be an ( n −
1) dimensional hyperplane on which u vanishes. Weconsider coordinates y = ( r, z ) ∈ R n , where z ∈ R n − and r = 0 ⇐⇒ y ∈ H. Let us now define the function (cid:101) u ( r, z ) := (cid:40) u ( r, z ) ( r, z ) ∈ Ω − u ( − r, z ) ( r, z ) (cid:54)∈ Ω . With this definition, (cid:101) u is anti-symmetric with respect to H . Since u satisfies theLaplace eigenvalue equation on R n and is real analytic, the same is true for (cid:101) u .Moreover, on ∂ Ω, u and (cid:101) u have the same Cauchy data; both functions vanish andhave the same normal derivative. Consequently, by standard uniqueness theory ofpartial differential equations [7], [8], u = (cid:101) u . Therefore u is also anti-symmetric withrespect to H . (cid:3) We are now poised to prove the first implication in Theorem 2.
Proposition 1.
Assume that Ω is a polytope in R n , and the first eigenfunctionsatisfies the first condition of Theorem 2. Then Ω strictly tessellates R n . RYSTALS, POLYTOPES, AND EIGENFUNCTIONS 9
Proof.
Let Ω be a polytope in R n as in the statement of the proposition. If n = 1,then Ω is a segment and may be written as ( a, b ) for some real numbers a < b . Wehave computed the eigenfunctions explicitly in this case. They are u k ( x ) = sin (cid:18) x − ab − a kπ (cid:19) . The first eigenfunction in particular satisfies the hypotheses of Theorem 2, and wehave also shown that all one dimensional polytopes strictly tessellate R . Hencethe proposition is proven in one dimension. So let us assume that n ≥
2. By theVanishing Planes Lemma, for an affine subset P which contains a boundary face ofΩ, all eigenfunctions of Ω vanish on P . Since the first eigenfunction never vanishesin the interior of Ω by Theorem 3, it follows that all of the hyperplanes whichcontain the boundary faces of Ω have empty intersection with the interior of Ω.Near a boundary face of Ω which is contained in the hyperplane P , we use thecoordinates ( r, z ) as in the previous Theorem. Without loss of generality, we mayassume that for r > r, z ) inside Ω. Since Ω is adomain, it is an open set. By Lam´e’s Fundamental Theorem, u ( − r, z ) = − u ( r, z ).Consequently, reflecting Ω across P , we see that u is also an eigenfunction in thisreflected copy of Ω which satisfies the Dirichlet boundary condition. Since the firsteigenfunction does not vanish on the interior of Ω, it also does not vanish on theinterior of the reflection of Ω across P . This shows that every hyperplane whichcontains a boundary face of Ω has empty intersection with this reflected copy ofΩ. We repeat this argument, reflecting across all of the boundary faces of Ω andthen reflecting across the boundary faces of the reflected copies of Ω. This showsthat the hyperplanes which contain the boundary faces of the reflected copies of Ωmust always have empty intersection with the interior of all reflected copies of Ω.Consequently, repeating indefinitely, we obtain a strict tessellation of R n by Ω. (cid:3) Root systems, alcoves, and strictly tessellating polytopes
In 1980, Pierre B´erard showed that a certain type of bounded domain in R n ,known as an alcove, always has a complete set of trigonometric eigenfunctionsfor the Laplace eigenvalue equation with the Dirichlet boundary condition. Todefine alcoves, we must first define root systems. The concept of a root systemwas originally introduced by Wilhelm Killing in 1888 [16]. His motivation was toclassify all simple Lie algebras over the field of complex numbers. In this section, wewill see how our analytic problem, the study of the Laplace eigenvalue equation, isconnected to these abstract algebraic concepts from Lie theory and representationtheory. Definition 4. A root system in R n is a finite set R of vectors which satisfy: (1) 0 is not in R . (2) The vectors in R span R n . (3) The only scalar multiples of v ∈ R are ± v . (4) R is closed with respect to reflection across any hyperplane whose normalis an element of R , that is v − u · v || u || u ∈ R, ∀ u , v ∈ R ; (5) If u , v ∈ R then the projection of u onto the line through v is an integer orhalf-integer multiple of v . The mathematical formulation of this is that u · v || v || ∈ Z , ∀ u , v ∈ R. The elements of a root system are often referred to as roots.
Remark.
There are different variations of the definition given above for a rootsystem depending on the context. Sometimes only conditions 1–4 are used todefine a root system. When the additional assumption (5) is included then the rootsystem is said to be crystallographic.
In other contexts, condition 3 is omitted, andthey would call a root system which satisfies condition 3 reduced.
We will need the dual root system to define the eigenvalues of the polytope whichwill be naturally associated to the root system.
Definition 5.
Let R be a root system. Then for v ∈ R the coroot v ∨ is defined tobe v ∨ = 2 || v || v . The set of coroots R ∨ := { v ∨ } v ∈ R . This is called the dual root system, and mayalso be called the inverse root system. The dual root system is itself a root system. We associate a Weyl group to a root system. These Weyl groups are subgroupsof the orthogonal group O ( n ). Definition 6.
For any root system R ⊂ R n we associate a subgroup of the orthogo-nal group O ( n ) known as its Weyl group.
This is the subgroup
W < O ( n ) generatedby the set of reflections by hyperplanes whose normal vectors are elements of R . For v ∈ R reflection across the hyperplane with normal vector equal to v is explicitly σ v : R n → R n , σ v ( x ) = x − v · x ) || v || v . A × A A B G Figure 7.
Here are four root systems in R . Below each rootsystem is the name of its Weyl group. The name of the Weylgroup may also be used as the name of the root system.By the definition of a root system, the associated Weyl group is finite. To explainwhat was proven in [3] by B´erard, we require the notion of Weyl chamber. Definition 7.
For a root system R ⊂ R n for each v ∈ R , let H v denote thehyperplane which contains the origin and whose normal vector is v . In particular H v := { x ∈ R n : x · v = 0 } . RYSTALS, POLYTOPES, AND EIGENFUNCTIONS 11
Let H = { H v } v ∈ R . Then R n \H is disconnected, and each connected open componentis known as a Weyl chamber. A Figure 8.
Extending the shaded area to infinity it is a Weyl cham-ber of the Weyl group A .To obtain an alcove, we not only wish to consider reflecting across the hyper-planes H v , but also across a discrete set of parallel translations of these hyperplanes. B H α,k A α
Figure 9.
This shows an alcove, A , corresponding to the rootsystem with Weyl group B . For α ∈ B , the hyperplanes H α,k for k ∈ Z are the parallel hyperplanes which have normal vector equalto α . Note that A is an isosceles right triangle. Definition 8.
Let R be a root system. Denote by H v the hyperplane in R n whichcontains the origin and whose normal vector is equal to v for v ∈ R . Let H v ,k = { x ∈ R n : v · x = k } , for k ∈ Z . Then H v , = H v . For k (cid:54) = 0 , the hyperplane H v ,k is parallel to H v . Wedefine an alcove to be a connected component of R n \ (cid:91) v ∈ R,k ∈ Z H v ,k . We note that the definition of an alcove immediately implies that it is a polytopein R n . An example of an alcove is shown in Figure 9. Proposition 2 (Proposition 9, p. 181 [3]) . Let Ω ⊂ R n be an alcove. Then Ω hasa complete set of trigonometric eigenfunctions for the Laplace eigenvalue equationwith the Dirichlet boundary condition. For readers who understand French and read [3], you may notice that the state-ment of Proposition 2 is not the English translation of [3, Proposition 9, p. 181].B´erard proved a stronger result; he specified the eigenvalues and correspondingeigenfunctions. To understand what B´erard proved, let R be a root system. Let C ( R ) denote a Weyl chamber, and let D ( R ) denote an alcove which is contained inthe Weyl chamber C ( R ). Consider the dual root system R ∨ . The vertices of theclosures of the alcoves associated to R ∨ create a lattice. Let us denote this latticeby Γ. The dual lattice Γ ∗ := { x ∈ R n : x · γ ∈ Z , ∀ γ ∈ Γ } . B´erard referred to the points contained in this dual lattice as ‘the group of weightsof R ’ (le groupe des poids de R) [3]. He proved that the eigenvalues for the alcove D ( R ) are given by { π || q || : q ∈ Γ ∗ ∩ C ( R ) } . The multiplicity of the eigenvalue λ = 4 π || q || is equal to the number of vectors q ∈ Γ ∗ ∩ C ( R ) which satisfy λ = 4 π || q || . The eigenfunctions are certain linearcombinations of e πi x · w ( q ) , where w ( q ) is in the affine Weyl group of R . The affineWeyl group of R is the semi-direct product of the Weyl group and the lattice Γ.Combining our Proposition 1 with B´erard’s Proposition 2 we obtain the followingcorollary which states that every alcove is a strictly tessellating polytope. Corollary 2.
Let Ω ⊂ R n be an alcove. Then Ω is a polytope which strictlytessellates R n . In the following proposition we prove the converse: every strictly tessellatingpolytope is an alcove of a root system.
Proposition 3.
Let Ω ⊂ R n be a polytope which strictly tessellates R n . Then Ω isan alcove.Proof. We will build a root system, R , using the fact that Ω strictly tessellatesspace. The main idea is to collect all the normal vectors to all faces of Ω in thetessellation. Since we may translate the entire tessellation, we will collect thesenormal directions by translating each vertex to the origin. We will specify thelengths of the vectors later, using the fact that the tessellation is strict.So, begin with a vertex at the origin. For each of the boundary faces whichcontains this vertex at the origin, reflection across the boundary face is an elementof the orthogonal group O ( n ). Since the tessellation is strict, and reflection fixesthe origin, after finitely many reflections across the boundary face and its imagesunder reflection, we must return to the original copy of Ω. We begin to build theroot system R by defining it to contain ± v , the normal vectors to this boundaryface, and to also contain all the images of ± v under the repeated reflection until wereturn to the original copy of Ω. The lengths of the vectors will be specified later.Since this repeated reflection returns to the original copy of Ω after finitely manyreflections, the set of vectors defined in this way is finite. Repeat this for all theboundary faces which contain the origin, and include any new normal directions ± u in R . Next, consider a different vertex in Ω. Translate the entire tessellation RYSTALS, POLYTOPES, AND EIGENFUNCTIONS 13 so that this vertex is at the origin and repeat, including any new normal directions ± w in the set R . Do this for all vertices. Each time we include at most finitelymany vectors. Since the number of vertices is also finite, the total resulting set ofvectors, denoted by R is a finite set of vectors. We define this set so that if v ∈ R then − v ∈ R as well. This specifies the directions of all vectors, but not theirlengths. The polytope is a bounded, connected, open set with boundary consistingof flat faces. The collection of normal vectors to the faces of Ω therefore span R n ,for if not, Ω would be contained in a k -dimensional hyperplane in R n and thus notan open set in R n . Since the set R contains all the normal vectors to all boundaryfaces of Ω, the set R spans R n . As we have defined it, 0 (cid:54)∈ R .Now fix a tessellation by Ω. Without loss of generality at least one vertex inΩ is at the origin. The tessellation defines hyperplanes in R n which contain theboundary faces of the copies of Ω in the tessellation. The way we have defined theset R , it contains all the normal directions of all these hyperplanes. Since R is afinite set, and there are countably many hyperplanes defined by the tessellation,this means that for each v ∈ R , we may enumerate the hyperplanes whose normaldirection is ± v by H v ,k for k ∈ Z . These hyperplanes are parallel. Consider those v ∈ R such that there is a hyperplane, denoted by H v , , whose normal vector is ± v and which contains the origin. We will define the length of v so that the closestparallel hyperplanes to H v , are H v , ± = { x ∈ R n : x · v = ± } . Since the hyperplanes H v , and H v , are parallel, there is a point y v ∈ R n with y v = c v v for some c v ∈ R , where v is the normal vector of (currently) unknownlength. This point y v is the unique point in H v , which is closest to the origin.Since y v ∈ H v , we have by definition y v · v = c v || v || = 1 . We know the location of H v , since it is given by the tessellation, so we also knowthe length of y v . Using the above equation, || y v || = | c v ||| v || , y v · v = c v || v || = 1 = ⇒ c v = 1 || v || = ⇒ || y v || = 1 || v || , so we conclude that || v || = 1 || y v || . Geometrically, || y v || = the distance between the hyperplanes H v , and H v , ± . We also have || v || = 1 || y v || , y v = v || v || . A schematic image of this is given in Figure 10.To define the lengths of all the elements of R , take each vertex in a single copyof Ω, and translate the entire tessellation so that the vertex is at the origin. Repeatthe above argument to define the lengths of all the vectors in R .For v ∈ R , reflection across a hyperplane in the tessellation whose normal direc-tion is that of v may only permute the ensemble of hyperplanes by virtue of thestrict tessellation. Let w ∈ R . By possibly translating the entire picture, assumethat there is a hyperplane in the tessellation with normal direction ± w and which Ω v Figure 10.
Given a polytope Ω, we construct the hyperplanes H v , , here in black dotted lines, and the normal vectors v . The set { H v ,k } includes the gray dotted lines.contains the origin, such that the origin is a vertex of a copy of Ω in the tessellation.Thus H w , is a hyperplane in the tesssellation. Consider the reflection with normaldirection v , denoted by σ v , that is σ v ( x ) = x − x · v || v || v . Then σ v (0) = 0. Consequently, σ v ( H w , ) is another hyperplane in the strict tessel-lation which also contains the origin, thus it is H u , for some u ∈ R . Similarly, wealso have σ v ( H w , ) = H u ,j for some j ∈ Z . Since σ v preserves the scalar product,for x ∈ H w , , by definition x · w = 1 = ⇒ σ v ( x ) · σ v ( w ) = 1 . Since σ v sends x to a point in H u ,j we also have σ v ( x ) · u = j. Since σ v ( H w , ) = H u , , we must have that σ v ( w ) = α u for some α ∈ R . Thereforecombining with the above we obtain1 = x · w = σ v ( x ) · σ v ( w ) = ασ v ( x ) · u = αj = ⇒ α = 1 j . So we have proven that σ v ( w ) = 1 j u . Now let us consider what happens to y w when we reflect by σ v . The vector y w is orthogonal to the hyperplanes H w , and H w , . The vector goes from the originin H w , and since its length is the distance between H w , and H w , , the end of thisvector sits in H w , . When this vector is reflected by σ v , it will again start from theorigin and have its endpoint lying on one of the parallel hyperplanes, by virtue ofthe strict tessellation. We compute explicitly that σ v ( y w ) = y w − y w · v v || v || = y w − y w · v ) y v . RYSTALS, POLYTOPES, AND EIGENFUNCTIONS 15
On the other hand, since σ v ( w ) = j u , we compute using the definitions of y w , y v ,and y u that σ v ( y w ) = σ v (cid:18) w || w || (cid:19) = 1 || w || σ v ( w ) = 1 || w || j u . Now, since || u || = j || w || = ⇒ σ v ( y w ) = j (cid:18) u || w || j (cid:19) = j y u . Combining these calculations we obtain σ v ( y w ) = y w − y w · v ) y v = j y u = ⇒ y w · v ) y v = y w − j y u . The vector y w goes from the origin to H w , , while the vector − j y u goes fromthe origin to H u , − j . By vector addition and the strict tessellation, the sum y w − j y u must go from the origin and end precisely at one of the parallel hyperplanes.Consequently, the vector 2( y w · v ) y v must be an integer multiple of y v because it goes from the origin in the directionof y v and lands at one of the parallel hyperplanes H v ,k for some k ∈ Z . Therefore,2( y w · v ) = k ∈ Z . By the definitions of y w and v ,2( y w · v ) = 2 w · v || w || = k ∈ Z . In a similar way, reversing the roles of w and v , we also obtain2 v · w || v || ∈ Z . Since w , v ∈ R Ω were arbitrary, this shows the final condition needed for R Ω to bea root system in Definition 4 is satisfied. We conclude that R Ω is a root systemand that Ω is one of its alcoves. (cid:3) The proofs of Theorem 2 and Corollary 1 will now follow from Propositions 1and 3 and B´erard’s Proposition 2.3.0.1.
Proof of Theorem 2.
In Proposition 1 we proved that if Ω is a polytope, thenthe statements in Theorem 2 satisfy: 1 = ⇒
2. In Proposition 3, we proved thatthe statements in Theorem 2 satisfy: 2 = ⇒
3. Finally, by B´erard’s Proposition2, statement 3, that Ω is an alcove, implies that all the eigenfunctions of Ω aretrigonometric. All trigonometric eigenfunctions are real analytic on R n and satisfythe Laplace eigenvalue equation on R n . Consequently, 3 = ⇒
1. The threestatements are therefore equivalent. (cid:3)
Proof of Corollary 1.
If the first eigenfunction of a polytope in R n satisfies thehypotheses of Theorem 2, then the polytope is an alcove. By B´erard’s Proposition2, all of the eigenfunctions of the polytope are trigonometric. (cid:3) Concluding remarks and conjectures
We have now answered the analysis question: when does a polytope in R n havea complete set of trigonometric eigenfunctions for the Laplace eigenvalue equation?In geometric terms, the necessary and sufficient condition for a polytope to have acomplete set of trigonometric eigenfunctions is that the polytope strictly tessellates R n . In algebraic terms, in the language of Bourbaki, the equivalent necessary andsufficient condition is that the polytope is congruent to a fundamental domain of acrystallographic Coxeter group [6, VI.25 Proposition 9 p. 180], [3, p. 179]. Return-ing to the analysis problem, it is interesting to note that it is enough to know thatthe first eigenfunction is real analytic and satisfies the Laplace eigenvalue equationon R n to conclude that it is a trigonometric function and moreover, all the eigen-functions are trigonometric. This is a remarkable fact. Moreover, the equivalenceof analytic, geometric, and algebraic statements shows that these different areasof mathematics are intimately connected. The Fuglede conjecture similarly bringstogether different areas of mathematics in the study of a single question.4.1. The Fuglede Conjecture.
To state the Fuglede Conjecture, we introduce afew concepts.
Definition 9.
A domain Ω ⊂ R d is said to be a spectral set if there exists Λ ⊂ R n such that the functions { e πiλ · x } λ ∈ Λ are an orthogonal basis for L (Ω) . The set Λ is then said to be a spectrum of Ω, and (Ω , Λ) is called a spectral pair.To relate these notions to our work here, we observe that if a domain Ω were tohave all its eigenfunctions for the Laplace eigenvalue equation of the form e πiλ · x ,then these functions would comprise an orthogonal basis for L (Ω). Consequently,knowing that the eigenfunctions are precisely of this form implies that the domainis a spectral set. However, the converse is not true, in the sense that if Ω is aspectral set, then its eigenfunctions are not necessarily individual complex expo-nential functions. If Ω is a spectral set, then the eigenfunctions must be linearcombinations of the e πiλ · x , since these are a basis for L (Ω). However, the linearcombinations could have countably infinitely many terms, so it is not clear whatprecise form the eigenfunctions will take. Conjecture 1 (Fuglede [12]) . Every domain of R n which has positive Lebesguemeasure is a spectral set if and only if it tiles R n by translation. Fuglede proved in 1974 that the conjecture holds if one assumes that the domainis the fundamental domain of a lattice [12]. Only several years later, in 2003, wasfurther progress made by Iosevich, Katz, and Tao [14] who proved that the Fugledeconjecture is true if one restricts to convex planar domains. In the following year,Tao proved that the Fuglede Conjecture is false in dimension 5 and higher [27]. In2006, the works of Farkas, Kolounzakis, Matolcsi and Mora [9], [17], [18], [21] provedthat the conjecture is also false for dimensions 3 and 4. In 2017, Greenfeld andLev proved that Fuglede’s Conjecture is true if one restricts attention to domainswhich are convex polytopes, but only in R [13]. In 2019, Lev and Matolcsi provedthat Fuglede’s Conjecture is true if one restricts attention to convex domains, inany dimension [20]. Interestingly, the Fuglede Conjecture is still an open problem RYSTALS, POLYTOPES, AND EIGENFUNCTIONS 17 for arbitrary domains in dimensions one and two. Here we make the followingconjecture which is related to yet independent from Fuglede’s.
Conjecture 2.
Let Ω be a domain in R n . Then Ω has a complete set of trigonomet-ric eigenfunctions for the Laplace eigenvalue equation with the Dirichlet boundarycondition if and only if Ω is a polytope which strictly tessellates R n . Equivalently, Ω has a complete set of trigonometric eigenfunctions for the Laplace eigenvalueequation with the Dirichlet boundary condition if and only if Ω is an alcove. -10 -5 0 5 10-10-50510 Figure 11.
This figure shows the null set of the function u ( x, y ) =sin( x ) + sin( y ) + sin(( x + y ) / √
2) in a square shaped region of R .The null set includes the line y = − x as well as the other curves inthe region. Consequently, by uniqueness, this function is the firsteigenfunction of the connected, open domains which are boundedby these curves, since it vanishes on the boundary but not on theinterior and satisfies the Laplace eigenvalue equation. Hence thefirst eigenfunction satisfies the first condition of Theorem 2, butwe do not obtain any further conclusions because the domain isnot a polytope.The difficulty in treating arbitrary domains is that we do not have a replace-ment for Lam´e’s results which are central to our proof. Moreover, it is possible toconstruct linear combinations of trigonometric functions which vanish on curvedregions; an example is given in Figure 11. Consequently, we cannot immediatelyconclude that domains which have trigonometric eigenfunctions have flat boundaryfaces, and hence they are polytopes. A domain with a curved boundary could havea few trigonometric eigenfunctions. What is reasonable to expect, however, is thatit does not have a complete set of trigonometric eigenfunctions.4.2. The crystallographic restriction theorem and a geometric approachto the Goldbach Conjecture.
The vertices of the strict tessellation given by apolytope which is an alcove are in fact the set of points in a full rank lattice. Wenote that two different polytopes may give rise to the same lattice; for example an isosceles right triangle and the square obtained by two copies of that triangle willproduce the same lattice. For any discrete group of isometries of R n , an element g of such a group has finite order if there is an integer k > g composedwith itself k times is the identity. The minimal such k is the order of g . To state thecrystallographic restriction theorem, we define a function which is like an extensionof the Euler totient function. For an odd prime p and r ≥ ψ ( p r ) := φ ( p r ) , φ ( p r ) = p r − p r − . Above, φ is the Euler totient function. The Euler totient function of a positiveinteger n counts the positive integers which are relatively prime with and less thanor equal to n . So, for example, for an odd prime p , the positive integers which arenot relatively prime with p r are p , 2 p , 3 p , . . . p r − p = p r . There are p r − of these.All other positive integers are relatively prime with p r , hence φ ( p r ) = p r − p r − .The function ψ is further defined as follows: ψ (1) = ψ (2) = 0 , ψ (2 r ) := φ (2 r ) for r > , and for m = (cid:89) i p r i i , ψ ( m ) := (cid:88) i ψ ( p r i i ) . Theorem 5 (Crystallographic Restriction I) . For any discrete group G of isome-tries of R n , for n ≥ , the set of orders of the elements G which have finite orderis equal to Ord n = { m ∈ N : ψ ( m ) ≤ n } . The crystallographic restriction theorem is connected to the mathematics ofcrystals when we reformulate the theorem in the context of lattices. A full-ranklattice is a set of points in R n of the form(1) Γ = { p ∈ R n : p = L x , L ∈ GL( n, R ) , x ∈ Z n } . Above, GL( n, R ) is the set of n × n invertible matrices with real entries, and Z n arethe elements of R n whose entries are integers. We say that the matrix L generatesthe lattice Γ. The generating matrix L is not unique, because for any M ∈ GL( n, Z )the set of points in (1) is equal to { p ∈ R n : p = LM x , x ∈ Z n } . Here GL( n, Z ) is the group of invertible n × n matrices whose entries are integers.Note that to be a group, this requires the determinant of all elements of GL( n, Z )to be equal to ±
1. Two matrices L , L ∈ GL( n, R ) generate the same lattice if andonly if there is an M ∈ GL( n, Z ) such that L = L M . For a matrix M ∈ GL( n, Z ),we identify it with the isometry of R n which maps x ∈ R n to M x . The matricesin GL( n, Z ) can therefore be identified with the group of symmetries of the crystalwhose atoms lie on the points of the lattice. Hence, the order of M is equal tothe smallest positive integer k such that M k is the identity matrix. It turns outthat the set of orders of the elements of any discrete group G of isometries of R n which have finite order is equal to the set of orders of the elements of GL( n, Z ).Consequently, the crystallographic restriction may be reformulated as follows. Theorem 6 (Crystallographic Restriction II) . For any n ≥ , the set of orders ofthe elements of GL( n, Z ) is equal to Ord n = { m ∈ N : ψ ( m ) ≤ n } . RYSTALS, POLYTOPES, AND EIGENFUNCTIONS 19
In [2], Bamberg, Cairns and Kilminster proved that one may reformulate theStrong Goldbach Conjecture in terms of the orders of elements of GL( n, Z ). Conjecture 3 (Strong Goldbach) . Every even natural number greater than six canbe written as the sum of two distinct odd primes.
Theorem 7 (Theorem 3 of [2]) . The following statements are equivalent: (1)
The strong Goldbach conjecture is true; (2)
For each even n ≥ there is a matrix M ∈ GL( n, Z ) which has order pq for distinct primes p and q , and there is no matrix in GL( k, Z ) of order pq for any k < n . The Goldbach Conjecture is an extremely difficult problem. Difficult, long-standing open problems have sometimes been solved by translating the probleminto a different field of mathematics. The proof of Fermat’s Last Theorem, alsoa statement in number theory, was achieved using newly developed techniques inalgebraic geometry [28], [29]. To approach the Goldbach Conjecture geometrically,we ask
Question 1.
Is there a geometric reason for the existence of a symmetry for fullrank lattices in R n , with n ≥ pq for two odd primes p (cid:54) = q such that p + q = n + 2?The condition that there is no matrix in GL( k, Z ) of order pq for any k < n is equivalent to requiring p + q = n + 2. This follows from the CrystallographicRestriction Theorem 6 which states that the orders of the elements of GL( k, Z ) isequal to the set of non-negative integers m with ψ ( m ) ≤ k . In order to guaranteethat ψ ( pq ) = p + q − > k for all k < n, but ψ ( pq ) ≤ n = ⇒ ψ ( pq ) = p + q − n. Consequently, the symmetry of order pq would correspond to a matrix M ∈ GL( n, Z ) which does not admit a diagonal decomposition into two matrices ofsmaller dimensions. Geometrically, this matrix would not arise as a product ofsymmetries of R k and R n − k for any k = 1 , . . . , n −
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