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On the number of waves arriving at the vertices of Platonic solids

The problem of counting the number of waves arriving at the vertex of a polyhedron is motivated by study of wave propagation on the manifolds with flat metric and finitely many singular points. In the article it was solved for the case of Platonic solids using three nontrivial results from number theory. This growth turns out to be subexponential.

Let us consider a polyhedron. At the initial moment of time, a concentric wave emerges from the selected vertex S along the surface of the polyhedron with a unit velocity. When it comes to any vertex, a new concentric wave appears. The problem is to understand how many waves will arrive at the selected vertexrecorder A at a given time t (see [4] for details of the same dynamical system on a hybrid Riemannian manifolds). Earlier questions that are related to the properties of the Laplace operator were considered (see [12], [13] for details and references).
Obviously the motion of waves on such a surface is determined by geodesics that emerges from the vertex S and arrive at the vertex A. We have a twodimensional flat surface with a finite number of conical singularities and we are considering geodesics joining two points. Such objects were studied in the well-known papers [1], [7]. Recent results of [2], [3], [10] allow us to solve this problem for regular tetrahedron, octahedron, icosahedron and cube. The number of waves in the vertex is determined by the number of broken geodesic lines, the length of which does not exceed the given parameter. If we know the distribution of components of these broken lines, then using the results from analytical number theory (theorems on the distribution of abstract primes, see [5], [6] for details), we can find the growth of the number of waves.

Geodesics on cube and the regular tetrahedron/octahedron/icosahedron
It is well known that the geodesics between the vertices of cube or regular tetrahedron correspond to segments between lattice points in the plane for square or triangular lattice.
Let us call a point p of square (respectively triangular) lattice irreducible if the segment from the origin to p does not contain any other lattice point. For square lattice the equivalent condition is that the coordinates of p are coprime integers.
Proposition 1.1. Fix some vertex v of the tetrahedron. Then all geodesics from v to the vertices of the tetrahedron (including v itself ) are in one-to-one correspondence with the set of irreducible points of triangular lattice inside a sector of angle 4π 3 . Fix some vertex v of the cube. Then all geodesics from v to the vertices of the cube (including v itself ) are in one-to-one correspondence with the set of irreducible points of square lattice inside a sector of angle 3π 2 . Thus we know the asymptotics as l → ∞ of the number of geodesics from one vertex to others with length less than l -it coincides with the asympotics of irreducible lattice points in the sector of fixed angle inside the circle of raduis l, which is 3π 4 l 2 for cube and 2π 3 l 2 for tetrahedron. The next natural question here is to understand how much of these geodesics connect v to each of the vertices individually rather than all together. This question is answered by [2] for tetrahedron and cube and by [3] for all platonic solids. Is is shown that each of the vertices gets asymptotically a fixed fraction of all the geodesics from v.
But we need to find asymptotics of the number of lengths of geodesics, not of the number of geodesics. These can be different since different geodesics may have equal length, e.g. points with coordinates (11, 3) and (9, 7) on a square lattice have equal distance √ 130 from the origin. Therefore, our question is to find the set of all possible norms of irreducible lattice vectors. Here again arises a question how much of these norms correspond to geodesics from v to some individual vertex, but we do not know the answer. We will only be interested in the asymptotics of these norms for geodesics to any vertex of our polyhedron. Such norms are exactly the numbers of the form x 2 + y 2 for square lattice and of the form x 2 + xy + y 2 for triangular one with integer x, y. Denote these sets of numbers by A and A . Proposition 1.2. The set of lengths of geodesics between vertices of cube is exactly the set A of lengths of vectors from square lattice. The set of lengths of geodesics between vertices of tetrahedron (or octahedron, or icosahedron) is exactly the set A of lengths of vectors from triangular lattice.
The asymptotics of such numbers is credited to Landau [8] and Ramanujan [9]. From their results (see [10]) follows that Analogously for triangular lattice: a (l) The constants γ and γ are defined by where product is over the set of primes.
Then the lengths of broken geodesics with vertices at the vertices of regular tetrahedron/octahedron/icosahedron (respectively cube) are exactly the sums of the numbers from A (respectively A ) with non-negative integer coefficients.
Note that different such sums may have equal values, even sum numbers from A and A may be integer multiples of others. Let us drop these values and consider B and B to be the sets of values from A and A which are not positive integer multiples of other values. It is clear that squares of values from B and B are exactly square-free positive integers of the form x 2 + xy + x 2 and x 2 + y 2 respectively.
The asymptotics of these numbers can be shown [11] to be π 2 6 times less than the previous asymptotics: ln l . The lengths of broken geodesics with vertices at the vertices of cube (respectively tetrahedron, or octahedron, or icosahedron) are still the sums of the numbers from B (resp. B ) with non-negative integer coefficients. Any two such sums are different since the set { √ n | n is a square-free positive integer} is linearly independent over Q.

Abstract prime number theorem
We will use the abstract additive prime number theorem. Let us consider an arithmetical semigroup G generated by products of the sequence p 1 , p 2 , ... of distinct elements of a semigroup (called abstract primes). If we have a norm on a semigroup then we could introduce two functions: counting function of the elements of a semigroup (i.e. abstract integers) N G (x) that are smaller than x and a counting function of abstract primes π G (x). Now we could investigate how assumptions about the asymptotic behaviour of one of these functions as x → ∞ influences that of the other. In the general setting, any theorem of this kind could be called an "abstract prime number theorem" (see [5]).
John Knopfmacher (see [5]) wrote: "The abstract additive prime number theorem is closely related to classical theorems of additive analytic number theory due to Hardy and Ramanujan, and concerning arithmetical functions of the same general type as the classical partition function p(n). In fact, the theorem itself may to a large extent be regarded as a reformulation of a certain well-known 'Tauberian' theorem of Hardy and Ramanujan". Theorem 2.1 (Additive Abstract Prime Number Theorem). Let G denote an additive arithmetical semigroup such that there exist positive constants C and κ, and a real constant ν, such that Then, as x → ∞,

Waves on regular tetrahedron and cube
From the results of first section we see that for regular tetrahedron and cube we have κ = 2 and ν = −1/2.
Let us substitute this to the theorem 2.1. We obtain that Then, as x → ∞,  Now we are ready to formulate the theorem for cube and regular tetrahedron. • regular tetrahedron/octahedron/icosahedron: 6 (ln t) , • cube: 6 (ln t) .
Here γ is the Landau-Ramanujan constant and ζ(s) is the Riemann zeta function.
Note that for the case of a cube we have c G 1.8690 and for the case of a regular tetrahedron we have c G 1.7617.
We see that the number of arriving waves growth slower than the exponent which is in perfectly fits the results of computer experiments.

Geodesics on dodecahedron and other pentagonal surfaces
For pentagonal surfaces -the ones which can be glued from unit regular pentagons -the situation is a bit more complicated. That is because there is no plane lattice containing all vertices of the pentagon. However the trajectory unfolding construction (see Fig. 1) yields that the vector corresponding to a geodesic connecting the vertices of our surface can be represented as integer linear combination of sides of the regular pentagon. Moreover, we may restrict to only positive combinations:  Proof . Let us unfold the metric along v. We obtain a sequence of pentagons of which any consecutive ones share a common edge. Now we choose a broken line path from the beginning of v to its end going along the edges of our sequence of pentagons. This path can be chosen such that its projection to v goes monotonously (see Fig. 2). We just show step-by-step that in each pentagon the vertices of edges intersecting with v are reachable by such monotonous broken lines. The existence of such broken line implies that the desired decomposition of vector v exists.  Proof . There is a the finite number of orientations e 1 , . . . , e 5 of edges. Let us fix any of them and estimate the number c(l) of vectors v with length less than l decomposable as v = n 1 e 1 +· · ·+n 5 e 5 with non-negative n i with extra condition that the dot products (v, e i ) are positive. Consider some v 0 from these vectors. Then (v, v) ≥ 1 (v0,v0) (v, v 0 ) 2 = 1 (v0,v0) ( i n i (e i , v 0 )) 2 > i n 2 i for some > 0 independent of v. Therefore the number of v's of length less than l is less than the number of sequences (n 1 , . . . , n 5 ) such that i n 2 i < 1 l 2 . Obviously, this is less than γl 5 for some γ. for some c.

Further work
The upper bound for the dodecahedron could be improved and the asymptotics could be obtained, as for other Platonic solids. For polyhedra with "rational" angles one could use the covers by translation surfaces and Siegel-Veech constants (see [1] for details). Also it is interesting to consider wave propagation on polyhedra in non-Euclidean spaces.