Plane integral drawings of the platonic solid graphs with triangle faces∗

The planar graphs of the platonic solids the tetrahedron, octahedron, and icosahedron can be drawn as triangulations of the plane. Such drawings are called primitive integral plane graphs if the edges are noncrossing straight line segments of integer lengths and if the greatest common divisor of the lengths is one. It is proved that for each of these three solids, there exist infinitely many primitive integral plane graphs. The simpler cases of the cube and dodecahedron are mentioned.


Introduction
Every planar graph can be drawn in the plane with noncrossing edges which are straight line segments (Steinitz and Rademacher, Wagner, Fáry, Stein -see a short proof in [7]). It is an open problem [2,4] whether the edges also can be straight line segments of integral lengths. Nevertheless, one can try to construct for each planar graph G such an integral plane drawing, denoted by D(G). Moreover, it can be asked for the minimum diameter d of D(G), where d denotes the largest edge length of a D(G).
For the five platonic solids, the tetrahedron, octahedron, cube, dodecahedron, and icosahedron, the minimum diameters of their integral plane graphs have been determined in [5] to be 17, 2, 13, 2, and 159, respectively. Those three of them which are triangulations of the plane, that is, where the corresponding polyhedra have triangular faces, namely the tetrahedron, octahedron, and icosahedron, occur in connection with generalized matchstick graphs in [1].
It is asked in [1] for the minimum number n 0 (r, d) of vertices of an integral plane drawing (matchstick graph) of an r-regular planar graph (r = 3, 4, or 5 due to the Eulerian polyhedron formula) with given diameter d. Since n 0 (3, d) = 4 for a plane tetrahedron graph, n 0 (4, d) = 6 for a plane octahedron graph, and n 0 (5, d) = 12 for a plane icosahedron graph are determined in [1] for many values of d, we will discuss here whether infinitely many values of d are possible, that is, whether there exist infinitely many primitive integral plane drawings of these three planar platonic solid graphs.

Tetrahedron
The smallest integral plane drawing of the tetrahedron graph, that is, with the smallest diameter d = 17, can be seen in Figure 1 (see [1,5]). Theorem 2.1. There exists an infinite number of primitive integral plane tetrahedron graphs determined by any pair of primitive pythagorean triangles.
Proof. Primitive pythagorean triangles are used with legs of lengths 2mn and m 2 −n 2 , and the hypotenuse of length m 2 +n 2 for parameters m, n with (m, n) = 1 and m ≡ n (mod 2) (see [11]).
Consider any two such triangles (a, b, c) and (d, e, f ) with a < b and e < d for the pairs of legs a, b and d, e, respectively. Then a multiplication of (a, b, c) by d and (d, e, f ) by a, and a reflection at the side of length bd leads to the tetrahedron graph in Figure 2  Then, from any n ≥ 3 pairs of opposite points on the circle, we always can choose three points such that the center of the circle is inside of the triangle determined by the three points. Figure 3 shows an example for R = 5 2 , which after multiplication by 2 implies the desired tetrahedron graph in Figure 4.
By computer search (see [1]) all 499 integral plane tetrahedron graphs with diameter d ≤ 100 have been found and it can be checked that d = 48 is the smallest diameter if the inner edges are of equal length as in Figure 4.  Proof. We use the following result of [9, p. 48] (see also [6] and [3, D21]): "Any pair of rational numbers (p, q) = (0, 0) in determines four points with pairwise rational distances where any of these four distances can serve as the side lengths of an equilateral triangle, and then the fourth point has the remaining distances to the vertices of this triangle." Choosing s = u we obtain Furthermore, we may choose p = 4t and q = 2t + 1 for t ≥ 1 to get u = 3280t 4 + 1696t 3 + 472t 2 + 72t + 5, If now x serves as the side length of the equilateral triangle, then it can be checked that for t ≥ 1, the lengths u, y, and z are less than x 2 √ 3, so that the fourth point lies inside the triangle. For example, for t = 1, the tetrahedron graph in Figure  5 is determined.
It was checked by computer in [9] that d = 112 is the smallest diameter for an equilateral triangle as shown in Figure  6. It may be remarked that due to a computer search it was conjectured in [1] that a primitive integral plane tetrahedron graph exists for any d ≥ 68.

Octahedron
For the octahedron graph the smallest diameter was determined in [1,5] to be d = 13 (see Figure 8). Proof. We use the primitive integral 120 • -triangles where the largest side is of length m 2 + mn + n 2 and where the legs of the angle of 120 • are of lengths 2mn + n 2 and m 2 − n 2 with parameters m, n for m ≡ n (mod 3) and (m, n) = 1 (see [8]).
For such a triangle (a, b, c), where a and b, with a > b, are the legs of the angle of 120 • , we consider an equilateral triangle of side length 2a + b. Incident to its vertices we insert three pairs of triangles (a, b, c), as in Figure 7. Then x = a + b − 2b = a − b and after the deletion of the inner six edges of length b the desired integral octahedron graph is complete.  As examples, for m = 2, n = 1, the minimum case of Figure 8 (see [1,5]) is given, and for m = 5, n = 1, the graph in Figure 9 is obtained.
In [1] for diameters up to 100 all 22 primitive integral plane octahedron graphs are listed.
By comparison of the coefficients we obtain a 4 s 4 + a 3 s 3 = m 2 s 4 + 2mns 3 , a 2 = n 2 + 2mT , and a 1 = 2nT . Insertion of the values for a γ , 0 ≤ γ ≤ 4, yields This gives a rational solution (s 0 , R 0 ) yielding a rational r. However, since s 0 is negative a geometrical realization is impossible. If we then replace s by x + s 0 in the equation for R 2 we obtain Since a 0 = R 2 is a square number again we can use the same procedure as above to obtain , and .
gives a desired rational solution (s 1 , R 1 ) yielding a positive rational r for any sufficiently large j ≡ 2 (mod 3). After multiplication with the least common denominator a proof is complete.
Of course, these constructed examples have extremely large integers. However, there are many other types than the minimum one and corresponding generalizations. For d ≤ 250 we know 20 icosahedron graphs with diameters d = 159, 160, 168, 205, 209, 218, and 247 with 2 and 13 different drawings for 247 and 205, respectively. With the denotations of Figure 11 these drawings are presented in Table 1.

Concluding remarks
For the two remaining platonic solids, the cube and the dodecahedron, infinitely many primitive integral plane drawings can be constructed as follows (see Figures 12 and 13). We start with a square or a regular pentagon of side length a together with edges of integer lengths b and c as in Figures  12 and 13, respectively. Now consider the dashed square or pentagon of side length x, (x, a) = 1, being concentric and with sides parallel to the first square or pentagon, respectively. Then lift it up parallel to the plane as far as the distances y become an integer y, then turn it about the common centerpoint to screw it back into the plane preserving the rigid edges of length y, and the desired drawing is constructed. Altogether, we have the following result.
Theorem 5.1. There exist infinitely many primitive integral plane drawings for each of the five platonic solid graphs.
In general, we may expand the open problem [2,4] mentioned in the Introduction in the following way. For the cube and dodecahedron d 0 = 1 follows from the above constructions.