Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 35 (1994), No. 1, 91-94.

What is the True Number of Semiregular (Archimedean) Solids?

Peter Schreiber

Fachrichtungen Mathematik/Informatik, Ernst-Moritz-Arndt-Universität, Friedrich-Ludwig-Jahn-Straße 15a, D-17487 Greifswald

Pappos told us in his ”Collectio” that Archimedes knew 13 semiregular solids (i.e. all of those known at the beginning of our century, besides the two families of the prisms with regular basis and of the Archimedean antiprisms which Archimedes did not mention). Only in 1930 Miller (according to Fejes-Tóth [3], p. 111, cf. also [2], p. 137) found a ”new” Archimedean solid by a slight modification (Fig. 1) of the rhombocuboctahedron (Fig. 2).

Figure 1. Figure 2.

1957 Aškinuse rediscovered this solid independently [1] without knowledge of Miller’s priority and tried to show by the study of all possible corner configurations that now all possible Archimedean solids (i.e. 2 infinite families + 14 singles) are known^{1}. The Miller-Aškinuse polyhedron is the only one that can not be generated by symmetric truncation of a regular solid. Today we understand very well that it has the property of a counterexample: It shows that the classical definition of semiregular solids ”to have regular faces and

1 Also Miller seems to have a forerunner: Malkevitch in [4] reports that as early as in [5] there is a Schlegel diagram of the Miller-Aškinuse polyhedron. I couldn’t see [5], in each case this paper had no resonance in the following.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1994 Heldermann Verlag, Berlin

pairwise congruent corner configuations” is not sufficient for ensuring the transitivity of the symmetry group of the solid upon the set of the corners (as all the other Archimedean solids suggest). Therefore, if we adopt the definition of semiregularity ”to have a symmetry group transitive upon the set of the corners” the number of single Archimedean solids remains at the state of 13.

Now let us discover two further Archimedean solids! The two solids shown in Figs. 3,4 are the only ones not to have a symmetry plane (and only ones of the 14 singles not so obvious derivable from regular solids).

Figure 3. Figure 4.

Figure 5. Figure 6.

Therefore, if we reflect them with respect to any plane we obtain two more Archimedean solids (Figs. 5,6). Of course they are combinatorically equivalent and even isometric to their counterimages but there is no proper motion that the one maps onto the other.

From a very elementary point of view there are at least two reasons why the solid shown in Fig. 3 is different from that shown in Fig. 5:

a) As Figs. 3,5 show, the ”side quadrangles” have an inclination and its direction is independent from what quadrangle is on the top. But this inclination is in Fig. 5 in the converse direction with respect to that in Fig. 3.

b) If you walk upon the surface of the body (Fig. 3) and want to go from the quadrangle on the top through some edges to one of the side quadrangles you must go first into a neighbouring triangle and then through the neighbouring triangle of that triangle at the right hand or else through two neighbouring triangles at the left hand (as the net in Fig. 7 shows). If you walk upon the body shown in Fig. 5 you must go from the neighbouring triangle through two triangles at the right hand or else through one neighbouring triangle at the left hand (Fig. 8).

Figure 7. Figure 8.

For the solids shown in Figs. 4,6 analogous reasons hold true. So, if we adopt the explanation that two Archimedean solids are ”the same” only if they are equivalent by a proper similarity mapping (i.e. without the use of reflections) then we have two families and 16 single Archimedean solids.

Let us conclude with the remark that in the same manner exactly one of the plane semiregular tilings, namely the one shown in Fig. 9, has a twin (Fig. 10).

Figure 9. Figure 10.

References

[1] Aškinuse, V.G.: On the number of semiregular solids (Russian). Matematičeskoe Prosveščenie 1 (1957), 107-118.

[2] Ball, W.W. Rouse: Mathematical Recreations and Essays (rev. by Coxeter). New York, MacMillan 1967.

[3] Fejes-Tóth, L.: Reguläre Figuren. Leipzig-Budapest: Teubner and Akadémiai Kiadó 1965.

[4] Malkewitch, J.: Milestones in the history of polyhedra. In: Shaping Space. Eds. M. Senechal and G. Fleck, Boston: Birkhäuser 1988, 80-92.

[5] Sommerville, D.M.Y.: Semiregular networks of the plane in absolute geometry. Transact. Royal Soc. Edinburgh 41 (1905), 725-747, plates I-XII.