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The strange case of the Bilinski dodecahedron and developments arising from it
Fifty years ago Stanko Bilinski showed that the enumeration of convex polyhedra having congruent rhombi as faces is incomplete, although it had been accepted as valid for the previous 75 years. The dodecahedron he discovered will be used here to document errors by several mathematical luminaries. It also prompted an examination of the largely unexplored topic of analogous non-convex polyhedra, which led to unexpected connections and problems.
1. Background.
In 1885, In 1885 Evgraf Stepanovich Fedorov published the results of several years of research under the title "Introduction to the Study of Figures" [F1] in which he defined and studied a variety of concepts that are relevant to our story. This book-long work is considered by many to be one of the milestones of mathematical crystallography. For a long time this was, essentially, inaccessible and unknown to Western researchers except for a summary [F2] in German.
Several mathematically interesting concepts were introduced in [F1]. We shall formulate them in terms that are customarily used today, even though Fedorov's original definitions were not exactly the same. First, a parallelohedron is a polyhedron in 3-space that admits a tiling of the space by translated copies of itself. Obvious examples of parallelohedra are the cube and the Archimedean six-sided prism. The analogous 2-dimensional objects are called parallelogons; it is not hard to show that the only polygons that are parallelogons are the centrally symmetric quadrangles and hexagons. It is clear that any prism with a parallelogonal basis is a parallelohedron, but we shall encounter many parallelohedra that are more complicated.
Another new concept in [F1] is that of zonohedra. A zonohedron is a polyhedron such that all its faces are centrally symmetric. All Archimedean prisms over even-sided bases are zonohedra, but again there are more interesting examples. A basic result about zonohedra is:
Each convex zonohedron has a center.
This result is often attributed to Aleksandrov [A1] (see [B3]), but in fact can be immediately deduced from a more general theorem of Minkowski [M1, p. 118]. Even earlier, this was Theorem 23 of Fedorov ([F1, p. 271], [F2, p. 689]), although Fedorov's proof is rather convoluted and hard to follow.
We say that a polyhedron is monohedral (or a monohedron) provided its faces are all mutually congruent. The term "isohedral" used by Fedorov [6] and Bilinski [2] nowadays indicates the more restricted class of polyhedra with the property that their symmetries act transitively on their faces. The polyhedra of Fedorov and Bilinski are not (in general) "isohedra" by definitions that are customary today. We call a polyhedron rhombic if all its faces are rhombi. It is an immediate consequence of Euler's theorem on polyhedra that the only monohedral zonohedra are the rhombic ones.
One of the results of Fedorov ([F1, page 267], [F2, page 689]) is contained in the claim:
There are precisely four distinct types of monohedral convex zonohedra: the rhombic triacontahedron T, the rhombic icosahedron F, the rhombic dodecahedron K, and the infinite family of rhombohedra (rhombic hexahedra) H.
These polyhedra are illustrated in Figure A; they are sometimes called isozonohedra. The polyhedra T and K go back at least to Kepler [K1], while F was first described by Fedorov [F1]. I do not know when H was first found it probably was known in antiquity.
An additional important result from Fedorov [F1] is:
Every convex parallelohedron is a zonohedron of one of the five combinatorial types shown in Figure B. Conversely, every convex zonohedron of one of the five combinatorial types in Figure B is a parallelohedron.
Fedorov's proof is not easy to follow; a more accessible proof of Fedorov's result can be found in [A2, Ch. 8].
2. Bilinski's rhombic dodecahedron.
Fedorov's enumeration of monohedral rhombic isohedra (called isozonohedra by Fedorov and Bilinski, and by Coxeter [C1]) mentioned above claimed that there are precisely four distinct types (counting all rhombohedra as one type). Considering the elementary character of such an enumeration, it is rather surprising that it took three-quarters of a century to find this to be mistaken. Bilinski [B1] found that there is an additional isozonohedron and proved:
There are precisely five distinct types of isozonohedra.
The rhombic monohedral dodecahedron found by Bilinski shall be denoted B; it is different from Kepler's dodecahedron, denoted K. Bilinski also proved that there are no other isozonohedra. To ease the comparison of B and K, both are shown in Figure C.
Bilinski's proof of the existence of the dodecahedron B is essentially trivial, and makes it even more mysterious that Fedorov missed it. It is based on two observation: (i) That all faces of every convex zonohedron are arranged in zones, that is families of faces in which all members share parallel edges of the same length; and (ii) That all edges of such a zone may be lengthened or shortened by the same amount while keeping the polyhedron zonohedral. In particular all such edges on one zone can be deleted (shrunk to 0). Performing such a zone deletion a process mentioned by Fedorov starting with Kepler's rhombic triacontahedron T yields (successively) Fedorov's icosahedron F, Bilinski's dodecahedron B, and two rhombohedra, the obtuse Ho and the acute Ha. This family of descendants of the triacontahedron is shown in Figure D. The proof that there are no other isozonohedra is slightly more complicated and is not of particular interest here.
The family of "direct" descendants of Kepler's rhombic dodecahedron K is smaller; it contains only one rhombohedron H*o, see Figure E. However, one may wish to include in the family a "cousin" H*a consisting of the same rhombi as H*o, but in an acute conformation.
One of the errors in the literature dealing with Bilinski's dodecahedron is the assertion by Coxeter [5, p. 148] that the two rhombic dodecahedra Kepler's and Bilinski's are affinely equivalent. To see the affine nonequivalence of the two dodecahedra (easily deduced even from the drawings in Figure C), consider the long (horizontal) body-diagonal of Bilinski's dodecahedron (Figure C(b)). It is parallel to four of the faces, and in each face to one of the diagonals. In two faces this is the short diagonal, in the other two the long one. But in the Kepler dodecahedron the corresponding diagonals are all of the same length. Since ratios of lengths of parallel segments are preserved under affinities, this establishes the non-equivalence.
If one has a model of Bilinski's dodecahedron in hand, one can look at one of the other diagonals connecting opposite 4-valent vertices, and see that no face diagonal is parallel to it.
By the theorems of Fedorov mentioned above, since Bilinski's dodecahedron B is zonohedron combinatorially equivalent to Keplers, it is a parallelohedron. This can be easily established directly, most simply by manipulating three or four models of B. It is strange that Bilinski does not mention the fact that B is a parallelohedron.
In this context we have to mention a serious error committed by A. Schoenflies [S1, pages 467 and 470] and very clearly formulated by E. Steinitz. It is more subtle than Coxeter's, who may have been misguided by the following statement of Steinitz [S2, page 130]:
The aim [formulated previously in a different form] is to determine the various partitions of the space into congruent polyhedra in parallel positions. Since an affine image of such a partition is a partition of the same kind, affinely related partitions are not to be considered as different. Then there are only five convex partitions of this kind. [My translation]
How did excellent mathematicians come to commit such errors? The confusion illustrates the delicate interactions among the concepts involved, considered by Fedorov, Dirichlet, Voronoi, and others. A correct version of Steinitz's statement would be:
Every convex parallelohedron P is affinely equivalent to a parallelohedron P' such that a tiling by translates of P' coincides with the tiling by the Dirichlet-Voronoi regions of the points of a lattice L'. The lattice L' is affinely related to the lattice L associated with one of the five Fedorov parallelohedra P". But P' need not be the image of P" under that affinity. Affine transformations do not commute with the formation of Dirichlet-Voronoi regions.
As an illustration of this situation, it is easy to see that Bilinski's dodecahedron B is affinely equivalent to a polyhedron B' that has an insphere (a sphere that touches all its faces). The centers of a tiling by translates of B' form a lattice L' such that this tiling is formed by Dirichlet-Voronoi regions of the points of L'. The lattice L' has an affine image L such that the tiling by Dirichlet-Voronoi regions of the points of L is a tiling by copies of the Kepler dodecahedron K. However, since the Dirichlet domain of a lattice is not affinely associated with the lattice, there is no implication that either B or B' is affinely equivalent to K.
A simple illustration of the analogous situation in the plane is possible with hexagonal parallelogons (a parallelogon is a polygon that admits a tiling of the plane by translated copies). As shown in Figure F, the tiling is by the Dirichlet regions of a lattice of points. This lattice is affinely equivalent to the lattice associated with regular hexagons, but the tiling is obviously not affinely equivalent to the tiling by regular hexagons.
3. Non-convex parallelohedra.
The two selfintersecting triacontahedra
Unkelbach's hexecontahedron
Bilinski's completion of the enumeration of isozonohedra needs no correction. However, it may be of interest to examine the situation if nonconvex isozonohedra are admitted. Moreover, there are various reasons why one should investigate more generally non-convex parallelohedra. Fedorov [6, 83] briefly mentioned the possibility of removing the assumption that the polyhedra considered are convex but his treatment was superficial and seems not to have elicited any follow-up investigations.
Two non-convex rhombic isohedra have been described in the nineteenth century, see Coxeter [C1, pages 102 103, 115 116]. Both are triacontahedra, and are selfintersecting. This illustrates the need for a precise description of the kinds of polyhedra we wish to consider here.
Convex polyhedra discussed so far need little explanation, even though certain variants in the definition are possible. However, now we are concerned with wider classes of polyhedra regarding which there is no generally accepted definition. Unless the contrary is explicitly noted, in the present note we consider only polyhedra with surface homeomorphic to a sphere and adjacent faces not coplanar. We say they are of spherical type. There are infinitely many combinatorially different rhombic monohedra of this type to obtain new ones it is enough to "paste together" along common faces two or more appropriate smaller polyhedra. This will interest us a little bit later.
The two triacontahedra mentioned above are not accepted in our discussion. However, a remarkable non-convex rhombic hexecontahedron of the spherical type was found by Unkelbach [U1]; it is shown in Figure U. Its rhombi are the same as those in Kepler's triacontahedron T. It is one of a score of rhombic hexecontahedra described in the draft of [G2]; however, all except U are not of the spherical type.
For a more detailed investigation of non-convex isozonohedra, we first restrict attention to rhombic dodecahedra. We start with the two convex ones Kepler's K and Bilinski's B and apply a modification we call indentation. An indentation is carried out at a 3-valent vertex of a isozonohedron. It consists of the removal of the three incident faces and their replacement by the three "inverted" faces that is the triplet of faces that has the same outer boundary as the original triplet, but fits on the other side of that boundary. This is illustrated in Figure G, where we start from Kepler's dodecahedron K shown in (a), and indent the nearest 3-valent vertex (b). It is clear that this results in a non-convex polyhedron. Since all 3-valent vertices of Kepler's dodecahedron are equivalent, there is only kind of indentation possible. On the other hand, Bilinski's dodecahedron B in Figure H(a) has two distinct kinds of 3-valent vertices, so the indentation construction leads to two distinct polyhedra; see parts (b) and (c) of Figure H.
Returning to Figure G, we may try to indent one of the 3-valent vertices in (b). However, none of the indentation produces a polyhedron of spherical type. The minimal departure from this type occurs on indenting the vertex opposite to the one indented first; in this case the two indented triplets of faces meet at the center of the original dodecahedron (see Figure G(c)). We may eliminate this coincidence by stretching the polyhedron along the zone determined by the family of parallel edges that do not intrude into the two indented triplets. This yields a parallelogram-faced dodecahedron that is of spherical type (but not a rhombic monohedron); see Figure G(d). A related polyhedron is shown in a different perspective as Figure 121 in Fedorov's book [6].
It is of significant interest that all the isozonohedra in Figures G and , even the ones we do not quite accept, shown in Figures G(c) and H(e) are parallelohedra. This can most easily be established by manipulating a few models; however, graphical or other computational verification is also readily possible.
To summarize the situation concerning dodecahedral rhombic monohedra, we have the following polyhedra of spherical type:
Two convex dodecahedra (Kepler's and Bilinski's);
Three simply indented dodecahedra (one from Kepler's polyhedron, two from Bilinski's)
One doubly indented dodecahedron (from Bilinski's polyhedron).
We turn now to the two larger isozonohedra, Fedorov icosahedron F and Kepler's triacontahedron T. Since each has 3-valent vertices, it is possible to indent them, and since the 3-valent vertices of each are all equivalent under symmetries, a unique indented polyhedron results in each case (Figure I).
The icosahedron F admits several non-equivalent double indentations, only one of which has been investigated in some detail (see Figure J). There are many distinct multiple up to sixfold indentations; their precise number has not been determined. No multiple indentations of the triacontahedron T have been investigated.
The double indentations of F shown in Figure J is quite surprising and deserves special mention: It is a parallelohedron but while zones in convex parallelohedra have at most six faces, all zones of this one have 8 faces . Again, the simplest way to verify that this is a parallelohedron is by using a few models, and investigating how they fit. This contrasts with the singly indented icosahedron which is not a parallelohedron. None of the other isozonohedra obtainable by indentation of F or T seem to be parallelohedra.
A different construction of isozonohedra is through the union of two or more given ones along whole faces; clearly this means that all those participating in the union must belong to the same family of rhombic monohedra either the family of the triacontahedron, or of Kepler's dodecahedron, or of rhombohedra (with equal rhombi) not in either of these families. Besides a brief notice of this possibility by Fedorov, the only other reference is to the union of two rhombohedra mentioned by Kappraff [Ka, p. 381].
For an example of the construction, by attaching two rhombohedra one can obtain three distinct decahedra, see Figure Z. Each is chiral, that is, comes in two mirror-image forms. For another example, from three acute and one obtuse rhombohedra of the triacontahedron family, that share an edge, one can form a decahexahedron. It is chiral, but it has an axis of 2-fold rotational symmetry. Each of the isozonohedra mentioned in this paragraph happen to be parallelohedra as well.
While the number of examples non-convex isozonohedra and parallelohedra could be increased indefinitely, in he next section we shall propose an explanation of the situation.
4. Remarks.
(i) The parallelohedra in Section 3 lack a center of symmetry, which was generally taken as present in parallelohedra and more generally in zonohedra. Convex zonohedra have been studied extensively; they have many interesting properties, among them central symmetry. However, the assumption of central symmetry (of the faces, and hence of the polyhedra) amounts to putting the cart before the horse if one wishes to study parallelohedra that is, polyhedra that tile space by translated copies.
In fact, the one and only immediate consequence of the assumed property of polyhedra that allow tilings by translated copies is that their faces come in pairs that are translationally equivalent. For example, the octagonal prism in Figure K in not centrally symmetric, and its bases have no center of symmetry either. But even so, it clearly is a parallelohedron. The dodecahedra in Figures G(b) and H(b,c) have no center of symmetry although their faces are rhombi and have a center of symmetry each. On the other hand, the doubly indented polyhedron is Figure H(d) has a center.
It is worth mentioning that by Fedorov's Definition 24 (page 285 of [F1], page 691 of [F2]) and earlier ones, a parallelohedron need not be convex, nor does it need to have centrally symmetric faces.
The term "gleichflchig" (= with equal surfaces) was quite established at the time of Fedorov's writing, but what it meant seems to have been more than the word implies. As explained in Edmund Hess's second note [H3] excoriating Fedorov [F4], the interpretation as "congruent faces" (that is, monohedral) is mistaken. Indeed, by "gleichflchig" Hess means something much more restrictive. Hess formulates it in [H3] very clumsily, but it amounts to symmetries acting transitively on the faces, that is, to isohedral. It is remarkable that even the definition given by Brckner (in his well-known book [B2, page 121], repeating the definition by Hess in [H1] and several other places) states that "gleichflchig" is the same as "monohedral" but Brckner (like Hess) takes it to mean "isohedral". Fedorov was aware of the various papers that use "gleichflchig", and it is not clear why he used "isohedral" for "monohedral" polyhedra. In any case, this led Fedorov to claim that his results disprove the assertion of Hess [H1] that every "gleichflchig" polyhedron admits an insphere. Fedorov's claim is unjustified, but with the rather natural misunderstanding of "gleichflchig" he was justified to think that his rhombic icosahedron is a counterexample. This, and disputed priority claims, led to protests by Hess (in [H2] and [H3]), repeated by Brckner [B2, page 162], and a rejoinder by Fedorov [F4]. Neither side pointed out that the misunderstanding arises from inadequately explained terminology; from a perspective of well over a century later, it seems that both Fedorov and Hess were very thin-skinned, inflexible and stubborn.
In different publications Fedorov uses different notions of "type". In several (for example, [F2], [F3]) he has only four "types" of parallelohedra, since the rhombic dodecahedron and the elongated dodecahedron ((c) and (b) in Figure B) are of the same type in these classifications. Since we are interested in combinatorial types, we accept Fedorov's original enumeration illustrated in Figure B.
A possible explanation is in a tendency that can be observed in other enumerations as well: After some necessary criteria for enumeration of objects of a certain kind have been established, the enumeration is deemed complete by providing an example for each of the sets of criteria without investigating whether there are more than one object per set of criteria. This has occurred countless time in the enumeration of Archimedean solids; details of this "enduring error" are described in [G1].
Many different classes of non-convex polyhedra have been defined in the literature. It would seem that the appropriate definition depends on the topic considered, and that a universally accepted definition is not to be expected.
In carrying out this construction we need to remember that adjacent faces may not be coplanar. This excludes the "semicrosses" of Stein [S3] and other authors, although it admits the (1,3) cross. For more information see [S4].
It is worth mentioning that Fedorov did not require any central symmetry in the definition of zonohedra ([F1, page 256], [F2, page 688]). However, he switched without explanation to considering only zonohedra with centrally symmetric faces. As pointed out by Taylor [T1], this has become the accepted definition.
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