-21- Naming Archimedean and Catalan Polyhedra and Tilings The book in which Archimedes enumerated the polyhedra that have regular faces and equivalent vertices is unfortunately lost; however, its contents were reconstructed by Kepler, from whom the tradi- tional names descend. In this chapter we explain these “Keplerian” names for the Archimedean and Catalan solids and extend them to the analogous tessellations of two- and three-dimensional Euclidean space. We shall describe the polyhedra in dual pairs indicated by the ar- rows, and at the same time give our abbreviations for them, starting with the five Platonic (or regular) ones: TC↔ OD↔ I Tetrahedron Cube Octahedron Dodecahedron Icosahedron Etymologically, the Greek stem “hedr-” is cognate with the Latin “sede-” and the English “seat,” so that, for instance, “dodecahedron” really means “twelve-seater.” (opposite page) The Archimedean solids and their duals can be nicely arranged so that their edges are mutually tangent, at their intersections, to a common sphere. 283 © 2016 by Taylor & Francis Group, LLC 284 21. Naming Archimedean and Catalan Polyhedra and Tilings Truncation and “Kis”ing These are followed by their “truncated” and “kis-” versions. Here, truncation means cutting off the corners in such a way that each regular n-gonal face is replaced by a regular 2n-gonal one. The dual operation is to erect a pyramid on each face, thus replacing a regular m-gon by m isoceles triangles. These give five Archimedean and five Catalan solids: truncated truncated truncated truncated truncated Tetrahedron Cube Octahedron Dodecahedron Icosahedron tT tC tO tD tI kT kC kO kD kI kisTetrahedron kisCube kisOctahedron kisDodecahedron kisIcosahedron The names used by Kepler for the Catalan ones were rather longer, namely, triakis tetrakis triakis pentakis triakis Downloaded by [University of Bergen Library] at 04:55 26 October 2016 tetrahedron hexahedron octahedron dodecahedron icosahedron and were usually printed as single words. In these, “kis” meant “times,” so that Kepler’s “tetrakishexahedron” was literally a “(4 × 6)-seater.” We retranslate “kis” as “multiplied,” allowing us to ab- breviate this to “kiscube,” meaning “multiplied cube.”1 1Beware! This is not the same as the “mucube” of later chapters. © 2016 by Taylor & Francis Group, LLC Marriage and Children 285 Marriage and Children We marry a regular polyhedron P and its dual Q by placing them so that corresponding edges intersect at right angles. Then, their daughter polyhedron is the Archimedean polyhedron that is their intersection, whose dual, their son, is the Catalan one that is their Polyhedral Sex convex hull. This gives rise to only two new Archimedean and two new Catalan solids, which we can respectively truncate and “kis” to We call a polyhedron with get two more of each: V vertices and F faces male if V>F(its virility ex- cedes its feminity) and fe- male if F>V.Thosewith V = F are hermaphrodites. truncated truncated CubOctahedron IcosiDodecahedron CubOctahedron IcosiDodecahedron CO ID tCO tID R12 R30 kR12 kR30 Rhombic Rhombic kisRhombic kisRhombic dodecahedron triacontahedron dodecahedron triacontahedron Downloaded by [University of Bergen Library] at 04:55 26 October 2016 The daughter and son of two mutually dual tetrahedra are the reg- ular octahedron and cube, so that particular “marriage” leads to no new Archimedean and Catalan solids. However, we can incestuously marry the two Rhombic solids R12 and R30 to their respective duals CO and ID, producing two more Archimedean daughters and two more Catalan sons: © 2016 by Taylor & Francis Group, LLC 286 21. Naming Archimedean and Catalan Polyhedra and Tilings Trapezium and Trapezoid If you look in a large enough dictionary you will probably find the asser- tions that “trapezium” (Br.) = “trapezoid” (U.S.) while RhombiCubOctahedron RhombIcosiDodecahedron “trapezoid” (Br.) = “trapez- RCO RID ium” (U.S.)! Although only the first of T24 T60 these is still true, it is inter- Tetragonal Tetragonal esting to see how this curi- icosikaitetrahedron hexacontahedron ous situation came about. Proclus, a commentator on Euclid, used “trapezion” for a quadrilateral with (just) two parallel sides and “trapezoid” for a general four sided polygon with (typically) no parallel Strictly speaking, the solids we get by truncating CO and ID sides. All the European languages except English are not Archimedean since they have unequal edges, but they can have maintained this usage. be reformed to become so by distorting them suitably, and Kepler’s However, in English the names refer to the reformed versions. Their duals, the two Catalan words “trapezium” and “trapezoid” were acci- solids we have described as kisrhombic have traditionally been called dentally interchanged in the hexakis octahedron and icosahedron. We prefer to use “hexakis” Hutton’s Mathematical only for the replacement of a hexagon (rather than a triangle) by six Dictionary of 1800 and this switched usage has triangles. persisted in the U.S.A. but Since the traditional names here have often been misunderstood, was corrected in England we shall explain them. The prefix “rhombi-” in rhombicuboctahe- between 1875 and 1900. The end result has been dron, for instance, does not refer to a supposed operation of “rhombi- that British “trapezium” and truncation” but is an abbreviation for “rhombic dodecahedron,” U.S. “trapezoid” have sur- vived as synonyms, while one of the two polyhedra R12 and CO of which RCO is the British “trapezoid” and U.S. daughter. “trapezium” have been ob- In the names of the two Catalan solids, “icosi-kai-tetra” and Downloaded by [University of Bergen Library] at 04:55 26 October 2016 solete for more than a cen- tury, the word “quadrilat- “hexaconta” mean “twenty-plus-four” and “sixty,” respectively, and eral” having been reintro- “tetragonal” refers to the four-cornered-ness of the faces. Tradition- duced to replace them. ally, the adjective has been “trapezoidal,” but this is based on an obsolete meaning of the word “trapezoid” (see sidebar). The two “rhombi” solids have some square faces that come from their father “rhombic” polyhedra, and we can divide each such square into two triangles in such a way as to increase the number of faces at each vertex from four to five and then reform the resulting two solids so that all their faces are regular. © 2016 by Taylor & Francis Group, LLC Coxeter’s Semi-Snub Operation 287 Kepler’s term for the first of the resulting two solids was “cubus simus” in which the second Latin word means “rounded” or “flat- tened.” The “simian” apes are those with flattened noses. The traditional English names are snub cube (sC) and snub dodecahe- dron (sD), although they could equally be described as the snub octahedron and icosahedron. Their duals have respectively 24 and 60 pentagonal faces: snub Cube snub Dodecahedron sC sD P24 P60 Pentagonal Pentagonal icosikaitetrahedron hexacontahedron Coxeter’s Semi-Snub Operation Coxeter pointed out that since the snub cube is equally the snub octahedron, it would be more sensible to regard it as being derived from the cuboctahedron, by a new type of snubbing operation. We call this semi-snubbing, abbreviated to “ssnub,” because it is only Downloaded by [University of Bergen Library] at 04:55 26 October 2016 half of the operation that leads from the cube to its ordinary snub: C → CO → ssCO = sC. Regarded as the semi-snub cuboctahedron, the snub cube is ob- tained by distorting its daughter rhombicuboctahedron by twisting © 2016 by Taylor & Francis Group, LLC 288 21. Naming Archimedean and Catalan Polyhedra and Tilings the regularly embedded (regular) squares in one direction, say, clock- wise, which automatically twists the triangles counterclockwise and turns the non-regularly embedded (half-regular) squares into skew quadrilaterals that can be filled with two triangles. However, this is only a topological description; the canonical snub cube is obtained by a deformation that makes all the faces regular. (If we continue this twisting, until the skew quadrilaterals have no width at all, we obtain the cuboctahedron.) The semi-snub can be defined topologically for an even-valence polyhedron in a similar way, but its faces can all be made regular only for some very special polyhedra. A notable example is the semi-snub mucube of Chapter 23. Euclidean Plane Tessellations Not many of the corresponding tilings of the Euclidean plane have previously received names. However, quadrille (Q), which we can interpret as “quadrangular grille,” has been used for the standard tiling by squares. Based on this we propose deltille (Δ) for that by equilateral triangles and hextille (H) for that by regular hexagons. In these, the termination “tille” may be regarded as an amalgam of “tile” and “grille.” The following table gives the resulting “Ke- plerian” names for the “Archimedean” and “Catalan” tilings of the plane. We have encountered their figures already, in Table 19.1 on pages 262–263. Symbol Archimedean Face Code Catalan Symbol Tiling Tiling Downloaded by [University of Bergen Library] at 04:55 26 October 2016 Q quadrille 4444 quadrille Q Δ deltille 333333 hextille H H hextille 666 deltille Δ tQ tr. quadrille 488 kisquadrille kQ tH tr. hextille 31212 kisdeltille kΔ HΔ hexadeltille 3636 rhombille R∞ tHΔ tr. hexadeltille 4612 kisrhombille kR∞ RHΔ rhombihexadeltille 3464 tetrille T∞ 4 sQ snub quadrille 43343 4-fold pentille P∞ 4 isQ isosnub quadrille 44333 iso(4-)pentille i P∞ 6 sH snub hextille 63333 6-fold pentille P∞ © 2016 by Taylor & Francis Group, LLC Additional Data 289 Additional Data We pause for additional data on the Archimedean polyhedra and planar tilings. Vertex Figures To any vertex V of a polyhedron, we associate the vertex figure, which is customarily obtained by slicing the polyhedron by a plane suitably near to V and perpendicular to the line joining V to the center of the polyhedron. Usually, each edge of the vertex figure is marked with the number of sides of the corresponding face of the polyhedron.