**2 Dec 2018.** *Fix dead links, other minor improvements.*

**4 April 2017.** *Added Hugel, 1876. Other minor clarifications.*

Stellation is the process of extending the sides of a polygon, the faces of a polyhedron, and generally the cells of a higher-dimensional polytope, until they meet to form a new figure. Facetting, or faceting, is the process of removing parts of a polygon, polyhedron or polytope without creating any new vertices, so exposing new faces, or facets. The two processes are reciprocal, or dual: if we take one polyhedron and stellate it, then we can take the dual polyhedron and facet it reciprocally; the resulting stellation and facetting will still be a dual pair of polyhedra.

Developed slowly over many centuries, stellation theory has wandered a little off course in recent times, notably at the hands of J.C.P. Miller and H.S.M. Coxeter. The theory of facetting may be relatively recent, being traceable back only as far as 1858, and little studied - I have come across few references. Since the processes are reciprocal, the two theories *should* also be presentable in reciprocal forms. Or, to put it another way, they are two different expressions of a common underlying mathematical truth. Unfortunately, most of the 20th Century studies of stellation ignored this aspect and are consequently inconsistent with it. One purpose of the (probably rather incomplete) history presented here is to help us pick out threads of continuity in the bigger picture, and not be led astray by seemingly neat, but not very soundly based, ideas.

Many individual references are given in the text. This history is also drawn from some more general sources:

- H.S.M. Coxeter;
*Regular Polytopes*, 3rd Edn, Dover (1973). - H.S.M. Coxeter et. al.;
*The fifty-nine icosahedra*, 3rd Edn, Tarquin (1999) - P. Cromwell;
*Polyhedra*, CUP (1999). - P. Messer; "Stellations of the rhombic triacontahedron and beyond",
*Symmetry: culture and science*,**11**(2000), pp 201-230. This is an updated version of the original paper from*Structural Topology*,**21**(1995), pp. 25-46. - K. H. Veltman;
*Geometric Games: A Brief History of the Not so Regular Solids*, web edition without illustrations, 1990. (Internet Archive) - My own researches, including private communications with many polyhedronists.

Along with the key highlights of each milestone I have included some commentary, even opinion, where I feel it is useful.

Any corrections, additions or other comments gratefully received.

**7th c. BC.** Pentagram depicted on the vase of Aristophanos (Aristophonus), found at Caere (modern Cerveteri) in northern Italy. It was used as a mystical symbol and as a secret recognition sign by the Pythagoreans.

**14th c. AD.** Thomas Bredwardine (Bredwardinus) records the first known systematic treatment of star polygons by extending the sides of another polygon.

**ca. 1430.** Small stellated dodecahedron depicted in a marble tarsia, or panel, on the floor of St. Mark's basilica, Venice. Attributed to Paulo Ucello.

**ca. 15th-16th c. AD.** Charles de Boulles investigates star polygons.

**1509.** In his book *De divina proportione* Luca Pacioli augments (adds pyramids to) the octahedron to obtain the figure that Kepler would later call the stella octangula. Leonardo da Vinci also includes a stellation of the dodecahdron among his illustrations for Pacioli's book.

**1568.** Wenzel Jamnitzer publishes his
Perspectiva Corporum Regularium, a series of woodcuts exploring the systematic cutting-away of the Platonic polyhedra. Among them are examples clearly showing that he understands the processes of stellation and facetting, though not their reciprocity. Some apparently regular star polyhedra are depicted, but it is evident from his long subtitle to the work that he does not realise they are regular.

**1619.** Kepler defines stellation first for polygons and then by extension for polyhedra, as the process of extending the edges (or faces) of a figure until they meet to obtain a new figure. He stellates the regular dodecahedron to obtain the small and great stellated dodecahedra. As we have seen, these figures have been described by earlier investigators, but Kepler is the first to realise that they are regular. He also recognises the properties of the stella octangula, as a regular compound which is both a stellation of the regular octahedron and a facetting of the cube (though he probably has no idea of the duality of these arrangements).

**1809.** Poinsot considers star vertex figures as well as star faces. He discovers the remaining regular stars: the great dodecahedron and great icosahedron, and also rediscovers Kepler's figures. His preliminary account of star polygons is somewhat confused, describing them as being would froma single piece of wire equally divided along its length, but then describing the double-wound hexagon as a compound of two distinct triangles (i.e. requiring two pieces of wire).

L. Poinsot, Mémoire sur les polygones et les polyèdres, *Journal del' École Polytechnique*, **4** (1810), pp. 16-49.

**1812.** Cauchy applies the principle of symmetry explicitly for the first time, and stellates the regular solids to prove that Poinsot's list is complete. Cauchy is also aware of the regular compounds, of five and ten tetrahedra and of five octahedra, as stellations of the regular icosahedron. He does not use words like "stellated" but talks about "polyhedra of a higher kind."

A.L. Cauchy, Recherches sur les polyèdres, *Journal de l'École Polytechnique*, **16** (1813), pp. 68-86.

English translation available **here** (PDF).

**1858.** Bertrand derives the regular star polyhedra more elegantly, by facetting the icosahedron and dodecahedron. He uses the term "étoilé" (starry or stellated) alongside Cauchy's terminology, in a manner which suggests that the term was already familiar to his audience. In a later note in the same journal, he mistakenly claims that Kepler did not appreciate that the faces of his star polyhedra were star pentagons.

J. Bertrand, Note sur la théorie des polyèdres réguliers, *Comptes rendus des séances de l'Académie des Sciences*, **46** (1858), pp. 79-82, 117.

English translation available **here** (PDF).

**1859.** Cayley modifies Euler's formula to try and account for the unusual counts of faces, edges and vertices that two of the regular stars have compared to a convex polyhedron (their relation to toroids was not yet understood). He gives the four regular star polyhedra their accepted English names, translating "étoilé" as "stellated."

**1876.** Hugel investigates many regular and semi-regular stars. For the first time he gives the correct geometrical form for the regular star hexagon {^{6}/_{2}} as a double-wound triangle in appearance, thus correcting a blunder made by Poinsot.

Theodore Hugel; *Die Regulären und Halbregulären Polyeder*, Gottschick-Witter, Neustadt a.d.H.

**1900.** Bruckner sets out the theory of polyhedral reciprocation in detail. He describes the known stellations, together with six more stellations of the icosahedron.

**1901.** Schläfli, having rediscovered Kepler's stars, denies that Poinsot's new ones are polyhedra because they do not conform to Euler's original formula.

**1924.** Wheeler applies group theory to the icosahedron and considers regions, or elements, within its face planes to discover nine more "forms" of the icosahedron, including two which consist of several discrete polyhedra symmetrically disposed in space. He introduces the idea of describing only the visible regions of a face - it is not clear why. He explicitly contrasts his approach to that of Kepler, and avoids the term "stellation" in describing it. Note rather his use in the paper's title of "higher polyhedra" after the manner of Cauchy.

A. H. Wheeler, Certain forms of the icosahedron and a method for deriving and designating higher polyhedra, *Proc. Internat. Math. Congress, Toronto, 1924*, **Vol. 1**, pp. 701-708.

**Ca. late 1920's?** J.C.P. Miller proposes five rules (presumably to Coxeter), for determining which stellations of the icosahedron should be considered properly significant and distinct. They are based on Wheeler's notion of considering only the visible regions, and make little reference to the traditional Keplerian understanding of stellation. Coxeter applies a refined variant of Wheeler's method to Miller's rules, to enumerate 59 "stellations." He unquestioningly accepts Wheeler's discrete forms as proper stellations.

Also around this time, Flather independently begins exploring the stellated icosahedra and making models of them.

**1932.** Coxeter and Flather meet. While some of Flather's existing models are discovered to be "non-Miller," he makes new models to complete his set of the 59.

**1938.** Du Val corroborates Coxeter's analysis of Miller's rules. Together with illustrations of the results drawn by J.F. Petrie and an arguably gratuitous acknowledgement to Flather, the list is published as *The fifty-nine icosahedra*. In the introduction, Coxeter reiterates the Keplerian definition of stellation, as extending the faces of a polyhedron until they meet again, before baldly stating Miller's rules - apparently unaware that they do not follow from the preceding discussion in any sensible way. Certainly, no rationale is offered for this abrupt change in direction. Coxeter's analysis goes on to (rightly) become an iconic symbol of the rebirth of modern geometry, burying for decades the question as to what it is actually analysing.

H.S.M. Coxeter, P. Du Val, H.T. Flather & J.F. Petrie, *The fifty-nine icosahedra* (1938).

**1947.** Coxeter dryly observes that stellation and faceting (sic) are reciprocal processes.

H.S.M. Coxeter, *Regular Polytopes* (1947).

**1957.** Luke examines the stellations of the rhombic dodecahedron. As with Coxeter et al, Luke considers only exposed face regions. A postscript signed H.M.C. (Martyn Cundy) suggests that Dorman Luke's stellations conform to the same rules as *The fifty-nine icosahedra*. A moment's examination shows that they do not: Luke has relaxed Miller's Rule (v), allowing concentric cell sets which meet only at points. Even so he still misses the figure formed by removing the second stellation from the third, as he does for other figures, but then substituting the first stellation in its place to obtain two concentric cell sets which join along edges such that any face of the completed solid runs indivisbly across both cell sets.

D. Luke, Stellations of the rhombic dodecahedron, *The Mathematical Gazette* **41** (1957), pp. 189-194.

**1958.** Ede enumerates the "main-line" stellations of the rhombic triacontahedron, which are those comprising each successive shell (fully enclosing layer of one or more types) of cells.

**Mid-1960's.** Bruce Chilton searches the facetting diagram of the dodecahedron and finds over 100 facettings, reputedly filling at least one notebook (unpublished). Unable to convince Coxeter that there can be more than 59. Chilton constructs a model of the deepest finite facetting, which he and George Olshevsky would later christen "Huitzilopochtli," after an Aztec deity.

**ca. 1970.** Conway distinguishes "stellation" as extending edges, "greatening" as extending faces, and (for 4D polychora) "aggrandizement" as extending cells. Based on this, he modifies the names of some well-known polyhedra. For example most stellations of the icosahedron instead become greatenings of it. The first published reference to Conway's scheme seems to be in Coxeter's
Regular Complex Polytopes (1974). In the world of polyhedra, the old meaning of "stellation" covering both edge stellation and face stellation has remained popular, and Conway's scheme has mainly been confined to the classification and naming of star polychora.

**1974.** Bridge enumerates what might be described as the more orthodox facettings of the regular dodecahedron and reciprocates them to discover various isomorphs of some stellated icosahedra, as well as stellated icosahedron **Df _{2}** that is a uniform dual but nevertheless not present among the 59 (being forbidden by Miller's rules). Coxeter is uninterested. Meanwhile Bridge himself rejects two facettings on the (spurious and unrelated) grounds that the reciprocal stellations would extend to infinity. Working more or less in isolation, he introduces the alternative spelling of the verb "facetting," along with the unusual noun "facetion."

N. J. Bridge, Facetting the dodecahedron,

**1975.** Pawley enumerates the "non-reentrant" stellations of the rhombic triacontahedron, which are those where all visible parts of a face are seen from the same side: there are no "undercut" regions. These are nowadays called "fully supported" stellations. The main-line stellations (1958) are a subset of them.

**1983.** Wenninger observes that 'hemi' polyhedra have duals which are infinite stellations (of the kind rejected by Bridge) and develops a method, based on reciprocation, for defining forms suitable for making models. The presentation of the theory is flawed, but it works in practice.

M.J. Wenninger, *Dual Models*, CUP (1983).

**1988.** Hudson and Kingston, apparently unaware of much recent work, provide a readable account based broadly on the approach of Coxeter et. al. Their stated rules for stellation are less strict and allow more stellations than Miller's, although it is not clear whether they understand this. Their rules still do not admit internal structure. In passing they give a definition of faceting which does not accord with the term as defined by Coxeter and employed by Bridge.

J.L. Hudson and J.G. Kingston, Stellating polyhedra, *Mathematical Intelligencer* **10** (1988), pp. 50-61.

**1989.** Messer defines "primary" stellations. The edges lie in reflection planes of the core (so asymmetric polyhedra do not have primary stellations). These are also a subset of the fully supprted stellations (1975).

**1998-2002.** Inchbald (the present author) uses a computer to discover several new stellations of the icosahedron. He writes a critique of the 59, in which he notes some of the shortcomings of Miller's rules and describes two of his new stellations.

Just as his paper is going to press, he stumbles across Coxeter's 1947 observation on reciprocity and realises that this is the key to a consistent theory.

G. Inchbald, In search of the lost icosahedra, *The Mathematical Gazette* **86** (July 2002) pp. 208-215.

**ca. 2000.** Robert Webb creates the *Great Stella* software, and adds tools for reciprocating polyhedra and for creating stellations from the cell diagram.

**ca. 2001.** Messer defines "monoacral" stellations (lit. single-peaked) having congruent vertices within a single symmetry orbit. First published use by Webb in *Great Stella*.

**2003.** Grünbaum describes four new noble polyhedra which exist in dual pairs and may be inscribed in (i.e. are facettings of) the dodecahedron. By duality they must also be stellations of the icosahedron. They have some rather curious features, and not everyone would accept them as proper polyhedra.

B. Grünbaum, Are your polyhedra the same as my polyhedra?, *Discrete and comput. geom: the Goodman-Pollack festschrift*, Ed. Aronov et. al., Springer (2003), pp. 461-488.

**2002-ca. 2006.** Inchbald begins the task of developing a modern theoretical framework, and continues to discover new facettings of the dodecahedron (including the hemi ones rejected by Bridge) and stellations of the icosahedron (including the infinite duals of the hemi facettings). Olshevsky contributes a few new icosahedral stellations of his own.

Olshevsky and Richard Klitzing embark on a project to methodically facet many other polyhedra and extend the process into higher dimensions.

Webb progressively introduces facetting tools into *Great Stella*, accelerating the work of these researchers and others. He adopts Wenninger's style of infinite duals for hemi polyedra.

**2006.** Inchbald rediscovers two models of "non-Miller" stellations of the icosahedron, preserved in the Department of Pure Mathematics & Mathematical Statistics, at the University of Cambridge, alongside Flather's famous models of the fifty-nine "Miller" icosahedra. He tentatively attributes these models to Flather. The implication is that he made these models before seeing Coxeter's results. Although they are forbidden by Miller's rules, they are allowed by Hudon & Kingston's less restrictive rules.

G. Inchbald, *Some lost stellations of the icosahedron*

**2009.** Hudson describes some new stellations according to his rules, which he now calls "external" stellations. Now recognising that star polyhedra have internal structure, he defines "internal" stellation as the process of removing outer cells to reveal parts of this structure. Like other cell-based approaches, it does not dualise cleanly to an equivalent for facetting.

J. Hudson, Further Stellations of the Uniform Polyhedra, *The Mathematical Intelligencer*, **31** No. 4 (2009), pp. 18-26.

Updated 2 Dec 2018