Johannes Kepler's Tessellations and Polyhedra

Johannes Kepler (1571-1630) who although is perhaps better known for his work in astronomy, most notably with his three laws of planetary motion, was also very much concerned with tessellations and polyhedra, and indeed, his interest in the latter very much influenced his astronomical thinking, discussed in more detail below, albeit his ideas were based upon a false premise. (Incidentally, Kepler has recently been in the news with a contemporary astronomical event, namely with a transit of Venus across the Sun of 8 June 2004, as he was the first astronomer to undertake this most difficult task of calculating and then predicting when this occurrence would then take place, the first such determination being of the year 1631. Cruelly Kepler was denied seeing this, as he died the previous year. Such a feature is is an exceptionally rare astronomical phenomenon (at least in terms of human life spans, as it occurs in pairs, with intervals of 105 to 122 years, followed by another 8 years after). Furthermore, ones location on Earth is critical for viewing, as the longest duration is of seven and a half hours, and so consequently this can occur at nighttime (and thus be missed) for those who find themselves in an inappropriate geographical place.) Indeed, he made some notable pioneering contributions and discoveries to both of these fields, of which he discussed at length in one of his major works Harmonice Mundi Books I-V (1619), in effect a synthesis of his ideas concerning the universe at large consisting of Geometry, Architecture, Harmonies, Metaphysics, Psychology, Astrology and Astronomy.

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Historical Implication
Kepler's study is of importance in a historical sense, in that for the first time a systematic analysis is undertaken, this in effect replacing earlier studies of an essentially sporadic nature. Although earlier examples of tessellation studies can indeed be found all over the world (the earliest dating back to Archimedes, albeit his work is lost), these cannot be recognised to the same degree of thoroughness, and so Kepler's study to all intents and purposes can be regarded as the starting point per se. More specifically, he first defines and illustrates the minimum set of tessellations known as Semi-regular or Archimedean, discussed in detail below. However, despite the subject finally being put on some solid foundations, somewhat disappointingly Kepler's work was then neglected, resulting in a whole host of mathematicians spending needless time composing tessellations that had already been studied. Furthermore, to add insult to injury they obtained spurious results, in effect adding erroneous tessellations to the above set, from which subsequent mathematicians repeated their errors. Indeed, upon the publication of Harmonice Mundi the subject per se basically went into a slumber, only really awakening in the latter part of the 19th century.

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Archimedean Tessellations

As mentioned above, Kepler is generally credited with the first systematic explanation of the set of tessellations known as Semi-regular or Archimedean, which are tessellations composed of the regular polygons, of which there are eleven distinct examples. An illustrated plate from Harmonice Mundi containing these diagrams is shown below, albeit it is not of the greatest clarity due to these being intermingled with other tessellating diagrams.

Star Polygon Tessellations

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Archimedean Solids

Although Kepler defined and illustrated this set in Harmonice Mundi (defined as the complete set of convex uniform polyhedra), he cannot be credited as the discoverer of these solids per se (although he believed himself to be so), as Archimedes is again is the first such reference (albeit his work is lost), hence their title. Kepler nonetheless made a major contribution in that for the first time he defined and systematically analyses all possibilities, proving that there are thirteen such solids. Furthermore, he also gave them the individual names by which these are known today, all illustrated in the plate from Harmonice Mundi below.

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Great Stellated Dodecahedron and Small Stellated Dodecahedron

Kepler can, to all intents and purposes be regarded as the discoverer of these two non-convex solids, shown as Ss and Tt in the plate from Harmonice Mundi below, as he undertook a systematic analysis for the first time. Although earlier examples �of sorts' can indeed be found (notably of a mosaic attributed to Uccello and a drawing by Jamnitzer respectively), these can be regarded as essentially non-systematic in their undertaking, and so as Kepler gave the first true mathematical explanation, they are thus known in his honour as Keplerian solids. However, even Kepler was not infallible in his analysis, as he missed two further examples of this particular category (of non-convex polyhedra), these being discovered by Louis Poinsot in 1809, (albeit an earlier reference by Jamnitzer again shows a drawing of the Great Dodecahedron) and so these are thus accordingly now known as Kepler-Poinsot polyhedra.

Semi Regular Prisms and Semi Regular Antiprisms

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Rhombic Triacontahedron

Kepler is generally credited with the discovery per se of this thirty-sided rhombic-faced solid, first describing it in De Nive Sexangula (1611) and with a published illustration in Harmonice Mundi (1619), shown below. Again, this has precursors, as Jamnitzer (1568) shows something similar, albeit with no mathematical background to his example. A further, more technical discussion of the background to this solid is in Kepler's Geometrical Cosmology by Judith V. Field, pages 202-219.

Rhombic Dodecahedron

Kepler made a special study of this twelve-sided rhombic solid, as he believed that this was previously unknown, albeit this was not actually the case - indeed, this solid had been known since pre-recorded times, more of which below. As such, forever looking for geometrical relations with polyhedra as applied to astronomy, the rhombic dodecahedron was thus utilised, in simple terms, for the purpose of �explaining' the four (newly discovered) satellites of Jupiter. Again, his work on such matters has subsequently proved spurious. Again, more details of a more technical nature are to be found in Kepler's Geometrical Cosmology, pages 218-219. Such astronomical matters aside, the rhombic dodecahedron is very interesting per se as it possesses many interesting properties. Firstly, it will fill three-dimensional space, of which only the cube and truncated octahedron will do so likewise. Furthermore, it also occurs in the natural world, as it occurs in the form of garnet crystals. As it is also related to the cube, by appropriately dissecting this solid into six pyramids, these can thus be reassembled to form a rhombic dodecahedron, of which this is, essentially, being illustrated on the plate.

Inscribed Polyhedra

An interesting feature of polyhedra is that of their inscription of each other. Some examples of Kepler's, from Harmonice Mundi, of an icosahedron inside a dodecahedron and of a octahedron inside a cube, are shown.

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Stella Octangular

Kepler named this solid in recognition of the fact that by noting that this could be formed by the process of stellating the octahedron, hence a stellated octahedron or Stella Octangular. Another interpretation of this solid is that it is formed by the inter penetration of two tetrahedra. Again, although all solids are interesting, some can be said to be of a more aesthetic nature than others are, of which the Stella Octangular undoubtedly is. Indeed, this particular solid has been the subject of a whole book, The Stella Octangular Activity Book, written for a popular audience. The example shown below, taken from my own collection, of which to emphasise the interpenetrating aspect of the tetrahedrons are thus shown in distinct colours.