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**Kim H. Veltman **

**Geometric Games A Brief History of the Not so Regular Solids **

**I. COSMOLOGY, THEOLOGY AND MATHEMATICS**

1. Ancient Roots 2. Mediaeval Developments 3. Regiomontanus 4. Piero della Francesca 5. Luca Pacioli 6. Leonardo da Vinci 7. Nürnberg Goldsmiths 8. Italian Popularizers 9. French Mathematics 10. Jesuits

1**. Ancient Roots**

In 1936 French archaeologists found a series of mathematical tablets at Susa, some 200 miles east of Babylon. These Babylonian tablets, dating c. 1800-1600 B.C. contain among the earliest known computations for regular polygons, triangle, square, pentagon, hexagon and heptagon.1 The earliest known semi-regular solids also came from Babylon and appear to have been used in connection with weights and measures2 (fig. 1.2-4). Their use remained practical. Neither the Babylonians nor the Egyptians developed mathematical or scientific theories about such solids.

In nature approximations of the cube and octahedron are found in pyrite crystals (FeS2) which occur in iron-ore deposits. In Switzerland and Northern Italy approximations of the icosahedron and dodecahedron are also found in such crystals in the valleys of the alps, particularly Traversella and Brosso, leading to Piedmont.3 Aside from the island of Elba this is the only region in the world where such crystals occur (cf. fig 1.1). It is significant, therefore, that interest in the twelve sided dodecahedron and twenty sided icosahedron developed in this part of Northern Italy during the iron age (c. 900-500 B.C.). At least 28 dodecahedra from this period have been recorded in museums.4 They were used as weights. Their sides were covered with number symbolism which may have come from Babylon, possibly in Egypt and/or the Phoenicians.5 The Etruscans and Celts also endowed these solids with religious symbolism which Pythagoras of Samos adopted when he came to Italy sometime between 540 and 520 B.C.6

All too little is known of his precise contribution. One tradition claims that Pythagoras studied only the cube, pyramid and dodecahedron and that Theaetetus subsequently studied the octahedon and dodecahedon. Another tradition claims that Pythagoras gave each of the five regular solids a symbolic meaning, associating the six sided cube or hexahedron with earth, the four sided pyramid or tetrahedron with fire, the eight sided octahedron with air, the twenty-sided icosahedron with water and the twelve sided dodecahedron with the universe, or the atom of the all embracing ether.7 There is evidence that he studied the construction of these regular solids in terms of triangles.8

The case of the dodecahedron is of particular interest. Each of its twelve sides is a regular pentagon. If lines are drawn to its consecutive corners a star pentagon is produced. The Pythagoreans called this star pentagon Health, and used it as a symbol of recognition for members of their school. Starting from this symbol the Pythagoreans could construct a pentagon using an isosceles triangle having each of its base angles double the verticle angle. The construction of this triangle involved the problem of dividing a line so that the rectangle contained by the whole and one of its parts is equal to the square on the other part. This is also known as the problem of the extreme and mean ratio or the problem of the golden section.9 In simple terms, the idea of squaring the sides of triangles, familiar from the Pythagorean theorem which we all learned at school, appears to have been linked with questions of using triangles to make pentagons in constructing dodecahedrons.

Pythagoras, who was referred to simply as HIM by members of his school did not write down his ideas. Probably the first to do so was his follower, Hippasus, who wrote a mathematical treatment of the dodecahedron involving its inscription in a sphere10 but, the story goes, then perished by shipwreck, for not acknowledging that everything belonged to HIM. Plato (428-348 B.C.) was more successful. He built on the Pythagorean associations in his *Timaeus*, claiming that fire (pyramid), air (octahedron) and water (icosahedron) were composed of scalene triangles and could be transformed into one another. Earth (cube), he claimed, was composed of isosceles triangles. The fifth construction (dodecahedron) ";which the god used for arranging the constellations on the whole heaven";11, Plato did not explain. Indeed he seems to have added nothing essentially new to the discussion. However, thanks to the enormous popularity, which the *Timaeus* subsequently enjoyed, the five regular solids are often referred to as the five Platonic solids.

Meanwhile the mathematician Theaethetus (fl. 380 B.C.) had written the first systematic treatise on all the regular solids.12 Some sixty years later, Aristaeus, the elder (fl. 320 B.C.) wrote a *Comparison of the five regular solids* in which he proved that ";the same circle circumscribes both the pentagon of the dodecahedron and triangle of the icosahedron when both are inscribed in the same sphere.";13 This work was one of the starting points for Euclid (fl. 300 B.C.) who dealt with these problems much more systematically in his *Elements*. In Book II Euclid dealt with the division of a straight line in extreme and mean ratio, on which the construction of the regular pentagon depends. In Book IV he gave a theoretical construction of the regular pentagon, probably on the basis of Pythagorean sources. Euclid, however, set the problem in a much larger framework. Book IV dealt systematically with polygonal figures in a plane, i.e. two-dimensionally. Euclid showed how to 1) inscribe a triangle, square, pentagon and hexagon in a circle, 2) circumscribe these around a circle and, in turn, 3) to circumscribe a circle around these forms. In the final proposition of this book he showed how to inscribe a fifteen-sided figure in a circle. In Book XII he described the construction of a 72 sided figure. In Book XIII, having pursued problems of dividing lines into extreme and mean ratios, Euclid explained how to inscribe a pyramid (tetrahedron), octahedron, cube (hexahedron), icosahedron and dodecahedron within a sphere, ending his book by comparing the relative sizes of the sides of these solids with one another. Theoretically, Euclid described the construction of three-dimensional versions of the five regular solids. However the diagrams which have come down to us are strikingly lacking in three dimensional qualities (fig. 2.1-5). Those of Pappus14 (fl. 340 A.D.) were more convincing (fig. 3.1-5). Yet it was not until the fifteenth century that Piero della Francesca (fig. 6.4-5) and then Leonardo da Vinci (fig.8-11) produced fully three dimensional versions of the five regular solids.

The fact that Euclid ended his book with the construction of the so-called Platonic figures led later commentators such as Proclus (c. 450 A.D.) to claim that the whole argument of the Elements was concerned with the cosmic figures.15 This was an exaggeration since the Elements provided a foundation for the study of geometry in general. Yet it points to important links between cosmology and mathematics. After Euclid, Apollonius of Perga (c. 262-180 B.C.) wrote a Comparison of the dodecahedron to the icosahedron showing their ratio to one another when inscribed within a single sphere.16 Not satisfied with this account, Basilides of Tyre and the father of Hypsicles made emendations.17 Hypsicles, in turn, wrote his own treatise on the subject in which he compared the sides, surfaces and contents of the cube, dodecahedron and icosahedron.18 This became the so-called Book XIV of the *Elements* which was often ascribed to Euclid himself.

Not everyone in Antiquity was happy with this abstract mathematical system of cosmology. Plato's best student, for instance, rejected it outright. Aristotle was committed to showing that nature could not have a vacuum. To accept the existence of regular polygonal shapes meant that there would be spaces between them. Hence he attacked Plato's cosmology on practical grounds:

In general the attempt to give a shape to each of the simple bodies (i.e. elements) is unsound

for the reason, first, that they will not succeed in filling the whole. It is agreed that there are

only three plane figures which can fill a space, the triangle, the square and the hexagon, and

only two solids, the pyramid and the cube. But the theory needs more than these because the elements which it recognizes are more in number.... From what has been said it is clear

that the difference of the elements does not depend upon their shape.19

Such good common sense did not suffice, however, to do away with the Pythagorean ideas which Plato had adopted. The matter continued to be debated. Even so there is evidence that the practical tradition gained importance. For example, Archimedes (287-212 B.C.), the one who yelled Eureka when he discovered the principle of specific gravity in his bathtub, studied truncated versions of the regular solids and thus discovered the thirteen semi-regular shapes which are today remembered as Archimedeian solids One of earliest systematic records of these is from a manuscript now in Trieste (fig 4-6, cf. Appendix 1).20 Practical concerns with the measurement of regular shapes also developed. Hero of Alexandria (fl. 150 A.D.), for instance, measured the relative sizes of the icosahedron and dodecahedron in his *Metrics*.21 So too did Pappus of Alexandria (fl. 340 a.D.).22

**2. Mediaeval Developments**

Practical uses of these regular shapes probably go back to earliest times. There is evidence that they were sometimes used in games of dice.23 In Antiquity glass and bronze jewels were made in the form of a cube-octahedron (fig. 1.5) and other semi-regular solids.24 This practice continued in the Middle Ages as is attested by their frequent occurrence in graves in Hungary and Northern Europe particularly in the fifth century A.D.25

Euclid was not forgotten. For instance, Isodorus of Miletus (fl. 532), the architect of Hagia Sophia in Constantinople (i.e. Istanbul) and one of his students, added a so-called Book XV of the *Elements*, which dealt with further problems relating to the regular solids.36 In the Arabic tradition, Ishaq b. Hunain (d. 901 A.D.), in his translation of the *Elements*, improved by Thabit b. Qurra (d. 910 A.D.), included Books XIV (by Hypsicles) and XV (by Isidorus) as if they had been written by Euclid. In the preface Ishaq explained that he had given his own method of inscribing the spheres in the five regular solids and developed the solution of inscribing any one of the solids in any other, noting those cases where this could not be done.27 In the twelfth century when Gerard of Cremona translated the *Elements* back into Latin he too assumed that Euclid had written all fifteen books.28 So too did Campanus of Novara in the thirteenth century when he made his translation of the *Elements*.29 To late Mediaeval scholars it thus appeared as if Euclid had devoted three books of his *Elements* to regular solids, and since the last of these dealt with cosmology and metaphysics, it seemed as if Euclid was concerned with much more than arithmetical proportions and geometrical features. His fascination with regular solids offered a key to Nature's regularity, the structure of the elements and the cosmos itself.

Two factors greatly increased the significance of this interpretation. The original Greek term for geometry had literally meant ";measurement of the earth."; Notwithstanding emphasis on practical applications by the Romans, ancient geometry remained largely an intellectual exercise involving abstract figures from a world of ideas. The Christian tradition changed this. It began from a premise of what AuerbacH40 has called ";creatural realism,"; that the natural world is real, because God created it. Hence when Boethius31 (480-524) revived the notion of geometry as a measurement of the earth, it gradually acquired an entirely new meaning. For the earth was no longer a poor imitation of a world of ideas. It was a testament to God's creation and geometry was no longer a purely intellectual exercise. It involved practical comprehension of the physical universe. The Arabic tradition, particularly the strands that came to the West helped to reinforce this approach.32

Meanwhile the metaphysical interpretation of Euclid's geometry and Plato's cosmology had also been integrated within the Christian tradition, such that God himself was seen as the Divine Geometer33 and knowledge of geometry was now a means of understanding God. So practical and intellectual knowledge became interdependent and both were linked with religion. Knowing more helped one to believe more. This approach was already firmly established by the eleventh century. From the twelfth century onwards as translation of ancient texts both from the original Greek and via Arabic versions expanded into a systematic venture, all this became more significant. Cumulative dimensions of knowledge became important. For instance, Aristotle's objections to Plato's cosmology had not been forgotten. At Cordoba, the Arabic scholar Averroes (1126-1198) wrote a long commentary on the relevant passages in Aristotle's On the Heavens. Two and a half centuries later, Aristotle's passage and Averroes' commentary in turn provoked Regiomontanus to write a treatise, which set the stage for our story.

**3. Regiomontanus**

Regiomontanus, whose real name was Johannes Müller, was a rather amazing figure.34 He studied with Peurbach, professor of astronomy at Vienna, who had the greatest collection of scientific instruments at the time. When Cardinal Bessarion, who commuted between Rome and Venice was trying to arrange a first edition of Ptolemy's Geography he was unable to find anyone in Italy. So he went to Peurbach. When Peurbach died Regiomontanus took over. He lectured at Padua but soon moved to Nürnberg to start the world's first publishing press for scientific books. He was one of the greatest astronomers of his day, was a pioneer in trigonometry and much involved in the reform of the Gregorian calender which, it is rumoured, led him to be poisoned in Rome at the age of 40. Regiomontanus is of special interest to us because he wrote a treatise *On the Five Equilateral Bodies, Commonly Called Regular, Namely, Which of Them will Fill a Natural Place and Which of Them do not*. *Against the Commentator on Aristotle, Averroes*.35 Regiomontanus was concerned with much more than the construction of the five regular solids. He wished to demonstrate their systematic transformation from one into another. For instance, he described how to change a cube into a tetrahedron, an octahedron and a dodecahedron. He then measured these bodies.36 Next he described how one could increase the size of a cube using square roots and cube roots. He ended the chapter by demonstrating that twelve cubes did not circumscribe a thirteenth and that twelve tetrahedrons (pyramids) did not fill up a space entirely. In the next two chapters he discussed the relation of diameter to circumference in a circle and the use of these properties in transformations into circles of different sizes. A further chapter was addressed to the volume and area of circles. This was based specifically on Archimedes' work *On the Sphere and Cylinde*r. Chapters on the measurement of irregular bodies and binomials followed. Regiomontanus went on to explain how a systematic development of the regular solids could lead to an ";unlimited"; number of regular irregular (i.e. semi-regular) solids.37 In his final chapter he proposed how this could be done methodically.

The original manuscript is lost so we know nothing specific about its illustrations. What we know of the text is largely because Regiomontanus summarized its arguments in another work.38 From an important history of Nürnberg mathematicians in the early eighteenth century we also know that its themes were still familiar in Nürnberg at that time.39 Since Regiomontanus lectured, worked and died in Italy it is very possible that he took the manuscript with him and that mathematicians there became aware of his work either directly or indirectly. This would account for parallels between his work and that of Piero della Francesca. In any case Regiomontanus' development of Euclid's geometry in connection with regular solids and cosmology helped to set the stage for Piero della Francesca (fig. 7.1 ), Pacioli, Jamnitzer, and others.

4**. Piero Della Francesca**

In Italy one of the key figures in these developments was Piero della Francesca. Born in San Sepolcro sometime around 1410, Piero studied painting with Domenico Veneziano in Florence and became one of the great painters of the Renaissance. Today he is most famous for his fresco cycle showing the *Legend of the True Cross* (Arezzo), his *Brera Altar* (Milan), Baptism (London, National Gallery), *Flagellation* (Urbino) and *Resurrection* (San Sepolcro. He was also involved in inlaid wood (intarsia) with architectural scenes and his name is frequently associated with those three famous views of idealized cities, the Baltimore, Berlin and Urbino panels.

Characteristic of his work was a mathematical rigour and clarity. This reflected his profound interest in mathematics, on which he wrote three treatises dedicated to the Duke of Urbino. The earliest of these was a *Treatise on the Abacus*. This stood firmly in a tradition that went back to the 1220's when Fibonacci--as in Fibonacci numbers--went to North Africa, studied algebra and practical geometry with the Arabs and incorporated their rules in his* Book of the Abacus*. This had served as a starting point for an abacus school, which eventually became the Renaissance version of a business school. Leonardo da Vinci, for instance, learned his basic mathematics at one of these schools. Piero's *Treatise of the Abacus* contained practical problems such as interest rates and measurement of the volume of wine barrels. It also dealt with measurement of the regular solids, a theme that Piero pursued in his *Booklet on the Five Regular Bodies*. Part one dealt with two dimensional figures: triangles, squares, pentagons, hexagons, octagons and circles; part two, with measurement of the five regular bodies contained in a sphere (fig. 6.3-5). Part three dealt with measurement of one regular body placed within another. As Daly Davis40 has shown, parts one and two were largely based on Piero's earlier *Treatise on the abacus*. Part three, by contrast, was based on Book XV of the *Elements*, but rearranged so that the solids in which they were contained, beginning with the tetrahedron and ending with the icosahedron were in order of complexity. In part four of his Booklet, Piero cited the work of Archimedes (287-212 B.C.) and also described five of the thirteen semi-regular bodies which history has remembered as the Archimedeian solids.

More was involved than a simple revival of ancient mathematics. Euclid, for instance, had dealt with the five regular solids as a construction problem using square roots to determine the relative lengths of their respective sides, but appears to have had no interest in either their physical reconstruction or their representation in three dimensional terms. Piero della Francesca, by contrast, was concerned with representing both the Euclidean and Archimedeian solids in spatial terms. Piero was of course working in a tradition. A generation earlier Leon Battista Alberti had written *On Mathematical Games41* in which he had dealt with problems of geometrical transformation such as quadrature of the circle and perspectival transformations of shape. Alberti had also written On Painting, the first extant treatise on perspective. Piero, in turn, wrote his third treatise, *On the Perspective of Painting* (c. 1478-1482). Ironically, this milestone in the conquest of visual space was finished after he had gone blind. In this work Piero demonstrated the perspectival foreshortening of two dimensional polygons, namely, a triangle, square, pentagon, hexagon, octagon and a sixteen sided figure, as well as a three dimensional cube.42 Piero also described the geometrical transformation of a three dimensional sphere into an egg so that one could draw an egg which appeared as a sphere when viewed from a given point.43 Here he was codifying a principle of trick perspective or anamorphosis which he had used in his *Brera Altar* (fig. 7.2) .44

Piero's egg offers a beautiful example of Renaissance symbolism. Ostrich eggs filled with perfumed salts were used as deodorants over doorways where persons took off their shoes in the mosques of Constantinople. If you go to the Blue Mosque in Istanbul you can still see them today. Piero, working about two decades after the fall of Constantinople presumably knew of this practice. Putting one in his painting added an exotic touch, possibly even an ecumenical note. Meanwhile, as scholars have noted, there was a mystical tradition which linked the ostrich egg with the womb of the Virgin and with the birth of Christ.45 For Piero, however, it also had a third meaning. As an egg which when seen correctly from below (fig. 7.3) transformed itself into a sphere, it was a symbol of the universe demonstrating the power of perspective not only to represent objects three dimensionally but also to transform them systematically. As such it permitted an observer to re-enact optically a version of Cusa's game of the globe in which God uses geometrical transformation to play with the universe at once spiritually and physically.

**5. Luca Pacioli**

Piero's works were not published in his lifetime. Manuscript copies of his three treatises entered the library of the Duke of Urbino who, apparently made them available to a Franciscan friar, Luca Pacioli, who had been born in the same small town of San Sepolcro as Piero. Pacioli became extremely interested in the regular and semi-regular bodies. By 1489 he had commissioned various models of these bodies. In 1494, as Daly Davis46 has shown, Pacioli used *Piero's Treatise of the Abacus* as the basis for a section of his *Summa[ry] of Arithmetic, Geometry, Proportion and Proportionality*. Two years later when he arrived as a guest of Duke Sforza at the court of Milan, Pacioli began work on his most famous text *On Divine Proportion* which he finished in 1498 and published in 1509. On the surface it is not original. Pacioli cites Plato's cosmology and Euclid's geometry as a starting point for his discussion of the regular and semi-regular solids. Scholars have discovered that Pacioli also borrowed heavily from Piero's *Booklet on the Five Regular Solids*.47 Hence it has become fashionable to dismiss him as a plagiarist. But this does not do him justice.

As mentioned earlier, Plato in his *Timaeus* described the composition of all five regular solids, but believed that only three could be changed into one another.48 Pacioli believes that all five are interchangeable. So too does Leonardo da Vinci49 Plato's *Timaeus* as it has come down to us, had no illustrations. Euclid's text, as we have already noted, had diagrams which were spatially unconvincing (fig. 2.1-5). Pacioli, by contrast, commissioned a magnificent set of illustrations by Leonardo da Vinci (fig. 8-11, pl.1,3,5). The opening lines of his preface confirm that this was not merely a decorative flourish. Pacioli cites Aristotle to claim that sight is the beginning of wisdom50 and to strengthen his case he uses another of Aristotle's phrases which the mediaeval philosophers had used quite differently: that there is nothing in the intellect which was not previously in the sense.51

It is quite true of course that Aristotle tended to praise sight above the other senses. But neither Aristotle nor any of the ancient philosophers made clear distinctions between sight a) in a mental sense of something in the mind's eye and b) is a physical sense of things seen by the eyes. In Pacioli's interpretation the focus is clearly on the second of these, i.e. on visual information and then in a rather special sense. For as Pacioli presents it, optics and perspective, that is, vision and representation are fully interdependent. Pacioli suggests, moreover, that there are connections between visual demonstration and mathematical certitude, which leads him, in the next chapter, to propose a new version of the seven liberal arts. The mediaeval tradition had favoured three arts (grammar, rhetoric and dialectic) and four sciences (the so-called *quadrivium *of arithmetic, geometry, astronomy and music). Disciplines such as optics, architecture and geography were seen as dependent upon these or classed simply as mechanical sciences. Pacioli pleads that perspective in the sense of both optics and linear perspective should become the fourth science, and that in terms of importance it deserves to be put into third place, directly after arithmetic and geometry.52

Hence while citing Plato, Aristotle, Euclid and other standard authorities, Pacioli emphasizes perspectival demonstration in a way they could not have imagined. His reasons for writing are also very different. First the unity of proportion, and its indivisible nature is a symbol of God. Second the three terms of proportion symbolize the Trinity. Third the irrationals of proportion reflect the mysteries beyond the rational involved in God. Fourth, God's immutability is reflected in the unchanging laws of proportion which apply to quantity be it discrete or continuous, large or small. Like the ancients, Pacioli sees proportion as the basis for construction of the five regular solids and thus as a key to the nature of trhe four elements on earth and the ether of the heavens above53. But whereas Plato stopped at five bodies, Pacioli consciously refers to ";infinite other bodies";54 dependent on these.

The actual number that Leonardo illustrates (fig. 8-11) is somewhat less: 40 to be precise plus an appendix with twenty-one variations of columns and pyramids. Nonetheless, the systematic approach that underlies their presentation is striking. Thirty-four of the figures relate to the five regular bodies, arranged in the order pyramid, cube, octahedron, icosahedron and dodecahedron. In each case both a solid and an open version is given, first of the regular body, followed by its truncated form and then its stellated form. In the case of the cube and dodecahedron the stellated versions are truncated in turn. Among the 34 shapes thus produced are five of the semi-regular Archimedeian solids. The next shape is a twenty-six sided figure, technically called a rhombicuboctahedron, which is again one of the Archimedeian solids. Its appearance both here and in Pacioli's portrait (fig. 42.1. ) is probably no coincidence. Since ancient times this shape had mystical associations. In the museum at Aquileia, for instance, there is an antique rhombicuboctahedron so constructed that light in the shape of a crescent moon appears at its surface (fig. 1,5). Finally there is, in the *Divine Proportion,* a seventy-two sided figure which Euclid had described in Book XII, proposition 10 of his Elements and which symbolized perfection during the Renaissance.55

If Pacioli's ideas were borrowed no one before him had ever presented them so clearly, systematically or eloquently. What had previously been an obscure philosophical matter now became a topic of interest at court. Copies of the manuscript went to members of the duke's family. Physical models of the solids were made. In 1504 the town council of Florence commissioned Pacioli to make models for them. In August 1508 in Venice, Pacioli even gave a sermon on proportion to leading noblemen and scholars56, which he published the following year as the preface to Book V of Euclid's *Elements*. Polyhedraphilia had begun.

**6. Leonardo Da Vinci**

The popularity of Pacioli's book was due largely to Leonardo da Vinci's illustrations. Leonardo's preparatory drawings for many of these have survived57 and offer some insight into how he worked. In some cases the sketches are so rough that he is clearly visualizing the object as he goes. In other cases his drawings are so polished that he very probably had a physical model in front of him and he may well have used the same perspectival window that he employed in drawing objects such as the armillary sphere (fig. 25.1).

There was a little more to Leonardo, however, than a person who made pretty pictures following someone else's instructions. We find, for example, that the notebooks contain various other solids not included by Pacioli. Among them are preparatory sketches of the seven other Archimedeian solids (see Appendix II). This means that Leonardo had represented all thirteen of the Archimedeian solids a full century before Kepler, who is frequently given credit for being the first to do so. Leonardo is also the first known to have made ground-plans of the regular solids or nets to use the modern technical term, a practice that was taken up in a *Modena manuscript* of 1509, then published by Dürer, Hirschvogel (fig. 16.1), Cousin (fig. 23) and has remained a standard aspect of regular solids ever since.

Leonardo's deeper contribution lies in changing the whole context of the discussions. He was well aware of traditional links between vision, perspective and geometrical play. He had almost certainly read Alberti's *On Painting* and he cited Alberti's *Mathematical Games* directly. Leonardo worked with Francesco di Giorgio Martini, who used surveying instruments to demonstrate basic principles of perspective. Around 1490 Leonardo began a systematic study of these principles. This led him to discover the inverse size/distance law of perspective which states that if one doubles the distance of an object its size on the picture plane is half; if one trebles the distance, the object's size is one-third and so on. Leonardo recorded his findings in a thirteen page treatise that is now a section of *Manuscript A* at the Institut de France in Paris.58

Leonardo's demonstrations involved a surveyor's rod, a perspectival window and other instruments, with the aid of which the geometrical properties of visual pyramids would be systematically recorded. Intersections of the pyramid demonstrated perspectival effects in the manner of conic sections.59 In Alberti's *Mathematical Games* transformations were purely a matter of geometry. In Leonardo's *Manuscript A* these transformations remained geometrical but related to visual experience, measurement by instruments and perspectival representation. They were no longer mental abstractions. They could be seen, measured, recorded and represented.

Euclid's version had been with two-dimensional lines. Leonardo's version meant that Euclid's propositions could be expressed three dimensionally. The Pythagorean theorem, for instance, was no longer an abstract geometrical idea: it involved perspectival versions of triangular and square boxes. Euclid had catalogued static lines. Leonardo set out to catalogue the dynamic properties of three dimensional shapes: a 3-D version of the geometrical game.60 His plan emerges slowly. Hints of it are found in his earliest notebooks. But he is 53 before he writes his first serious treatise *On the Geometrical Game* 61 in 1505. It has three books, we would say chapters. It is written in mirror script. As far as a modern reader is concerned even the pagination goes backwards. But we need merely glance at a few pages of this text, with its neat paragraphs and numbered illustrations to recognize that Leonardo is working systematically (fig. 12-13). He is concerned with equivalent areas of pyramids, cubes and rectangles which leads him a few pages later to show how one transforms cubes to pyramids and pyramids into dodecahedrons and conversely (fig. 14.1-2). This leads in turn to an amazing list of twenty-eight kinds of geometrical transformations62 the first twelve of which are simple, i.e. where two changes leave another aspect unchanged, while the remaining sixteen are composite in which all aspects change.

This treatise now in the Victoria and Albert Museum in London marks a first step in a much more ambitious programme that dominates the next eleven years of his life. The manuscript that resulted is lost, but there are enough hints in his preparatory notebooks to give us a vivid idea of his activities. For some time Leonardo remains undecided about a title so we find him referring to a book of equations63 in the sense of equivalent shapes, or a book of transmutations64 in the sense of transformation. He continues to refer to a work *On the Geometrical Game* 65 but its contents change with time. As noted above his treatise of 1505 had listed 28 kinds of transformations. In 1515 he describes the geometrical game as a ";process of infinite variety of quadratures of surfaces of curved sides.";66 About a year later this has evolved into a treatise in its own right of 113 chapters with 33 different methods of squaring the circle67, which he intends to use as an introduction to his work *On the Geometrical Game*. In his own words:

Having finished giving various means of squaring the circle, i.e. giving quadrates of equal

size to those of the circle, and having given rules to proceed to infinity, at present I am

beginning the book on the geometrical game and shall once again give the means of infinite

regression.68

By this time, however, *On Squaring the Circle* and *On the Geometric Game* have both become part of a magnum opus on the *Elements [of Mechanical Geometry*] with a second volume on the *Elements of Machines*.69 This second volume was not simply an afterthought. It was again something on which he had been working for almost thirty years. Initially, as an engineer he had become struck how machines involved a surprisingly limited number of parts such as gears, screws and rivets. He catalogued 21 of the 22 parts known today. While doing so Leonardo became convinced that these must be governed by more fundamental principles or powers. By 1492 he was convinced that there were four basic powers: weight, force, motion and percussion.70 To explore their properties he made experiments with weights and balances, pulleys and other mechanical devices and discovered that the powers had proportional variations.

As a theologian, Pacioli had been interested in proportion mainly as a stimulus for religious meditation, as a key to understanding God himself. By contrast, Leonardo, as a scientist, was concerned with proportion as a means for understanding God's creation: the natural world. For a time he pursued this goal in terms of two separate research programmes. One focussed on pyramidal proportion and involved perspective, optics, transformational geometry, surveying and painting. A second programme involved proportions in mechanics and physics. As the 1490's progressed he gradually hit upon the idea that pyramidal proportion offered a key to both programmes. As he put it in a note in 1500:

All the natural powers have to be or should be said to be pyramidal, that is, that they have

degrees of continuous proportion towards their diminution as towards their growth. Look at

weight, which in every degree of descent, as long as it is not impeded, acquires degrees in

continuous geometrical proportion. And force does the same in levers.71

For Leonardo proportion was ";not only in numbers and measures but equally in sounds, weights, tones and sites and every power that exists.";72 Further experiments convinced him that the pyramidal proportions of perspective involved a pyramidal law that applied to all dynamic situations falling into two basic categories: first, changes in shape as in transformational geometry; second, changes in weight, motion, force and percussion (including optics, acoustics, and heat) in mechanics and physics. By 1516 these two classes had inspired his two volumes on the *Elements [of Mechanical Geometry*] and the *Elements of Machines.* As far as Leonardo was concerned these works were his version of a unified theory.

At the level of practice two other projects had also defined his life's work. As a young man Leonardo had set out to write a basic work on the microcosm which blossomed into his anatomical studies. He also planned a companion work on the macrocosm. Based on his optical and astronomical studies this was intended to offer a new cosmology. He envisaged that his *Elements of Geometry* and *Elements of Machines* could serve as a theoretical foundation for his *Anatomy* and *Cosmology*, but unfortunately he died before he could publish his new encyclopaedic synthesis.

A generation earlier Cusa, building on Plato's Timaeus and Euclid's *Elements* had used proportion and regular solids as a means of understanding God. Pacioli shared Cusa's views of mathematical theology with the exception that where Cusa relied on intellectual images of geometrical transformation, Pacioli insisted on perspectival examples of the regular and semi-regular solids on which these transformations were based. Leonardo drew the images that Pacioli envisaged. He also changed the context in which they were seen. The regular solids, the geometrical game, the whole of Euclidean geometry became part of a new approach to science that was visible, quantitative and reversible. Indeed, geometrical transformation now became synonymous with science itself. As Leonardo put it:

If a rule divides a whole in parts and another of these parts recomposes such a whole

then one and the other rule is valid. If by a certain science one transforms the

surface of one figure into another figure, and this same science restores such a

surface into its first figure then such a science is valid. The science which restores a

figure to the original shape from which it was changed is perfect.73

For Leonardo's predecessors the regular solids were stimuli for religious meditation.

For Leonardo they became building blocks of reality, revealing the structure of the universe, accounting not just for static objects, but for all changes of shape therein. The regular solids were no longer merely abstract symbols linked with a world of ideas. They were intimately connected with the physical world, were models of reality74 and as such could be physically represented, reconstructed and measured. Even their transformations could be computed mechanically. This, as we shall see was exactly what happened in the generations following Leonardo.

A first reaction to these innovations was simply to make physical models of the regular solids. In (fig. 43.1) the famous anonymous portrait of Pacioli, for instance, we see a model of a dodecahedron in the lower right. In the upper left there is a glass model of the twenty-six sided figure, or rhombicuboctahedron which Leonardo also illustrated in Pacioli's book *On Divine Proportion* (pl. 1). In both cases the model is suspended. But in the painting it is also half filled with water and as Dalai-Emiliani75 has acutely noted involves unexpected optical effects. If we look at a detail (pl. 2), we see at the upper left of the rhombicuboctahedron a reflection of a window in a Renaissance palace. It is almost certainly Urbino since Pacioli is shown with the young Duke of Urbino looking over his shoulder. This image of a window is reflected a second time on the surface of the water whence it is refracted to the lower right hand surface.

The Duke's interest in the subject apparently extended well beyond looking over Pacioli's shoulder. There is a story still told by the priests of Urbino today that he chose a stellated figure of the dodecahedron (pl. 3) as his personal symbol and had lamps made in this form. There is one in the basement of the cathedral. Thus far I have been unable to find documentary evidence so the story may well be apocryphal. Nonetheless, there is a shop in Urbino, which thrives in selling beautiful reproductions of the so-called ducal stellations.

Others were also fascinated by these forms. Brother Giovanni of Verona, a monk who was among the leading masters of inlaid wood in his day adopted a number of Leonardo's illustrations for his own purposes. For instance, in the choir stalls of Monte Oliveto Maggiore, near Siena, he included both a stellated dodecahedron (pl. 4) and a seventy-two sided figure (pl. 6). Later in the sacristy of Santa Maria in Organo76 in his native city of Verona he again used this seventy-two sided figure, this time a combination with a twenty-sided icosahedron and its truncated form (pl. 7).

If all these solids were taken directly from Leonardo's illustrations in Pacioli's *On Divine Proportion*, their meaning was based on Pacioli himself. These were symbols intended to inspire religious meditation, and this remained the norm in Italy during the first decades of the sixteenth century. It was elsewhere that Leonardo's interests in practical and scientific transformation were first appreciated.

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