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 Auerbach30 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 Cylinder.  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.

 

 

 

 

7. Nürnberg Goldsmiths

 

            In early sixteenth century Germany it was mainly the practical aspects of regular solids that aroused interest.  Bits of evidence confirm that Leonardo had some influence on Albrecht Dürer in this respect.  Several of Leonardo's anatomical drawings recur in Dürer's notebooks.  Leonardo's perspectival window (fig. 25.1) recurs in Dürer's Dresden Sketchbook in 1513, in the London Sketchbook in 1515, and then in his published Instruction of Measurement of 1525.  More important for our purposes, Leonardo's drawing for a dodecahedron is published by Dürer in the same book.  But neither here nor elsewhere is Leonardo acknowledged.  Had they actually met, as some scholars have rumoured, then one would have expected Dürer to report on Leonardo's scientific world view.

 

            Indeed in his Instruction of Measurement, Dürer cites only Euclid in connection with the regular solids.  However, what had been abstract problems of mathematical construction for Euclid, are concrete and practical for Dürer.  He describes, for instance, how each of the five solids can be constructed physically, how to construct a sphere within which these solids can be inscribed, fit models of one solid within another and add pyramids to their sides to produce their stellations.  His instructions for semi-regular solids are even more vivid:

 

            Much more beautiful bodies can also be constructed which can again be inscribed with all their corners within a hollow sphere, but these have uneven sides.  Some of these I shall draw in plan completely, such that anyone can put them together.  Whoever wants to make his own should draw them larger on a cardboard of double thickness and cut them with a sharp knife such that one cuts through the one layer through to the next.  When one then folds the body together it can readily be bent at the cuttings.  Hence pay attention to the following drawings since such things are of manifold use.78

 

            Dürer then describes and illustrates the nets of six Archimedeian solids and three variants (cf. Appendix I) adding that:

 

            if one uses sharp scissors and cuts away the corners from these examples and then cuts

away the remaining corners one can in this way construct a number of other bodies. 

From these things one can make a good number of objects where one part is placed on top

of another, which is useful in sculpting columns and their decorations.79

 

            Where Leonardo was concerned with principles for constructing and representing solids, Dürer provides the equivalent of a how to do it book.  For Leonardo the solids were the key to a scientific understanding of the universe.  For Dürer they provide practical hints for architectural decoration.  To be sure the Instruction of measurement is also about many other subjects.  It deals with measurement in the context of architecture, instruments, transformational geometry and principles of perspective.  In a larger sense it is about mechanical means of construction and representation.  It had a seminal influence.  It was soon translated into Latin and subsequently into French.80

 

            Some of the work connected with Dürer's workshop remained unpublished.  Some drawings illustrated, for example, how the method of truncation, as described by Dürer in the passage just cited, actually worked when applied to a tetrahedron (fig. 15.1). Other drawings confirmed that his workshop was also experimenting with a) how different orientations and shadings could be used to produce different effects in a given shape (fig. 15.2); b) how new shapes could be generated by careful geometric play (fig. 15.3) and c) were using anamorphosis, i.e. deliberately distorted versions of solids such as a truncated cube to create new effects81 (fig. 15.4).  This is of considerable interest for our purposes because all these developments are used in the masterpieces of inlaid wood a few decades later (pl. 53-54).

           

Six years after the first edition of Dürer's Instruction of Measurement, an anonymous Beautiful Useful Booklet 82 (1531), edited by Rodler set out to popularize Dürer's ideas.  But this scarcely mentioned the regular solids.  Twelve years passed.  Then came 1543, which was an important year for the history of science.  Tartaglia published the first vernacular edition of Euclid.  Copernicus' Revolutions of the Heavens and Vesalius' Fabric of the Human Body also appeared that year.  More relevant for our purposes, Augustin Hirschvogel published his work on Geometry with the curious subtitle:  The book geometry is my name, all liberal arts at first from me came.  Architecture and perspective together I bring.83  It contained some basic propositions on perspective but was mainly about polyhedra.  Besides the regular solids, Hirschvogel considered seven of the thirteen Archimedeian solids (Appendix** ).  His novelty lay mainly in his presentation.  Where Dürer gave only ground plans, Hirschvogel provided them in combination with different views, thus correlating two- and three-dimensional versions of an object (fig. 16.1-3).  Even so his emphasis remained on the practical architectural applications of these solids.  Lautensack's treatise on perspective (1564), which again contained both regular and semi-regular solids (fig. 17.1-4) continued this tradition.84

           

This changed with the publication of Jamnitzer's Perspective of the Regular Solids. Wenzel Jamnitzer85 (1508-1585) had been born in Vienna and arrived in Nürnberg in 1534. In the decades that followed he became one of Nürnberg's leading citizens86, and Europe's most famous goldsmith.  While clearly interested in the practical applications of these forms in making jewelry and ornamental vases, Jamnitzer was also fascinated by the cosmological aspects of the solids as he indicated in his long subtitle:

 

            that is, a diligent exposition of how the five regular solids of which Plato writes in the

Timaeus and Euclid in his Elements are artfully brought into perspective using a

particularly new, thorough and proper method never before employed.  And appended to this a fine introduction how out of the same five bodies one can go on endlessly making many other bodies of various kinds and shapes.87

 

            Jamnitzer never managed to to publish his fine introduction. Nonetheless, his method was clear.  Part one had five sections, each headed by one of the five vowels (a, e, i, o. u), corresponding to one of the five regular solids and one of the five basic types of matter: earth, air, fire, water, heavens (pl. 9-13).  Each section contained 24 illustrations, i.e. a solid followed by 23 truncations and stellations.  Scholars have suggested that Jamnitzer's 24 variations were an allusion to the 24 letters of the Greek alphabet88, in which case these shapes are a metaphorical alphabet of Nature's forms.  Later writers such as Lencker and Halt were explicit about these comparisons.  Part two opened with five pages again headed by five vowels, each showing two transparent regular solids mounted on a stand (e.g. pl. 14-15).  Six pages of variations on the seventy-two sided figure followed (e.g. pl. 16-17).  The first of these was based directly on Leonardo's illustration (pl. 5) which Brother Giovanni had used in Verona (pl. 7).  Four pages of pyramids (pl. 18-19) and three pages of cylinders (e.g. pl. 20) completed the book.  Jamnitzer's work remained popular, was reprinted and went through three pirate editions in the early seventeenth century.89

           

Hans Lencker, Jamnitzer's younger contemporary, was also a goldsmith.  His first work was a Perspective of Letters 90 (1567) which represented all the letters of the alphabet perspectivally as if they were semi-regular solids. (pl. 21-22).  The idea had evolved ever since Pacioli included letters as an appendix to the solids in his Divine Proportion.  Albrecht Dürer and Geoffrey Tory92 also explored both themes together. However, it was Lencker who first represented the whole alphabet three-dimensionally.  His first edition (1567) contained only illustrations.  The second edition (1595) added a preface noting that these forms were the true elements and first principles which one needed to learn all other disciplines.93  A generation later Peter Halt (1625) took this reasoning further: just as one could not speak without vowels, one could achieve nothing in terms of perspectival drawing without the regular solids.94  This spelled out, so to speak, Jamnitzer's earlier use of the five vowels and the five solids.  In Lencker's book these perspectival letters took up the first thirteen folios of the book. These were followed by striking examples of semi-regular forms in combination.

 

Where Jamnitzer had concentrated on making visible the programmatic elements of his method, Lencker concentrated on demonstrating his perspectival prowess.  His first figure showed three cylinders learning around a pyramid, each cylinder mounted by a cross on top of which were balanced, in trapeze artist style, a series of seven six-sided stars, the central one of which had poised on it a diamond shaped object (pl. 23).  A third showed interesting rings on a stand (cf pl. 30).  Another showed an open cube inscribed within an open twelve sided shape balancing on a stand (pl. 24).  The final illustration was a stellated shell (pl. 25).  Lencker's main work, Perspective (1571) contained examples of a truncated cube, an octahedron and a dodecahedron on a stand.  In the tradition of Dürer and Lautensack, he was interested in the regular solids mainly in terms of architectural applications.  Lencker's originality, as we shall see later, was in inventing new instruments for representing and constructing these regular solids.

 

            Closely related to both Jamnitzer and Lencker is an anonymous manuscript now in the Herzog August Bibliothek in Wolfenbüttel. Many of its 36 regular and semi-regular solids are so clearly derived from Jamnitzer that Franke (1972) has attributed the manuscript to him. Several factors argue against this. Since Jamnitzer and Lencker were rivals it is highly unlikely that Jamnitzer would simply have copied poorly a figure based on Lencker (pl.30) without improving on it. The execution of the figures, while exquisite lacks the sharpness of line characteristic of Jamnitzer (pl. 9-13) or his engraver Jost Amman (pl8). Moreover, as we have seen Jamnitzer had metaphysical concerns which would not have been in keeping with the playfulness of these shapes (pl. 30-31), nor with the entertaining animals that accompanied them (e.g. pl. 27, 32), which are more than a little reminiscent of the side panels of the Münster cabinet (pl. 57-58). Was the anonymous author of the manuscript from Augsburg and possibly from the same workshop that produced the cabinet?

 

            Younger than both Jamnitzer and Lencker was a third individual about whom we know all too little.  We know that Lorenz Stoer was a painter and illustrator, that he was active in Nürnberg until 1557 when he moved to Augsburg where he died around 1621.  He is credited with one book, Geometry and Perspective 95(1567) which involved a series of eleven woodcuts showing combinations of the regular solids in a landscape with ornamental shapes (pl. 33-36).  These were intended as designs for inlaid wood.  The work was popular enough to go through a second edition.  While I have not yet found any furniture, which used precisely these motifs, the remarkable chest now in Münster was clearly inspired by Stoer's ideas (pl. 56-58).

           

Not generally known is that Stoer produced two other illustrated manuscripts. One of these, without a title page, is a collection of 33 hand painted illustrations now in the Herzog August Bibliothek in Wolfenbüttel96. The first folio contained only two line drawings of a cube, the second a cruciform figure. The remaining 31 folios each contained semi-regular solids. A majority of these were truncations resulting in sphere-like shapes. Some were stellations (pl.37-38). Three were variants on cylindrical forms and two were based on pyramidal shapes. Two were composite with either a series of regular and semi-regular solids (pl.39) or in combination with a series of crosses (pl.41). The other manuscript, a magnificent collection of 336 folios with over 640 solids  of such illustrations, is now in the University Library at Munich.  This is in six parts, beginning with The Five Regular Solids Cut in Various Ways 97 which deals systematically with the five regular solids in various truncations and stellations.  Using a related method Jamnitzer had produced 120 solids (pl. 9-13).  Stoer's method generated  ** solids (e.g. pl. 43).  Part two, entitled Geometrical and Perspectival Bodies 98 included a series of further truncations and stellations (e.g. pl. 44).  A short third section focussed on conic and cylindrical shapes99 (pl. 48).  More truncations and stellations followed in a fourth section on Geometry and Perspective.100  A fifth section showed these objects piled two, three and fourfold on top of one another101 (pl. 45-47).  A final section entitled Geometrical and Perspectival Regular and Irregular Bodies 102 contained further dramatic examples (pl. 42, 49-50).

 

            Stoer's manuscript was actually a compilation of over three decades' work ranging from 1562 through 1599.  The novelty of his remarkable effects lay mainly in his combination of earlier techniques.  We have noted, for example, that Dürer's workshop explored the use of shading to enhance the spatial effects of these solids.  Jamnitzer and Lencker developed this technique using narrow banks of colour to accentuate the borders of these shapes.  Stoer added a feature of his own.  He used the surfaces of his solids as spaces in which to inscribe further polygonal shapes.  Frequently he combined both techniques.  In the case of a dodecahedron, for instance, he outlined the boundaries of its twelve surfaces with bands of colour.  In each of these he then inscribed a pentagon.

 

            The beauty and effectiveness of Stoer's shapes depended in part on their colour, which may well explain why these manuscripts were never published.  There was no colour printing at the time.  It also helps us to understand why these motifs became popular in furniture.  Here narrow bands of colour could readily be adapted as stripes of wood, ivory, ebony, etc., which is almost precisely what an anonymous master craftsman did when he produced an inlaid writing desk top now at the Museum for Decorative Arts in Frankfurt (pl. 58).  Indeed, although clearly not a copy of Stoer's illustrations, it is obviously a complex variation of his themes (cf. pl. 57.1). Where Stoer had a central shape topped by an inverted pyramid, the desk has a variant shape, topped by a semi-regular solid and a complex inverted pyramid.  The intersecting square shape in the upper left of Stoer's illustration recurs in the upper right of the inlaid panel. The hollow cubic shape in the lower right hand corner of Stoer's illustration--which Leonardo had used two generations earlier, recurs in the upper left of the inlaid desk.

 

            These principles were further developed in a cabinet now in the Museum of Applied Arts in Cologne (pl. 59).  At first sight the two central compound scenes surrounded by thirteen drawers of three solids and flanked by at least twenty figures on each of the side panels creates an impression of overwhelming complexity (pl. 60).  On looking more closely we find that the two central panels are effectively mirror images of one another, in terms of both shape and colour, i.e. a black strip on one side usually has a corresponding light strip on the other side.  Moreover, we find that the two upper innermost figures in the central panels recur as the upper outermost figures in the flanking side panels.  Regularity is in fact the underlying theme.  The two rows of three solids in the upper central section are identical in shape.  In the right hand section of the central panel the upper two rows have a recurrent pattern which is repeated on the left in the second from the top row and the bottom row.  Indeed it is likely that the bottom drawer was originally in the top left.  In which case both the two upper left and upper right drawers would have been symmetrical, as would all three drawers in the bottom row.  A simple switch of the drawers in rows two and three on the left side would make them symmetrical with their counterparts on the right.  In other words only five basic patterns underly the thirteen drawers and it is simply through an ingenious play of colour that these five rows of three look like 39 different solids.

 

            The side panels each have two complex scenes flanked by three rows of four semi-regular bodies.  The shapes in the left panel are again mirrored by those on the right, with a corresponding alternation of light and dark.  The dodecahedron in the complex scene (pl.57.2) is again reminiscent of Stoer's illustration (pl. 43.2) except that where Stoer had only one set of inscribed pentagons this panel has two.  In the upper right of the left hand panel is a cross composed of seven cubes.  This shape, now called a hypercube was another of those which Leonardo had explored two generations earlier (fig. 18.1).  It recurred in Stoer's manuscript, is used four times on these panels (e.g. fig. 18.2) recurred in seventeenth century texts (fig. 18.3-4) and English gardens of the eighteenth century before being rediscovered as a symbol of the fourth dimension in the twentieth century.  If one opens the drawers there are further scenes and if one opens these in turn there are even more scenes all in inlaid wood: noble figures, hunting scenes, allegories, scenes of towns, landscapes, animals.  A full analysis would require a monograph in itself.  For our purposes it is enough to note how these complex plays of semi-regular solids became one layer of a complex Mannerist game of hidden images.

 

            The cabinet is not dated but is likely to have been made between the mid 1560's when Jamnitzer was active in this field and 1585 when there was a plague in Nürnberg.  Both Jamnitzer and Lencker were victims of that plague.  So too were Jamnitzer's papers.  Indeed enthusiasm for the regular and semi-regular solids was never quite the same thereafter.  One exception was Paul Pfintzing103. Comparison of manuscripts in Nürnberg (fig. 19.1) and in Bamberg (fig. 19.2-3) allows us to trace how he developed his illustrations. Pfintzing, who was an artist also active as surveyor and map maker, set out to write a summary history of perspective focussed on technical developments in Nürnberg: Dürer, Jamnitzer, Lencker and an otherwise unknown artist Hans Hayden.  His imaginative drawings (pl. 52-56) show various perspectival instruments and some of the shapes they were used to produce.  Pfintzing planned his book for private distribution to friends.  Several printed drafts with manuscript additions now in Wolfenbüttel and Bamberg 104 confirm that he changed his mind several times.  The result was a book with highly unusual illustrations105, showing these technical instruments for perspective perched precariously on regular solids.

 

            In the next generation a perspective book by Brunn (1615) contained some regular solids.  But the focus of attention shifted to Augsburg and Ulm where there were new editions of Lencker107, Pfintzing108 and Stoer109 in the years 1615 and 1617.  There were edited by Stephan Michelspacher who clearly intended a revival of this genre.  But in 1618 the thirty years war began.  In 1625 there was one last important work by Peter Halt, entitled The Art of Perspectival Drawing.  Inspired by Jamnitzer's association between the five vowels and the five regular solids.  Halt's title page literally showed each of the vowels precariously balanced on its corresponding solid.  His woodcuts were frequently original. Yet there was a rough and ready aspect to their execution that was not simply a reflection of difficult times (pl. 69-71).  The fascination with these shapes as symbols of Divine perfection remained, but their representation had become less important. Before we can understand this paradox we need to examine developments in Italy and France.

 

8. Italian Popularizers

 

            In Italy publication of works on perspective proceeded more slowly than one might have expected.  Indeed, after Pacioli's Divine Proportion in 1509, there was nothing new until Serlio in the 1540's and as a practicing architect Serlio had little interest in regular solids. The first to pursue these themes was Daniele Barbaro in his Practice of Perspective 111 in 1568.  Barbaro's book was in nine parts with sections on optics, based mainly on Euclid: planes, solids, architecture, anamorphosis, the planisphere, light and shade, the human body and instruments.  Part three, by far the largest section of his book (pp. 43-138), while ostensibly dealing with representation of three dimensional bodies was actually a treatise on the regular and semi-regular solids, ending with a detailed study of the mazzocchio which also served as a leitmotif for his chapter headings (figs. 20.1-6).  Barbaro began with an outline of three basic methods of rendering objects perspectivally, then provided plans, elevations and three dimensional versions of the five regular solids, descriptions of nine Archimedeian and ten other semi-regular solids, plus ten stellations three of which he illustrated (Appendix III).

 

            Barbaro's presentation differed from his predecessors.  Piero della Francesca, for instance, in the tradition of Euclid, had focussed on a mathematical description of how the solids could be constructed.  His diagrams functioned almost as an afterthought to the text.  In Pacioli's Divine Proportion mathematical description remained, but as we have seen, Leonardo's diagrams vied with the text in importance.  Dürer, by contrast, had assumed no mathematical knowledge, and emphasized a hands on approach, describing how each body could be constructed physically.  Barbaro's approach was somewhere in between.  He implied that the bodies could be physically constructed, but presented them in so abstract a way that a mathematical approach was also implied although not officially required.  In some cases, as with the twenty-six sided figure112, he provided a series of viewpoints of a single object (fig. 21.1-5, each assuming an optical approach, yet lacking the spatial presence of Leonardo's version (cf. pl. 1).  At other times Barbaro provided only a ground plan which did not always correspond to his description.  In his text, for instance, he described an irregular object with 36 squares, 24 hexagons, 6 octagons and 8 twelve sided figures.  However, his accompanying diagram showed only 8 squares, 7 hexagons, 3 octagons and 2 twelve sided figures.113  In his text Barbaro emphasized the value of short cut methods and since he no doubt assumed that most readers would use these, he did not worry about his diagrams being accurate.

 

            Any suspicions that Barbaro was not capable of a painstaking mathematical approach quickly disappears if we turn to the manuscripts on which the printed text is based.114  One manuscript contains an unpublished section of over 100 pages dealing with three semi-regular solids in what looks in retrospect as a fanatical attention to detail.115  Indeed we discover that Barbaro had two quite different approaches to illustrations:  one was simply to give a general impression of the principles involved, the second was a step by step method using instruments to arrive at a mathematically based visual image.

 

            A generation later this second approach gained a brief supremacy but with two paradoxical results.  As the means for technical reproduction of perspectival images were perfected emphasis shifted to principles of reproduction and abstract mathematical figures.116  Egnatio Danti's commentary to Vignola's Two Rules of Practical Perspective117 (1583) is a good example. Danti dwelt at some length on the principles for producing regular polygons.118  He specifically defined the perspectivist's task as the transformation of geometrical bodies119 and yet he gave no three dimensional examples.  To be sure the regular solids did not disappear entirely.  A second result of these developments was a focus on stock examples, as in Sirigatti's treatise120 of 1596, which was dominated by variations on the sphere and mazzocchio (pl. 64-68).  Some attention to combinations of these solids in producing architectural effects continued in Sirigatti.  An unpublished manuscript by Vasari, Jr., the son of the man who wrote the famous Lives of the Artists, contained even more striking examples of this121: rows of cylinders and mazzocchio figures, stairs, architectural vaults and columns, multifaceted spheres, regular solids, stellations, and combinations in architectural settings (e.g. fig. 22-23), including semi-regular shapes such as crosses, musical instruments, chairs and barrels.  Most of these shapes were not new: they were popularizations of shapes invented by Nürnburg goldsmiths or even earlier by Leonardo.

 

            The supremacy of the mathematical approach soon had further consequences.  Even in books on perspective abstract diagrams triumphed over three-dimensional illustrations.  Accolti's Deception of the Eye 122 (1625), for instance, dealt only with the five regular solids, the sphere and the twenty-six sided figure, and while due reference was still given to model making123, the emphasis was on short cuts in arriving at geometrical outlines124, and references to Euclid's Elements.  By the 1670's the situation in Italy was much the same as in Nürnberg. Fascination with representing the multiplicity of semi-regular shapes had all but disappeared.  What had been a realm of theology and art was now increasingly a branch of abstract mathematics. This was partly due to developments in France.

 

9. French Mathematicians

 

            In France, as elsewhere, the interpretations of Euclid were many.  One was theological and mystical.  Two years after Pacioli's Divine Proportion (1509), Charles de Bovelles125 published a treatise On Geometrical bodies (1511).  He was influenced by Nicholas of Cusa's approach whereby geometrical symbolism provided mathematical guidance to the Divine.  He probably knew of Pacioli's work.  But where Pacioli had been content to make passing analogies between proportion and the Trinity, Bovelles set out to show systematically that the geometry of the polyhedra provided a symbolic demonstration of the power of one in three and three in one, and as such offered a mathematical guide by means of which one could contemplate the mystery of the Trinity.

 

            A second interpretation was practical.  This grew naturally out of a mediaeval tradition whereby geometry was treated literally as measurement of the earth.  We have already noted that this occurred all over Europe.  What set the French examples apart, however, was their emphasis on the regular solids.  In 1544, for instance, Oronce Finé‚ published a treatise On Practical Geometry or on The Practice of Lengths, Planes and Solids, that is of Lines, Surfaces and Solids in Quantities and Other Mechanical Operations as a Corollary to the Demonstrations of Euclid's Elements.126  In this work Finé‚ included a chapter to show that polygons and multilateral figures could be measured127 and another chapter on the measurement of the rest of the regular bodies.128  That same year Finé‚ published a series of small treatises on quadrature of the circle, measurement of the circle, all the polygons and on the planisphere.129  This contained a booklet On the Absolute Description of All Straight Lines and Multiangular Figures (which are Called Regular) Both Inside and Outside a Given Circle and on a Flat Line.130  He pursued this theme in 1556 with a Corollary on the description of the regular solids131 in a given circle using isoceles triangles, followed by a section on volumetric measurement of cubes, rectangles, pyramids and the like. Regular polygons in two dimensions and regular solids in three dimensions were becoming a key for abstract planimetric and stereometric measurement.

 

             Even so links with practical problems continued. Cousin, for instance, illustrated models of the five regular solids on the title page of his Book of Perspective 132 (1560), and described in detail their three-dimensional construction with the aid of compasses.  The royal mathematician, Claude de Boissière in The Art of Arithmetic Containing all Dimension both for the Military Art and Other Calculations133 (1561) also dealt with the regular solids in terms of construction, measurement and transformation of one form into another using shapes (fig. 14.3-4) more than slightly reminiscent of Leonardo da Vinci (fig. 14.1-2), who had been his predecessor at the court of the King of France.  Boissière's treatment of regular solids was followed by a general rule on the quantity of all barrels.134 Here, once again, measurement of ideal solids served as a basis for measuring less regular shapes in the everyday world. Moreover, as the editor of this work, Lucas Tremblay, pointed out, all this was part of a larger plan to provide a synopsis of arithmetic and geometry with a view to uniting discrete and continuous quantities.135  The Greeks, it will be recalled, had dealt with the discrete quantity of arithmetical numbers separately from the continuous quantity of geometrical lines.  By the latter sixteenth century the challenge loomed of dealing with both types of quantity together. Both Fermat and Descartes found a solution in the 1630's in what we now call analytic geometry.

 

10. The Jesuits

 

            The construction and representation of the solids as well as their religious connotations continued to hold a certain fascination.  We have seen how Piero della Francesca's egg in the Brera Altar used anamorphosis for both religious and mystical connotations.  Holbein further explored these possibilities with the anamorphic skull in his Ambassadors, which became such a famous motif that it was included among the instruments of the French Academy of Sciences.136  Even so it was not until the 1630's that the didactic potentials of anamorphosis were considered seriously.  In 1638, a young Franciscan friar, Jean François Nicéron, produced (fig. 24.1-3) a treatise entitled Curious Perspective.144  He was 25 at the time. This served as the basis for an expanded version named Optical Magic137 (1642).  Almost immediately some of his ideas were adapted by the Jesuit Father, Jean Dubreuil in his Practical Perspective 141, a massive three volume compendium (1642-1649).  Dubreuil also dealt with the regular solids as if they were models, thus reviving sixteenth century traditions.  It was a popular work and a controversial one. It plagiarized ideas from Desargues, getting some of them wrong in the process. It was attacked by the experts, and needless to say it was a great success. It was translated into Latin, German, Dutch and English going through more than 20 editions.  Significantly, however, the translations were often abridged versions in which treatment of the regular solids was all but omitted.  The tradition did not die out entirely. Subsequent treatises very often included the five regular solids and some other examples. Nilson, in his Introduction to Linear Perspective (1812), included more. One of these, reproduced on our frontispiece, was a stellated form framed by a hemispherical niche. The majority of the others were in a garden setting and included regular solids stacked on one another, semi-regular shapes functioning as sundials, a stellated form on a pedestal of a pyramid resting on a cube (pl. 72), a truncated cubic form (pl. 73) based on Jamnitzer (cf.pl. 11.2 top left) again resting on a cube and a more complex stack of semi-regular shapes (pl. 74) again based on a Nürnberg precedent (cf. pl. 32). Nilson, however, was an exception, indeed perhaps a last example of serious interest in the theme. The sixteenth century passion for these shapes had gone.  Why this happened is the subject of our next chapter.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

II         CRYSTALLOGRAPHY, MATHEMATICS AND ART

 

1. Introduction  2. Instruments  3. Models of the Universe  4. Nature's Models  5. Abstract Mathematics  6. Art  7.Visual Mathematics.

 

1. Introduction

 

            In chapter one we explored how cosmological, philosophical, mystical and religious aspects of the regular solids linked with a type of metaphysical mathematics to produce a geometrical game based on transformations of shapes. This game had neo-Platonic tendencies, in the sense of being linked with the world of ideas more than the physical world.  As such it could have become the ultimate mind game of the Renaissance and the three-dimensional illustrations by artists would have represented mere bravados of abstraction.

 

            This did not occur because as we have already mentioned there was also a practical side to these geometrical games:  the challenge of actually measuring their volume and determining how a given volume was affected by changes in size and shapes.  Already in Antiquity Archimedes had explored such problems in his work On the Sphere and the Cylinder.1  Hero of Alexander2 had pursued them.  Subsequently they had been taken up by Leonardo of Pisa3 and had become part of the abacus tradition, which was partly why the regular solids played so prominent a role in Piero della Francesca's Book of the Abacus.  He too was concerned with measuring their volume.  Pacioli's Divine Proportion also stood within this practical tradition.  Only 34 of his 61 illustrations dealt with the five regular solids (figs. 8-11).  Four illustrations involved a 26 sided shape, two illustrations involved a 72 sided shape, both of which he explicitly discussed in terms of their practical applications to architecture.4 The final 21 shapes were all variants of columns and cylinders.  Here again Pacioli emphasized their practical applications for architecture adding a chapter in which he outlined how they could be measured,5 citing Archimedes' work on Quadrature of the Circle6 and alluding to a treatise on measurement of the regular solids7 which he had dedicated to Guido Ubaldo, the Duke of Urbino. The sixteenth century continued this practical tradition and also transformed it by developing a series of instruments for both the representation and measurement of these solids.

 

2. Instruments

 

            The simplest of these was the perspectival window which was probably invented by Brunelleschi in the early fifteenth century.  Leonardo's drawing of a perspectival window in recording an armillary sphere is the first extant example of this device (fig. 25.1).  Dürer published a version of this window in his Instruction of Measurement (1525).  So too did Rodler (1531) in the popular version that he edited.  Thereafter it became a stock image in perspective treatises.  In Nürnberg, Dürer also developed variants of the perspective window.  These were improved upon by both Jamnitzer and Lencker and were subsequently published by Pfintzing (fig. 19).  For our purposes they are of interest because they illustrate how perspectival instruments led to a new type of measured drawing.  This is all the more significant because both Jamnitzer and Lencker were also involved in the development of universal measuring devices.  Jamnitzer, for instance produced a special instrument for systematic measurement of various metals, accompanying which he wrote a treatise dedicated Prince August of Saxony in 1565. In 1585 he wrote a more comprehensive treatise on various instruments and surveying practices now in the Victoria and Albert Museum.  Lencker also developed his own instruments, which he illustrated and described in his Perspective (1567). Measured representation and systematic volumetric measurement thus went hand in hand.

 

            The development of these universal instruments included the measurement of two-dimensional polygons and three dimensional solids. These methods evolved gradually. In the case of polygons Fin‚8 had outlined several systematic approaches in his booklet of 1544 without instruments.  Implicit in his approach was the principle that different polygons subtended different angles within a circle.9  Danti10 in his commentary to Vignola's Two Rules of Practical Perspective (1583) described how this principle could be applied to a circular surveying instrument, an idea that Coignet11 and others carried out in practice.  The brothers Fabrizio and Gaspare Mordente found another solution.  One could record the diameter of a circle as a line and then mark off the relative lengths of a triangle, square, pentagon, hexagon, etc. inscribed within this circle.12  According to their own account13 they developed this while on a voyage to India in 1554.  It was written up in 1578 and published in 1584. A third approach was to record this information on a sector specifically designed for this purpose. This idea is usually associated with Guidobaldo del Monte14 in Urbino around 1569, although the idea of using lines on a sector dated back to 1509.15  The first published version of the Guidobaldo type sector was by Gallucci16 in 1594 (cf. fig. 26.1-2).

 

            Ever since Antiquity practical concerns had prompted three-dimensional volumetric measurement.  From the thirteenth century onwards the wine trade provided a special stimulus involving the measurement of wine barrels and led to an independent body of literature on gauging.17 Some cities such as Nürnberg and Antwerp even had a gauging master. By the sixteenth century gauging rods specifically designed to measure the volume of barrels had been developed.  These rods usually had lines involving square roots or cube roots. Problems of volumetric measurement also arose in the military with cannonballs of varying sizes and different metals, which led to the development of calipers for these purposes. By the 1550's there were efforts to find a single instrument which would solve all problems of measuring lines, surfaces and volumes.18  Mordente's compass and ruler were an early attempt.  Besides dealing with polygons, they dealt with volumetric measurement of pyramids, cubes, spheres and with the transformation of spheres into cubes.

 

            The reduction compass was also used in these efforts (fig. 26.3).  It had originally been invented in Antiquity.19  Around 1500 it was developed by Leonardo da Vinci who referred to it specifically as a proportional compass.20  In the period 1560-1580 Wilhelm IV further developed this instrument in terms of seven operations, the last of which is of particular interest for our purposes:

 

            1. To divide a given straight line with a given proportion.

            2. To divide a given circular line into various parts.

            3. To multiply or diminish a given surface into a surface of the same shape.

            4. To multiply or diminish a given body into a body of the same shape.

            5. To find the ratio of any diameter to its circumference.

            6. To transform some circular or square surface into another one.

            7. To transform a given sphere and the five regular solids into one another.21

 

            Wilhelm IV was the Landgraf of Hesse and passionately interested in science.  In 1561 he started the world's first modern astronomical observatory at Kassel.  Both Tycho Brahe and Kepler had connections with his court.22  In 1579 Jobst Bürgi joined the court as an instrument maker.  He developed a reduction compass which carried out the operations listed above.23

 

            The seven operations in Wilhelm IV's instrument are the more intriguing because they recur in a manuscript entitled Perspective attributed to Hans Lencker mentioned earlier.24  Although it has the same title as his treatise of 1571 the manuscript contains 47 additional pages of handwritten text and illustrations.  Lencker was based in Nürnberg, but he also travelled.  From 1572 through 1576 he was mainly in Dresden where he taught Prince Christian I at the court of Saxony.  In 1574 Lencker also had commissions for the courts of Munich and Kassel.  It is likely that he would have learned about the Landgraf's manuscript at that time.

 

            The manuscript attributed to Lencker was not simply a copy of the Kassel manuscript.  It listed the same seven operations but then discussed them in connection with both a reduction compass (fig. 26.3) and a sector (fig.27.1).  The text was more detailed.  Some of the diagrams such as those relating to comparative weights of metals were new.  Others relating to volumes of spheres showed principles familiar from the gauging literature.  A number of the diagrams were clearly based on the Kassel manuscript including the illustrations of the regular solids.  In 1604 Levinus Hulsius25 published a report of Bürgi's reduction compass which borrowed diagrams from the manuscript ascribed to Lencker.  In 1605 Horcher published the principles underlying this compass.26  In 1606 Galileo published his own version of the sector27 and claimed precedence for the invention.  So too did others.  This led to a lawsuit. Galileo won. But the debates continued.  Neither the details of these debates nor the contents of the 120 books published on the subject and many related instruments (cf. fig. 27.2) in the century that followed need concern us here.28

 

            Important for our purposes is how these new instruments effectively mechanized the basic processes of the geometric game: two-dimensional quadrature of the circle, three-dimensional cubature of the sphere, problems of doubling the volume of a cube or transforming one regular solid into another were now operations which could be analyzed quantitatively.  They were physical, mechanical problems which could be reduced to numerical ratios and these could happen without the aid of three-dimensional representations.  Hence it was paradoxically the very study of the regular and irregular solids as concrete physical models that brought about a new level of abstraction, which resembled the earlier neo-Platonic interpretation but was in fact fundamentally different because it assumed a new mechanistic view of the universe.  Indeed, where the geometrical game had been an intellectual play of geometrical forms in mediaeval times, it now involved nature itself.

 

3. Models of the Universe

 

            If this change occurred slowly, it is fascinating to note that it involved precisely the individuals whom we have been studying, notably, Pacioli, Leonardo, Jamnitzer, Lencker, and Kepler.  Pacioli was fully aware of Aristotle's objections to Plato and was probably aware of mediaeval debates on the subject.  But whereas his predecessors saw the regular solids as the source of contention, Pacioli interpreted them as a solution to the debate.29  For him the fact that the regular solids could be snugly fitted inside one another presumably resolved the problem of a potential vacuum and it may well be that he began building models in 1489 partly by way of demonstration.  Two decades later when Leonardo da Vinci30 challenged Plato's ideas in the Timaeus, he did so on the basis of experiments with actual solids. He had discovered that pyramids (tetragons) were more difficult to roll than cubes (hexagons): i.e. they were more stable.  For this reason he claimed that the pyramid should symbolize the most stable element earth, while the cube should symbolize fire, thus reversing Plato's order.

 

            While Dürer was very much concerned with physical models of the regular solids he did not discuss how this related to their symbolic nature. Jamnitzer, by contrast, is said to have deliberately improved Dürer's perspectival instrument in order that he could represent the regular and irregular solids more accurately. As we have seen Jamnitzer specifically associated the solids with the elements and the heavens, quoting Plato, but meaning something very different. For whereas Plato was referring to something in the world of ideas, Jamnitzer was concerned with something physical and his three-dimensional record of these physical models was his way of getting at their truth. The models were no longer symbols or even models in the sense of replicas. They corresponded somehow to reality itself.

 

            Indeed it is difficult to imagine what other incentive could have prompted him to go to such pains.  For we are told that both Jamnitzer and Lencker, after they had made their drawings, carefully coloured them and frequently arranged them in striated panels (tabulas striatas).31  Scholars have interpreted this to mean that they employed them for anamorphic effects.32 Of this there is no evidence.  But there is a more obvious interpretation, namely, that the striated panels were panels of inlaid wood.  In this context the close parallels noted earlier between Stoer's manuscript and the inlaid wood of the desk at Frankfurt or the cabinet in Cologne become the more significant.  For if the manuscript has painted strips, the furniture literally has striated panels (tabulas striatas).

 

            If we look more closely at Stoer's painted examples (e.g. pl. 51.1) we discover that the colours are not just ornamental.  They enable us to see and distinguish how various solids are nested within one another.  Contemporary sources report that Jamnitzer and Lencker developed these techniques and that the painted interiors which resulted were so masterfully arranged in their precision and foreshortenings that the sight of them caused hallucinations.33  Whether modern readers will see them as psychedelic pictures is not our concern.  What interests us here is why artists at the time made these tremendous efforts, which becomes understandable when we recognize that these solids were intended to represent the elements of the cosmos, models of the universe which they believed corresponded to reality itself.  Hence the resemblance between these three-dimensional drawings, Kepler's model of the universe in terms of the five regular solids nested within one another and physical models of the spheres (fig. 28.1-3) was no coincidence.  In the minds of the Nürnberg artists these were lessons in cosmology.  Indeed, in one of Stoer's examples we can clearly see a dodecahedron (the heavens) within which is nested an icosahedron (water) and other regular solids (pl. 51.1).

 

            In this context the term striate panels (tabulas striatas) takes on new meaning.  For we find that the term stria 34 recurs in Kepler's description of nature, although the Oxford Dictionary, which defines striate as "marked or scored with striae, showing narrow structural bands, striped, streaked, furrowed," claims that "the earliest examples relate to the hypothesis of Descartes as to the striate or channeled condition of the constituent particles in nature."35  Were these geometrical games of the Nürnberg artists sources for Kepler's and Descartes' geometrical views of nature?  We know that pirate editions of Jamnitzer appeared in 1608, 1618 and 1626 in Amsterdam where Descartes lived.  Did Jamnitzer's work then have a direct effect on Descartes?

 

4. Nature's Models