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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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