# The Platonic Solids

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In Greek antiquity, it was already known that there are five and only five regular polyhedra, that is, three-dimensional geometrical figures with all identical equilateral faces, which are also called "Platonic" solids. A cube is the most common example of a regular solid. How can one tell that there are four and only four possible others?

Start by thinking about how one might construct a regular solid, starting with the sides around one point. There have to be at least three sides; otherwise, a three-dimensional body cannot be formed. Arrange three squares around a point, and then fold them up to form a three-sided figure, which is half a cube. An identical figure attached to the first will complete the cube with six sides. Equilateral triangles will fold up more tightly, leaving a space at the top the same size as the other sides. Attach one more side, and one has a tetrahedron with four faces. Pentagons will fold up like a shallow dish, but additional pentagons can be attached to the edges. If one adds more sides to these and then more sides to the next set of edges, a dodecahedron with twelve identical pentagonal faces eventually will be formed. Hexagonal faces will not work. Three hexagons meet in a flush plane, and they cannot be folded up to form the sides of a solid.

If we go back to triangles, we see that we could try four equilateral triangles around a point. Fold them up, and one has a pyramid shape. An identical pyramid joined to the top forms a regular octahedron with eight faces. Five equilateral triangles will also fit around a point. Fold them up, and the figure is very shallow, but if one keeps adding on sides, a regular twenty-sided icosahedron eventually will be formed. Six equilateral triangles will form a flush plane which cannot be folded into a three-dimensional figure. Four squares will also form a flush plane. Nor can any other combination of regular polygons be fit around a single point. Therefore, these are the only possible regular solids.

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The use of plane geometry was unsatisfactory, and he quickly realized that he would have to use solid geometry. The universe is three dimensional, after all. With three dimensions, he would have to work with spheres instead of circles and regular solids instead of polygons. It had been known to mathematicians since antiquity that there are five and only five regular solids, the tetrahedron (four-sided), the cube (six-sided), the octahedron (eight-sided), the dodecahedron (twelve-sided), and the icosahedron (twenty-sided). As soon as Kepler remembered that, the answer became clear to him. Later, in the Preface to his book Mysterium cosmographicum, he quoted the proposition just as it had come to him at that moment:

The earth's circle is the measure of all things. Circumscribe a dodecahedron around it. The circle surrounding it will be Mars. Circumscribe a tetrahedron around Mars. The circle surrounding it will be Jupiter. Circumscribe a cube around Jupiter. The surrounding circle will be Saturn. Now, inscribe an icosahedron inside the earth. The circle inscribed in it will be Venus. Inscribe an octahedron inside Venus. The circle inscribed in it will be Mercury.

This detail of a plate from Kepler's Harmonice mundi shows the construction of the Platonic solids: the tetrahedron (top left), the octahedron (Oo), icosahedron (Pp), cube (Qq), and dodecahedron (Rr).

The spacing of the planets within the polyhedra seemed just about right. More importantly, Kepler knew immediately why there are six and only six planets. Since there were only five possible regular polyhedra, they could be inscribed between only six different spheres. The discovery he made on July 20, 1595, was so profound he wept tears of joy. As he wrote in a letter to Maestlin, he regarded his discoveries as "stupendous miracles of God."

By October 1595, Kepler had resolved to publish his findings in a book. It would be, as he saw it, a physical proof of the truth of Copernicus s heliocentric system and, at the same time, a testament to God's glory. In so making known God's plan of the world, Kepler found a way to make meaningful the assignment he had been given to become a mathematician. As he wrote in a letter to Maestlin at the beginning of October,

I am in haste to publish, dearest teacher, but not for my benefit ... I am devoting my effort so that these things can be published as quickly as possible for the glory of God, who wants to be recognized from the Book of Nature . . . Just as I pledged myself to God, so my intention remains. I wanted to be a theologian, and for a while I was anguished. But now, see how, God is also glorified in astronomy through my work.

There would be many details to be ironed out before he was ready. Among other things, there was another fundamental question about the Copernican system to address: why did the planets have their particular periods? Here, Kepler's thinking took a very important turn. Ever since he was a student, he had thought that the reason the nearer planets go around faster was their proximity to the sun, which is somehow the source of the force that makes them go around. Now, he tried to derive a mathematical formula based on his physical intuition that would relate the planets' periods to their distances. There were two effects to take into account. The first is just geometry: the further a planet is from the sun, the longer its orbit will be and the longer it will take to get around. But in addition, the further away it is, the weaker the planet-moving force will be. So he added these effects to come up with the formula: from one planet to another, the increase in the period will be twice the difference of their distances. He himself later realized the formula was incorrect, but remarkably it yielded planetary distances that were similar to those derived from the polyhedral hypothesis. Again, he wept tears of joy and excitedly wrote to Maestlin about his new hypothesis, "Behold how near the truth I have come!"

Kepler sent his first outline of the two main arguments he would include in his book to Maestlin in October 1595. Throughout that cold winter, he filled out that outline with a number of auxiliary arguments. Since the polyhedral hypothesis was founded on the idea that God had rationally structured the universe based on the five regular solids, Kepler turned his attention to seeing what meaning he could discern in the particular arrangement of the solids. In the process, he ended up having a lot more to say about the polyhedral hypothesis than the planet-moving force hypothesis, but he did come up with one additional argument based on the planet-moving force hypothesis that would be extremely influential in his later thinking about planetary theory.

Around March 1596, when he was putting the finishing touches on his manuscript, he noticed a very interesting application of the planet-moving force hypothesis. Previously, he had only seen the planet-moving force as a way to relate the periods and distances of different planets to one another. After some more thought, he realized that it could be applied to a single planet as it moved on its own orbit around the sun. As the planet approached nearer to the sun, the planet-moving force would be stronger and the planet would move more quickly. Later on in its orbit, as it receded from the sun, the force would be weaker and the planet would slow down. This general change of speed of a planet with its distance from the sun had been built into

Ptolemy's and Copernicus's mathematical models of the motion of the planets, but neither of them had interpreted this change in speed physically.

This idea was, in fact, the single element of the book that would worry Maestlin. He later admonished Kepler not to make too much of this planet-moving force hypothesis "lest it should lead to the ruin of astronomy." What troubled Maestlin was that Kepler seemed to be trampling on a delicate division line between two parts of astronomy. In the sixteenth century, astronomy was widely regarded to consist of a physical part, which dealt with the nature and structure of the universe, otherwise known as cosmology, and a mathematical part, devoted to producing accurate mathematical theories of the planets' motion. Everything else in Kepler's book seemed to fall into the physical part. But by saying that his planet-moving force could explain certain mathematical details of Ptolemy's and Copernicus's planetary theories, Kepler seemed to be importing physical reasoning into mathematical astronomy. As far as Maestlin was concerned, it seemed this would only mess up the theories of the planets.

In January 1596, Kepler received word from home that both his grandfathers were ailing, and at the end of the month, he left Graz to visit them. Sadly, old Sebald died during his visit home. While Kepler was in Württemberg, he took the opportunity to promote his new hypotheses. In February, he traveled up to the capital, Stuttgart, to try his luck at the ducal court.

The aristocracy were patrons of science and the arts generally, but Kepler had a curiosity to market: a model of his new system of nested polyhedra in silver. Or, if something really splashy was desired, Kepler outlined how the model could be realized in the form of a huge punch bowl. The spaces between the different planetary spheres could be filled with various beverages, and by means of hidden pipes and valves, the party guests could fill their glasses from seven taps spaced around the rim. The duke was skeptical at first, but after seeing a paper model Kepler had painstakingly constructed and consulting with his astronomical expert (Maestlin), he advanced Kepler some money to fabricate the more restrained silver model.

The next three months were a frustrating disaster. Kepler was stuck in Stuttgart pestering the goldsmith, and the project hardly got anywhere. In the end, he had to go back to Styria, leaving the project in the goldsmith's hands. Although the matter dragged out for a few years, the entertaining model of Kepler's polyhedra was never built. It would have been wondrous to see.

In the meantime, Kepler had the opportunity to travel to Tübingen, visit with Maestlin, and begin negotiations with a printer to publish his book. None of the printers back in Graz were competent to print a complex astronomical book, but Tübingen had a serious printer named Gruppenbach. Gruppenbach agreed to publish the book on condition that it be approved by the university senate. The senate asked Maestlin for his expert opinion of the astronomical content, and he responded enthusiastically. The only part that the theological faculty demanded to be removed was Kepler's chapter on how to reconcile heliocentrism with passages in the bible that seemed to support geocentrism, such as Psalm 104:5, which states that God "laid the foundations of the earth that it should not be removed for ever." The real meaning of Holy Scripture was not Kepler's business. As he was admonished in a letter from Matthias Hafenreffer, a professor of theology, Kepler was to restrict himself to "playing the part of the abstract mathematician." It was frustrating to Kepler, since he had conceived his work as a physical proof of the truth of heliocentrism. How was he to glorify God speaking only hypothetically? But he obediently went along with the Lutheran authorities.

When Kepler returned to Graz in August 1596, there was some damage to repair from his long absence. To begin with, he had received leave for two months, and he had

These miniature portraits ofJohannes Kepler and his wife Barbara date from around the time of their wedding in 1597.

been absent for seven. But he carried a letter from the duke of Württemberg asking for Kepler's superiors' forgiveness since Kepler had been delayed in his service. This was excuse enough. Unfortunately, Kepler's neglect of his love life would not prove so easy to repair.

As early as the previous December, Kepler had made the acquaintance of a young woman with whom he quickly fell in love. Her name was Barbara Müller. Among other things, we know that she was pretty, plump, and extremely fond of cooked tortoise. She was the eldest daughter of a wealthy mill owner and entrepreneur, Jobst Müller, who resided on an estate about two hours south of Graz. Although she was only 23, Barbara had recently become a widow for the second time. Both of Barbara's previous husbands had been significantly older than she was—both were 40—which was not an uncommon state of affairs in days when family and community played so great a role in determining whom one was to marry. An older man would have shown his capacity to be a success and to provide for his family. By contrast, Kepler was scarcely 24 when he began to woo her. Although he had a university education, he was still only a schoolteacher with unknown prospects. It would not prove easy to convince Herr Müller that Kepler was a suitable match for her. Herr Müller was a businessman who kept his eye on the bottom line. Barbara had financial assets. Kepler was a penniless scholar.

Probably as early as January 1596, a delegation of respectable members of the Protestant community was assembled to present and recommend Kepler to Jobst Müller as a suitor for Barbara. Kepler left his matrimonial affairs in their hands when he left for his long trip to Württemberg. In June, during his stay there, he received word that they had been successful. He was advised to hurry home, but not before purchasing silk (or at least double taffeta) wedding clothes for himself and his fiancee on the way in Ulm.

As Kepler's failing attempt to construct the model of his celestial discovery dragged out through the summer, the arrangements for the wedding also fell through. In his absence, Herr Müller had become convinced that he could do better for his daughter. When Kepler returned in the fall, he learned that his longed-for union had been canceled. Fortunately, he received support from his school and church, which weighed in on his behalf. Before he had left for Württemberg, he had given Barbara his word. By the middle of January, Kepler appealed to the church: either it must get involved and convince Barbara's father, or Kepler needed to be released from his promise. In short order, the church had set things right again. A solemn promise of marriage was celebrated on February 9 and the wedding on April 27, 1597.

For a while at least, joy reigned supreme in the Kepler household. Kepler received a silver cup as a wedding gift from the school authorities, as well as a raise of 50 florins, to 200 florins a year, to accommodate his move out of the school grounds. Kepler loved his seven-year-old stepdaughter Regina. Barbara quickly became pregnant and bore him a son on February 2, 1598. He was christened Heinrich, Kepler's father's and brother's name. Kepler cast a horoscope for his firstborn son. He would be like his father, only better—charming, noble in character, nimble of body and mind, with mathematical and mechanical aptitude. It was a crushing blow when after only two months of life, his little son Heinrich became ill and passed away. "The passage of time does not lessen my wife's grief," Kepler wrote, quoting Ecclesiastes, "the passage strikes at my heart: 'O vanity of vanities, and all is vanity.'"

The first happy days of Kepler's marriage saw the arrival of the first copies of his book, whose complicated printing was not finished until March 1597. Although the volume was slim, its title was long. It read Prodromus dissertationum cosmographicarum, continens mysterium cosmographicum, de admirabili proportione orbium coelestium, deque causis coelorum numeri, magnitudinis, motuumque periodicorum genuinis & propri-js, demonstratum per quinque regularia corpora geometrica, or in English, The Forerunner of Cosmographical Essays, Containing the Cosmographical Secret: On the Marvelous Proportion of the Celestial Spheres, and on the True and Particular Causes of the Number, Size, and Periodic Motions of the Heavens, Demonstrated by Means of the Five Regular Geometric Bodies. It is known by the abbreviated Latin title, the Mysterium cosmographicum, which translates roughly as The Secret of the Universe. Kepler called it a "forerunner" because he foresaw writing a series of treatises on the Coper-nican system. This book contained his premier discovery, and so he wanted to get it out first and see how people responded to it.

He now began sending copies of the book to astronomers for their opinions. The two copies he dispatched blindly to Italy found their way into the hands of a then little-known mathematics professor at the University of Padua. The man confided to Kepler in a letter that he too had been a Copernican for many years and had been collecting physical proofs of the motion of the earth but had kept them to himself, "terrified as I am by the fortune of our teacher Copernicus himself, who although he earned

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