• Tetrahedron - made of 4 equilateral triangles
   
• Cube - made of 6 squares
   
• Octahedron - made of 8 equilateral triangles
   
• Dodecahedron - made of 12 regular pentagons
   
• Icosahedron - made of 20 equilateral triangles

Johannes Kepler If this [the Mysterium cosmographicum] is published, others will perhaps make discoveries I might have reserved for myself. But we are all ephemeral creatures (and none more so than I). I have, therefore, for the Glory of God, who wants to be recognized from the book of Nature, that these things may be published as quickly as possible. The more others build on my work the happier I shall be.   — Johannes Kepler [Letter to Michael Maestlin (3 Oct 1595)]

His first publication in 1597 was called Mysterium Cosmographicum (literally, the Cosmic Mystery).  Mysterium, Kepler attempted to show that planetary orbits were arranged in accordance with the regular, or Platonic polyhedra . In this way, for example, Kepler argued that the ratio of orbital radii for Venus and Mercury was the same, or nearly so, as the ratio of the radii for the circumscribed and inscribed spheres of an octahedron.

As he indicated in the title, Kepler thought he had revealed God’s geometrical plan for the universe. His subsequent main astronomical works were in some sense only further developments of it, concerned with finding more precise inner and outer dimensions for the spheres by calculating the eccentricities of the planetary orbits within it. In 1621 Kepler published an expanded second edition of Mysterium, half as long again as the first, detailing in footnotes the corrections and improvements he had achieved in the 25 years since its first publication.

Through their letters, Tycho and Kepler discussed a broad range of astronomical problems, dwelling on lunar phenomena and Copernican theory (particularly its theological viability). But without the significantly more accurate data of Tycho's observatory, Kepler had no way to address many of these issues.

Instead, Keplar turned his attention to chronology and "harmony," the numerological relationships among music, mathematics and the physical world, and their astrological consequences. By assuming the Earth to possess a soul (a property he would later invoke to explain how the sun causes the motion of planets), he established a speculative system connecting astrological aspects and astronomical distances to weather and other earthly phenomena.

Mysterium Cosmographicum: the cosmic mystery

Johannes Kepler believed in Copernicus’ picture. Raised in the Greek geometric tradition, he believed God must have had some geometric reason for placing the six planets at the particular distances from the sun that they occupied. He thought of their orbits as being on spheres, one inside the other. One day, he suddenly remembered that there were just five perfect Platonic solids, and this gave a reason for there being six planets - the orbit spheres were maybe just such that between two successive ones a perfect solid would just fit. He convinced himself that, given the uncertainties of observation at the time, this picture might be the right one.

The 5 solids as drawn in Kepler's Mysterium Cosmographicum. A platonic solid (also called regular polyhedra or Pythagorean solid) is a convex polyhedron whose vertices and faces are all of the same type.  There are only five regular figures have many remarkable properties; All of their vertices lie on a sphere. They all have equal edges, equal faces, and equal angles.  That these solids ordered our solar system was Keplar's great initial insight, which informed and supported his later breakthrough discoveries, accords much to geometry as a principle which orders the universe and all its creatures. Although best known as Platonic solids their importance in ordering the universe was first brought to the attention by Pythagoras [570-c. 495 BC] whose education was distinguished by decades of study in Egypt and other ancient centers of learning and initiation.

Some of the greatest achievements of Plato's Academy came in the areas of mathematics and astronomy. Heraclides Ponticus discovered the axial revolution of the earth and the revolution of Venus and Mercury around the sun. Eudoxus gave us the theory of proportion, the method of exhaustion, and the concentric spheres of the Ptolemaic cosmology. Theaetetus was the inventor of solid geometry and the general theory of incommensurables. He also constructed the regular polyhedra, demonstrated that each of them could be inscribed in a sphere, and proved that there could only be five. During a trip to Sicily in 367 B.C.E., Plato may have first learned of the regular polyhedra from Archytas, the last of the Pythagoreans. In the Timaeus Plato used the regular polyhedra as the basic elements of the universe. As a brilliant anticipation of modern physics, the "receptacle" of the Timaeus is combination of space, matter, and energy. Classical physics followed the Greek atomists in separating matter and energy, but modern physics of the 20th century has had to return to the inseparability first proposed by Plato.

Kepler's geometric scheme for the relative orbits of the planets in our visible solar system which eventually led to his formulation of the Laws of Planetary motion. Kepler's first book; The Cosmic Mystery argued that the distances of the planets from the Sun in the Copernican system were determined by the five regular solids, if one supposed that a planet's orbit was circumscribed about one solid and inscribed in another. Except for Mercury, Kepler's construction produced remarkably accurate results. Tycho Brae impressed with Kepler's talent and insight as a mathemetician asked Kepler to calculate new orbits for the planets from Tycho's observations. Kepler moved to Prague in 1600. According to Keplar, the orbits of the six planets known at the time—Mercury, Venus, Earth, Mars, Jupiter and Saturn—could be arranged in spheres nested around the five Platonic solids: octahedron, icosahedron, dodecahedron, tetrahedron and cube. The Platonic polyhedra arranged in this order, coinciding circumspheres for a given polyhedron and inspheres for the next polyhedron gave a fair approximation for the relative sizes of planetary orbits around the Sun. Kepler never rejected this model but his strive for increased accuracy and explanations for variances would occupy his career. Kepler was convinced  "that the geometrical things have provided the Creator with the model for decorating the whole world." In the tradition of the legendary Greek philosopher Pythagoras (6th century BC), Kepler did not view science and spirituality as mutually exclusive, he saw his work reconciling the emerging vision of a Sun-centred planetary system with the ancient Pythagorean concept of armonia, or universal harmony.

 Pythagoras discovered that the pitch of a musical note depends upon the length of the string which produces it. This allowed him to correlate the intervals of the musical scale with simple numerical ratios. When a musician plays a string stopped exactly half-way along its length an octave is produced. The octave has the same quality of sound as the note produced by the unstopped string but, as it vibrates at twice the frequency, it is heard at a higher pitch. The mathematical relationship between the keynote and its octave is expressed as a 'frequency ratio' of 1:2. In every type of musical scale, the notes progress in a series of intervals from a keynote to the octave above or below. Notes separated by intervals of a perfect fifth (ratio 2:3) and a perfect fourth (3:4) have always been the most important 'consonances' in western music. In recognizing these ratios, Pythagoras had discovered the mathematical basis of musical harmony.

The musicologist Joscelyn Godwin comments, "...the celestial harmony of the solar system... is of a scope and harmonic complexity that no single approach can exhaust. The nearest one can come to understanding it as a whole is to consider some great musical work and think of the variety of analytical approaches that could be made to it, none of them embracing anything like the whole."

In Harmony, Keplar attempted to explain the proportions of the natural world—particularly the astronomical and astrological aspects—in terms of music. The central set of "harmonies" was the musica universalis or "music of the spheres," which had been studied by Pythagoras, Ptolemy and many others before Kepler. Soon after publishing Harmonices Mundi, Kepler was embroiled in a celebrated exchange with Robert Fludd, who had recently published his own harmonic theory. Their ideas continue to resonate in the 21st century as the call grows louder for a recognition of the value of their sort of alchemical or intuitive science that is capable ofn embracing the concept of multiple levels of soul and its calling card of synchronicity.

"Geometry is unique and eternal, a reflection from the mind of God. That mankind shares in it is because man is an image of God." ---Johannes Kepler   4000 year old Carved Stone Balls with symmetrical form of the five Platonic solids

Five of nearly 400 Neolithic Stone Balls (ca. 2,000 BCE) now kept at Ashmolean Museum at Oxford University.

These rounded stones with regularly spaced bumps are from largely from Northeast Scotland. Some degree a high level of craftsmanship and their use is not known and it may have been oracular or as venerated ritual art objects. High points of each bump mark the vertices of each of the regular polyhedra. The stone balls also appear to demonstrate the duals of three of the regular polyhedra. The lack of balls found in graves may indicate that they were not considered to belong to individuals. 'Sink stones' found in Denmark and Ireland have some slight similarities, these artifacts being used in conjunction with fishing nets.

1600: Tycho hires Kepler
 

By 1599, however, Keplar again felt his work limited by the inaccuracy of available data—just as growing religious tension was also threatening his continued employment in Graz. Because of his talent as a mathematician, displayed in this volume, Kepler was invited by Tycho Brahe to Prague to become his assistant and calculate new orbits for the planets from Tycho's observations in December of that year. Tycho invited Kepler to visit him in Prague; on January 1, 1600 (before he even received the invitation), Kepler set off in the hopes that Tycho's patronage could solve his philosophical problems as well as his social and financial ones.

On August 2, 1600, after refusing to convert to Catholicism, Kepler and his family were banished from Graz. Several months later, Kepler returned, now with the rest of his household, to Prague. Through most of 1601, he was supported directly by Tycho, who assigned him to analyzing planetary observations and writing a tract against Tycho's deceased rival, Ursus. In September, Tycho secured him a commission as a collaborator on the new project he had proposed to the emperor: the Rudolphine Tables that should replace the Prutenic Tables of Erasmus Reinhold.

Kepler was Imperial Mathematician, the most prestigious appointment in mathematics in Europe until 1612, when his patron, Emperor Rudolph II was deposed. In Prague Kepler published a number of important books. In 1604 Astronomia pars Optica ("The Optical Part of Astronomy") appeared, in which he treated atmospheric refraction but also treated lenses and gave the modern explanation of the workings of the eye; in 1606 he published De Stella Nova ("Concerning the New Star") on the new star that had appeared in 1604; and in 1609 his Astronomia Nova ("New Astronomy") appeared, which contained his first two laws (planets move in elliptical orbits with the sun as one of the foci, and a planet sweeps out equal areas in equal times). Whereas other astronomers still followed the ancient precept that the study of the planets is a problem only in kinematics, Kepler took an openly dynamic approach, introducing physics into the heavens.

In 1610 Kepler heard and read about Galileo's discoveries with the spyglass. He quickly composed a long letter of support which he published as Dissertatio cum Nuncio Sidereo ("Conversation with the Sidereal Messenger"), and when, later that year, he obtained the use of a suitable telescope, he published his observations of Jupiter's satellites under the title Narratio de Observatis Quatuor Jovis Satellitibus ("Narration about Four Satellites of Jupiter observed"). These tracts were an enormous support to Galileo, whose discoveries were doubted or denied by many. Both of Kepler's tracts were quickly reprinted in Florence. Kepler went on to provide the beginning of a theory of the telescope in his Dioptrice, published in 1611.

During this period the Keplers had three children (two had been born in Graz but died within months), Susanna (1602), who married Kepler's assistant Jakob Bartsch in 1630, Friedrich (1604-1611), and Ludwig (1607-1663). Kepler's wife, Barbara, died in 1612. In that year Kepler accepted the position of district mathematician in the city of Linz, a position he occupied until 1626. In Linz Kepler married Susanna Reuttinger. The couple had six children, of whom three died very early.

In Linz Kepler published first a work on chronology and the year of Jesus's birth, In German in 1613 and more amply in Latin in 1614: De Vero Anno quo Aeternus Dei Filius Humanam Naturam in Utero Benedictae Virginis Mariae Assumpsit (Concerning the True Year in which the Son of God assumed a Human Nature in the Uterus of the Blessed Virgin Mary"). In this work Kepler demonstrated that the Christian calendar was in error by five years, and that Jesus had been born in 4 BC, a conclusion that is now universally accepted. Between 1617 and 1621 Kepler published Epitome Astronomiae Copernicanae ("Epitome of Copernican Astronomy"), which became the most influential introduction to heliocentric astronomy; in 1619 he published Harmonice Mundi ("Harmony of the World"), in which he derived the heliocentric distances of the planets and their periods from considerations of musical harmony. In this work we find his third law, relating the periods of the planets to their mean orbital radii.

In 1615-16 there was a witch hunt in Kepler's native region, and his own mother was accused of being a witch. It was not until late in 1620 that the proceedings against her ended with her being set free. At her trial, her defense was conducted by her son Johannes.

1618 marked the beginning of the Thirty Years War, a war that devastated the German and Austrian region. Kepler's position in Linz now became progressively worse, as Counter Reformation measures put pressure on Protestants in the Upper Austria province of which Linz was the capital. Because he was a court official, Kepler was exempted from a decree that banished all Protestants from the province, but he nevertheless suffered persecution. During this time Kepler was having his Tabulae Rudolphinae ("Rudolphine Tables") printed, the new tables, based on Tycho Brahe's accurate observations, calculated according to Kepler's elliptical astronomy. When a peasant rebellion broke out and Linz was besieged, a fire destroyed the printer's house and shop, and with it much of the printed edition. Soldiers were garrisoned in Kepler's house. He and his family left Linz in 1626. The Tabulae Rudolphinae were published in Ulm in 1627.

Kepler lived in an era when there was no clear distinction between astronomy and astrology, but there was a strong division between astronomy (a branch of mathematics within the liberal arts) and physics (a branch of natural philosophy). Kepler also incorporated religious arguments and reasoning into his work, motivated by the religious conviction that God had created the world according to an intelligible plan that is accessible through the natural light of reason. Kepler's most important inspiration was a model to explain the relative distances of the planets from the Sun in the Copernican System.

Johannes Kepler was born in Weil der Stadt in Swabia, in southwest Germany. His paternal grandfather, Sebald Kepler, was a respected craftsman who served as mayor of the city; his maternal grandfather, Melchior Guldenmann, was an innkeeper and mayor of the nearby village of Eltingen. His father, Heinrich Kepler, was "an immoral, rough and quarrelsome soldier," according to Kepler, His father left the family when Johannes was five years old and is believed to have died in the Eighty Years' War in the Netherlands. His mother Katharina Guldenmann, an inn-keeper's daughter, was a healer and herbalist. From 1574 to 1576 Johannes lived with his grandparents; in 1576 his parents moved to nearby Leonberg, where Johannes entered the Latin school. In 1584 he entered the Protestant seminary at Adelberg, and in 1589 he began his university education at the Protestant university of Tübingen. Here he studied theology and read widely. He passed the M.A. examination in 1591 and continued his studies as a graduate student.

 

On 19th July 1595, a sudden revelation changed the course of Kepler's life. In preparation for a geometry class he had drawn a figure on the blackboard of an equilateral triangle within a circle with a second circle inscribed within it. He realized that the ratio of the two circles replicated the ratio of the orbits of Jupiter and Saturn. In a flash of inspiration, he saw the orbits of all the planets around the Sun arranged so that regular geometric figures would fit neatly between them. He tested this intuition using two-dimensional plane figures — the triangle, square, pentagon, etc. — but this didn't work. As space is three-dimensional, he went on to experiment with three-dimensional geometric solids.  Life had taken Kepler by surprise and ordered he move toward the unknown.

For six years, Kepler taught arithmetic, geometry, Virgil, and rhetoric. In his spare time he pursued his private studies in astronomy and astrology. In 1597 Kepler married Barbara Müller. In that same year he published his first important work, The Cosmographic Mystery, in which he argued that the distances of the planets from the Sun in the Copernican system were determined by the five regular solids, if one supposed that a planet's orbit was circumscribed about one solid and inscribed in another. Except for Mercury, Kepler's construction produced remarkably accurate results. 

Kepler was the first to state clearly that the way to understand the motion of the planets was in terms of some kind of force from the sun. However, in contrast to Galileo, Kepler thought that a continuous force was necessary to maintain motion, so he visualized the force from the sun like a rotating spoke pushing the planet around its orbit.

... retreat before the general ignorance and not to expose ourselves or heedlessly to oppose the violent attacks of the mob of scholars and in this you follow Plato and Pythagoras our true perceptors. But after a tremendous task has been begun in our time first by Copernicus and then by many very learned mathematicians and when the assertion that the Earth moves can no longer be considered something new would it not be much better to pull the wagon to its goal by our joint efforts now that we have got it under way and gradually with powerful voices to shout down the common herd which really does not weigh the arguments very carefully?    --- Kepler to Galileo [more] I wish, my dear Kepler, that we could have a good laugh together at the extraordinary stupidity of the mob. What do you think of the foremost philosophers of this University? In spite of my oft-repeated efforts and invitations, they have refused, with the obstinacy of a glutted adder, to look at the planets or Moon or my telescope. ---Galileo to Kepler O telescope, instrument of knowledge, more precious than any sceptre. — Johannes Kepler Letter to Galileo (1610). Galileo and Kepler corresponded and Kepler was among Galileo's most prominent public defenders, but while Galileo defended Copernican astronomy he never wrote about Kepler's model.Galileo may have been repelled by Kepler's mysticism as he ridiculed Kepler's correct speculation on tidal action being influenced by the moon as "occult phenomena." More likely, Galileo, as a one of science's greatest self-promoters, was not inclined to share credit in his field with any contemporary.

"Among the great men who have philosophized about [the action of the tides], the one who surprised me most is Kepler. He was a person of independent genius, [but he] became interested in the action of the moon on the water, and in other occult phenomena, and similar childishness. I wish, my dear Kepler, that we could have a good laugh together at the extraordinary stupidity of the mob. What do you think of the foremost philosophers of this University? In spite of my oft-repeated efforts and invitations, they have refused, with the obstinacy of a glutted adder, to look at the planets or Moon or my telescope. --- Galileo Galilei Galileo Galilei (1564-1642) Galileo's most original contributions to science were in mechanics where he helped clarify concepts of acceleration, velocity, and instantaneous motion. One of the first to record observations from the recently invented telescope, was an aggressive popularizer of Copernican viewpoint and satirist of Aristotelian physics.

Newton probably believed, like most alchemists, that he was rediscovering the lost knowledge of Moses and of Hermes Trismegistus. These were the legendary sages to whom it was believed had been revealed divine knowledge. Both lived in Egypt, where much of their learning had been preserved in the Great Library of Alexandria, until its destruction. That knowledge had since been steadily lost over time; the ancient Greeks had been a 'golden age' when much of it had still been preserved, but since their time more and more had been lost. For this reason, alchemy had become a library-research discipline, not one given to experiment. Knowledge of the 'hidden arts' was to be obtained through the discovery and deciphering of lost and mysterious ancient texts. Newton, of course, took the opposite tack - although he accepted that he was rediscovering lost knowledge, he meticulously checked and noted down his findings by means of chemical experiment. The brazier in his room burned constantly for years at a time. More serious alchemists adopted codenames for their correspondence. Thus, Newton dubbed himself Jeova Sanctus Unus - Latin for 'One Holy God', and simultaneously a nod to Newton's Arian beliefs (that God was a single, indivisible entity and not a Trinity) and a near-anagram of his Latinised name, Isaacus Neuutonus. It was under this moniker that Newton corresponded with other members of the Hartlib Circle, including Henry More and Boyle.

In the beginning of the year 1665 I found the Method of approximating series & the Rule for reducing any dignity of any Bionomial into such a series. The same year in May I found the method of Tangents of Gregory & Slusius, & in November had the direct method of fluxions & the next year in January had the Theory of Colours & in May following I had entrance into ye inverse method of fluxions. And the same year I began to think of gravity extending to ye orb of the Moon & (having found out how to estimate the force with wch [a] globe revolving within a sphere presses the surface of the sphere) from Keplers rule of the periodic times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about wch they revolve: & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them answer pretty nearly. All this was in the two plague years of 1665-1666. For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more then than at any time since. — Sir Isaac Newton Quoted in Richard Westfall, Never at Rest: A Biography of Isaac Newton (1980), 143.

Isaac Newton initiated classical mechanics in physics. Built in part on Kepler's concept of Sun as center of solar system, planets move faster near Sun. Inverse-square law. Once law known, can use calculus to drive Kepler's Laws. Unification Kepler's Laws; showed their common basis.

Albert Einstein

"Imagination is more important than knowledge."

"But before mankind could be ripe for a science which takes in the whole of reality, a second fundamental truth was needed, which only became common property among philosophers with the advent of Kepler and Galileo. Pure logical thinking cannot yield us any knowledge of the empirical world; all knowledge of reality starts from experience and ends in it. Propositions arrived at by purely logical means are completely empty as regards reality. Because Galileo saw this, and particularly because he drummed it into the scientific world, he is the father of modern physics -- indeed, of modern science altogether."
-Albert Einstein, Ideas and Opinions

"Cosmology -The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built by pure deduction "

'I believe in Spinoza's God who reveals himself in the orderly harmony of what exists, not in a God who concerns himself with the fates and actions of human beings.'

"The release of atom power has changed everything except our way of thinking...the solution to this problem lies in the heart of mankind. If only I had known, I should have become a watchmaker."  ---Albert Einstein

"Education is what remains after one has forgotten everything he learned in school." 
 -Albert Einstein