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# Kepler's Cosmographic Mystery

Heliocentrism and the Platonic Solids

by

Tweet## Steffen Mathis

on 10 April 2013#### Transcript of Kepler's Cosmographic Mystery

Heliocentrism and the Platonic Solids Kepler's Cosmographic Mystery Kepler's Three Questions 1)Why are there the number of planets that there are?

2)Why are the planets spaced as they are?

3)Why do they move with the speeds they do? The polyhedral hypothesis: -Kepler set out to prove Copernicus' model of the universe. He asked if there was an ordered sequence of the solids that would preserve the known relative distances from the sun. In Copernicus' system, the relative sizes of the orbits were fixed.

-Kepler proposed his own sun-centered solar system in Mysterium Cosmographicum, first published in 1596.

-For Kepler, nested polyhedra incribed and circumscribed inside and around the orbits of the planets answered his first two questions.

-Kepler first had the empirical data and then tried to fit it to an aesthetic account. Plato's Cosmos Kepler was influenced by the cosmos that Plato described in his work Timaeus.

The outermost boundary of Plato's cosmos was spherical and turned around the Earth every 24 hours.

The zodiac was inclined at an angle to the the axis of rotation of the celestial sphere.

The planets move in regular uniform motions in circular paths.

Plato believed that the triangles of the elements combined to make different substances. Kepler's Platonic Cosmos -Kepler ordered the polyhedra based on certain relationships to the sphere. The mathematical relationship of each polyhedron to the sphere represents the distinction between Creator and created. The Mean Sun -Copernicus used the 'mean' position of the sun in his calculations. This assumed a stationary point near the sun as well as a circular orbit for the sun.

-Kepler instructed Maestlin to calculate the position of the true sun and use this data for his model. "This important modification went against the whole history of astronomy," (Martens 46). Although this did not provide a much closer fit for the polyhedral hypothesis, it did pave the way for Kepler's use of empirical data to develop his laws of motion.

-Kepler wanted to bring astronomy into cosmogony with calculations based on empirical data. He made a distinction between himself as a cosmogonist and Copernicus as an astronomer. The Five Platonic Solids Nested Polyhedra The Order of Kepler's Solids Why do the planets move at the speed they do? Kepler's Three Laws Kepler's Accomplishments Contemporary Applications Kepler Said: "My aim is to show that the heavenly machine is not a kind of divine, live being, but a kind of clockwork ... insofar as nearly all the manifold motions are caused by a most simple magnetic, and material force, just as all motions of the clock are caused by a simple weight. And I also show how these physical causes are to be given numerical and geometrical expression." Letter to J.G. Herwart von Hohenburg, Feb. 16, 1605 Cube: 6 squares, EARTH

Octahedron: 8 equilateral triangles, AIR

Tetrahedron: 4 equilateral triangles, FIRE

Icosahedron: 20 equilateral triangles, WATER

Dodecahedron: 12 equilateral triangles, COSMOS Octahedron, Icosahedron,

Dodecahedron, Tetrahedron, Cube

Primary Class: Cube, Tetrahedron, Dodecahedron. Each of these have a different type of face. The vertices of each of these solids join three faces. Stands on a face.

Secondary Class: Octahedron, Icosahedron

These have the same type of face, a triangle. Their vertices join four or five faces. Stands on a vertex.

-The Earth's orbit divides the primary and secondary solids. -Gregory, Frederick. Natural Science in Western History.

-Martens, Rhonda. Kepler's Philosophy and the New Astronomy.

-Zakai, Avihu. "The Rise of Modern Science and the Decline of Theology as the 'Queen of Sciences' in the Early Modern Era."

-http://www.keplersdiscovery.com/MeanSun.html -Plato was also aware that the changes in Mars's orbit may be apparent rather than actual. It has also been reported that Copernicus did suggest that the planets may move in elliptical orbits, but this suggestion was conspicuously absent from his De Revolutionibus where it is stated it appeared in III.4 in a pre-printer manuscript.

-Kepler hypothesized that the sun must be involved in the motion of the planets and found that an ellipse fit the data for the orbit of Mars. Kepler's first and second laws were formulated from this conclusion. He was able to keep his system both aesthetic and mathematically pleasing.

-In 1618, Kepler wrote The Harmony of the World. He began applying the ratios of whole numbers found in the musical scale to the varying angular velocities of the planet as they go around the sun.

-From exploring the ratios among whole numbers, he was able to explain the relationship between the periods of revolution to their mean distance from the sun, Kepler's Third Law. -Geometric and aesthetic order preserved, like Plato.

-Quantity is a form of matter and the source of its definition.

-The archetypes (forms) are final causes in their divine state, formal causes in their material state, and God created physical forces such that the movements would express them. The correspondence between the final, formal, and efficient causes is a 3-tiered universe. The geometrical links the physical to the archetypical by serving as a spatial translation of the divine. Kepler called this "architectonic harmony," (Martens 49). 1) The planets go around the sun in elliptical orbits and they do so because the sun governs them. The sun is at one focus of the ellipse.

2)The line joining the planet to the sun sweeps out equal areas in equal times as the planet travels around the ellipse.

3) The squares of the periods of revolution of the planets were directly proportional to the cubes of their mean distances from the sun. This "mean distance" can also be described as the semi-major axis (i.e. the radius of the orbit of the planet at its two furthest points). -Kepler created a "celestial physics" and laid the foundations for Newton's work by developing what later came to be known as his laws of nature.

-Astronomy became integrated into cosmology by the efforts of Kepler to maintain that the universe was created according to a Divine plan and that man was able to understand this plan through the expression of the mind of God in the geometric relationships in Nature.

-Kepler combined the archetypal world of Plato with the use of empirical data. His work yielded to his empirical observations while maintaining a belief in a Divinely organized cosmos. -Kepler's work in establishing laws of nature greatly influenced Newton. Newton used Kepler's third law to conclude that an object's force (and therefore also the acceleration) would diminish proportionately to 1/r-squared, i.e. to formulate his inverse square law.

-Dense particle packings in engineering and materials studies use the Platonic and Archimedean solids. Kepler's sphere conjecture of the Platonic solids (his model of the spacing of the planets based on polyhedron's inscribed and circumscribed within or around spheres) leads to the densest packing.

-Is the universe itself a dodecahedron?:

http://physicsworld.com/cws/article/news/2003/oct/08/is-the-universe-a-dodecahedron

Some scientists in US and France claimed that a universe with the same shape as the twelve-sided polygon can explain measurements of the cosmic microwave background – the radiation left over from the big bang – that spaces with more mundane shapes cannot (J-P Luminet et al. 2003 Nature 425 593).

Full transcript2)Why are the planets spaced as they are?

3)Why do they move with the speeds they do? The polyhedral hypothesis: -Kepler set out to prove Copernicus' model of the universe. He asked if there was an ordered sequence of the solids that would preserve the known relative distances from the sun. In Copernicus' system, the relative sizes of the orbits were fixed.

-Kepler proposed his own sun-centered solar system in Mysterium Cosmographicum, first published in 1596.

-For Kepler, nested polyhedra incribed and circumscribed inside and around the orbits of the planets answered his first two questions.

-Kepler first had the empirical data and then tried to fit it to an aesthetic account. Plato's Cosmos Kepler was influenced by the cosmos that Plato described in his work Timaeus.

The outermost boundary of Plato's cosmos was spherical and turned around the Earth every 24 hours.

The zodiac was inclined at an angle to the the axis of rotation of the celestial sphere.

The planets move in regular uniform motions in circular paths.

Plato believed that the triangles of the elements combined to make different substances. Kepler's Platonic Cosmos -Kepler ordered the polyhedra based on certain relationships to the sphere. The mathematical relationship of each polyhedron to the sphere represents the distinction between Creator and created. The Mean Sun -Copernicus used the 'mean' position of the sun in his calculations. This assumed a stationary point near the sun as well as a circular orbit for the sun.

-Kepler instructed Maestlin to calculate the position of the true sun and use this data for his model. "This important modification went against the whole history of astronomy," (Martens 46). Although this did not provide a much closer fit for the polyhedral hypothesis, it did pave the way for Kepler's use of empirical data to develop his laws of motion.

-Kepler wanted to bring astronomy into cosmogony with calculations based on empirical data. He made a distinction between himself as a cosmogonist and Copernicus as an astronomer. The Five Platonic Solids Nested Polyhedra The Order of Kepler's Solids Why do the planets move at the speed they do? Kepler's Three Laws Kepler's Accomplishments Contemporary Applications Kepler Said: "My aim is to show that the heavenly machine is not a kind of divine, live being, but a kind of clockwork ... insofar as nearly all the manifold motions are caused by a most simple magnetic, and material force, just as all motions of the clock are caused by a simple weight. And I also show how these physical causes are to be given numerical and geometrical expression." Letter to J.G. Herwart von Hohenburg, Feb. 16, 1605 Cube: 6 squares, EARTH

Octahedron: 8 equilateral triangles, AIR

Tetrahedron: 4 equilateral triangles, FIRE

Icosahedron: 20 equilateral triangles, WATER

Dodecahedron: 12 equilateral triangles, COSMOS Octahedron, Icosahedron,

Dodecahedron, Tetrahedron, Cube

Primary Class: Cube, Tetrahedron, Dodecahedron. Each of these have a different type of face. The vertices of each of these solids join three faces. Stands on a face.

Secondary Class: Octahedron, Icosahedron

These have the same type of face, a triangle. Their vertices join four or five faces. Stands on a vertex.

-The Earth's orbit divides the primary and secondary solids. -Gregory, Frederick. Natural Science in Western History.

-Martens, Rhonda. Kepler's Philosophy and the New Astronomy.

-Zakai, Avihu. "The Rise of Modern Science and the Decline of Theology as the 'Queen of Sciences' in the Early Modern Era."

-http://www.keplersdiscovery.com/MeanSun.html -Plato was also aware that the changes in Mars's orbit may be apparent rather than actual. It has also been reported that Copernicus did suggest that the planets may move in elliptical orbits, but this suggestion was conspicuously absent from his De Revolutionibus where it is stated it appeared in III.4 in a pre-printer manuscript.

-Kepler hypothesized that the sun must be involved in the motion of the planets and found that an ellipse fit the data for the orbit of Mars. Kepler's first and second laws were formulated from this conclusion. He was able to keep his system both aesthetic and mathematically pleasing.

-In 1618, Kepler wrote The Harmony of the World. He began applying the ratios of whole numbers found in the musical scale to the varying angular velocities of the planet as they go around the sun.

-From exploring the ratios among whole numbers, he was able to explain the relationship between the periods of revolution to their mean distance from the sun, Kepler's Third Law. -Geometric and aesthetic order preserved, like Plato.

-Quantity is a form of matter and the source of its definition.

-The archetypes (forms) are final causes in their divine state, formal causes in their material state, and God created physical forces such that the movements would express them. The correspondence between the final, formal, and efficient causes is a 3-tiered universe. The geometrical links the physical to the archetypical by serving as a spatial translation of the divine. Kepler called this "architectonic harmony," (Martens 49). 1) The planets go around the sun in elliptical orbits and they do so because the sun governs them. The sun is at one focus of the ellipse.

2)The line joining the planet to the sun sweeps out equal areas in equal times as the planet travels around the ellipse.

3) The squares of the periods of revolution of the planets were directly proportional to the cubes of their mean distances from the sun. This "mean distance" can also be described as the semi-major axis (i.e. the radius of the orbit of the planet at its two furthest points). -Kepler created a "celestial physics" and laid the foundations for Newton's work by developing what later came to be known as his laws of nature.

-Astronomy became integrated into cosmology by the efforts of Kepler to maintain that the universe was created according to a Divine plan and that man was able to understand this plan through the expression of the mind of God in the geometric relationships in Nature.

-Kepler combined the archetypal world of Plato with the use of empirical data. His work yielded to his empirical observations while maintaining a belief in a Divinely organized cosmos. -Kepler's work in establishing laws of nature greatly influenced Newton. Newton used Kepler's third law to conclude that an object's force (and therefore also the acceleration) would diminish proportionately to 1/r-squared, i.e. to formulate his inverse square law.

-Dense particle packings in engineering and materials studies use the Platonic and Archimedean solids. Kepler's sphere conjecture of the Platonic solids (his model of the spacing of the planets based on polyhedron's inscribed and circumscribed within or around spheres) leads to the densest packing.

-Is the universe itself a dodecahedron?:

http://physicsworld.com/cws/article/news/2003/oct/08/is-the-universe-a-dodecahedron

Some scientists in US and France claimed that a universe with the same shape as the twelve-sided polygon can explain measurements of the cosmic microwave background – the radiation left over from the big bang – that spaces with more mundane shapes cannot (J-P Luminet et al. 2003 Nature 425 593).