The Life Of Johannes Kepler Essay, Research Paper

Johannes Kepler

(1571-1630)

HIS LIFE

Johannes Kepler was a German astronomer and mathematician ho discovered that planetary motion is elliptical. Early in his life, Kepler wanted to prove that the universe obeyed Platonistic mathematical relationships, such as the planetary orbits were circular and at distances from the sun proportional to the Platonic solids (see paragraph below). However, when his friend the astronomer Tycho Brahe died, he gave Kepler his immense collection of astronomical observations. After years of studying these observations, Kepler realized that his previous thought about planetary motion were wrong, and he came up with his three laws of planetary motion. Unfortunately, he did not have a unifying theory for these laws. This had to until Newton formulated his laws of gravity and motion.

PLATONIC SOLIDS

A platonic solid is a solid having similar, regular polygonal faces. There are five Platonic solids: the icosahedron, tetrahedron, octahedron, dodecahedron, and cube. They are characterized by the fact that each face is a straight-sided figure with equal sides and equal angles:

Tetrahedron: 4 triangular faces, 4 vertices, 6 edges

Cube: 6 square faces, 8 vertices, 12 edges

Octahedron: 8 triangular faces, 6 vertices, 12 edges

Dodecahedron: 12 pentagonal faces, 20 vertices, 30 edges

Icosahedron: 20 triangular faces, 12 vertices, 30 edges

Many people wonder why there should be exactly five Platonic solids, and whether there is one that has not been found yet. However, it is easy show that there must be five and that there cannot be more than five.

At each vertex, at least three faces must come together, because if only two came together they would collapse against one another and a solid would not be created. Secondly, the sum of the interior angles of the faces meeting at each vertex must be less than 360?, because if they didn’t, they would not all fit together.

Each interior angle of an equilateral triangle is 60?, therefore we could fit together three, four, or five of them at a vertex, and these correspond to the tetrahedron, the octahedron, and the icosahedron. Each interior angle of a square is 90?, so we can fit only three of them together at each vertex, giving us a cube. The interior angles of the regular pentagon are 108?, so again we can fit only three together at a vertex, giving us the dodecahedron.

That makes five regular polyhedra. However, what would happen if we had a six-sided figure? Well, its interior angles are 120?, so if we fit three of them together at a vertex the angles add up to 360?, and therefore they lie flat. For this reason we cannot use hexagons to make a Platonic solid. In addition, obviously, no polygon with more than six sides can be used either, because the interior angles just keep getting larger.

The Greeks, who had to find religious truth in mathematics, found the idea of exactly five Platonic solids very compelling. The philosopher Plato concluded that they must be the fundamental building blocks of nature, and assigned to them what he believed to be the essential elements of the universe. He followed the earlier philosopher Empedocles in assigning fire to the tetrahedron, earth to the cube, air to the octahedron, and water to the icosahedron. To the dodecahedron, Plato assigned the element cosmos, reasoning that, since it was so different from the others, because of its pentagonal faces, it must be what the stars and planets are composed of.

Although this might seem odd to us, these were really very powerful ideas, and led to real knowledge. As late as the 16th century, Johannes Kepler was applying a similar intuition to attempt to explain the motion of the planets. Early in his life, he concluded that the distances of the orbits, which he assumed were circular, were related to the Platonic solids in their proportions. Only later in his life, after his friend the great astronomer Tycho Brahe gave him a his collection of astronomical observations, did Kepler finally realize that this model of planetary motion was mistaken, and that in fact planets moved around the sun in ellipses, not circles. It was this discovery that led Newton, less than a century later, to formulate his law of gravity and gave us our modern conception of the universe.

HIS LAWS

The German mathematician and astronomer Johannes Kepler (1571 – 1630) was a Platonist, and set out early in his professional career to show that the motion of the planets was circular, and that they could be described in terms of the Platonic solids. However, he was also a friend and assistant to the great Danish astronomer Tycho Brahe, who used the newly invented telescope to make precise observations of the planets and stars. When Tycho Brahe died, in 1601, Kepler inherited this enormous collection of data and studied it. After studying this data for 20 years, Kepler realized that his earlier assumptions about planetary motion were wrong, and that if an earth-centered (Ptolemaic) understanding of the universe were abandoned for a sun-centered (Copernican) model, then the motion of the planets was clearly elliptical.

From this basis, Kepler generated his three “laws” of planetary motion:

1.The orbit of each planet is an ellipse with the sun at one focus.

2.The line segment joining a planet to the sun sweeps out equal areas in

equal time intervals.

3.The square of the period of revolution of a planet about the sun is

proportional to the cube of the semimajor axis of the planet’s elliptical

orbit.

These laws are shown in the following diagram:

:

These laws imply that the speed of revolution of a planet around the sun is not uniform, but changes throughout the planet’s year. It is fastest when the planet is nearest the sun (called the perihelion) and slowest when the planet is farthest away (called the aphelion). A circle is also an ellipse and the orbits of most planets are far more nearly circular

than the diagram would suggest. However, they are not circles nonetheless; they are ellipses with non-zero eccentricity.

The third law means that if Y is the length of a planet’s year, and if A is the length of the semimajor axis of the planet’s orbit, then the quantity Y2/A3 is the same for every planet (and comet, and other satellites) in the solar system. Thus, if a planet’s orbit is known, the length of it’s year can be immediately calculated, and vice versa.

Kepler’s laws were derived strictly from careful observation and had no theoretical basis. However, about 30 years after Kepler died, the English mathematician/physicist Sir Isaac Newton derived his inverse square law of gravity, which says that the force acting on two gravitating bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. Kepler’s laws may be derived from this theoretical principle using calculus.

BIBLIOGRAPHY

1) Calculus: A First Course

James Stewart, Thomas Davison, Bryan Ferroni

McGraw-Hill Ryerson Limited

Copyright 1989

2) Applied Physics Third Edition

Arthur Beiser

McGraw-Hill Limited

Copyright 1995

3) http://www.mathacademy.com

4) http://www.encyclopedia.com/articles/06915.html

5) http://www.letsfindout.com/subjects/space/kepler.html

Johannes Kepler

(1571-1630)

WHY I CHOSE THIS TOPIC

I chose to do my independent study on the astronomer Johannes Kepler because I want to be an astronomer. I thought that if I was going to have to do research on a mathematicians’ life I might as well choose a field that I am going to get in to. I think that Kepler is the founding father in astronomy. I believe this because he was the first to state that the planets travel in elliptical motion. If it wasn’t for Kepler’s constant devotion towards the planets than Newton would not have been able to come up with his law of gravity and our concept of the universe might not be the same.

HIS LIFE

Johannes Kepler was a German astronomer and mathematician ho discovered that planetary motion is elliptical. Early in his life, Kepler wanted to prove that the universe obeyed Platonistic mathematical relationships, such as the planetary orbits were circular and at distances from the sun proportional to the Platonic solids (see paragraph below). However, when his friend the astronomer Tycho Brahe died, he gave Kepler his immense collection of astronomical observations. After years of studying these observations, Kepler realized that his previous thought about planetary motion were wrong, and he came up with his three laws of planetary motion. Unfortunately, he did not have a unifying theory for these laws. This had to until Newton formulated his laws of gravity and motion.

PLATONIC SOLIDS

A platonic solid is a solid having similar, regular polygonal faces. There are five Platonic solids: the icosahedron, tetrahedron, octahedron, dodecahedron, and cube. They are characterized by the fact that each face is a straight-sided figure with equal sides and equal angles:

Tetrahedron: 4 triangular faces, 4 vertices, 6 edges

Cube: 6 square faces, 8 vertices, 12 edges

Octahedron: 8 triangular faces, 6 vertices, 12 edges

Dodecahedron: 12 pentagonal faces, 20 vertices, 30 edges

Icosahedron: 20 triangular faces, 12 vertices, 30 edges

Many people wonder why there should be exactly five Platonic solids, and whether there is one that has not been found yet. However, it is easy show that there must be five and that there cannot be more than five.

At each vertex, at least three faces must come together, because if only two came together they would collapse against one another and a solid would not be created. Secondly, the sum of the interior angles of the faces meeting at each vertex must be less than 360?, because if they didn’t, they would not all fit together.

Each interior angle of an equilateral triangle is 60?, therefore we could fit together three, four, or five of them at a vertex, and these correspond to the tetrahedron, the octahedron, and the icosahedron. Each interior angle of a square is 90?, so we can fit only three of them together at each vertex, giving us a cube. The interior angles of the regular pentagon are 108?, so again we can fit only three together at a vertex, giving us the dodecahedron.

That makes five regular polyhedra. However, what would happen if we had a six-sided figure? Well, its interior angles are 120?, so if we fit three of them together at a vertex the angles add up to 360?, and therefore they lie flat. For this reason we cannot use hexagons to make a Platonic solid. In addition, obviously, no polygon with more than six sides can be used either, because the interior angles just keep getting larger.

The Greeks, who had to find religious truth in mathematics, found the idea of exactly five Platonic solids very compelling. The philosopher Plato concluded that they must be the fundamental building blocks of nature, and assigned to them what he believed to be the essential elements of the universe. He followed the earlier philosopher Empedocles in assigning fire to the tetrahedron, earth to the cube, air to the octahedron, and water to the icosahedron. To the dodecahedron, Plato assigned the element cosmos, reasoning that, since it was so different from the others, because of its pentagonal faces, it must be what the stars and planets are composed of.

Although this might seem odd to us, these were really very powerful ideas, and led to real knowledge. As late as the 16th century, Johannes Kepler was applying a similar intuition to attempt to explain the motion of the planets. Early in his life, he concluded that the distances of the orbits, which he assumed were circular, were related to the Platonic solids in their proportions. Only later in his life, after his friend the great astronomer Tycho Brahe gave him a his collection of astronomical observations, did Kepler finally realize that this model of planetary motion was mistaken, and that in fact planets moved around the sun in ellipses, not circles. It was this discovery that led Newton, less than a century later, to formulate his law of gravity and gave us our modern conception of the universe.

HIS LAWS

The German mathematician and astronomer Johannes Kepler (1571 – 1630) was a Platonist, and set out early in his professional career to show that the motion of the planets was circular, and that they could be described in terms of the Platonic solids. However, he was also a friend and assistant to the great Danish astronomer Tycho Brahe, who used the newly invented telescope to make precise observations of the planets and stars. When Tycho Brahe died, in 1601, Kepler inherited this enormous collection of data and studied it. After studying this data for 20 years, Kepler realized that his earlier assumptions about planetary motion were wrong, and that if an earth-centered (Ptolemaic) understanding of the universe were abandoned for a sun-centered (Copernican) model, then the motion of the planets was clearly elliptical.

From this basis, Kepler generated his three “laws” of planetary motion:

1.The orbit of each planet is an ellipse with the sun at one focus.

2.The line segment joining a planet to the sun sweeps out equal areas in

equal time intervals.

3.The square of the period of revolution of a planet about the sun is

proportional to the cube of the semimajor axis of the planet’s elliptical

orbit.

These laws are shown in the following diagram:

:

These laws imply that the speed of revolution of a planet around the sun is not uniform, but changes throughout the planet’s year. It is fastest when the planet is nearest the sun (called the perihelion) and slowest when the planet is farthest away (called the aphelion). A circle is also an ellipse and the orbits of most planets are far more nearly circular

than the diagram would suggest. However, they are not circles nonetheless; they are ellipses with non-zero eccentricity.

The third law means that if Y is the length of a planet’s year, and if A is the length of the semimajor axis of the planet’s orbit, then the quantity Y2/A3 is the same for every planet (and comet, and other satellites) in the solar system. Thus, if a planet’s orbit is known, the length of it’s year can be immediately calculated, and vice versa.

Kepler’s laws were derived strictly from careful observation and had no theoretical basis. However, about 30 years after Kepler died, the English mathematician/physicist Sir Isaac Newton derived his inverse square law of gravity, which says that the force acting on two gravitating bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. Kepler’s laws may be derived from this theoretical principle using calculus.

Bibliography

1) Calculus: A First Course

James Stewart, Thomas Davison, Bryan Ferroni

McGraw-Hill Ryerson Limited

Copyright 1989

2) Applied Physics Third Edition

Arthur Beiser

McGraw-Hill Limited

Copyright 1995

3) http://www.mathacademy.com

4) http://www.encyclopedia.com/articles/06915.html

5) http://www.letsfindout.com/subjects/space/kepler.html

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