An impossible dream
Kepler, the guy who discovered that planets go in ellipses around the Sun, was in love with geometry. Among other things, he tried to figure out how to tile the plane with regular pentagons (dark blue) and decagons (blue-gray). They fit nicely at a corner... but he couldn't get it to work.
Then he discovered he could do better if he also used 5-pointed stars!
Can you tile the whole plane with these three shapes? No!
The picture here is very tempting... but if you continue you quickly run into trouble. It's an impossible dream.
However, Kepler figured out that he could go on forever if he also used overlapping
decagons, which he called 'monsters'. Look at this picture he drew:https://plus.maths.org/issue45/features/kaplan/kepler.gif
If he had worked even harder, he might have found the Penrose tilings, or similar things discovered by Islamic tiling artists. Read the whole story here:
• Craig Kaplan, The trouble with five, https://plus.maths.org/content/trouble-five
How did Kepler fall in love with geometry? He actually started as a theologian. Let me quote the story as told in the wonderful blog The Renaissance Mathematicus
:Kepler was born into a family that had known better times, his mother was an innkeeper and his father was a mercenary. Under normal circumstances he probably would not have expected to receive much in the way of education but the local feudal ruler was quite advanced in his way and believed in providing financial support for deserving scholars. Kepler whose intelligence was obvious from an early age won scholarships to school and to the University of Tübingen where he had the luck to study under Michael Mästlin one of the very few convinced Copernican in the later part of the 16th century. Having completed his BA Kepler went on to do a master degree in theology as he was a very devote believer and wished to become a theologian. Recognising his mathematical talents and realising that his religious views were dangerously heterodox, they would cause him much trouble later in life, his teacher, Mästlin, decided it would be wiser to send him off to work as a school maths teacher in the Austrian province.Although obeying his superiors and heading off to Graz to teach Protestant school boys the joys of Euclid, Kepler was far from happy as he saw his purpose in life in serving his God and not Urania (the Greek muse of astronomy). After having made the discovery that I will shortly describe Kepler found a compromise between his desire to serve God and his activities in astronomy. In a letter to Mästlin in 1595 he wrote:I am in a hurry 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 recognised 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 efforts.So what was the process of thought that led to this conversion from a God glorifying theologian to a God glorifying astronomer and what was the discovery that he was so eager to publish? Kepler’s God was a geometer who had created a rational, mathematical universe who wanted his believers to discover the geometrical rules of construction of that universe and reveal them to his glory. Nothing is the universe was pure chance or without meaning everything that God had created had a purpose and a reason and the function of the scientist was to uncover those reasons. In another letter to Mästlin Kepler asked whether:you have ever heard or read there to be anything, which devised an explanation for the arrangement of the planets? The Creator undertook nothing without reason. Therefore, there will be reason why Saturn should be nearly twice as high as Jupiter, Mars a little more than the Earth, [the Earth a little more] than Venus and Jupiter, moreover, more than three times as high as Mars.The discovery that Kepler made and which started him on his road to the complete reform of astronomy was the answer to both the question as to the distance between the planets and also why there were exactly six of them: as stated above, everything created by God was done for a purpose.On the 19th July 1595 Kepler was explaining to his students the regular cycle of the conjunctions of Saturn and Jupiter, planetary conjunctions played a central role in astrology. These conjunctions rotating around the ecliptic, the apparent path of the sun around the Earth, created a series of rotating equilateral triangles. Suddenly Kepler realised that the inscribed and circumscribed circles generated by his triangles were in approximately the same ratio as Saturn’s orbit to Jupiter’s. Thinking that he had found a solution to the problem of the distances between the planets he tried out various two-dimensional models without success. On the next day a flash of intuition provided him with the required three-dimensional solution, as he wrote to Mästlin:I give you the proposition in words just as it came to me and at that very moment: “The Earth is the circle which is the measure of all. Construct a dodecahedron round it. The circle surrounding that will be Mars. Round Mars construct a tetrahedron. The circle surrounding that will be Jupiter. Round Jupiter construct a cube. The circle surrounding it will be Saturn. Now construct an icosahedron inside the Earth. The circle inscribed within that will be Venus. Inside Venus inscribe an octahedron. The circle inscribed inside that will be Mercury.”
This model, while approximately true, is now considered completely silly! We no longer think there should be a simple geometrical explanation of why planets in our Solar System have the orbits they do.
So: a genius can have a beautiful idea in a flash of inspiration and it can still be wrong
But Kepler didn't stop there! He kept working on planetary orbits until he noticed that Mars didn't move in a circle around the Sun. He noticed that it moved in an ellipse! Starting there, he found the correct laws governing planetary motion... which later helped Newton invent classical mechanics.
So it pays to be persistent - but also not get stuck believing your first good idea.
Read The Renaissance Mathematicus
can you tile the plane with shapes, each of which has at least the symmetry group of a regular pentagon?
So, regular pentagons and decagons are allowed, and so are regular 5-pointed stars, and many other things... but not Kepler's monsters. The tiling itself does not need to repeat in a periodic way. #geometry #astronomy