# 5 regular polyhedrons to run them all - Mysterium Cosmographicum

in MES Sciencelast year

There are no photos because my posts exceeedes 65xxx bytes after them. On Steemit there was no problem with that.

If something in science has Latin name, it is probably worth to know. Previously I posted about Theorema Egregium - today about Mysterium Cosmographicum.

On a plane, You have infinitely many regular polygons - for every natural number greater or equal 3.

But in three dimensions, You have only 5 regular polyhedrons.

It was known even in ancient Greece. Practically, they knew about 4 of them in times of Pitagoras.

Tetrahedron, Cube, Octahedron and Icosahedron.

The fifth and last, Dodecahedron, was also found by Pitagoreans, but we know about it from Plato.

Aristotle said that 4 regular polyhedrons are related with 4 elements.

Tetrahedron - Fire

Cube - Earth (maybe that inspired the creator of Minecraft xD ).

Octahedron - Wind

ICosahedron - Water.

What about Dodecahedron? Aristotle said that this is ether - something which fills the spce between the stars and planets.

I will not bore You with proof that there are no more regular polyhedrons.

You've probably heard about 3 Laws of Johannes Kepler, German astronomist, physicist and mathematician.

There is also something called "4th Law of Kepler", although this is not a law. This is funny observation, a serendipity.

It is probably quite a coincidence, but he did not know about it.

Kepler measured the medium rays of orbits of 6 planets (Mercury, Venus, Earth, Mars, Jupiter, Saturn).

Imagine You draw a sphere with a ray of orbit of Mercury. You describe octahedron on that sphere. That octahedron is in the same moment inscribed in the sphere with ray of orbit of Venus (everything centered in the same point). Then You describe iscoahedron on Venus's sphere. It is inscribed in Earth's sphere. Dodecahedron on Earth's sphere is inscribed in Mars's sphere. Tetrahedron described on Mars's sphere is inscribed in Jupiter's sphere. Finally, cube described on Jupiter's sphere is described in Saturn's sphere.

Of course this is not magic. Perfect objects does not exist. There is some error, and rays of the planets are not constant. But, generally that construction is satisfying.

He thought of it as a law of nature. But now we know, that we have more planets - Uran and Neptun. And we don't have more regular polyhedrons.

I found on Internet you can buy such models for ~40 dollars.