Real Mathematics – Geometry #10&11

Q: How many regular polygons are there?

In the previous articles I have shown how to construct an equilateral triangle using Euclid’s method which means using a compass and an unmarked ruler. An equilateral triangle is a regular polygon as it consists of equal sides and angles. We can even conclude that an equilateral triangle is the smallest regular polygon.

Then let me continue with increasing the number of sides to construct more polygons. A regular polygon with four equal sides and angles… Hmmm… A square!

A regular polygon with five sides: Pentagon.

A regular polygon with six sides: Hexagon.

regular-polygons-6-638

A regular polygon with fifty sides: Pentacontagon.

There are no limits for the number of sides of regular polygons we can create. Although after certain number of sides it is almost impossible to distinguish a regular polygon from a circle.

Pentacontagon (left) and its comparison with a regular polygon that has 200 sides.

Regular polygons exist in two dimensional worlds. What if we try to construct regular objects in three dimensions?

Q: How many regular polyhedrons are there?

There are two specific properties for regular polyhedrons: Each of their faces is the same regular polygons and there are same numbers of regular polygons meeting at each corner.

Let me give you the answer right away: There are five different regular polyhedrons. Exactly five!

The very first time I heard about it, I thought it was bizarre to have only five different regular polyhedrons while there are infinite number of regular polygons. How can coming to three dimensions from two dimensions changes so much?

School of Broad Shouldered

Around 427 BC a baby named Aristocles was born in Athens. When he grew up, Aristocles had wide shoulders which resulted with him adopting the nickname “Plato” that meant “broad” in Greek. (This story is from C.J. Rowe’s Plato.)

plato
Plato (427 BC. – 347 BC.)

Majority of the people have no idea that Plato was an integral individual for mathematics’ development. His persistence for clearer explanations for proof and hypothesizes evolved mathematics completely. Although his biggest contribution, not just to mathematics but to all branches of science, was the school he founded.

In 387 BC he attempted to build a school in Athens, on the land of a guy named Academas. He gave the school his name: Academy. For over 900 years Plato’s Academy was the home of countless philosophers. (Just for comparison to its worth: University of Bologna was built 1400 years after Academy was founded.)

geometri-bilmeyen-giremez
Above Academy’s door: “Let no man ignorant of geometry enter here.”

Platonic Objects

Polyhedrons are also called Platonic objects because he was the first to explain that there are exactly five of them.

  1. Tetrahedron: An object formed by four equilateral triangles.
  2. Cube: An object formed by six squares. I guess I didn’t need to explain it.
  3. Octahedron: An object formed by eight equilateral triangles.
  4. Dodecahedron: An object formed by twelve pentagons.
  5. Icosahedron: An object formed by twenty equilateral triangles.

platonic_solids

Why Five?

When examined carefully one might see the properties that are exclusive to the regular polyhedrons. These properties both explain and prove why there are exactly five of them.

I’ve already mentioned that regular polyhedrons are three dimensional objects. If you study a regular polyhedron, you’d see that at least three regular polygons meet at a point (corner). Let’s try it with the smallest regular polyhedron.

Take any corner on a tetrahedron. You’ll see that three equilateral triangles meet at that corner. When it is reduced to two dimensions, it would look like as follows:

IMG_6348

The corner has 360 degree around itself. One equilateral triangle has 60 degrees of internal angle on this specific point which means there are

360 – (60+60+60) = 180

degrees of empty space around the corner. By means of this empty space it is possible to construct a three dimensional object. If there were no space, then this shape would not have the flexibility and thus it would not be turned into a three dimensional object.

We are getting close. Let’s use induction and go from tetrahedron to general.

In order to construct a polyhedron we must follow these rules:

  • Draw a point on a paper.
  • Draw three regular polygons which meet at that point.
  • Add the internal angles of the regular polygons. If it doesn’t exceed 360 degrees, then it is possible to construct a regular polyhedron with these regular polygons.
  • Also one can continue adding same regular polygons as long as summation never exceeds 360 degrees.
  • If the summation of the internal angles of the regular polygons is 360 degrees or more, then this shape can’t be converted into three dimensions. This means a regular polyhedron can’t be constructed with such regular polygons.

Example 1: Tetrahedron.

Take a point and draw three equilateral triangles around it. One internal degree of equilateral triangle is 60, which make 60*3=180 degrees around the point. It is less than 360, that is why we know it is possible to construct a Platonic solid with them: Tetrahedron.

Example 2: Tetrahedron + 1 more equilateral triangle.

If we add one more equilateral triangle, internal angles around the point will make 240 degrees, which is still less than 360. Hence it is possible to construct a Platonic solid with four equilateral triangle: Octahedron.

Example 3: Tetrahedron + 2 more equilateral triangles.

We continue to add more equilateral triangles. Now we have five of them around a point, which gives 300 degrees. We are still under 360 degrees, thus we can construct a Platonic solid with them. This is our third Platonic solid and we constructed them with equilateral triangles: Icosahedron.

Example 4: Tetrahedron + 3 more equilateral triangles.

We add the sixth triangle. There is 60*6=360 degrees around the point. It is impossible to construct a Platonic solid with these triangles. Actually this shape stays only in two dimensions.

Example 5: Cube and adding a square to it.

We are done with equilateral triangles, hence we move to the next regular polygon: Square. Internal degree of a square is 90, which gives 90*3=270 degrees around the point. It is less than 360 degrees, and that is why it is possible to construct Platonic solid with three squares. This is the fourth Platonic solid: Cube.

If we add one more square, then internal angles around the point will add up to 90*4=360 degrees. It is impossible to construct a Platonic solid in this case.

Example 6: Dodecahedron and adding one more pentagon.

Since we are done with squares, we move to the next regular polygon: Pentagon. A pentagon has internal degrees 108 at its corners. Three pentagons around a point will give 108*3=324 degrees which is less than 360 degrees. So it is possible to construct a Platonic solid with three pentagons meeting at a point. This is our fifth Platonic solid.

Although if we a forth regular pentagon, internal degrees will be 108*4=432 degrees. It is more than 360, which makes the pentagons overlap. Hence it is impossible to construct a Platonic solid with these.

One wonders…

Continue to the next polygon: Hexagon. Examine why it is impossible to construct a Platonic solid with them.

M. Serkan Kalaycıoğlu

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