Plato's Studies in other Subjects

Even though I’m trying to focus on Plato’s contributions to math, he studied many other subjects as well. His most famous subject is politics. Plato recorded The Republic, which is his most famous dialogue. In The Republic he talked to his students about what makes an ideal government, and that a government should be centered around justice (Kraut, 311).

Plato, though it may seem weird, studied dance, music, and drama also (O'Connor, J. J., and E. F. Robertson)!

Platonic Figure Nets

I decided that I wanted to make nets of the Platonic Figures.

 Tetrahedron Net

 Octahedron Net

 Dodecahedron Net

 Icosahedron Net

 Tetrahedron

 Octahedron

Dodecahedron

Icosahedron

Plato Survey

So, I took a survey to see what people knew about Plato. Here’s what I found.

Who is Plato?

- 17 people said that he was the guy who invented Play doh… -.-

-a Greek philosopher

-Greek

-a mathematician

-Greek, an astronomer, a philosopher, and did a bunch of stuff

- mathematician and philosopher

-mathematician and Greek

-an “old scientist guy” who is dead

- invented something in math

What are some of the things Plato studied?

-19 people said geometry, astronomy, math, or philosophy

- stars and animals

-how to make things with Play- Doh

-anatomy and geography

-architecture and science

-astrology

-shapes

What are some of the things that Plato contributed to math?

-16 people said that he made shapes…well...they weren't wrong…

-invented the pyramid

-how to make a triangle

-did something with a triangle

-made theorems and postulates

-made geometry stuff

-invented the number 0

-invented algebra

- two people said they had no idea

I guess people don’t know enough about Plato...

Platonic Figures

Platonic solids are regular polyhedra that were made by Plato, a greek mathematician from Athens. There are five platonic figures, all unique and different (Boswell, Lee, and Larson).

The first is a cube, which is a very well-known shape.  It’s six faces are squares and if you flattened the shape into a net, it would look like a cross. It has 6 faces, 8 vertices, and 12 edges (Boswell, Lee, and Larson).

The second is a tetrahedron. When I showed a picture of a tetrahedron to my friends, they insisted it was a triangular pyramid. Both names are correct, but tetrahedron sounds cooler. Its net looks like a Triforce, or, if you don’t know what that is, it looks like an equilateral triangle split by its mid segments. It has 4 faces, 4 vertices, and 6 edges (Boswell, Lee, and Larson).

The third is an octahedron. This shape is popular for making dice, because is has 8 congruent sides. The faces are equilateral triangles. The net is a bit too complicated to explain, but you can see the picture of the net that I made! :D It has 8 faces, 6 vertices, and 12 edges.

The fourth is an icosahedron, which is like an octahedron on steroids. It has 20 equilateral triangle faces. This shape is amazing for making dice for Dungeons and Dragons or Magic the Gathering….*sigh* It has 20 faces, 12 vertices, and 30 edges (Boswell, Lee, and Larson)

The fifth one is a dodecahedron, which was a pain to make a net for. Its net is made of 12 pentagonal faces. It has 12 faces, 20 vertices, and 30 edges.

Surface area of a Tetrahedron:

So, to find the surface area, we first need to find the area of one of the faces. In some cases you could have the height of the triangle, but if you don't have that, and only have the length of the sides, you could use the area formula above.

A= 1/4 (square root of 3) s^2

A= 1/4 (square root of 3) (2) ^2

A= 1/4 (square root of 3) (4)

A= (1) (square root of 3)

A= (square root of 3)

Now you just multiply this number by the number of faces. In this case, its 4.

SA= 4(square root of 3) inches squared

Now, let's try finding the surface area of a cube.

A=bh

A= (1) (1)

A= 1

Now multiply it by the number of sides, 6.

SA= 6(1)

SA= 6 square inches

Polyhedra

What is a polyhedron? Polyhedra are solids whose faces are polygons. So the faces could be anything from squares to the “L” shaped figures in Tetris. As long as the 3-D shape has all straight edges and flat faces, its a polyhedron (Boswell, Lee, and Larson)

This is a cube. Cubes are polyhedrons.

Of course, there are some shapes that are not polyhedra. These are any shapes that have a curved or any not-flat face. Some examples are cylinders and cones. Their circular base is not a polygon, and its lateral surface isn’t flat (Boswell, Lee, and Larson).

This is a cylinder. Cylinders are not polyhedrons.

All polyhedra have three features that define the shape; faces, vertices, and edges. Faces are the shapes that make up the shape itself. For example, a face of a cube is a square. Edges are the line segments made when two faces meet. Last, a vertex is a point where at least three edges meet. In some circumstances you might need to find out the number of each of these features a certain polyhedron has. You could just count them manually, but that’s just a pain. There is a theorem that can make it a lot easier. It is called Euler’s Theorem, and it makes it really simple to figure out the number of faces, edges or vertices (Boswell, Lee, and Larson).

Euler’s Theorem: F + V= E + 2

F= faces

E= edges

V= vertices

Basically, the sum of the number of faces and the number of vertices is equal to the number of edges added with two.

Here are some examples of how to use this theorem.

You have a figure with 90 edges and  32 faces. Find the number of vertices.

F + V = E + 2 Write the equation

(32) + V = (90) + 2 Substitute in what you know

32 + V = 92 Combine like terms

-32 -32 Subtract 32 from both sides of the

equals sign

V= 92 + (-32) After subtracting from both sides

V= 60 Combine like terms

There are 60 vertices in this shape.

See, you don’t need to see the shape to figure out the number of faces, edges, and vertices.

There are other terms you can use to categorize polyhedra.

One is regular or irregular. Regular polyhedra are polyhedra whose faces are all the same and congruent. Unlike this, a non regular polyhedron’s faces are not all the same (Boswell, Lee, and Larson).

This is an icosahedron. They are regular.

This polyhedron is not regular.

The next is convex or concave. Convex shapes are shapes that look full and complete. “A polyhedron is convex if any two points on its surface can be connected by a segment that lies entirely inside or on the polyhedron. (Boswell, Lee, and Larson)” Concave 3-D shapes kind of look like there was a chunk punched out of it, and cave in.

This shape is concave.

Plato's Biography

I had to ask a ton of my friends who they thought Plato was, and the number one response was, “Oh, is it that one yummy toy I played with as a kid?”, and I just gave them a cold stare. Really? So today, I’m going to tell you more about Plato, so that people will no longer think that he is a mere toy!

This is a painting of Plato, who was a mathematician, politician, and scientist.

Plato was the youngest son of Ariston and Perictione, two very wealthy people in Athens at the time. he was still young when his dad died, and his mother remarried to a guy named Pyrilampes. In fact, he did a lot of his growing up in the house of Pyrilampes. When he was still a young man, Plato was taught by Cratylus, who was famous for his studies in cosmology. At this time, historians are almost certain that he met Socrates, because his uncle was a close friend to him. Socrates eventually became his teacher at one point (O’Connor and Robertson).

This is a statue of Socrates, who was Plato's teacher at one point

Plato also served in the Peloponnesian War for 5 years. Even after fightin for this long, he didn’t feel very happy, because he wanted to be a politician more than a soldier. After five eyars of service he joined a form of government ruled by multiple families and people, called an oligarchy. This oligarchy was called the Thirty Tyrants. (O’Connor and Robertson)The name says it all. They really abused the people they ruled in search of loot and possessions. Here are some examples: The thirty had a henchman, and he was recorded to have kidnapped someone in order to gain his family's wealth. People would even rip gold earrings from people’s ears in order to get loot. The thirty were eventually taken down, but before that, Plato left the oligarchy because of one of the member’s violent actions (Martin).

Democracy was returned to Athens, and Plato was able to do what he loved most, politics. However, the more influential he was as a politician, the more irritated the authorities became. They considered philosophers and politicians like him a bad influence on the younger populations of Athens. Certain events made him seriously want to stop being a public politician. One of them was that his philosopher friend died because his friend was a bit too influential. That would make you rethink your priorities a bit. Socrates, Plato’s friend was forced to either leave Athens, or drink hemlock. And if you don’t already know, hemlock isn’t a fun plant. It kind of looks like a pinecone, but its really poisonous (Grieve). Socrates had the choice to either leave Athens or drink hemlock (Nails). Fun stuff, isn’t it? Anyways, Plato took a little break from politics and took an awesome vacation to Egypt and Italy. In Egypt, he learned about water clocks, and later, introduced to Greece. In Italy, he learned about the works of Pythagoras, and figured out that he really liked math (O’Connor and Robertson).

Later in his life, Plato actually went into the military again. Many people think that he started writing his famous dialogues at this time. If you don’t know what a dialogue is, it’s basically a conversation recorded on paper. He made a lot of these over the course of his life (O’Connor and Robertson).

A sculpture of Plato's head.

At about 386 BC, he founded his academy in Athens, which lasted even after he died. In this academy, he taught philosophy, math, science, and politics, but unfortunately, way after Plato died, it was shut down in 529 AD (O’Connor and Robinson). This academy lasted for a really long time. Like 915 years. Thats a really long time. To put it into perspective, the famous Trinity College in Ireland was founded in 1592 (Trinity College Dublin, Facts and Figures). That was a while ago. This was around the time of the renaissance. You know, Shakespeare, Michelangelo, The original Martin Luther, and King Henry the Eighth. Trinity College is 422 years old. Plato’s college: 915. See what I’m saying? His school was really old.

The most important contributions Plato made to knowledge, was in politics. Most of his dialogues were on politics. His most famous dialogue is The Republic, where he discussed what makes the ideal government (O’Connor and Robinson).

Many of the mathematical contributions he made are found in the dialogue, Timaeus. This is where the knowledge of Platonic solids comes from. Platonic solids include the cube, icosahedron, octahedron, tetrahedron, and dodecahedron (Plato, Timaeus).

Plato died in 347 BC, when he was attending a wedding. What a wedding gift; A dead guy (O’Connor and Robinson).

Well, if you made it to the end of this blog post, you must have actually payed attention to it. Now you know that Plato was so much more than a toy. He was a very influential politician, scientist, teacher, and mathematician.