Tenzin Menrinetsang
Platonic Solids
In a game of Monopoly, there’s the game board, money, little houses, and Chance cards. The objects allowing the character to move sit right in the middle, the dice. Each die has an equal number of faces, edges, vertices. These dice are an example of Platonic solids. A Platonic solid is a regular, convex polyhedron. When an object is regular, it means that all sides and all angles are equal. Referring to Figure 1, the first hexagon is a regular hexagon, because all angles are equal to each other and all sides are equal to each other. The second hexagon is not regular because not all sides are equal and not all angles are equal. When an object is convex, a line segment between any two points on the shape will always lie within it. In Figure 2, the first shape is convex because, as shown, any line segment drawn connecting any two points in the shape will always lie within the shape. However these line segments drawn in shapes that are not convex, concave shapes, will extend outside the shape. Dice, which are cubes, are not the only Platonic solids. In fact, the Greek mathematician Euclid proved that there are five different Platonic solids. These solids are the regular tetrahedron (made up of five equilateral triangles), the cube (made up of six squares), regular octahedron (eight equilateral triangles), regular dodecahedron (twelve regular pentagons), and the regular icosahedron (twenty equilateral triangles).
The true discoverer of Platonic solids is unknown. Many people believe that Pythagoras, a Greek philosopher, may have known and discovered three Platonic solids- the tetrahedron, cube, and dodecahedron. However, it is widely accepted and proven that ancient Egyptians knew of at least three Platonic solids, which may have encouraged Pythagoras’ work. Euclid, the "father of geometry", stated that Theaetetus, an Athenian mathematician, was the first person to introduce the octahedron and icosahedron. Despite whoever first discussed Platonic solids, they owe their name to Plato, a Greek philosopher. In his dialogue Timaeus, Plato associated each Platonic solid with the basic elements. He assigned the tetrahedron with fire because of its sharp edges and small volume, the cube with earth due to its stability, octahedron with air because of its ability to spin boundlessly when held by two opposite vertices, and icosahedron with water because of its high volume. The last platonic solid, the dodecahedron, made up of 12 faces, was matched to the heavens, with its 12 constellations.
When proving that there are only five Platonic solids, the first thing to know is that if the faces add up to 360° at one vertex, its flat. Therefore, in order to be a platonic solid, the faces have to have a sum of less than 360° at one vertex. It's impossible for a Platonic solid to have 2 faces at each vertex because they either fold over each other or they form gaps when connected with other faces. The best way to understand this proof is to think of it as paper being cut and folded together. All faces are regular, so each angle measure is the same within a face. Connecting the gaps in the three triangles at one vertex will get a tetrahedron, without the base. If 4 triangles meet at each point, the result is half of an octahedron (without a base) so two sets have to be made and conjoined together to create a full octahedron. If 5 triangles are placed around a vertex, when all faces and connected, they form a pentagonal pyramid, without a base. This is one half of an icosahedron (without a base), so duplicates would need to be made, rotated 36° so that they line up and then connected. If six triangles were placed around a common point, that will create a flat surface because each angle in an equilateral triangle is 60 °and since there six triangles sharing a point, there’s a total of 360°. Remember that the faces cannot have a sum of 360° at a vertex because that would form a flat surface, it wouldn't become a 3D figure, thus not creating a Platonic solid. It's also known that any number of triangles larger than 6 would also not work because the angle measures are too big and cannot overlap. Moving onto square faces, if 3 squares that meet at one point and connect them, the result would be half a cube. Therefore, a double must be made and the cube is formed. If there are four squares, the surface would be flat because four 90° angles will get a sum of 360°. Thus, it does not have more than four squares as well. To create a dodecagon, because it is somewhat harder to imagine, an alternative method is available. If there is a regular pentagon that will serve as a base, and five other regular pentagons attached to each edge, connecting, half of a dodecagon is formed. After creating a duplicate and rotating it 36° so that they’d line up, they could be placed on top of one another and a dodecahedron is made.
Leonhard Euler created a formula that relates the number of edges, vertices, and faces on any planar graph, as well as polyhedra. His formula stated that the difference in the number of vertices (V) and edges (E) added to the number of faces (F) would have a total of 2, or as a formula, V-E+F=2. To test this formula out, one must focus on the cube first. The cube has 6 faces (F=6), 8 vertices (V=8), and 12 edges (E=12). Substituting these values into the formula, it becomes 8-12+6=2, which is true. The tetrahedron has 4 faces (F=4), 4 vertices (V=4), and 6 edges (E=6). So, the formula with the tetrahedral values would be 4-6+4=2, which is also true. The icosahedron has 20 faces (F=20), 12 vertices (V=12), and 30 edges (E=30). Replacing the variables with these values, the formula would become 12-30+20=2, which also works out to be correct. The octahedron has 8 faces (F=8), 6 vertices (V=6), and 12 edges (E=12). The formula for the octahedron would become 6-12+8=2, which works out. Continuing on with the final platonic solid, the dodecahedron, which has 12 faces (F=12), 30 edges (E=30), and 20 vertices (V=20), the formula would also be accurate because 20-30+12 will have a total of 2. However, this formula isn’t only for Platonic solids. Table 1 shows properties that the five Platonic solids have. It’s now understood that F means faces, V means vertices, and E stands for edges. In the table, p represents the number of sides each face has and q shows the number of edges meeting at each vertex. The formulas pf=2e and qv=2e are used to prove if a solid is a platonic solid, or just to prove there are only 5. Simplifying these, the formulas become f=2ep and v=2eq, respectively. When subbing in these values into Euler’s formula, it turns into 2eq-e+2ep=2. When e is factored out the formula then becomes e(2q-1+2p)=2. Because Platonic solids are made up of polygons as faces, the minimum number of sides each face has is 3, p3. Also, because the number of edges meeting at each vertex has to be at least three as well, q3. Because Platonic solids cannot have a negative or no amount of edges, e is greater than 0. Therefore, it can be said that 2q-1+2p>0. After solving for1q, the formula becomes 1q>12- 1p. Substituting p for 3, because that is our lowest value for p, we know that the minimum value for 1qis 16, which marks the conclusion q<6. After doing the same thing for p, it also becomes known that p<6 as well. Putting together everything, the final equations 3q<6 and 3p<6 are produced. From this equation, 5 possibilities are derived: when p=3, q could equal 3, 4, or 5, when p=4, q=3, and when p=5, q=5. Because there are only five solutions, there are only five Platonic solids.
Why are Platonic solids important? Questioning the importance of Platonic solids is like asking why shapes are important. They may not be some mathematical formula that we use every day, but that doesn’t mean they’re nonexistent in people’s daily lives. Just like normal shapes, everyone will encounter at least one Platonic solid throughout their entire lives. Platonic solids are everywhere, places nobody even thinks of. There are crystals and gemstones that are shaped like octahedrons (figure 3). Modern-day satellites orbit with each other in a tetrahedral shape, so that it’s easier to make 3D observations and study them up close. Dice and beads are commonly shaped like cubes. Focusing more on dice, the shape of dice can be any of the Platonic solids because every face, angle, and edge are all equal, giving each face an equal chance of being landed on. There are also some molecules that are shaped like the different Platonic solids (figure 4). Platonic solids are all around, serving different purposes with each form.
Platonic solids are 3D figures that are convex and regular. They’re polyhedra, but they stand out and carry unique qualities. The Platonic solids were each assigned to the basic elements, air, earth, water, fire, and the universe itself. These solids have distinct properties that can be proven and found using the Euler’s formula. Through this, it’s possible to prove that there are only five Platonic solids in the 3-dimensional world. Platonic solids are everywhere, overlooked and never noticed. From outer space to a simple board game, Platonic solids are put into use.