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//********************* Platonic Solids - April 5th, 2019 *******************//
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- Chicago: land of the sea-like pizzas (and Carl Sandburg)
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- So, we were talking about regular polygons yesterday: polygons that have equal sides, equal angles, AND are convex (*squints suspiciously at pentagram*)
- "Pentagrams are great for summoning demons in games, but not much else"
- As it turns out, there's a nifty mathematical problem called REGULAR PLANER TILING
- The problem is this: suppose we want to fill an infinite plane using just one type of regular polygon, all the same size, without any gaps or overlaps
- (I suppose mathematicians just have a cabal of infinite warehouses somewhere, in desperate need of interior decorating)
- A SQUARE TILING is easy - it's just a bunch of squares all put together!
- Here, each of the tiles has 4 sides (of course) and each vertex has a valence of 4
- We can also do a DUAL TILING, where we take the regular tiling, swap the vertices and faces (i.e. put a vertex in the middle of each polygon, put a polygon over each vertex), and then rotate the edges to connect the vertices
- This will result in the valences and the number of sides swapping values
- In the square tiling case, we just end up with a SELF-DUAL: the polygons have 4 faces, and the vertices have valence 4, so all that happens is that the tiling gets shifted a bit by the dualing operation
- Not all the tilings, though, are so trivial:
- A TRIANGULAR TILING has 3 sides, but results in a valence of 6; we basically put the triangles in a bunch of hexagonal shapes!
- So, 3 sides, valence of 6 - if we "dual" this tiling, we'll actually end up with a hexagonal tiling, with valence 3
- Why don't we see this tiling used in more apartments? Well, it's probably due to practical ceramic reasons; the triangle tips are sharper, and more likely to break
- So, a HEXAGONAL TILING then will conversely have 6 sides and a valence of 3, and will "dual" into a triangular tiling
- This looks like a honeycomb, so mathematicians, in their infinite wisdom, have nicknamed this the "Honeycomb tiling"
- (Professor Turk takes this opportunity to pass out a bunch of plastic models of platonic solids)
- "I want to give out all of them now, so you all have a chance to hold a platonic solid"
- This leads us into the first PLATONIC SOLID we'll talk about: the CUBE!
- As we know, a cube (or a "hexahedron") has 6 faces, 8 vertices, and 12 edges
- ...what if we tried our "dual" tiling operation on this 3D cube, though? Let's try it!
- So, we put a vertex in the middle of each face, connect them with lines, and what do we end up with? An OCTAHEDRON - the next platonic solid!
- This is a solid with 8 faces, 6 vertices, and 12 edges- huh! The number of edges seems the same, and the numbers of faces/vertices have flipped!
- So, the cube/octahedron are duals of each other
- It looks like dualing a platonic solid just swaps the number of edges/vertices - cool!
- As it turns out, there's an even simpler platonic solid: the TETRAHEDRON, with 4 sides, 4 vertices, and 6 edges
- This looks like a pyramid with a triangular base, and could be constructed by taking 4 "opposite" vertices of a cube and connecting them
- What's the dual of a tetrahedron, though? Let's see...
- As it turns out, it just makes ANOTHER tetrehadron; like the square tiling, it's a self-dual!
- So, there are two more platonic solids; let's consider them
- From the top-down, a tetrahedron looks like a "circle" of 3 triangles, and an octahedron looks like one with 4 triangles - what happens if we make one with 5 triangles?
- Well, we'd end up with an ICOSAHEDRON, composed of triangles and with a total of 12 vertices, 20 faces, and 30 edges
- If we dual THIS, we end up with the 5th and final platonic solid...
- A DODECAHEDRON, composed of pentagonal faces and with 20 vertices, 12 faces, and 30 edges
- ...and as it turns out, that's it; there aren't any more platonic solids
- If we add a 6th triangle to our "fan," it'll just lie flat; it won't form a valid 3D shape
- Similarly, there's
- So, 5 platonic solids - what are they used for? Well, lots!
- They're VERY useful for many-sided dice
- A soccer ball is a close relative of the dodecahedron/icosahedron; it's known as a "truncated icosahedron"
- Radiolarans, a microscopic creature, actually have an icosahedral body
- Geodesic domes come from the nice property that icosahedrons can be subdivided into a sphere
- Cubes are also very useful since they're the ONLY platonic solid that can tile a 3D space
- If you're allowed 2 solids, though, an "octet" of tetrahedrons/octahedrons will also tile in 3D, and actually form a very strong structure sometimes used for building struts
- In chemistry, you find the patterns of molecules adhering to one another tend to form platonic-solid-shaped lattices
- ...and in computer graphics, they tend to be very good "starting blocks" for building other shapes
- Now, let's talk about a very famous mathematical equation:
V + F = E + 2
- Where "V" is the number of vertices, "F" is the number of faces, and "E" is the number of edges
- This is EULER'S EQUATION, and actually holds not just for platonic solids, but for a whole host of other shapes
- In fact, it'll hold for any 3D solids that don't have "handles" - it's pretty general!
- Why "+2"? That doesn't seem very clean, but remember: vertices are 0-dimensional, edges are 1-dimensional, and faces are 2-dimensional
- So, the equation is basically saying "the number of even-dimension geometric objects (vertices and faces) is equal to the number of odd"
- So, the 2 is the number of volumes - there's an inside and an outside for each 3D solid!
- "...I'd like the platonic solids to come back, please"
- All right, that's the day!