Day 20, MAT115

1. Complete graphs: the completely social simple graph (everyone connected). The formula for number of edges possible (i.e., the case of the complete graph) is a triangular number, which you can get out of Pascal's triangle: \[ \frac{n(n-1)}{2} \]

   Try to use symmetry to make these beautiful, and to help you to see that you've got all the edges.

   Every vertex is the same in a complete graph -- has the same degree, the same connectivity as every other.

2. Simple graphs:
   • 1 simple graph with one vertex; 2 simple graphs with two vertices; 4 simple graphs with three vertices.
   • Simple graphs, four vertices: 11
   • Simple graphs, five vertices: 34
   • Simple graphs, six vertices: 156 of them! Good thing I didn't ask you to draw those!
   • 1, 2, 4, 11, 34, 156: the pattern's not obvious, but they appear to be growing quickly!
3. For your house diagrams, I want you to think like Euler: he turned each land mass into a vertex, and each bridge into an edge. We associate each room to a vertex, as though it is a "land mass"; and doors connect "land masses" like bridges. So each door is one edge, connecting two land masses. (One room may be a hallway, if your house is like mine. All the bedrooms open onto a hallway, which can be considered a room, which then connects up with the living room.)

   Often in mathematics we're trying to formulate the correct (or best) analogy.

4. The "enemy of my enemy" reading is all about triangles, so I'm surprised to see people making graphs without any triangles.

   My point in the previous problem #3 is that one should try to create the best analogy -- and I think that in this case you want to continue to think about triangles!

Here's a key.

Note that, although you are to draw some links and knots for this assignment, these will not be on the exam.

It will feature everything from Fibonacci and golden spirals on. In particular we learned a little of the Egyptian's mathematics (multiplication and fractions), then about symmetry, the Platonic solids, and more on graphs. Ending with planar graphs.

Make sure that you've done all the associated readings, since you'll see some questions about those as well.

Emmy asked: I would love one to be posted, maybe in two parts, just a general review video, a worksheet with a little bit from everything we've covered (and with maybe some questions from readings and stuff) with how it will be on the test, and then a second video reviewing the worksheet (so maybe we can do it and check our answers).