Day 21, MAT115

1. The Earth as Icosahedron
2. Hexagonal faces can't be folded up from a tiling (or tesselation) of the plane; Octagonal faces can't even be put together to tile the plane.

   Put two hexagons (or octagons) together along an edge. Now try to slip another hexagon (or octagon) in at a vertex along the edge. It can't be done without flattening out the hexagons (and it can't be done at all for the octagon).

   So you'll not be able to create a platonic solid from these (or any other regular polygons).

   Another answer.

3. Logos:
   • Mitsubishi
   • Game Cube
   • Citgo
   • PNC bank
   • Flextegrity

4. here they are, the Platonic graphs.
5. All have F-E+V=2.
6. The soccer ball has 32 faces, 60 vertices, and 90 edges. F-E+V=2.

• One focus of this reading is negative numbers. It was the Indians (of Asia) who put negative numbers on a firm foundation (as well as zero), in about the 6th century AD.

  "...you can't see negative 4 cookies and you certainly can't eat them -- but you can think about them...." ("and you have to", says our author).

• One of the confusing aspects of negative numbers from Strogatz's perspective is that the product of two negative numbers is a positive number.

• But then Strogatz goes on to use the context of social interactions (negative and positive) to discuss how we maintain stable relationships.

  And the key to understanding stability in three-way social relationships is that the product of two interactions (signified by either +1 or -1) must be equal to the other: so that if two legs are positive, the third in the triangle must be positive; if one leg positive, and the other negative, then the third leg must be negative as well.

  

  Strogatz sums up the second case above in the familiar saying that "The enemy of my enemy is my friend".

  The following (two) graphs are unbalanced:

  
  Why is the all negative graph also unbalanced? Because "the enemy of my enemy is my friend!"

  Finally Strogatz shows how historical relationships settled down into this pattern of stability: in "...the run-up to World War I. The diagram that follows shows the shifting alliances among Great Britain, France, Russia, Italy, Germany, and Austria-Hungary between 1872 and 1907."

  

  The bottom right graph (complete!) is the only stable configuration, "...balanced, but on the brink of war."

This section describes how Google uses the "PageRank" algorithm to determine the importance of a webpage (and hence where it falls in the search results for a particular topic).

Graphs provide a useful way of illustrating how pages interact. If there's a link between two pages, then a directed arrow indicates it. Here's the graph of the "toy web" Strogatz considers, with the final rankings:

He justifies this ranking in a series of graphs, and a set of equations on page 195. Let's see how these equations work (we'll use this Excel spreadsheet).

Notice that I've written the equations with an index, , rather than with the primes. That's because we keep updating the values to get them at the stage, and we update based on the previous stage's () values.

We just "do it again", over and over....

This strange algebra is from the field of mathematics called "linear algebra".

In this reading, we discover how to rotate and flip our mattress over the course of the year, so that the mattress gets even wear (while avoiding having to flip the mattress end-over-end the long way -- the "death-defying vertical flip").

"Spin in the spring, flip in the fall" is what some manufacturers suggest.

There is a certain amount of curious algebra here:

means that a horizontal flip followed by a rotation is equivalent to a vertical flip. Using only horizontal flips and rotations, we can achieve each of the four possible configurations of a matress. So over the course of two years, you'll have visited all four configurations of your mattress. Bon Nuit!

Standard diagram of the Borromean rings

A realization of the Borromean rings as ellipses

Coat of arms (of the municipality of Hallsberg, Sweden) showing padlocks in Borromean rings configuration

(Thanks Wikipedia!)

• Use symmetry!
• How far off golden are we?
• Do you see the icosahedron?

• I always think of them in the context of this quote:

  "We must all hang together, or assuredly we shall all hang separately." Benjamin Franklin, at the signing of the Declaration of Independence.

• Led Zeppelin, in the album sleeve of their album Led Zeppelin IV used it:

  

  "Drummer John Bonham's symbol, the three interlocking rings, was picked by the drummer from [Rudolf Koch's Book of Signs]. It represents the triad of mother, father and child, but also happens to be the logo for Ballantine beer." (from the Wikipedia article on Led Zeppelin IV).

  

  (thanks Matt!).

• If you want to draw the Borromean rings, you'd draw three circles, as in the first figure above:	but you'd want to indicate, somehow, that one ring is below another ring (aka the "Irish Trinity"):

  One more:

  

  Scene from Stora Hammar stone

  with a closeup of the rings: