Symbols
Geometry is full of symbols. Here are a few of the important ones:
Networks and Graph Theory
This is not necessary to review before looking at other units, as it is kind of a tangent, but it is fun to learn about, and doesn't necessarily fit into any other unit.
Graph theory was first developed by Leonhard Euler, who was posed a problem about the Seven Bridges of Konigsberg.
Graph theory was first developed by Leonhard Euler, who was posed a problem about the Seven Bridges of Konigsberg.
Photos courtesy of Wikimedia Commons
Seven Bridges of Konigsberg
A king wants to take a walk around the city of Konigsberg, to the left. He only wants to cross each bridge once and only once. Can he do it?
To do this problem, we have to think of a more abstract problem.
To do this problem, we have to think of a more abstract problem.
This is a network. The numbered points are called nodes, while the lines are called paths. They can model things like the city of Konigsberg.
This network is transversable, meaning that you can cross each path once and only once (sound familiar? The paths in the Seven Bridges of Konigsberg are bridges). A transversable network has 2 or 0 odd nodes, meaning that either all nodes or all nodes except two nodes have an even number of paths coming out of them. The network above has 2 odd nodes- one where you start your trip (around the city, say), and one where you end your trip. If there are no odd nodes, then you start and end your trip at the same, even node.
This network is transversable, meaning that you can cross each path once and only once (sound familiar? The paths in the Seven Bridges of Konigsberg are bridges). A transversable network has 2 or 0 odd nodes, meaning that either all nodes or all nodes except two nodes have an even number of paths coming out of them. The network above has 2 odd nodes- one where you start your trip (around the city, say), and one where you end your trip. If there are no odd nodes, then you start and end your trip at the same, even node.
The network above represents the Seven Bridges of Konigsberg (the paths are bridges and the nodes are islands). The king is in essence asking if the network above is transversable.
So, can you figure it out? Can the king take his walk without crossing any bridge twice?
So, can you figure it out? Can the king take his walk without crossing any bridge twice?