Euler Article

The foundations of topology are often not part of high school math curricula, and thus for many it sounds strange and intimidating.

A simple way to describe topology is as a 'rubber sheet geometry' - topologists study those properties of shapes that remain the same when the shapes are stretched or compressed.

The 'Euler number' of a 'network' like the ones presented later in this discussion is an example of a property that does not change when the network is stretched or compressed.

Solution


Problem 2
Suppose they had decided to build one fewer bridge in Konigsberg, so that the map looked like this:



Now try to solve the problem.

His work in this field was inspired by the following problem:


The Seven Bridges of Konigsberg
In Konigsberg, Germany, a river ran through the city such that in its center was an island, and after passing the island, the river broke into two parts.

A crude map of the center of Konigsberg might look like this:



The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once.

Sketch the above map of the city on a sheet of paper and try to 'plan your journey' with a pencil in such a way that you trace over each bridge once and only once and you complete the 'plan' with one continuous pencil stroke.

A crude map of the center of Konigsberg might look like this:



The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once.

His work in this field was inspired by the following problem:


The Seven Bridges of Konigsberg
In Konigsberg, Germany, a river ran through the city such that in its center was an island, and after passing the island, the river broke into two parts.

A simple way to describe topology is as a 'rubber sheet geometry' - topologists study those properties of shapes that remain the same when the shapes are stretched or compressed.

Sketch the above map of the city on a sheet of paper and try to 'plan your journey' with a pencil in such a way that you trace over each bridge once and only once and you complete the 'plan' with one continuous pencil stroke.

A simple way to describe topology is as a 'rubber sheet geometry' - topologists study those properties of shapes that remain the same when the shapes are stretched or compressed.

A simple way to describe topology is as a 'rubber sheet geometry' - topologists study those properties of shapes that remain the same when the shapes are stretched or compressed.

Sketch the above map of the city on a sheet of paper and try to 'plan your journey' with a pencil in such a way that you trace over each bridge once and only once and you complete the 'plan' with one continuous pencil stroke.

A simple way to describe topology is as a 'rubber sheet geometry' - topologists study those properties of shapes that remain the same when the shapes are stretched or compressed.

The foundations of topology are often not part of high school math curricula, and thus for many it sounds strange and intimidating.

Sketch the above map of the city on a sheet of paper and try to 'plan your journey' with a pencil in such a way that you trace over each bridge once and only once and you complete the 'plan' with one continuous pencil stroke.

One of these areas is the topology of networks, first developed by Leonhard Euler in 1735.

Topology is one of the newest branches of mathematics.

One of these areas is the topology of networks, first developed by Leonhard Euler in 1735.

Sketch the above map of the city on a sheet of paper and try to 'plan your journey' with a pencil in such a way that you trace over each bridge once and only once and you complete the 'plan' with one continuous pencil stroke.

A simple way to describe topology is as a 'rubber sheet geometry' - topologists study those properties of shapes that remain the same when the shapes are stretched or compressed.

A simple way to describe topology is as a 'rubber sheet geometry' - topologists study those properties of shapes that remain the same when the shapes are stretched or compressed.

The 'Euler number' of a 'network' like the ones presented later in this discussion is an example of a property that does not change when the network is stretched or compressed.