Topology Topics

 

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What is topology? It is a branch of mathematics that deals with the way geometrical objects are made and connected, but doesn't deal with their dimensions. A topologist would ask a questions such as 'how do they connect?' or 'does it have holes in it and how are they made?' rather than 'how long is their connection' or 'how big are the holes'.

Sometimes we have to solve problems which can not 'fit' into two or three dimensions - which are easy for us to study or depict. We would like to be able to draw a ten-dimensional graph sometimes, to explain various influences between objects and connections between them, but we don't have enough dimensions to do that. It would also be very difficult to try and imagine that because our intuition is used to only three dimensions! We can perhaps sometimes use algebra, but then we loose the visual insight which is one of the most important and useful tool we humans have in understanding relationships. In this case we use topology to consider all the possible relationships and connections so that we can study them.

Here are some illustrations of the application of topology:

Mathematical topology - understanding the principles of relationships between objects, and the way the objects themselves are made

 

Möbius strip

Klein bottle

 

Königsberg bridge problem

 

Application of topology to communications, information dissemination and the Internet

 

Topological maps - the London underground map

Communication systems maps - like this one of the Internet (click on it to see bigger picture)

 

Computer network maps

 

Some famous mathematicians who contributed to the development of topology are

Euler (1707-1783)

Möbius (1790-1868)

Listing (1808-1882)

 

Riemann (1826-1866)

 

Poincaré (1854-1912)

 

Hilbert (1862-1943)

 

 

 

   

See some topological topics

Möbius strip

click on the number man for a worksheet on Möbius strip

Königsberg bridge problem

The Four Colour Theorem

See topological atlas of cyberspace here for some amazing pictures of the Internet and communication networks.

 

 

 
 

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