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## Math And Geometry - Math's Relation To Geometry

Geometry is concerned with the properties of space and the shapes and relationship of things in it. It is an important topic for enlightened discussion. It is also amusing how much of mathematics is relative to geometry. If algebra equations can be any set of objects characterized by abstract symbols and a set of rules, the only standard is that the algebra is consistent, has no contradictions, no requirement to represent reality, so why does so much of it have the interpretation of geometry?

Counting may seem more abstract and fundamental than measuring, but many of the most fundamental concepts all seem to be a study in geometry first. The traditional geometry that many of us learnt at school involves two dimensional constructions with points, lines between points, angles between lines and trigonometry. This type of geometry is very useful and we use a lot of it in various situation. We owe a debt of gratitude to Euclid who was a Greek mathematician from around 300 BC who unified the known geometry and derived results from five postulates. (Euclid's theory is straight line, a finite straight line, a circle, a right angle and two straight lines meeting.)

Over the many years we learnt about other types of geometry, known as non-Eucliden geometries, which are derivatives by changing the postulates. When we rearrange the fifth postulate, the parallel postulate, we get hyperbolic geometry. Along with this there is the vector space which has direction and magnitude and is useable to specify the position of points relative to the origin of the coordinate system. Using vectors can do different operations such as transforming a shape much more automatically by transforming all the vertices of the shape.

We can also detail physics principles using vectors. Such operations on vectors are specific using vector algebra; alternatively linear transforms on vectors can be specific using matrices or quaternions. There could be advantages if we have the option of other geometry approaches available too. Looking at things in dimensions seems to show us an underlying structure of things properties of 3D space that seem arbitrary when studied alone seems to fall into a similar pattern. Clifford algebra is one way of working in dimensions and group theory helps us categorize the properties.