Adi's Mathematics

Subtitle

Adi's Mathematics

Adrian Cox

Educated at the Open University and received a B.Sc, (Open) degree in Mathematics.

Adrian Cox also plays guitar and has music under the name of EMO Adi

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Music by EMO Adi - Do What You Want

Applications And Non Euclidean Geometry

by Adi Cox 29th April 2013

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A curious thing is that the area of a circle with radius

one has an area of pi. Is this true if a circle is

measured by a unit larger than a galaxy, or some tiny

quantum unit smaller than an atom, I wonder if space is

really that consistantly euclidean.

Non Euclidean geometry is something I was taught at the

Open University and this is a subject I very often find

myself thinking about. I have often thought about

exploring this subject and so here I am writing this

small piece.

Geometry is generally considered to be a pure

mathematical subject but I argue here that really

Eucldean geometry could be seen as a more applied

mathematics because it replicates the space we live in

around us. Whereas Non Euclidean geometry is a litte

more exotic.

Although Non Euclidean geometry is easy to find

applications for finding the areas of curved surfaces

like second order quadrics in a real three dimentional

space with equations like the following:

Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz

+ Gx + Hy + Iz + J = 0

This world we live in is very much three dimensional. Is

there anything that is not three dimensional in the real

world? Even shadows are projections onto uneven surfaces

with three dimensional irregularity. The truth is that a

two dimensional Non Euclidean area needs three

coordinates.

So my final point here is that to take Non Euclidean

geometry up another dimension we will be out of this

physical space because we shall require third order

quadrics in a four dimensional space with equations like

the following:

Aw^3 + Bx^3 + Cy^3 + Dz^3

+ Ewxy + Fwxz + Gwyz + Hxyz

+ Iw^2 + Jx^2 + Ky^2 + Lz^2

+ Mwx + Nwy + Owz + Pxy + Qxz + Ryz

+ Sw + Tx + Uy + Vz + W = 0

Three dimensional Geometry in a four dimensinal space is

more abstract and trully a pure mathematics because it

takes us into a mathematical place that is not

immediately representative of the physical world around

us.

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