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Geometries????


To use geometry in every-day life, we rely on hundreds of small assumptions and laws.

Take the SOH CAH TOA, area of a circle= pi x r^2 and that a tangent to a circle is at an angle 90 degrees to its centre. But have you ever tried to prove these from first basis? I reckon if you tried, you would land on another level of assumptions.


For example: area of triangle = 1/2absinC


Where to start?


Instinct might say to jump to trig (sin cos and tan), but do we know how to prove them?


A more seasoned mathematicians might point you onwards to the unit circle, where sin and cos can be proved to be the x y coordinates of points on a circle centre (0,0) and radius 1. But this relies on the geometry of lines intersecting circles - how have we proven from the most basic definition of a circle that it will intersect a line.


What is intersection? WHAT IS A CIRCLE????


Further and further down the rabbit hole we go.


Rather than purpose this rather helpless form of logical deduction, the most ancient mathematicians tried to build mathematical principles by identifying the bare minimum that must be defined without a deeper level of explanation. I will dedicate another post to explaining the difference between axioms and postulates, but to assume these are the author's best attempts at 'universal truths' is good enough for now. For the natural numbers, these were Peano's axioms (see Analysis!!)


For geometry, Euclid's postulates are as follows:


Postulate 1 : A straight line may be drawn from any one point to any other point.


Postulate 2 : A terminated line can be produced indefinitely. day terms, the second postulate says that a line segment can be extended on either side to form a line


Postulate 3: A circle can be drawn with any centre and any radius.


Postulate 4 : All right angles are equal to one another.


Postulate 5 : If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

(in other words, given a line A and a point b not on line A, there is only one line that goes through b and is parallel to A)


He of course prefaced these with a number of definitions, such as


1. A point is that which has no part.

2. A line is breadth-less length.

3. The ends of a line are points.

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.

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This all seems very simple, but isn't it fantastic that all of geometry has been derived off of these principles??!! They all seem pretty straight forward, like you could have thought of them yourself. But it is a surprisingly difficult exercise to identify what ideas come from simply being what we call 'true', and what ideas are logical implications from these truths.


Well, not ALL that straight forward. 5 is lookin' a little complicated, not very elegant, and not completely obviously true. Mathematicians tried to prove it for ages and some have gone MAD thinking about it. Carl Friedrich Gauss (the coolest person ever), Janós Bolyai, and Nicolai Ivanovich Lobachevski actually found that it was not necessarily true in all dimensions.

Consider a sphere. If you were to draw two straight lines on a globe, they could obey all the rules for a straight line and still intersect twice. Furthermore, as shown in the diagram below, you can have triangle which obey all previous postulates, yet have internal angels adding to more than 180 degrees!!



The two main areas of weird geometry (or non-Euclidean geometry) can be thought of in this context.


Hyperbolic Geometry - There are two or more lines which go through b and never intersect A.


Elliptical (spherical) Geometry - There are no lines which go through b and never intersect A.


Cool right??!! Two very fancy looking ideas, rooted in a small and simple to understand problem.





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