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euclidean

Non Euclidean Geometry V – Pseudospheres and other amazing shapes

Non Euclidean geometry takes place on a number of weird and wonderful shapes.  Remember, one of fundamental questions mathematicians investigating the parallel postulate were asking was how many degrees would a triangle have in that geometry- and it turns out that this question can be answered depending on something called Gaussian curvature.

Gaussian curvature measures the nature of the curvature of a a 3 dimensional shape.  The way to calculate it is to take a point on a surface, draw a pair of lines at right angles to each other, and note the direction of their curvature.  If both curve down or both curve up, then the surface has positive curvature.  If one line curves up and the other down, then the surface has negative curvature.  If at least one of the lines is flat then the surface has no curvature.

Positive curvature:

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A sphere is an example of a shape with constant positive curvature – that means the curvature at every point is the same.

Negative curvature:

 

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The pseudosphere is a shape which is in some respects the opposite of a sphere (hence the name pseudo-sphere).  This shape has a constant negative curvature.  It is formed by a surface of revolution of a called called a tractrix.

Zero curvature:

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It might be surprising at first to find that the cylinder is a shape is one which is classified as having zero curvature.  But one of the lines drawn on it will always be flat – hence we have zero curvature.  We can think of the cylinder as analogous to the flat plane – because we could unravel the cylinder without bending or stretching it, and achieve a flat plane.

So, what is the difference between the geometries of the 3 types of shapes?

Parallel lines

Firstly, given a line m and a point p not on m, how many lines parallel to m through p can be drawn on each type of shape?

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A shape with positive curvature has no such lines – and so has no parallel lines.  A shape with negative curvature has many such lines – and so has many parallel lines through the same point.  A shape with no curvature follows our normal Euclidean rules – and has a single parallel line through a point.

Sums of angles in a triangle and other facts

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Triangles on shapes with positive curvature have angles which add to more than 180 degrees.  Triangles on shapes with negative curvature have angles which add to less than 180 degrees.  Triangles on shapes with no curvature are our familiar 180 degree types.  Pythagoras’ theorem no longer holds, and circles no longer have pi as a ratio of their circumference and diameter outside of non-curved space.

Torus

The torus is a really interesting mathematical shape – basically a donut shape, which has the property of of having variable Gaussian curvature.  Some parts of the surface has positive curvature, others zero, others negative.

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The blue parts of the torus above have positive curvature, the red parts negative and the top grey band has zero curvature.  If our 3 dimensional space was like the surface areas of a 4 dimensional torus, then triangles would have different angle sums depending on where we were on the torus’ surface.  This is actually one of the current theories as to the shape of the universe.

Mobius Strip and Klein Bottle

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These are two more bizarre shapes with strange properties.  The Mobius strip only has one side – if you start anywhere on its surface and follow the curvature round you will eventually return to the same place having travelled on every part of the surface.

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The Klein bottle is in someways a 3D version of the Mobius strip – and even though it exists in 3 dimensions, to make a true one you need to “fold through” the 4th dimension.

The shape of the universe

OK, so this starts to get quite esoteric – why is knowing the geometry and mathematics of all these strange shapes actually useful?  Can’t we just stick to good old flat-plane Euclidean geometry?  Well, on a fundamental level non-Euclidean geometry is at the heart of one of the most important questions in mankind’s history – just what is the universe?

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At the heart of understanding the universe is the question of the shape of the universe.  Does it have positive curvature, negative curvature, or is it flat?  Is it like a torus, a sphere, a saddle or something else completely?  These questions will help determine if the universe is truly infinite – or perhaps a bounded loop – in which if you travelled far enough in one direction you would return to where you had set off from.  It will also help determine what will happen to universe – will it keep expanding?  Slow down and stop, or crunch back in on itself?  You can read more on these questions here.

 

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IB HL Paper 3 Practice Questions (120 page pdf)

IB HL Paper 3 Practice Questions 

Seventeen full investigation questions – each one designed to last around 1 hour, and totaling around 40 pages and 600 marks worth of content.  There is also a fully typed up mark scheme.  Together this is around 120 pages of content.

Available to download here.

IB Maths Exploration Guide

IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Includes:

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Modelling Guide


IB Exploration Modelling Guide 

A 50 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Modelling Guide includes:

Linear regression and log linearization, quadratic regression and cubic regression, exponential and trigonometric regression, comprehensive technology guide for using Desmos and Tracker.

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Statistics Guide

IB Exploration Statistics Guide

A 55 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Statistics Guide includes: Pearson’s Product investigation, Chi Squared investigation, Binomial distribution investigation, t-test investigation, sampling techniques, normal distribution investigation and how to effectively use Desmos to represent data.

Available to download here.

IB Revision Notes

IB Revision Notes

Full revision notes for SL Analysis (60 pages), HL Analysis (112 pages) and SL Applications (53 pages).  Beautifully written by an experienced IB Mathematics teacher, and of an exceptionally high quality.  Fully updated for the new syllabus.  A must for all Analysis and Applications students!

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