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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.

 

Hexaflexagons – Amazing Shapes Investigation:

Hexaflexagons look at first glance to be somewhat prosaic origami shapes – but like mobius strips they hide some surprises.  Flexagons are paper shapes that can be folded to reveal hidden faces – and hexaflexagons themselves have six sides.  What’s remarkable about hexaflexagons is that during folding, some faces are harder to find than others – which is somewhat counter-intuitive.  Surely, you would expect to rotate through the faces equally?

Princeton University student Arther Stone discovered flexagons in 1931.  After moving from Britain, he discovered that his paper no longer fit his new American style binders – and so after cutting his paper down to size he started to play around with the left over bits of paper.  In doing so he discovered a new structure to be explored in the mathematical field of topology (study of shapes and space).

You can experiment further by making tetrahexaflexagons (four sided shapes) and different types of hexaflexagons – which have some remarkable properties.  The Vi Hart video at the top of the page serves as an introduction to this topic – and the one below goes into the maths a little more:

If you would like to try and make your own then the template and instructions are available from here.

Some ideas for investigations into flexagons are taken from a fantastic book on the subject by The Art of Mathematics – available to read as a pdf here. I’ve copied some of their text below:

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To do some mathematical investigation you first need to learn how to “flex” the flexagon correctly:

1) Hold your Hexahexaflexagon flat.
2) Fold one creased edge up, into a mountain fold
3) Fold the adjacent creased edge down, into a
valley fold
4) Repeat the two previous steps each twice more so you have mountain and valley folds alternating around the six creased edges.
5) If you push the valley folds together, your flexigon will have folded up into a 3-pointed star; a shape that looks almost like a Y from above.
6) With your thumbs grab the point closest to you where the three valleys have come together. You should see that if you pull this point away from the center that the edges fold back down into a flattened hexagon.

That sound pretty difficult – but watch the videos again – it should make sense.

The book also provides some interesting starting questions for investigation:

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1) As you flex your hexa-hexaflexagon (henceforward simply referred to as a flexagon), what happens to some of the faces?
2) What is happening to the flexagon that allows the faces to disappear and reappear like this?
3) How many different faces do you seem to find as you first start flexing your flexagon?
4) Do you think that there might be more faces that can be found simply by flexing? Try to find some and then list those that you have found by listing the markings on the faces.
5) You should see several patterns in the markings on the faces that you have found. Describe these patterns.
6) Look back at the original strip that you used to make your flexagon. Based on the markings on this original strip, can you guess what other faces should be possible to make by flexing your flexagon?
7) Return again to the original strip you used to create your flexagon. Count how many faces make up this strip. Compare this number with the number of triangular faces that appears on each face of the flexagon.
8) Can you explain why the proper name for this flexagon is a hexa-hexaflexagon based on your investigations so far? Explain.
9) In a given state, how many different ways are there that you can flex your flexagon?
10) In a given state, how many layers thick, including the top layer, are the sections under each triangular face in your flexagon? (Count the two glued layers as a single layer).

If you enjoyed this post you might also like:

Imagining the 4th Dimension. How mathematics can help us explore the notion that there may be more than 3 spatial dimensions.

Wau: The Most Amazing Number in the World? A great video by Vi Hart – see if you can spot the twist!

Exploration Guide

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