Shapes

There are lots of names for shapes which you are supposed to know, such as "circle", "triangle" and "square". I don't need to tell you what those three shapes look like, but there are different kinds of triangle, and different kinds of quadrilateral (four-sided shapes) whose names you ought to learn to recognise. I'm going to start by showing you a square, but I'm not going to bother with fancy graphics for the first ones because they cost more money (I actually have to pay money to get this stuff to you over the Internet).
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The angles between the sides of a square are called "right angles" or "90 degree angles" (90 angles), though they can sometimes be described just by using the word "square" (as in "this line is square to that line"). If the angles of a quadrilateral are not right angles but the sides are all the same length, then you have a diamond, although the correct name for it is "rhombus":-
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The next shape is a rectangle, but it can also be called an "oblong".
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If the angles aren't square, it becomes a parallelogram:-
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There first triangle we need to look at is the equilateral triangle: "equi" means "equal" and "lateral" means "sided", so all the lengths are the same, and each of the angles is 60:-
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If only two of the sides are the same length, then you have an isosoles triangle (meaning "equal legs"!) which may look like this:-
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...or this:-
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There are also "right-angled triangles" which have a 90 angle at one of the corners, as seen below:-
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There are popular drawing tools used in design called "set squares" which are triangular in shape, and their name comes from the fact that one of their angles is "square". The other two angles may both be 45 (as in the diagram above), or else one will be 30 and the other 60 (as shown below):-
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A "polygon" can be any shape with three or more straight sides, but we normally only use the word when we're talking about shapes with five or more sides. The shape below is a pentagon:-
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...and the next is a hexagon:-
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Before we look at three-dimensional shapes, we have a couple of flat ones left to do. I'm sure you know what a circle looks like, but here's one anyway:-
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When a circle is stretched, it turns into an elipse, as you can see below. If you close one eye and look at the screen from the side, it will turn back into a circle:-
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Running tracks are oval rather than eliptical. The name "oval" means "egg-like", and it covers a variety of shapes including elipses, so it's not such a precise term. Running tracks are actually made up of two straight lines and two semi-circles, as shown below. If you close one eye and look in from the side, it doesn't turn into a neat circle in the way that a true elipse does:-
_ # # # # # # # # # # # # # # # # # # # _
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We'll now move on to some three-dimensional shapes. I can't show you a sphere (or a ball) very well because it looks just like a circle regardles of which direction you look at it from, but here it is anyway:-
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A cube is much easier to show. All the sides are supposed to be the same length, though you can't tell that from the diagram (if you measure them on the screen they will not seem the same at all):-
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The rectangular equivalent of a cube is called a "cuboid":-
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Three kinds of regular three-dimensional shapes can be made out of equilateral triangles. The simplest is the "tetrahedron" and it is made from four triangles:-
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.#########################################.
This should not be confused with the square-based pyramid which is made using four triangles round a square base. When two square-based pyramids are joined together by their square sides they turn into an octahedron: the two squares then become part of the inside and you can build the whole structure out of eight equilateral triangles:-

Diagram needed

The icosahedron, shown below, is made using twenty triangles. These pictures are actually stereo, but you'll need to cross your eyes to see any stereo effect: hold your thumb near the screen between the two pictures; bring it very slowly towards your nose, keeping your eyes on the tip of your thumb the whole time; and when it's a little bit nearer to your face than it is to the screen you should see the background images suddenly merge together into a 3D picture. It may take a while for you to get used to doing this, but the trick is to relax your eyes and not try to force things: just get your thumb into the right place and keep it there, then look at it and try to keep your focus on its tip. Your eyes will eventually tire of focussing on your thumb and they'll let their focus drift to the shape in the background instead:-

icosahedronicosahedron

If you're still struggling to see it in stereo, try getting a sheet of paper and making a round hole in it about 4cm across. You can then hold this sheet of paper half way between your face and the screen and very slowly bring it towards you: the 3D icosahedron should suddenly appear as if it's floating in space in the hole in the middle of the sheet of paper. The picture below shows the same model, but With twelve extra rods connecting in to the centre: you can now see that the whole thing can be built out of tetrahedrons:-

icosahedronicosahedron

This way of making stereo images works for moving pictures on ordinary televisions too: I don't know why it's never been used, because there are many occasions when diagrams and models of things in TV programs could be made much clearer by showing them in 3D. There are also lots of silly books full of strange patterns which contain hidden 3D shapes which you can only see by moving your eyes further apart than normal: it's much harder to do that than using the cross-eyed method, and it can't be used for photos if they're bigger than the distance between your eyes (and indeed it actually takes a mammoth struggle just to get photos half that size to merge). Anyway, we'll finish now by looking at my two favourite shapes. The dodecahedron is made out of twelve pentagons:-

Diagram needed

I wish Rubik would do the same with that as he did with the cube: what a fantastic puzzle that would be! Anyway, we move on now to our final shape. It is not possible to make a three-dimensional shape purely out of hexagons because they will only ever fit together to make a flat surface, but if you use a mixture of 12 pentagons and 20 hexagons, you can make a football (and I mean a proper football: not the fat-banana thing that Usanians call a football):-

Photo needed

If you look at it carefully, however, you may be able to see that it is really just a dodecahedron in disguise. Even so, the extra panels enable it to stretch out into a near-perfect sphere when it's inflated.