Cosmology for Beginners

Ellipses
Spirals

Shapes of galaxies

(Irregular galaxy source http://starchild.gsfc.nasa.gov/docs/StarChild/universe_level2/ngc6822.html
Elliptical galaxies source http://www.gsfc.nasa.gov/scienceques2002/20030131.htm, picture credits Digital Sky Survey and NASA/CXC/UVa/CSarazin et al
Spiral galaxy source http://starchild.gsfc.nasa.gov/docs/StarChild/universe_level2/ngc2997.html http://www.spacetoday.org/images/Hubble/HubbleBeauty/NGC7742SpiralGalaxyNASA.jpg picture credit NASA/AURA/STSCI
NASA copyright free policy )

A galaxy is a cluster of stars, dust, and gas which is held together by gravity. They vary considerably in size and are to be found throughout the universe, either alone or in groups, known as superclusters. Galaxies are classified according to their shape and appearance. Irregular galaxies (top left) have an undefined shape and are made up of young stars, dust, and gas. Spiral galaxies (bottom photos) are disc-shaped with spiral arms which spiral outwards as the galaxy rotates. They tend to contain more middle-aged stars plus gas and dust. Elliptical galaxies contain older stars and very little gas and dust. Elliptical galaxies (middle and top right) vary in their shape from round to flattened, elongated spheres.

Ellipses and spirals are mathematical shapes, which you can investigate further in this project.

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Ellipses

Drawing ellipses

You can use string and blu-tak or drawing pins to draw an ellipse too. For this you will need:

• piece of string, say about 50 or 60 cm long (or longer for really big ellipses!)
• two thumbtacks or drawing pins
• a piece of paper held down securely on a surface you can push pins into
• some coloured pencils

Start by forming a loop by tying the ends of your piece of string firmly together.

• Now take your piece of paper and make sure it is held down securely. Draw a straight line on it - the length of this line should be less than half the circumference of your loop.
• Press two drawing pins (thumbtacks) into the paper at the ends of the straight line, and mark the positions of the drawing pins (shown as A and B on the image to the right). Place the loop of string around the two drawing pins.
• Put the pencil inside the loop of string, and pull the string tight as shown. Keep the loop of string taut and then run the pencil around inside the loop so it makes a mark on the paper. Be careful that the string is always looped around both drawing pins and that it doesn't slacken at any point.
• Repeat this for different line lengths with the same loop of string.
• What do you notice about the shapes of the resulting ellipses?
• How do you think you might get a circle using this method?

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About ellipses

Other things to find out about ellipses:

• which planet has the most elliptical orbit (or least circular)?
• is an egg ellipse-shaped?
• what shape is the Whispering Gallery in St Paul's Cathedral in London?
• can you find out why it is called the Whispering Gallery, and what this has to do with ellipses?

Here is another activity you could try. For this one you will need:

• a sheet of paper
• compasses and pencil
• a ruler (optional)

1. Start by cutting out a large circle from a sheet of paper .
2. Now mark an X somewhere in the circle (but NOT at the centre).
3. Fold the paper so that the edge touches the 'X', making a crease.
4. Do this over and over again, making different folds in the paper (you might want to draw a line along the crease so it shows up well).

What shape do the creases make? How does where you put the 'X' affect the shape you get?

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Spirals

The image on the left shows a spiral which you can create either by drawing or by using Logo. To draw this image, start with a rectangle 10cm x 16.18cm. It is important that you draw this as accurately as you can.

Now mark off a square of side 10cm, and draw a quarter circle in it, with the corner of the square as the centre for the arc, as in the first image on the right. The repeat the process, as in the image on the far right. Mark off a square at the top of the right hand rectangle (which will have side 6.18cm), and draw another quarter circle, with the corner of the square as its centre, so that the arc is continuous with the previous arc. Continue this process until the square becomes too small to continue.

Alternatively, you could set up a Logo program to draw the spiral for you (which was how these images were obtained). You can download MSW Logo (freeware) if you do not have it available. Try to write a program for yourself. You can use the hints below if you are unfamiliar with Logo. If you get completely stuck, a full program is given in the answers to this project.

Here are some useful basic commands:

Use the Set menu to change the pen size, pen colour and screen colour.

You will need to start with a program for a square whose side length you can vary:

to square :side
repeat 4[fd :side rt 90]
end

Logo knows this is a program from the 'to' at its beginning and the 'end'. When you type this program into the Commander (the line at the bottom where you input commands), you should get the response 'square defined'. It is important that you include the spaces shown in this program and the colon. The colon in front of 'side' means it is a variable, and putting the 100 after 'square' when you want to make a square tells the Commander what value this should have. Now test that your program works by typing:

square 100

into the Commander. You should get a square of side length 100. If you don't, use the Edall button to edit your programme, then File > Save and Exit to return to the Commander. It is important that you have spaces and colons in the right places.

Try your square command for different values of the side length.

You will also need a program for a quarter circle.

Logo works in straight lines, but you can get what looks like a circle by making a polygon with a lot of sides. Here is a program for a polygon with a variable number of sides (n) and a variable side length (x). Check that you understand what Logo is doing in this program and that you can see how the polygon is constructed. Can you see why n needs to be a factor of 360 degrees?

to poly :n :x
repeat :n[fd :x rt 360/:n]
end

Again test your program to see that it works by typing in poly 12 20. This should draw a 12-sided polygon with a side length of 20 units. Now see what poly 72 5 looks like. This should look like a circle. However, we don't want a whole circle, but a quarter circle. Can you see how to edit the program to make a quarter circle? Test that it works. (Click here for help with this).

You now have the basic programs you need to construct the spiral. Before you can go further, however, you need to know how to use these programs in another program for the spiral, and how to make the sub and poly programs repeat themselves. Let's call the overall program 'to spiral'. This is how it should be structured:

to spiral :side :n
square :side
rt ?
poly :n :side*2*tan(180/:n)
lt ?
if side<1 [stop]
spiral :side* ? :n
ht
end

Let's have a look at this program in more detail. It starts by telling Logo that it will need two variables to be specified, with the names of 'side' for the side length and 'n' for the number of sides. The value for n needs to be big enough for the side length of the polygon you are using for your quarter circle not to distort the squares in the spiral, so you get a distorted spiral, try increasing the value of n.

The program then instructs Logo to construct a square of whatever length has been specified for the initial side length. At the end of this square, the turtle (cursor) will be pointing directly upwards. However, we do not want our first quarter circle to start going straight up - if you leave out the lines rt ? and lt ?, you will see why. Your first problem here is to work out what angle the correction needs to be. You can do this if you think about the connection between the angles at the centre of a polygon and the exterior angle (as in the diagram on the right). This angle will depend on the number of sides you have put in your polygon, of course. (Click here for help with this).

The program then draws the quarter circle, using the corrected program for to poly, so that it draws a quarter circle. The side length here is given to you, as you need to know trigonometry to calculate the side length of the polygon sides from the square side length. The next command is for the turtle to turn left - this angle correction is the same as the right correction made above, and puts the turtle back in position to draw the next square. To see why this is necessary, leave it out and see what happens.

The line starting with 'if' is very important. We are about to set up a recursion, which means the program will carry on for ever, unless it has some command which stops it. This line stops the program when the side length of the square falls below 1 unit - at which point it will be too small to distinguish. Then we tell Logo to repeat the whole spiral program again. So far we have made one square with one quarter circle in it. This time, however, we need to tell Logo how to decrease the length of the side of the square. The factor here is a number known as the golden ratio. The final command 'ht' instructs Logo to hide the turtle, so that we can see the final construction without the turtle hiding any of it.