Symmetry

This session uses Logo to demonstrate line and rotation symmetry

You are going to edit two routines called "shape and "shapeleft. Once you have programmed these routines into Logo you can then call them just like the normal Logo commands.

Get Logo running in the usual fashion and then call up the routine editor by entering edit "shape.

edit "shape

Enter this code to make a shape with only the most basic symmetry (order 1 rotation symmetry). You can make your own shape if you like. Remember to "save and exit" from the editor.

to shape
repeat 4 [fd 20 rt 90]
fd 80
repeat 3 [fd 20 rt 120]
bk 80
end

Now enter the command

shape

This draws a flag on a box. This has order 1 rotational symmetry and no lines of symmetry but you can use it to form patterns with more symmetry.

You can produce an image with order 2 rotational symmetry by using this code.

repeat 2 [ shape rt 180]



Task 1

Modify this program to produce other orders of rotational symmetry. Try to obtain orders 4, 5, 6, 8, 10 and 12.
Can you work out a general rule for rotational symmetry of order n? Order 3 is done below.

Order 3 rotational symmetry.

repeat 3 [ shape rt 120]

What is the general formula for rotation symmetry?



Reflection Symmetry with Logo

There are no easy commands to reflect things using Logo but you can produce reflection symmetry if you think carefully.

Make a mirror image of "shape. I have called mine "shapeleft.

edit "shapeleft

Now alter the code to look like this. Remember to "save and exit" once you have put it in.

to shapeleft
repeat 4 [fd 20 lt 90]
fd 80
repeat 3 [fd 20 lt 120]
bk 80
end

To draw "shapeleft with your new program type:

shapeleft

If you have made your left handed image correctly then if you type this code in you should have an object with a line of symmetry running vertically through it.

shape shapeleft

Now if you combine your new object with what you have learned about rotation symmetry then you should be able to produce images that have rotation and reflection symmetry.

repeat 3 [ shape shapeleft rt 120]

Task 2 Further Activities

Produce images with higher orders of reflrection and rotation symmetry.

Produce an image with order 5 rotation and reflection symmetry.

Does an image always have to have the same order of reflection and rotation symmetry?
Produce images that prove or disprove this statement.

Produce an edited version of "shape which changes colour for rotations greater than 360 degrees. can you invent higher orders of symmetry which work for total reflections round to 720 degrees (or more1)
What happens to reflection symmetry in your new system?