Symmetries

​Hi there. You trim these apple trees nicely. ​ ​Thanks, Lena. I'm trying to make them symmetric. ​ ​Synthetic? ​ ​No, symmetric. Like this. Imagine a line right in the middle of the tree.

One side of the tree must be a mirror image of the other. This is called the mirror symmetry. ​ ​Okay, but what about that tree? It doesn't look very symmetric. ​ ​No? Step back a bit and take a look at the next tree as well. Together they form the mirror symmetry. ​ ​Nice. ​ ​Come with me and you'll see a few other types of symmetry.

I've trimmed this boxwood tree in the shape of the letter C. Is it symmetric? ​ ​No, it's open on one side, but not on the other. ​ ​Actually, it also has the mirror symmetry, but the axis of symmetry is horizontal. The upper half mirrors the lower one, like when the surface of a still lake mirrors what's on the other side. ​ ​What about that bush, the one that looks like an H. It has a horizontal axis of symmetry too, doesn't it? ​ ​It does, but not just that. It actually has two axes of symmetry, one vertical and one horizontal. ​ ​Two axes of symmetry?

Do you have a bush with three axes? ​ ​No, not out here. Most plants only have only one axis of symmetry, usually vertical so that the right side mirrors the left. Most animals have this type of symmetry as well. Let's go inside and I'll show you something. Look at this painting.

Is it symmetric? ​ ​No. If you mirror it, the horses will run the other way. ​ ​Look what happens if I do this. ​ ​You turned it, but it looks the same?  ​ ​Exactly. This is also a type of symmetry called rotational symmetry. Patterns that you can turn and still get to the same pattern have a rotational symmetry. ​ ​What about the floor? What kind of symmetry is there? ​ ​Good example.

You can mirror this pattern both left-right and up-down and get the same pattern, and you can rotate it. You can even move it sideways and still get the same pattern. ​ ​What's that? Is that also a symmetry, if it can shift sideways? ​ ​Yes, it is. In geometry it's called a translation, when you simply move a figure without changing it ​ ​Like the tree lane outside. It also has a translational symmetry. ​ ​Yes, exactly.

If you move all trees one step towards or away from the stable, the pattern remains the same. By the way, do you see any other types of symmetry in the lane? ​ ​Yes, a mirror symmetry, one half is a mirror image of the other, and a rotational symmetry - you can turn it 180 degrees and get the same pattern. ​ ​Good. Now let's go and trim the remaining apple trees, shall we?