By Eckhard M.S. Hitzer, University of Fukui, Prof. Ryosuke Nagaoka (University of the Air, Japan)

I will soon show you some graphics. But let me make a remark about the use of the word invariant in linear algebra and in mathematics in general. Invariant does not mean, that all elements, every vector is staying fixed. But it means that the set, the whole space, or a subspace remains invariant even if the elements move or their positions change.

... Prof. Nagaoka ...

Yes so if you look now at the computer graphics, first you see an example of a motion, which is not invariant. So really this space is changing.

And next you see a motion, where the elements move, but the whole subspace in itself is preserved under this motion. So again the elements are not fixed, but the whole subspace is invariant under this motion.

Another example here rotations. So you see a set of points and this set of points is now rotated around this center O. And you see all these points they individually change their positions along these curves.

But if we now look at this disk here, we can also visualize some parts of the disk. And we see how these lines, which are on the disk they change by the rotation. But the whole disk itself has been preserved, it has been invariant under the rotation.

Next lets look at this square and we can rotate the square also around this center of rotation O. First if we perform a 60 degree rotation, we see it is not going back to its original shape, so it is not preserved - the whole square - under a 60 degree rotation. But it is preserved under a 90 degree rotation, as you can easily see.

But rotation symmetries are not limitted to just 90 degree rotations. You can also perform if you wish 60 degree rotations on figures, which are invariant. So here we have performed a 60 degree rotation on this figure and it has returned back to itself.

Next I show you how you can construct a figure, which is invariant under a 120 degree rotation. So we have this first part of the figure, we rotate it by 120 degrees, and rotate it one more time under 120 degrees. Then this whole figure, which results is invariant under a 120 degree rotation, as you see here.

And in general, when you remember, often Japanese families they have a sign, which is called "kamon". It is specific to the family. And often these signs have either rotation invariance, or some reflection invariance.

Soli Deo Gloria. Created by
Eckhard Hitzer (Fukui).

Last Modified 16 January 2004

EMS Hitzer and Prof. R. Nagaoka are not responsible for the content of external internet sites.