two-color rotational symmetry

We can only add a two-color symmetry to a rotational symmetry if the rotational symmetry is of even order. After some simple calculations, we get from the previous post a real mapping for selecting the color variants

where the d are real valued coefficients and the U are real functions.

If the dimension p of the embedding space is an even number then

and if p is an odd number then

These results are actually easy to get by intuition without needing the theory of the last post. The sign of W chooses one of the two different color variants. For the transition region of small W we better use a neutral color such as a dark grey to improve the image.

The effect of the two-color symmetry can be subtle, such as in this image with basic 8-fold rotational symmetry:

More drastic two-color symmetries are possible too. Here for an image of basic 10-fold rotational symmetry:

To create your own images have a look at my Github repository https://github.com/PeterStampfli/creatingSymmetries. You would not to download the current commit 68f2f1b and open warpingKaleidoscope.html in your browser. Be aware that this is a moving target.

This entry was posted in Anamorphosis, Kaleidoscopes, Quasiperiodic design, Tilings and tagged , , , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s