Cutting Corners

Domains

The two psychedelic designs up above arise from their simplistic ancestors we looked at last time by cutting off corners. These are still two conformal annuli that also satisfy a slightly complicated condition on the lengths of their edges, which makes them responsible for a variation of the Diamond surface:

 

 

Mathematica

If you cut either of the psychedelic shapes into quarters, using a vertical and a horizontal cut, you get four right angled octagons, with some right angles being exterior angles. Similarly, the marked symmetry lines on the surface up above cut the surface into eight right angled curved octagons, that correspond to the psychedelic octagons via a conformal and harmonic map.  

 

D5 deg1

There is a 1-parameter family of such critters. Above and below are larger portions of extreme cases that also show how the surface repeats.

D5 deg2

You can see in the image above pieces of the doubly periodic Karcher-Scherk surface reappearing. No surprise, its psychedelic polygons also arise by cutting corners in the polygons corresponding to the original Scherk surface.

Everything simple reappears.

Advertisements

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 )

Google photo

You are commenting using your Google 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 )

Connecting to %s