# Rectangle-like structures and their folding in biology

I've heard that mathematics helps to explain some biological problem. For example gömböc, which was a well hidden body from mathematicians, explains the body structure of some tortoises in relation to their ability to return to an equilibrium position after being placed upside down.

My question is an attempt to find an analogous application for a very special rectangle in biology: There exists a unique rectangle with sides $$a$$ and $$b$$, where $$a$$ less $$b$$, with two ways of being folded along a line through its center such that the area of overlap is minimized and each area yields a different shape – a triangle and a pentagon. The unique ratio of side lengths is $$\frac {a} {b}=0.8150237$$.

The special rectangle is like a bistable structure with some minimum of overlapping and folding angles: $$45^\circ$$ and $$\approx 24.66^\circ$$ (for over ratio of sides there is just one folding angle: for eg. long rectangle is always folded at $$45^\circ$$). (more details about folding rectangle and pictures in OEIS: A366185 )

Any ideas where the property may appear in biology (including molecular biology) are highly welcomed.

• An image showing the two ways of folding might be helpful in this case. I for one can't visualize getting a pentagon out of a single fold on a rectangle. Also is the a/b ratio correct? Not meant to be "0.8" rather than "08"?
– bob1
Apr 2 at 21:02
• @bob1 yes, correct "0.8", I also added a picture & OEIS link to an article (Folded rectangle) on how the special rectangle was found. Apr 3 at 4:24