Trisections of the plane

 

Yin-Yang Trisection

 

This curve is an embedding of the line into the plane that divides the plane into three parts. It does this by not being a closed curve, but instead approaches a limit cycle. For any points a, b, and c in sets A, B, and C, there is a path from a to b that crosses the line only once; but any line from a to c, or from b to c, must cross the line infinitely often.

 

Paisley and Pearl Trisection

 

This is like the Yin-Yang Trisection, but inverted through C’s infinite boundary.

Weak Scroll Trisection

 

This curve is an embedding of the line into the plane that divides the plane into three parts. It does this by not being a closed curve, but instead approaches two limit cycles. Unlike the Yin-Yang Trisection and the Paisley-and-Pearl trisection, the line here is a ‘weak’ boundary; it does not separate regions after a finite number of crossing; only after an infinite number of crossings.

Weak Inner Scroll Trisection

 

This is like the Weak Scroll Trisection, but inverted through A’s infinite boundary.