Pair-of-Pants surfaces, the math

A pair-of-pants is a surface that looks exactly like a pair-of-pants that you wear. Technically, it is topologically equivalent to a sphere which has been punctured three times, or a disk which had been punctured twice (shown below). It is an orientable surface of genus two having three boundary components. They are useful objects in topology, in that they give a different decomposition of surfaces. pair-pantsWe usually think of closed connected surfaces as spheres, where either handles or cross-caps have been added. More formally, recall the Classification of Surfaces Theorem: Any closed, connected surface is topologically equivalent to a sphere, a connected sum of tori, or a connected sum of projective planes.

It turns out that we can cut up just about any orientable closed surface into pairs of pants with simple closed curves. This is called a pants decomposition of a surface. Pants decompositions are not unique. For example, we can cut up a genus 2 surface (a sphere with two handles) in two different ways:


What happens in general?  Suppose our surface has \(g\) handles, where \(g\geq 2\). Then we can slice the surface with \( 3g-3\) “vertical” simple closed curves, which decomposes the surface into \(2g-2\) pairs of pants. The genus 3 case is shown below and illustrates the general idea.

Pair-pants-2Since a pair-of-pants is a subset of a thrice punctured sphere, it also admits a hyperbolic structure. Alternatively, simply construct a hyperbolic pair-of-pants by gluing together two right angled hexagons (hyperbolic) along alternate edges. More generally, it is relatively straightforward to show that there exists a unique hyperbolic pair of pants with cuff lengths \((l_1,l_2,l_3)\) , for any  \(l_1,l_2,l_3>0\). Here, cuff lengths refers to the lengths of the three boundary components. Even more can be said about hyperbolic surfaces and pants decompositions, but this will lead us too far astray.




A new challenge

In December, I was contacted by Professor Ricardo Nemirovsky from San Diego State University to design 3D printable surfaces for the  Taping Shape* exhibit at the Rueben H. Fleet Science Center in San Diego, California. The exhibit runs from January 30 through June 12, 2016.

The exhibit contains a structure made out of packing tape with
three interconnected regions: a torus, a topological
equivalent to Schwarz P surface, and a pair-of-pants
surface with the legs twisted. The structure is large enough for visitors to walk and crawl through. There are three “work tables” (one for each region), with materials, suggested activities, poster displays, etc. The 3D printed models will be a part of the work table and displays.

Ricardo requested I make pair-of-pants surfaces with caps that can be joined together in different ways, Schwarz P surfaces that can be joined together, and also a frame that allows the Schwarz P surface to be created as a soap film spanning the frame. The challenge was on!

In the following blog posts, I’ll explain a bit about the math behind the surfaces, and how we figured out how to build and print them.

*The Taping shape exhibit is part of the InforMath project funded by the National Science Foundation (DRL-1323587).  (The InforMath Project is a partnership between San Diego State University and several museums at the Balboa Park, including the Rueben H. Fleet Science Center .)