# sarah-marie belcastro

Director / Research Associate

MathILy / Smith College

Holyoke, Massachusetts, USA

I am a mathematician who knits as well as a knitter who does mathematics. It has always seemed natural to me to combine mathematics and knitting, whether that results in knitting a model of a mathematical object or in using mathematics to design a garment. Indeed, over my mathematical life both of these types of combinations have occurred. Most of the mathematical models I have created are only of aesthetic value and have no real function; it is rare that I am able to adapt a mathematical object for use as a garment or other practical use. (It is perhaps too much to hope that I could regularly combine artistry and function in addition to knitting and mathematics.)

Striped connected sum of projective planes

20 x 20 x 10 cm

Knitted sparkly wool yarn

2017

By the classification of topological surfaces, every non-sphere surface can be expressed as a connected sum of tori or a connected sum of projective planes. This striped knitted surface is a connected sum of projective planes. How many projective planes? That is left as a puzzle for the viewer. The piece was made for students to study, but it turns out that professional mathematicians also find it nontrivial to identify this surface, and thus an interesting challenge. Viewers are welcome to manipulate the surface as part of their investigations.

About the construction (possible clues?): Each crosscap is created intrinsically and the fabric texture is maximally symmetric. The stripes are parallel to homotopy generators.

About the construction (possible clues?): Each crosscap is created intrinsically and the fabric texture is maximally symmetric. The stripes are parallel to homotopy generators.