Amrita Acharyya

Statement

The first painting is based on intersecting a few curves drawn with polar co-ordinates. The cardioids $r=a+asin(\theta), r=a-asin(\theta), r=a+acos(\theta), r=a-acos(\theta)$ for any $a>0$ intersects the 2D plane in several regions. The central part with 4 petals is shown by yellow. The regions inside the outer boundary in pink, with additional curvy structures in red along with 4 yellow bulbs from 4 points of intersections. From the origin, we draw 4 circles $r=asin(\theta), r =-asin(\theta), r=acos(\theta), r=-acos(\theta)$ with further points of intersections. We draw additional structures like stairs in brown towards the center. We add tree branches as a background outside those bounded curves from 4 outer points of intersections.

Artworks

The painting is based on the Cayley graph of two-generators $a, b$ and one relator groups that play an important role in combinatorial group theory and geometric group theory as (counter)examples and test-cases. It is given by the group presentation $BS(m,n)=\langle a,b:ba^{m}b^{-1}=a^{n}\rangle$ for any integer $m,n$. For example, $BS(1, -1)$ is the fundamental group of the Klein bottle. We describe $BS(1,3)$. The cottages represents the vertices of the Cayley graph of the group and the horizontal connectors between two consecutive cottages represents $a$, whereas the vertical stair connectors corresponds to $b$. The groups were defined by Gilbert Baumslag and Donald Solitar in 1962 to provide examples of non-Hopfian groups.