Extremal Topological Animals

Matthew Kahle, Greg Malen & Érika Roldán

Here you can try to find Extremal Topological Animals, that is, polyforms with maximally many holes. A polyform is built by gluing together squares or triangles (in the case of this app) by their edges. And a hole in a polyform (that mathematicians call the first Betti number in this 2D case) is a finite connected component of the complement of the polyform. To get some intuition, just build a polyform with squares with 7 tiles and one hole, or a polyiamond with 9 triangles and one hole. Could you create one hole with fewer tiles in any of these two cases? 

Have a look at these papers for the maths behind this puzzle of finding polyforms with maximally many holes:

https://arxiv.org/pdf/1807.10231.pdf

https://www.combinatorics.org/ojs/index.php/eljc/article/view/v27i2p56/pdf

https://arxiv.org/abs/1906.08447v1