by Camille Clark, Layke Jones, Aekta Kallepalli, Maya Mukerjee, and Caroline Zhou.

Euler Characteristics

Euler characteristics are defined by the equation V- E + F = 2 where V = number of vertices, E = number of edges or nodes, and F = number of faces. Sometimes though, the equation V – E + F = 2 does not work for all situations because the solution can give various outcomes due to the dimensions and simplicity of the object. If 2 objects are topologically the same, they will have the same Euler characteristics. For all simple polygons, the Euler characteristics equal one. Figures with holes don’t follow these conventions as the holes in these figures add additional faces and edges not proportional to the formulas for simple figures.

Advanced Cutting and Pasting 

In general, we can take a polygon with n sides and, by gluing corresponding sides together, the product ends up often with a surface. For every distinct labeled edge, there is a partner edge to which it is attached. Each side is labeled with an edge label (a letter) and a direction (arrow). If in traversing the perimeter of the polygon, we traverse an edge in the direction of its arrow, we record the letter as is; if we traverse it opposite of the direction of its arrow, we write down the letter with an exponent of negative 1.

Homeomorphic Shapes

Two shapes are considered homeomorphic if they can be bent or morphed into one another without cutting, pinching, or collapsing any part of the shape. Homeomorphism happens when a function that has a one to one mapping between such sets that have both the function and its inverse that are continuous. It is located in a topology that exists for geometrical figures which can be transformed one into another by elastic deformation. Very roughly speaking, a topological space is a type of geometric object while the homeomorphism is a continuous stretching and bending of an object into a new shape. For example, a square and a circle are homeomorphic shapes while a sphere and torus are not homeomorphic shapes, when they are being compared.

The images featured in this post are from:
http://www.lix.polytechnique.fr/~maks/Verona_MPAM/TD/TD2/
https://mathstrek.blog/2013/12/02/from-euler-characteristics-to-cohomology-i/
http://math.columbia.edu/~hliu/shp/f18-geometry-and-topology.pdf
https://www.reed.edu/reed-magazine/articles/2014/kyle-ornsby-nsf.html