As stated in the introduction, Topology is defined as the study of geometric properties and spatial relationships unaffected by the continuous change of shape or size of figures. As such, the classic joke is that a topologist cannot tell the difference between a donut and a coffee cup. This is because the focus is much more granular than the appearance of the overall surface. The affordances of topological space allow for an exploration of surfaces through their properties instead of their appearances. Surfaces are described by the structure of the sets of points they are composed of rather than their overall form.
When looking into Topology one of the main concepts which I found intriguing was Homeomorphisms. A homeomorphisms is defined as a continuous transformation where the sets of points on two surfaces have a one to one correspondence and therefore are topologically equated. Due to the lack of metric or measure in topological space, the structure of the surface is defined by these sets of points. This is the root of the coffee cup joke.
An easy way to visualize this concept is to imagine a blob of stick putty or slime. When you first got the blob it was in the shape of a heart. You then push, pull, and stretch that same blob into a circle or some other fun shape. As long you you haven’t punctured, cut, or glued any part of the blob, those two forms are homeomorphic or topologically equivalent.
These concepts provide a new way of looking at three dimensional surfaces. Learning about the idea of homeomorphisms pushed me to think about forms in terms of their underlying structures and properties instead of how I might visually perceive them. Getting into the concepts of invariance, connectedness, boundaries, and transformation, things get even more interesting. These exciting notions could be employed in any number of art or design projects. The first set of experimental prototypes explores the visual representation of these concepts.
After the end of the first round of research and prototyping I began considering more options for input to define surfaces and so I looked toward data. My research led me to Topological Data Analysis or TDA. TDA is, as its name describes, a method of data analysis which uses techniques from Topology. This type of data analysis is particularly useful for high-dimensional, noisey, or incomplete data sets.
When it comes time to analyze a data set, the shape of that data set matters in choosing the modeling method. For example linear regression for a line, clustering for points, and so on. The advantage of TDA is that it’s a modeling method that works for all shapes that can be computed easily.
This is important in this time of big data where the information that needs to be analyzed is complex and comprehensive. TDA has been used in the analysis of sound, text, precision medicine, and much more. The value of this methodology, even more so than other modeling methods, is in its ability to reveal connections and patterns that aren’t immediately present. One specific technique within TDA that is particularly good at this is Persistent Homology.
Persistent Homology is a method for computing topological features at different resolutions. It is the process of the creation of simplicial complexes from different sized radii around each point in the data set. When the radii overlap a link is drawn between the corresponding points. When they intersect they are also recorded on a diagram which keeps track of the birth and the death of each of these features. The purpose of this is to identify those that persist throughout each level of scale. Examples of topological features that we would be keeping track of include components, clusters, and holes of different dimensions.
Identifying these features allows us to clearly define the structure of a data set. Defining the structure gives us insight into the behavior and properties of the data.
After completing this second bit of research, I began to experiment and develop my own methodology which is a synthesis of Python based TDA tools. I chose to use a data set from Information is Beautiful which contained the records of various data breaches throughout the last few decades. This data set is probably incomplete but still works for my purposes. While I am including an explanation of this data, I want to keep the focus on the tools themselves and less so on what is being modeled since this is still an experimental methodology.
The images on the right are from the analysis of one year of this data. To create these graphs I first cleaned up the data, separated it by year, and chose three variables which were representative of the scale and quality of the event which they corresponded to. Through the use of the Python libraries Scikit-TDA/Ripser.py and Matplotlib I generated these images. The top left and bottom left are 3D plottings of the data with and without triangular polygons. These represent the distribution in space. The TDA takes place in the generation of the diagram on the right. This is a persistent homology diagram which, as described in the TDA research section, keeps track of the features which are present in the 3D plotting of the data, specifically holes. As you can see the axes are the birth and death of each feature.
The next step for this project is to take the form of data, as described by these persistent homology diagrams, and bring it back into the context of homeomorphisms. What this means is that if I have a data set and I've identified its structure as that of a torus, I can then say that it will behave the same as if it took the form of a coffee mug.
The final presentation of this project is as a web “book” (which you are reading now). It reads horizontally to imitate the experience of reading a book and serves as documentation of this research and prototyping process. As the goals of this project are centered around making mathematics more approachable and reimagining methods of visualization, creating a web book gives perpetual access to viewers and allows me to continue to update my work. I hope that this will operate not only as a display of my achievements but also as a sort of manual for others. As a research project based in experimentation it is also extremely important to document the details of the methods and outcomes.
I had planned to have rendered animations of the surfaces developed through TDA as well. This project was slowed down dramatically due to the circumstances resulting from the current pandemic. I had hoped to have made further progress by this point but regardless I will be continuing my research and experimentation after this exhibition.
I have learned a tremendous amount while conducting this research and exploring the capabilities of Topology. Not only have I gained an understanding of Topology itself, but also an insight into how it can be applied or incorporated into design practices. While some knowledge of programming is necessary for the use of the tools I have used, the existence of libraries and software lessens the barrier for entry. As I continue to develop this project and expand my research, I will be able to draw more conclusions about the effectiveness of these methods.
The next steps will include exploring other TDA techniques and translating those into more kinetic forms. This project is by no means finished, especially due to the circumstances that we have found ourselves in. I have only scratched the surface of these topics and I plan to continue this work when I am able to.
