Introduction to Ramsey Theory: Students' Projects

Chapter 6 Ramsey Theory and Art

\(\textbf{Summary:}\) This project is a unique collaboration between an established artist, Nora Blanck, and Dr. Veselin Jungić. The result of this collaboration is an artwork inspired by an edge three–colouring of the complete graph on sixteen vertices.

This project shows us the importance mathematics has in other fields. This project has helped us understand Ramsey's Theory as a whole with a different perspective. The dinner party problem, while a mathematical concept, has shown its relevance in real–world scenarios and artistic expressions. It encourages us to think creatively and tackle both theoretical and practical challenges.

Watch the video below.

\(\textbf{Project:}\) The project focuses on a video that discusses the fascinating world of finding order in chaos and what inspired them to take up this incredible project. The project focuses on who Nora is, her previous artwork, her collaboration, and the overall process.

Due to sharing a common interest in interpreting the world's reality, Nora Blanck and Dr. Veselin Jungić published a paper based on an artwork of a sixteen vertices graph on a Douglas Fir Bark using three different colours of threads which aligned with Ramsey's theory named, “A threading path to a Ramsey number”. This paper explores through mathematical logic why a monochromatic triangle of either of the colours of thread is impossible.

\(\textbf{Modification of the Dinner Party Problem}\text{:}\) Let's say that we are observing a group of people. You can establish one of the following relationships between any two of these people in the group:

• Two people know and like each other.
• Two people know but do not like each other.
• Two people do not know each other.

This can be modelled mathematically using graph theory. We can imagine each person at the table is a point, their relationship is an edge, and we colour that edge one of three colours based on one of the situations.

The dinner party problem just mentioned is the same as colouring the edges of a “complete graph of 6 points” (6 points all connected to every other point). We can notice that at least three connected edges will be one colour or the other no matter what (meaning three people will share one of the relationships). This is a special case of Ramsey's Theorem, which tells us that there will always be some number of points to guarantee us any number of monochromatic interconnected edges (you can think of these like smaller complete graphs inside of the whole complete graph). The minimum number to form one of these monochromatic subgraphs is a Ramsey number. This forms the basis for Ramsey's theory as a whole, finding patterns when we have a large enough set.

One interesting fact that comes from Ramsey's theorem is that there will also be a number just below the Ramsey number where we can avoid a monochromatic set of interconnected vertices, but only barely (generally there are only a very few ways to do it).

For example, the Ramsey number for triangles of 3–colours \(R(3,3,3)\) is 17, so that means for 16 interconnected points, there is a way to connect them which avoids any monochromatic triangles at all.

Building on this captivating concept, the art piece we?re discussing is based on a colouring of the complete graph \(K_{17}\text{,}\) visualizing the interplay of relationships and patterns that emerge from Ramsey's theorem.

\(\textbf{Nora Blanck}\text{:}\) Nora Blanck is a Vancouver–based artist and educator who works with various mediums and themes. Nora's art is known in Canada and the United States for its exploration of change, repetition, and difference, themes that were especially highlighted in the “Shift: Working Through Repetition and Difference” exhibition at the Richmond Art Gallery in 2007. Her meticulous thread drawings delve into topics such as beauty, eugenics, as well as mathematics, and genetic engineering. Throughout her career, Blanck's contributions to the art world have been recognized by various groups.

Nora's approach to her artwork is deeply personal and innovative. She began using collaged leaf fragments, abutting them together, and combining them with ?threading?. This freeform and chaotic line work, eventually also geometrically configured, felt like suturing. She was connecting the dots\(\ldots\) and re–connecting with Mary Shelley's “Frankenstein”. These pieces became THE FABRIC OF NATURE: THE BEAUTIFUL FRANKENSTEINS.

\(\textbf{Challenges faced during the video making}\text{:}\) Creating the video presented its own set of unique challenges. Balancing the technical aspects of video making with the importance of clearly expressing complex mathematical concepts was a significant task that needed planning and execution.

Additionally, another challenge we faced was translating the abstract ideas from the project into a visual medium to ensure high–quality visuals and sounds while maintaining viewer engagement.

Despite the challenges, the process was a valuable learning experience and contributed to the overall success of the project.

\(\textbf{The outcome of the Project}\text{:}\) The outcome of our project is a comprehensive and visually engaging video that effectively explains the intersection between Ramsey's Theory and Art. The video shows an incredible collaboration showing how mathematics can blend with the creative part of art.

Our video breaks down complex mathematical concepts discussed in Ramsey Theory making them accessible and understandable to a broad audience. It does this by going through the use of clear explanations, examples, and compelling visuals.

Additionally, the video highlights the artistic process of Nora Blanck, providing our audience with a unique insight into how Nora uses art with the use of mathematics. It showcases her artwork over several arts including the process of a Ramsey art piece. In addition to explaining these concepts, the video also talks about the collaboration between Dr. Veselin Jungić and Nora Blanck. It highlights the experience Nora had working with Dr. Jungić leading to an approach to new insights through the lens of mathematics.

Overall, the video is an excellent educational resource that not only teaches viewers about a specific area of mathematics but also inspires them to think more broadly about the connections between different fields of study.