Sergio Ramos & Leonhard Euler: An introduction to Graph Theory
We are talking about geniouses. Both stand out as exceptional prodigies, shining with brilliance from an early age. They share the distinction of being remarkably precocious: one assumed the role of a professor at just 19, stepping into Bernoulli’s shoes following his passing, while the other, also at 19, joined the ranks of the White House team after Walter Samuel’s exit. But finding similarities between them is complicated.
The answer, as it couldn’t be otherwise, lies in the city of mathematicians, physicists, and philosophers: Königsberg. The former city in Prussia with orderly streets and Lutheran churches, where Kant was born. Although I’m almost certain that Kant would never have been a fan of Ramos.
Introduction
In my opinion, the best way of getting hooked to a topic is reading about it’s history and stories. In this case, in the world of mathematics, certain stories have the power to captivate our imagination. One such story that ignited my passion for mathematics was the intriguing narrative of the Bridges of Königsberg. Set during my very first year of studying mathematics, one of the teachers of my uni used this story to introduce the class to Graph Theory, a subject that deviates from conventional mathematical concepts. Let’s explore how this tale, centered around the city of Königsberg, played a pivotal role in launching the theory of graphs.
Problem statement
The story transports us to Königsberg, now recognized as Kaliningrad, located in East Prussia.
A part of being a city of mathematicians, physicists, and philosophers and the Kant’s birthplace. This city is globally renowned for two distinctive features: a massive graffiti artwork of Sergio Ramos, created in the 2018 World Cup to commemorate a fateful match between Spain and Morocco, and its division by the Pregolia River, resulting in the formation of two islands within its boundaries — Kneiphof and Lomse.
The river effectively segregates the terrain into four discernible regions, all interlinked by a network of seven bridges.
Inquisitive residents of Königsberg pondered the intriguing possibility of embarking on a journey across the city, traversing each of the seven bridges precisely once, and ultimately returning to their initial point of departure. The queen of Königsberg presented this problem as a mathematical challenge:
“Given the map of Königsberg, with the Pregel River dividing the plane into four distinct regions connected by seven bridges, can one take a walk starting from any of these regions, crossing all the bridges only once, and return to the same starting point?”
The solution to this puzzle was not realized until centuries later, when in 1736, the renowned mathematician working at the Prussian Academy of Sciences, Sergio Ramos… I mean Leonhard Euler (1707–1783), took a keen interest in this enigma and embarked on finding a solution.
Euler’s greatness stems from his groundbreaking contributions spanning calculus, graph theory, and physics in the 18th century. With 850 works, he shaped modern mathematics. Let’s dive into a snippet of graph theory to get a glimpse of the solution Euler proposed.
Graph Theroy introduction
In graph theory, a “graph” is composed of two primary components: “nodes” and “edges.” Nodes, also referred to as vertices, are individual points that represent distinct entities. Edges, on the other hand, are the lines connecting these nodes and signify the relationships or interactions between them.
The “degree” of a node is a pivotal concept in graph theory. It refers to the number of edges connected to a particular node. Nodes with higher degrees indicate stronger connections or greater interactions within the graph. A node with no edges connected to it is known as an “isolated” node.
Problem Resolution:
Euler demonstrated that for a successful route, each node must have an even number of edges. In Königsberg, four nodes had odd edges, leading to Euler’s Theorem formulation. The theorem states that in a connected graph with all nodes having even edges, a closed path traverses all edges. Alternatively, if exactly two nodes have odd edges, a closed path starts and ends at them.
Conclusion and Impact on Sports Analytics
Euler’s solution birthed graph theory, applied in domains like transport, social networks, and urban planning. In sports analytics, graph theory can be used to analyze player interactions, strategy optimization, and performance prediction.
Here we can see one clear examples. The famous passing maps, where we can see each player is a Node and the number of pases between them are the vertex.
So we can state that the Bridges of Königsberg puzzle sparked a crucial mathematical field, showcasing how creative problem-solving continues to have a lasting impact on diverse areas of science and technology.
