Konigsberg bridge problem and the evolution of Mathematics

harini-jayaraman — Fri, 29 Apr 2022 20:44:00 +0000

Well, In this blog I will discuss one of the problems that led to the development of new branches in mathematics like ‘Graph theory’ and ‘Topology’. Let’s dive into that interesting problem. What’s the Konigsberg problem? Konigsberg, a city you will find a hard time finding on the world map. However, it is one of the most famous cities in the world of Mathematics due to a particular quirk in its geography. The city of Konigsberg was set on both sides of the Pregel River, and included two large islands, which were connected to each other, or to the two mainland portions of the city, by seven bridges. In the 17th century, people were bothered with the question, whether they could walk around all 7 bridges crossing them only once. Let’s try it out! You could very well see other routes in this video and come up with even more routes. No matter how many ever routes you try you have to give up at some point because Euler answered the question in the negative and proved that there could be no route in Konigsberg that used all seven bridges exactly once and laid the foundation for Graph theory. How was it proved? Well, the answer he came up with, did not basically exist then, what he called the ‘Geometry of position’ , now known as the ‘Graph theory’ and which led to the fundamentals of ‘Topology’. Euler realized during his various trials that the geography or route taken to reach the island or riverbanks does not matter. This insight led to the simplifying of map further. What matters is the 4 landmasses and the relationship between them. We could represent each of the four landmasses here named as A,B,C and D respectively as you could see in the picture below, as a single point, which we now call as Nodes in Graph theory. We now have 4 nodes. We draw lines or edges between them to represent the bridges. Formally, we define a graph G = (N, E) as a pair where, The graph can be used to represent binary relations: (i,j) ϵ E if and only if the entity, i ϵ N and the entity j ϵ N interact with each other. The connectivity of nodes is vital for the structure of the graph. From the simplified graph, as you see in the above picture (Forgive my poor editing skills) with just nodes(Pink dots) and edges(Green lines) we can find the degree of the nodes which forms the basis of this solution. The degree of the node is nothing but the number of edges incident to it. So, in our case, it is the number of bridges that connects to each of the landmass. According to our challenge, person entering the landmass via one bridge should exit it through another bridge. Technically, edges leading to and from one node should occur in distinct pairs, meaning that the number of edges touching each node must be even. The only possible exceptions would be the locations of the beginning and the end of the walk. Looking at our graph, we can easily see that all the nodes have odd degree (in yellow in the below figure). Hence, no matter whichever path we choose we will end up crossing a bridge twice. Euler used this as a proof to formulate a general theory that apples to all graphs with two or more nodes. He defined something called Eulerian path that visits each edge only once and is possible only in one of the two scenarios. In the first scenario, all other nodes would be even and the starting point and ending point are different and should be nodes with odd degree. In the second scenario, starting and ending point would be the same. Why does it matter? In the history of mathematics, this is considered to be the first theorem of Graph theory. It basically forms the first proof for Network theory. Network theory in today’s world is a broad area of research. A network is a more general type of data structure that describes relations or interactions between entities. For instance, The Internet network is the physical infrastructure made of internet service providers, servers, routers (entities) and all the physical connections between them. We can never forget the Social networks (Friendships,relationships,work ties, social interactions) in today’s scenario. If time permits, I will discuss more on Social networks in my next blog soon. Current state of Konigsberg After the world war II two of its bridges has been destroyed and Eulerian path has been created. However you can’t find Konigsberg now. It is renamed as Kalingrad and is a part of Russia. I love this particular video explaining this problem in a simple way. You could well play this game based on this theory. Check if you can draw these line by going over each only once. https://www.transum.org/Maths/Activity/without/ Further Reading This lecture notes could quite give you an idea about Eulerian path and this blog could be a great starting point in understanding the world of Graph theory and Networks.

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My First term in STOR-i

harini-jayaraman — Sun, 13 Feb 2022 22:33:00 +0000

Despite all the anticipation, dreaming and planning after getting selected for your dream course, at times the reality hits you hard. My first term experience as a STOR-i MRes student was quite daunting. After all, when the world is struck in pandemic, no amount of thinking or planning can truly prepare you for what it is like being surrounded by uncertainities. I would like to share my experience having completed my first term recently after all the hardships. I am an international student and I am supposed to have a ATAS clearance and VISA to begin my studies. I had my interview in the last week of July and the course was supposed to begin in the first week of October,2021. I had applied for my ATAS clearance as soon as i decided to pursue this course, by the first week of August. ATAS clearance usually happens within 15-20 working days but due to pandemic and back logs, it took 57 working days in my case. I haven’t even applied for VISA then, as it requires ATAS Clearance. My course had already by that time but thankfully ��̽̽App, STOR-i director Jonathan Tawn and Idris Eckley helped me to begin the course without deferring as I didn’t want to waste a year. The first 5 weeks of the term, which was supposed to be a whirlwind of meeting my colleagues in MRes, PhD students and Professors went quiet in front of a computer screen as I tried to work and study on my own. Despite having an experience of an year of distance learning, this wasn’t easy by any means as I was the only one learning from distance in my cohort unlike previous year in my masters and I wasn’t able to attend the lectures. I had even missed the away day, a two-day team building retreat planned to ease us in the STOR-i atmosphere by getting to know various student and staff. However, once my ATAS came, I applied for a priority visa, that arrives in 5 working days and planned to reach Lancaster before the sprint sessions, which is a group learning module. It is a unique element in the course, where a new Statistics or Operations Research (OR) topic will be introduced in a 2 hour lecture at the beginning of the week and we will be researching the topic as a small group for the rest of the week and present our findings. Personally, Since I came to Lancaster only during that week, it was even more challenging to manage academics and settling down in a new place. I managed to find an accommodation before the christmas break and finally settled down. However, the sprint sessions were very rewarding as we explored a lot of new and significant topics. My classmates, professors and staff were very kind and helpful. They never made me feel out of place though I joined them late. I learnt a lot in terms of statistical knowledge, team work and presentation skills. Its been a fascinating experience overall in the first term with some hardships. I look forward to my journey in STOR-i in the next terms which I am sure would be more enriching. See you in my next blog post with some interesting topics in Statistics/OR I have come across in my learnings here.

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General – Harini Jayaraman

Konigsberg bridge problem and the evolution of Mathematics

My First term in STOR-i