MATHS

V – E + F = 2 🧮

Take a piece of paper, draw a bunch of dots on the paper and connect them up to split the page into regions, such that:

1. All vertices have a line going out of them, and

2. If any two lines cross, they intersect at a dot.

3. Any two dots can only have one line connecting them

Then, count the number of dots (vertices) and call this number V. Count the number of lines (edges) and call this number E. Count the number of regions (faces) (INCLUDING THE OUTSIDE REGION) and call this number F. Then either V-E+F=2, or you’ve miscounted. This is called Euler’s Characteristic. Let’s prove it!

Below is a proof by induction on the number of dots (vertices) V:

💡 Things to consider

• 30 Proofs!: This is a statement that has upwards of 30 different proofs which you can explore further online. See if, for example, you can prove the statement by induction on the number of edges or induction on the number of faces.

• The Euler Characteristic of a Surface: The fact that V-E+F is constant on a plane generalises to all 2D surfaces, e.g. spheres, toruses, two-holed toruses etc. But in these cases, V-E+F = 2 - the no. of holes in the object. So, it equals 2 for a sphere, 0 for a torus, -2 for a two-holed torus etc. These numbers are called the Euler characteristic of a surface.

  

  Me, a humanities student, editing this

• Stretching a polyhedron?!: We can imagine a stretching a polyhedron onto the plane and now the number of faces, edges and vertices on the plane corresponds to the number of faces, edges, and vertices of the original polyhedron. So, this then allows us to quickly prove there are 5 platonic solids… suppose:

  • The faces have the same number of edges around them, n>2: i.e. nF = 2E

  • The vertices have the same number of edges into them, m>2: i.e. mV = 2E

  Then plug these into Euler’s Characteristic: V-E+F=2 → 2E/m-E+2E/n =2 → 1/M+1/N=1/E+1/2 → 1/M+1/N > 1/2 for M,N>2

  These give (M,N)=(3,3), (3,4), (4,3), (5,3) and (3,5) which correspond respectively to: tetrahedrons, octahedrons, cubes, dodecahedrons and icosahedrons. These in fact, correspond in order to the diagrams drawn at the beginning of this scoop!

🔎 Find out more

• Euler's polyhedron formula

• Twenty-one Proofs of Euler's Formula

• An Introduction to Induction Proofs

🍒 The cherry on top

• 🕰️ Time Travel: Although a popular trope in science fiction, time travel is also a serious subject of discussion in Physics. Can changing the past create paradoxes, such as the famous "grandfather paradox," or are multiple realities possible? This article explores different views on time travel, making it a fun read for those into Physics or Philosophy.

• 🤒 The History of Pandemics: Throughout history, pandemics have reshaped societies. By studying past events such as plagues from the 16th and 17th century, we can gain insights into how pandemics spread and how we can better respond in the future. This video is perfect for anyone interested in History or Sociology.

• 🐶 Do Animals Think?: Have you ever looked at your dog and wondered what they may be thinking, or if they are thinking at all? This article offers some fascinating perspectives. A great read for those into Philosophy or Psychology.

👀 Keep your eyes peeled for…

• Wednesday 26th November

  • QMUL AI Taster Session

• Thursday 27th November

  • Balliol College Oxford Into Humanities (History) Webinar

  • UEA Geography Webinar

• Monday 1st December

  • Balliol College Oxford Insight into Science (Biology) Webinar

  • UEA Linguistics Webinar

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