When casually playing an RPG (Role-Playing Game🎮️), you might not notice it, but when it is pointed out to you, you may think “That make sense💡”.

Visualizing the World Map as a Torus

Imagine the square world map of an RPG🗺️. First, the top and bottom edges of the map can be identified as the same, so if you roll up the map and stick the top and bottom together, you can create a cylinder. Furthermore, the left and right edges of the map can also be identified as the same, so we can stick those edges together as well. This creates a doughnut-shaped surface🍩.

In mathematics, this surface is called a torus. In other words, the world of an RPG is not a sphere🌐 but a torus🍩.

Now, this next part is a bit more specialized. I’ll explain how a torus has the structure of a Lie group.

The Torus as a Lie Group

First, the circle \( S^1 \) is one of the simplest examples of a Lie group, and it is immediately apparent that it is isomorphic to the 1-dimensional unitary group \( U(1) \). Next, the 1-dimensional torus group \( \mathbb{T} \simeq \mathbb{R}/2 \pi \mathbb{Z} \) is a group that arises from identifying two elements in the additive group \( \mathbb{R} \) that differ by an integer multiple of \( 2 \pi \). The image is like wrapping a straight line around a cylinder whose circumference is \( 2\pi \). The torus group \( \mathbb{T} \) is isomorphic to \( U(1) \), and therefore, topologically it is homeomorphic to \( S^1 \), so in the end, \( \mathbb{T} \) is just \( S^1 \).

Now, since a doughnut can be represented by two \( S^1 \)’s, it can be written as \( \mathbb{T}^2 = S^1 \times S^1 \), which is called a 2-dimensional torus, or simply a 2-torus. Since \( S^1 \) is a 1-dimensional Lie group, their direct product is also a Lie group, and we see that the 2-torus is a 2-dimensional Lie group.