The general theory of relativity allows for the possibility that our universe is not “flat,” but instead some other, non-Euclidean geometry, such as the surface of a sphere (a “closed” geometry). How could we test if our universe exists in a closed or flat geometry? One way is to consider specific shapes and examine their properties when projected into different geometries.

In this demonstration, you create a triangle in flat geometry on the left — drag its vertices around to change the triangle’s shape. Note how its angles will always add up to 180° because it is on a flat surface. Then, the triangle is projected onto the surface of a sphere, stretching the shape and skewing the angles in such a way that the angles will always add up to more than 180°, helping to differentiate the two example geometries.

Try bringing the vertices close together and then far apart. How does the sum of the angles on the sphere change? What implications does this have for one’s ability to tell whether the geometry of one’s universe is flat?