The Curve in the Curvahedra
These are Curvahedra pieces:They can hook together to make all sorts of geometric objects. For example, take three pieces and make a triangle (or something triangle like with wiggly edges)
Taking a close look, each piece has five arms, and they are equally spaced around so the angle between two arms must be 360/5 or 72 degrees.
The interior angles of this triangle are all the same so we have 3*72 = 216. Yet from geometry we know that the the interior angles of a triangle always add up to 180. What has gone wrong?
Here is a 60 degree triangle (note the pieces have six arms, so the angle between neighbours is 360/6), can you see the difference?
Unlike the first triangle this lies flat on the table whereas the first curves away. The difference is clearer if we complete all the pieces around a corner for each.
Going further the five triangles come together to form a ball, while the six triangles would keep on spreading, we won’t be able to complete that sheet.
What we are seeing here is the curvature of the surface we are making. The triangle with 72 degree angles can be said to have an excess of 36 degrees. The greater the excess the more it curves. Look at this triangle with 90 degree angles (for a total of 270 degrees, an excess of 90 degrees), the curvature is very clear:
Completing this creates a smaller ball.
This new ball has eight faces, each with a 90 degree excess. Adding all these together gives a total excess of 720. The first model has 20 faces, with a 36 degree excess, and again a total of 720. Lets think about the model with just three triangles around a corner:
The total angle is 120*3 = 360, so the excess is 180 degrees. If the pattern holds we should need 4 of these triangles to make a ball, and indeed we do:
In fact if you take anything that is like a sphere, take the angle excess on every face you will always get 720. For a more complex example take this model:
This has eight triangles and eighteen squares, and all the angles are 90 degrees. For the square this is normal the total interior angle of a quadrilateral should be 360 and 4*90 is 360. So there is no angle excess. This leaves the eight 90 degree triangles once again giving 720. Also notice in the model the square faces are flatter with the curvature occurring at the triangles. This gives the model the shape of a cube with rounded edges, rather than a sphere.
This result is called Descartes’ Theorem and it is a special case of the Gauss-Bonnet theorem, both are closely related to the Euler Characteristic. These theorems stand at the heart of topology and differential geometry.
A natural follow up to this is to ask what happens with a shape with two little angle (an angle defect). For example the sum of the angles of a quadrilateral should be 360. What happens if we take a square (4-equal sides and angles) with 72 degree angles. The sum is now 72*4 = 288, which is less than 360. This creates a saddle:
The saddle is said to have negative curvature, and connecting up more and more squares, like this does not create a ball connecting up on itself. Instead it gives this wavy surface that grows faster and faster, modelling a hyperbolic plane, all these images are the same object!
Final note: The curvature discussed here is actually called Gaussian Curvature, and is a property of the surface itself not the way it fits in space. For example consider this cone:This is covered with equilateral triangles with 60 degree angles. So although it looks curved the geometry is flat, the triangles all have no angle excess. In other words if you investigated distances just on the surfaces of the model they would be the same locally as those on the locally flat plane of triangles given above. You can only detect the change if you loop back on yourself round the cone. The only exception is the tip of the cone. Here you can see a piece is left hanging.
The same thing happens when you bend a piece of paper, you change how that sheet lies in three dimensions, but not what happens on the sheet. You can even use this to work out how to best hold pizza. On the other hand the Gaussian curvature, discussed above, does change what happens on the surface. The angles of triangles can be measured without leaving the surface. In fact this might have been part of Gauss‘ motivation. He wanted to work out if the earth was a perfect sphere, but did not have access to space. In other words he had to take measurements just on the surface of the earth.
These ideas had even greater importance with the work of Einstein. General relativity assumes that the three dimensional space (or the four dimensional spacetime) that we live in is itself curved. In fact that curvature is related to gravity and explains how gravity acts at a distance. This huge idea fundamentally changed our understanding of the universe yet we can start to appreciate it with a simple toy, which you can get for yourself here. Another way to explore the geometry is with crochet from Daima Tamina’s beautiful book.