Curving and Gluing
By Asem Wardak, The University of Sydney
In primary school I remember taking nets of different shapes like cubes and pyramids, and gluing their sides together to get the three-dimensional shape. But it was only in the project I undertook this summer when I really thought about how the shape behaved at its edges and vertices. For example, the sum of the angles around the apex of a square pyramid, like the ones at Giza, isn’t 360 degrees like on a flat piece of paper, but 240 degrees (the 60 degree angles of the equilateral triangle multiplied by 4). If you tried to “flatten” the shape around this vertex, your shape becomes distorted. This is an example of a shape with positive curvature. On the other hand, if the sum of the angles were 360 degrees, then flattening the shape at the vertex would not distort the shape at all.
By analogy, if the sum of the angles around some vertex happened to be more than 360 degrees, then your shape would have negative curvature around that point; try visualising that in three dimensions!
In my project this summer, I looked at the corresponding case one dimension up. I looked at tetrahedra, which are like the pyramids but where the base is also a triangle, and examined how gluing them in different ways influenced its curvature properties. Here, your nets are three-dimensional tetrahedra glued together at faces. This gluing produces a shape in four-dimensional space, just as gluing together squares in a specific way at the edges produces a cube in three-dimensional space. The result is that you end up with shapes which do not necessarily fit in three-dimensional space. For example, imagine gluing one of those triangles of a Giza pyramid, to a triangle next to it, of the same pyramid. This would be like “folding in” one of the faces into the one next to it, which would considerably distort the shape of the pyramid. Then if you go around the edge shared by those two faces, you would only need to travel the dihedral angle between the two faces, which is considerably less than 360 degrees, and the shape would be positively curved around this edge. The curve produced by going around an edge in this way is called an edge link, which is comprised of pieces of circles.
This process can also be done with vertices, producing pieces of spheres glued together in what is called the vertex link. If you can show that there is no length-minimising loop, called a geodesic loop, with length less than 360 degrees in either the vertex links or the edge links, then you know that the original shape is nonpositively curved. Nonpositive curvature is an extremely important property spanning areas in physics (general relativity) and chemistry (knot theory of molecules), as well as mathematics.
Asem Wardak was one of the recipients of a 2016/17 AMSI Vacation Research Scholarship.