The Mathematical Voodoo of Triangles
When you try to build shapes out of too many triangles, some bizarre things start to happen.
By Jay Bennett
The five Platonic solids are the foundation for a number of geometric studies, and they encompass the only three-dimensional polyhedrons that are both regular and convex. In other words, they are 3D shapes that have regular polygons for faces, straight edges, and sharp corners called vertices. (The shapes are used in expanded dice sets for games such as Dungeons & Dragons because they form mathematically fair dice.)
The Mathematics of Dice
The five Platonic solids.
Named for Plato, who theorized that the classical elements were made of these regular polyhedrons, three of the Platonic solids are composed of equilateral triangles: the tetrahedron, octahedron, and icosahedron. The other two Platonic solids are a regular dodecahedron, made of 12 hexagons, and a regular hexahedron, more commonly called a cube.
As a new episode of the fantastic YouTube channel Numberphile explores, the Platonic solids made of equilateral triangles start to do some bizarre things when you attempt to cram in more triangles. The tetrahedron has 3 triangles connecting at each vertex, resulting in 4 total triangles, while the octahedron has 4 triangles connecting at each vertex (8 total triangles), and the icosahedron has 5 triangles connecting at each vertex (20 total triangles).
What happens if you connect 6 equilateral triangles at each vertex? You get a flat surface known as regular triangular tiling. The shape is now two dimensional, and it can spread out infinitely in any direction while maintaining its regular construction.
Regular triangular tiling.
Tomruen
The reason the shape changes from 3D to 2D when you get to six triangles has to do with the number of degrees in an equilateral triangle. There are 60 degrees at each vertex of the triangle, so when you have 3 triangles connecting, like in the tetrahedron, there is a total of 180 degrees at that point. For an octahedron, the angles of the 4 triangles add up to equal 240 degrees, and for an icosahedron there are 300 degrees where the 5 triangles meet at each vertex. For each of these less-than-360-degree angles you get positive curvature and the shapes bend around into regular three-dimensional polyhedrons—but when you have 6 triangles connecting at a vertex, the angle reaches 360 degrees (a circle) and the shape flattens into a two-dimensional plane.
So what happens when you go beyond 360 degrees by connecting 7 triangles? To find out, Oklahoma State University mathematician Henry Segerman used a 3D printer to build a shape that has 7 equilateral triangles connected at each vertex. It's not flat, and it's not a nice regular 3D shape either. It's kind of a chaotic mess where parts of the structure can be almost flat while other parts fold in on themselves. And yes, forcing 8 triangles to meet at a vertex just makes everything even messier.
This messy 7-triangle-per-vertex tiling is a good example of a shape that exists in what is known as hyperbolic space, or a shape that has constant negative curvature—it curves in opposite directions, rather than positive curvature where the curve flows in one direction, like a bowl.
Mathematicians still don't know if the strange 7-triangle-per-vertex tiling can continue infinitely without obstructing itself, like regular triangular tiling can, though they suspect that at some point the 7-triangle-per-vertex shape would reach a point where it can no longer grow larger.
Triangles are crazy. Who knew?
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