So now  I have the vertices of this triangle at the points  (3,0,0), (0,3,0) and (0,0,3).And now I take the  edges of the triangle, and I take the midpoint of  each of these edges and these midpoints are now  the vertices of the new set of triangles.So for  example, this midpoint here has coordinates  (1.5,1.5,0) and so on.And now I draw the edges of  the new set of triangles exactly in the same way  as before; I connect two vertices if the corresponding  faces are contained in each other.So for example  this one is connected to this one because this edge  is contained in the whole triangle; and this one  is connected to this one because this vertex is  contained in this edge and so on.So this is the  edgewise subdivision of this triangle.

Today’s video is about subdividing triangles and possibly indications of how such subdivisions happen in higher dimensional analogues of triangles.

So we start with a triangle and we use a subdivision that is classic in mathematics, the Barycentric subdivision.

  • (Brady: This could be any triangle, basically?) This could be any triangle, but for the time being let’s let’s be as an equilateral triangle. We want to subdivide the area of this triangle into smaller pieces. The barycentric subdivision proceeds as follows: in order to subdivide it we need more vertices first. And the vertices here we get by looking at the so-called faces of the object. So faces are the vertices, the edges, and the whole triangle. And for each of those we take the barycentre; which is like the centre point in a in a certain sense, the sum of the vertices divided by the number of vertices; and for the actual vertices this is just the vertex itself; for an edge it’s the midpoint of this edge; and then we have the barycentre of the whole triangle somewhere here.

And now we need to know how to draw the edges of the new triangle, and we need a rule for that. And the rule is that we connect two vertices of this new set of vertices if the corresponding faces are contained in each other. So for example, we keep this edge from here to here - so this is now a single edge - because this is the centre point of this edge which contains this vertex. We also have an edge from here to here because this vertex is part of the whole triangle. We have no edge from here to here because this vertex is not contained in this edge. so we continue with that so this one is we have an edge here because this edge is contained in the whole triangle and so on. So this is the barycentric subdivision of this triangle; we’ve got six new triangles. You can continue, you can now uh start subdividing again to a new refinement like- now you go to each of those triangles, you do the same thing, subdivide, you can do this to each of these six. Again subdivide and you get a finer and finer set of triangles which covers the original area of these triangles.

  • (You were telling me there might be a problem with this.) Well it depends on what purpose you are doing that for. I mean one purpose you could do is an applied purpose, you have many triangles that, say, try to approximate a shape - you model some monster in a movie. Your original set of triangles is too coarse, there are too many edges in your monster and you want to have it more smooth, and then you try to do this by having smaller and smaller triangles. Now if you do barycentric subdivision one of the disadvantages of this subdivision is that the angles in these triangles, they become more and more acute or obtuse. So here you have a very acute triangle and here even more and so on. So the angles become uneven, the triangles tend to be kind of lengthy, and that’s not what you want usually if you want to model.

For mathematical purposes the barycentric subdivision is perfectly fine, there are very many applications in mathematics of barycentric subdivisions of triangles, higher dimensional simplices, and so on. The alternative is the so-called Edgewise Subdivision. So now, again, we need first to create a new set of vertices and the way we create this new set of vertices here is a different one. So I will draw a little second picture, which now is supposed to be in 3-space. These are the coordinate axes of 3-dimensional space, and now I place exactly this triangle here in this position. So the coordinates now of these three points are 1 in the x Direction, We need a rule to connect the points in order to form the new triangles. This rule states that we should subtract the coordinates of each vertex from each other. For example, we take the vertex with coordinates 300 and the vertex with coordinates 111, we get 333 minus 123 which then is 3 minus 1 is 2, 3 minus 2 is 1, and 3 minus 3 is 0, yielding 210. This is not okay, as the difference should only use 1s, -1s and 0s. We should instead take the two vertices that yield 100; if we change the role of the 2, we would get -100, which is also okay. If we apply this rule, we will get the picture we would have guessed from the beginning. This picture yields nine triangles, all congruent if we had started from an equilateral triangle. Yeah, and presumably this is just a perfect thing, it can’t work out any other way that you ever get 1s and 0s that shouldn’t be joined and- That’s right, it’s an if and only if. And if you’d used a different number other than 3 and you had lots and lots of divisions along these lines - same rule applies? The same rule applies, just the number of triangles increases. So the number of triangles you get is always this number r squared. In- in general if you go into higher dimensions, like you do this and apply this to say a tetrahedron - that’s the next dimension - so that would be a 3-dimensional object - you would get r cubed many new tetrahedra. In general you get r to the d many new triangles or higher dimensional analogs of triangles.

Works a treat.And what- and the advantage of this is we seem to get more friendly triangles do we? That’s right, that’s right.We get more friendly triangles, it’s more evenly distributed, um you get- uh you get in 2 dimensions you get only - if you start with an equilateral triangle - you only get congruent triangles.In the next dimension you get a small number of congruent simplices, and that’s also so for higher dimensions, is you can control how many different incongruent objects you get and you also can control the deviation of their shape.

So you’ve shown me 2-dimensions triangles but you’ve also alluded to the fact this works in higher dimensions, we could get like a simplex, like a 3-dimensional object and divide that up into three- 3D pieces could we? That’s right.I mean the the 3D simplex, that’s the tetrahedron so- We’ll have four numbers for that will we? For the edgewise subdivision we have four numbers.For simplicity let’s choose here r equals to 2 otherwise we end up with too many dots and it becomes hard to visualise.So that means that now we place - in order to get this picture here - we place this tetrahedron in 4-space, so in a 4-dimensional space; and with the vertices being the unit coordinate vectors and then dilate this by 2. So we end up with this for example being 2000, this being 0200, 0020, 0002.And then we get these new vertices, like that one, for example that would be a 1001.In the middle of each line? In the middle of each line.And then because r is small, r is 2, we don’t get any additional point in the interior.Like here you get something in the interior of this triangle; and here you get nothing in the interior of each face and also nothing in the interior of the whole tetrahedron.If r is 3 we get some- an additional point in the interior of every these triangles, and if r is 4 we would get a point in the middle. But now let’s apply this rule, it’s already a bit more tricky.So here you do of course the same on each face, so that’s the easy rule, essentially we have seen in dimension 2; but now this is not yet a subdivision into tetrahedra.I mean, you see here there is a tetrahedron sitting on top, a tetrahedra sitting on the front, the tetrahedron sitting here - but then in the middle you have this object which is actually an octahedron and you need to further subdivide it and for that suddenly this edge becomes important, you need to connect those two vertices.And in order to get this to subdivide into tetrahedra you draw another edge here in the middle.This new edge now serves as an edge that subdivides this octahedron into four tetrahedron. It is possible to make a transformation of coordinates and subtract them to decide when to make a join. The resulting vector must be a 0,1 vector or a 0, -1 vector to connect them by an edge. It can have multiple 1s, but not 1 and -1. Barycentric subdivision is a classical object used in algebraic topology, while edgewise subdivision is used in combinatorics, algebraic topology, and other fields. It is also possible to combine the two subdivisions, or to make flips on the edges to create new ways of subdividing the original triangle. There are still open questions about the structure of the number of faces that appear when subdividing iteratively. New ways of subdividing may be useful if there are new mathematical questions that the two methods are not good for. Let’s go upstairs and sort out our superhero triangles! .

I have now given you permission to put clickbait thumbnails on this video. Get Captain America over here, I can’t believe it! Superhero triangles!

  • Which one is the actual centre of the triangle?
  • There is no actual centre, it depends on your viewpoint.