A triangle, but I drew lines in the corners going out, and did the same with the mini-shapes formed:
I have no idea what to call this thing.
For a square, it forms a diamond inside of it.
For a hexagon, it forms a hexagon inside of it.
(I don't have the image but I believe this is what happens.)
I don't know if anyone has discovered these before so please tell me if anyone has. This seems interesting so I'd like to know more about it.
Edit: Here are some more images. Also, apparently, there isn't supposed to be a kite in the triangle.
1 Answer
$\begingroup$When you draw the line segment from each corner, make sure that it ends at the midpoint of the opposite side. That will guarantee that all three lines have a common intersection. The result is called the first barycentric subdivision, which subdivides the original triangle into 6 smaller triangle.
Now repeat that same construction in each of those 6 triangles. That's called the second barycentric subdivision, which is a subdivision of the original triangle into $6^2=36$ triangles.
Continue for as long as you like, to get the 3rd, 4th, 5th, etc. barycentric subdivisions.
This is an important construction in topology, due (in part) to the fact that the maximum diameter of the $6^n$ triangles in the $n^{\text{th}}$ barycentric subdivision goes to $0$ as $n \to \infty$.
$\endgroup$ 3
