I put the recent snow day, and a snowball maker, to good use!
On a visit to the Lowline, I noticed an interesting application of mathematics above us.
The ceiling is a tiling of hexagons and equilateral triangles. But unlike a typical tiling of a flat bathroom floor, this tiling seems to create a curved surface! Here’s a closer look:
The underlying pattern is hexagonal, but when a hexagon is replaced with six small, hinged equilateral triangles, the surface gains the potential to curve.
It’s interesting to follow the “straight” line paths as they curve over the surface. And since this tiling is suspended from above, it’s interesting to think about what the surface would look like if it were lying on the ground. How “flat” would it be? Or a better question might be “How far from flat is it?”
Varignon’s Theorem is one of my favorite results in elementary geometry: connect the adjacent midpoints of the four sides of any quadrilateral, and a parallelogram is formed! It is a magical result that defies expectations, and it’s so much fun to play around with, explore, and extend.
Steven Strogatz shared his favorite proof of Varignon’s Theorem on Twitter yesterday, and so I felt compelled to share mine. This is a standard proof of Varignon, but it is so clean and elegant: it is an immediate consequence of the Triangle Midsegment Theorem and the transitivity of parallelism.
Strogatz’s vector proof is beautiful and efficient, but the power of transitivity really shines in this elementary geometric proof.
I created a a simple Desmos demonstration to explore Varignon’s Theorem. And like all compelling mathematical results, there are so many fascinating follow-up questions to ask!