Geometry

Ceilings of Curvature

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?”

By MrHonner, 10 yearsFebruary 26, 2017 ago

Geometry Photography

Math Photo: Right but Obtuse

I love the geometry of this tower crane as it swings around, creating obtuse projections of its perpendicularity. I wonder how accurately I could estimate my altitude from this picture?

By MrHonner, 10 yearsNovember 6, 2016 ago

Appreciation Geometry Photography

Math Photo: A Dodecagon of Octagons

I’d never looked closely at the Parachute Jump in Coney Island, but recently noticed the octagons at each vertex of the dodecagon 250 feet up in the sky. It really is a beautiful structure, but I’ll leave it up to you to decide whether or not it deserves to be called “the Eiffel Tower of Brooklyn”.

Geometry Teaching

Varignon’s Theorem

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!

Geometry Testing

Regents Recap — June 2016: Are These Figures Congruent?

Given the congruent triangles below, is the statement “Triangle ABC can be proved congruent to triangle ZYX” true, or false?

I imagine most will say that the statement is false, and argue that the correspondence of the triangles is incorrect. That is, segment AB is not congruent to segment ZY, and so on. I think this is a reasonable response.

However, a substantial part of me believes the statement is true. “Triangle ABC” references an object, as does “triangle ZYX“. These two objects are indeed congruent. Thus, how can it be said they can’t be proved congruent?

In other words, I don’t believe the statement “Triangle ABC can be proved congruent to triangle ZYX” entails a binding correspondence in the way that the statement

$\Delta ABC \cong \Delta XYZ$

does.

I was thinking about this because of this question from the June 2016 Common Core Geometry Regents exam.

According to the rubric, the correct answer is (3) reflection over the x-axis. The most common incorrect response, of course, was (1) rotation. But I’m not certain it’s really incorrect. I don’t think anyone would get this question wrong based on my objection, but since the question is designed to entice students to say rotation, I think it deserves some scrutiny.

Ceilings of Curvature

Math Photo: Right but Obtuse

Math Photo: A Dodecagon of Octagons

Varignon’s Theorem

Regents Recap — June 2016: Are These Figures Congruent?

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