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Archive of posts filed under the Geometry category.

Jason Merrill’s Lawnmower Puzzle

Jason Merrill recently posted a fun geometry puzzle inspired by his work on the Lawnmower Math activity for Desmos.  Here’s my paraphrase of the puzzle:

Suppose a lawnmower is tethered to a circular peg in the middle of the lawn. As the lawnmower moves along its spiral path, the rope shortens as its winds around the peg. At the moment the lawnmower contacts the peg, how much rope remains uncoiled?

When I first considered this problem it seemed hard.  After some thought, it seemed obvious.  Then, after some more thought, it seemed hard again.  That’s the sign of a compelling problem!

I enjoyed working out a solution, the heart of which I’ve included below.  Jason graciously included my solution in his post sharing his own, and he also does a wonderful job describing the journey of making simplifying assumptions, both mathematical and physical, that allow us to start moving toward a solution.  It’s the kind of work that often goes unmentioned in problem solving, especially in school mathematics, and this puzzle provides a nice opportunity to make that thinking transparent.

I highly recommend reading the puzzle and his solution at his blog.  Thanks for the fun problem, Jason!


Math Photo: Snowball Tetrahedron

I put the recent snow day, and a snowball maker, to good use!

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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?”

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?

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”.