I love the way the struts of this structure come together and align with the geometry of the city.
Challenge Geometry
When do Multiple Rotations Exist?
I recently profiled an erroneous high-stakes math exam question that had two correct answers.
Here, it is possible to map AB onto A’B’ using either a glide reflection or a rotation.
It’s interesting to note that there are actually two distinct rotations that map AB onto A’B’, as demonstrated below.
This raises an interesting question: given two congruent objects, under what circumstances will two distinct rotations exist that map one onto the other?
In a comment on the original post, Joshua Greene offered another interesting follow-up question:
Under what circumstance, if any, are two line segments of equal length not images of each other under rotation? In which of those cases, if any, are the two line segments images of each other under glide reflection?
With enough work, even erroneous exam questions are redeemable!
Application Appreciation Geometry
An Ancient Tiling Problem
This menacing creature is Glyptotherium texanum. Well, was Glypotherium texanum.
The armor caught my attention. It’s made of small plates that create an interesting tiling of a curved surface.
Notice how the tiles slowly change from quadrilaterals to pentagons to hexagons, and back again! I wonder what factors determine the shape of each tile as they grow.
Application Geometry Teaching
Pinscreen Approximations
I’ve always enjoyed playing around with pinscreens, but only recently did I realize what cool mathematical concepts they display!
For example, the image above shows an approximation of the volume of a hemisphere using right cylinders. A pinscreen Riemann sum!
And the images below suggest how we might approximate the areas of a circle and a square using pinscreens.
Compute the ratio of raised pins to total pins, and multiply by the total area of the pinscreen. A pinscreen Monte Carlo method!
Any other cool math hiding in there?
Appreciation Geometry Photography
Math Photo: Surprising Heptagon
It took several hundred encounters with this park bench before I realized it was a heptagon! I don’t see many regular, seven-sided figures in my experience, which made this a surprising discovery. I wonder what prompted this design choice.
Like most real-world instances of perfect geometric objects, it doesn’t exactly measure up. But what’s a few degrees between n-gons?