Appreciation

Happy Right Triangle Day!  Today we celebrate a favorite geometric object:  the 5-12-13 right triangle.

Of course, the sides of this triangle satisfy the Pythagorean Theorem

but one reason I like this particular right triangle so much is the role it plays in another favorite triangle.  The 5-12-13 triangle fits together perfectly with the 9-12-15 right triangle

to make the 13-14-15 triangle!

The 13-14-15 triangle is special in its own right:  it is a Heronian triangle, a triangle with rational side lengths and rational area.  In fact, this triangle has  integer side lengths and integer area, making it especially interesting!

Happy Right Triangle Day!  Be sure to marvel at some perpendicularity today.

Studying vector calculus tends to make you see space curves everywhere you go.  Here’s a conical helix (or a helical cone?).

A good way to understand the behavior of curves in space is to understand how their projections behave.  The sun does a nice job of showing us one such projection of this space curve.

This suggests a common mathematical practice:  trading a hard problem for an easier one.  Space curves can be difficult to analyze, but their projections are more easily understood.  And by understanding its projections, you can develop knowledge of the space curve itself.

Of course, it’s important to understand what information you lose through the projection, as well!

I feel like most manufacturers of octogonal tables would just put them in square boxes and shim the sides.  But not IKEA.

I appreciate the elegance of this approach, but I wonder if this also yields benefits in production, shipping, and storage costs.