Tag Archives: differential geometry

Bugles, Trumpets, and Beltrami

This essay began as an effort to prove Tanya Khovanova’s statement in her article “The Annoyance of Hyperbolic Surfaces” that her crocheted hyperbolic surface had constant (negative) curvature. I discussed Khovanova’s article in my previous essay “Exponential Yarn”. What I thought would be a fairly straight-forward exercise turned into a more concerted effort as I concluded that her crocheted surface did not have constant curvature. However, I found additional references that supported her statement, so I was becoming quite confused. I looked at other, similar surfaces to try to understand the whole curvature situation. This involved a lot of tedious computations (with my usual plethora of mistakes) that proved most challenging. But then I realized where I had gone astray. To cover my ignorance I claimed my error stemmed from a subtle misunderstanding. Herewith is a presentation of what I found. See Bugles, Trumpets, and Beltrami.

(Update 4/6/2019) Continue reading

Hyperboloid as Ruled Surface

When our daughter-in-law made wheat shocks as center-pieces for hers and our son’s fall-themed wedding reception, I naturally could not help pointing out the age-old observation that they represented a hyperboloid of one sheet. This was naturally greeted with the usual groans, but the thought stayed with me as I realized I had never proved this mathematically to myself. And so I did. See the Hyperboloid as Ruled Surface.