I was reading yet another book on the Scientific Revolution when I came across a discussion of the mathematical significance of the invention of perspective for painting in the 15th century Italian Renaissance. The main player in the saga was Leon Battista Alberti (1404 – 1472) and his tome De Pictura (On Painting) (1435-6), which contained the first mathematical presentation of perspective. Even though mathematics was advertised, it was not at the level of trigonometry I used in my post “The Perspective Map”, but rather entailed simple Euclidean plane geometry. So the discussion was largely historical rather than mathematical. Nevertheless, I became curious to learn how much Alberti was able to discover about perspective without a lot of math. This essay is the result.
See Alberti’s Perspective Construction
(Update 7/29/2019) I got a response! Continue reading
This is a challenging problem from Mathematical Quickies (1967).
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I was astonished that this problem was suitable for 8th graders. First of all the formula for the volume of a cone is one of the least-remembered of formulas, and I certainly never remember it. So my only viable approach was calculus, which is probably not a suitable solution for an 8th grader.
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 “
Over the years one of the subjects I return to periodically to study is Einstein’s Theory of Relativity, both the Special and General theories. Interest in the Special Theory focused on the derivation of the Lorentz transformations (or contractions). Why did objects appear with different lengths and clocks run at different speeds for observers moving relative to one another? Early on (late 60s) I came across a great explanation in the 1923 book by C. P. Steinmetz. He derived it from two general assumptions of special relativity: (1) that all motion is relative, the motion of the railway train relative to the track being the same as the motion of the track relative to the train, and (2) that the laws of nature, and thus the velocity of light, are the same everywhere. I did not follow his derivation completely, so I produced my own, which I will give here. See the 
I came across the following problem from an Italian high school exam on the British 
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.
This was a catchy, misleading title that I could not resist, since my essay is not about math vs. religion as one might expect from the title, but rather about math helping religion. Back in 2016 I was reading Dr. Bart D. Ehrman’s blog that he was writing in preparation for his eventual book, The Triumph of Christianity, in which he was considering Rodney Stark’s purely mathematical analysis of the growth of Christianity in the first three centuries. Neither Rodney Stark nor Bart Ehrman described explicitly the underlying mathematical models of exponential growth that they were using and exactly what was meant by a rate of growth. Given the natural audience for the subject, these omissions were not surprising. So I thought I would clarify the math and also offer some variations on the models, which eventually reflected the actual situation more faithfully. See