Archive for March, 2023

Christopher Wren and the Cycloid

Sir Christopher Wren [ image ]

The remarkable polymath Christopher Wren died in March 1723, just 300 years ago. Sarah Hart, Professor of Geometry at Gresham College, recently presented a lecture, The Mathematical Life of Sir Christopher Wren; a video of her presentation in available online (see sources below). The illustration above is from the Gresham College website.

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Bach and Euler chat in Frederick’s Court

Frederick the Great of Prussia, a devoted patron of the arts, had a particular interest in music, and admired the music of Johann Sebastian Bach. In 1747, Bach visited Potsdam, where his son Carl Philipp Emanuel was the Kapellmeister in Frederick’s court. When Frederick learned of this, he summoned ‘Old Bach’ to the palace and invited him to try out his collection of pianofortes. As they went from room to room, Bach improvised a new piece of music on each instrument  [TM243 or search for “thatsmaths” at].

Johann Sebastian Bach and Leonhard Euler.

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Sets that are Elements of Themselves: Verboten

Russell’s Paradox.

Can a set be an element of itself? A simple example will provide an answer to this question. Continue reading ‘Sets that are Elements of Themselves: Verboten’

Benford’s Law Revisited

{Probability for the first decimal digit {D_1} of a number to take values from 1 to 9.

Several researchers have observed that, in a wide variety of collections of numerical data, the leading — or most significant — decimal digits are not uniformly distributed, but conform to a logarithmic distribution. Of the nine possible values, {D_1=1} occurs more than {30\%} of the time while {D_1=9} is found in less than {5\%} of cases (see Figure above). Specifically, the probability distribution is

\displaystyle \mathsf{P}(D_1 = d) = \log_{10} \left( 1 + \frac{1}{d} \right) \,, \quad \mbox{ for\ } d = 1, 2, \dots , 9 \,. \ \ \ \ \ (1)

A more complete form of the law gives the probabilities for the second and subsequent digits. A full discussion of Benford’s Law is given in Berger and Hill (2015).

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