Another interesting property of the logarithmic spiral is revealed if you roll it along a horizontal line. This animation shows the curves traced by points on the spiral, and note that the very centre follows the path of a straight line. The angle between this line and the horizontal is called the pitch of the spiral, and for our spiral galaxy the pitch is around 12 degrees. [more] [code]
“As a young man, Kolmogorov was nourished by the intellectual ferment of post-revolutionary Moscow, where literary experimentation, the artistic avant-garde, and radical new scientific ideas were in the air. In the early 1920s, as a 17-year-old history student, he presented a paper to a group of his peers at Moscow University, offering an unconventional statistical analysis of the lives of medieval Russians. It found, for example, that the tax levied on villages was usually a whole number, while taxes on individual households were often expressed as fractions. The paper concluded, controversially for the time, that taxes were imposed on whole villages and then split among the households, rather than imposed on households and accumulated by village. “You have found only one proof,” was his professor’s acid observation. “That is not enough for a historian. You need at least five proofs.” At that moment, Kolmogorov decided to change his concentration to mathematics, where one proof would suffice.”
An article well-worth the read. Never having heard of Andrei Kolgomorov, this was a big dive into one of the most creative minds I have ever heard of, and how he survived as a mathematician in post-revolutionary Russia.
The 2,300-year-old matrix is the world’s oldest decimal multiplication table.
Feng Lisheng, a historian of mathematics at Tsinghua University in Beijing, along with his colleagues, announce the discovery of a ~2300 year old multiplication table inscribed on strips of bamboo. In fact, it is the world’s oldest multiplication table in base-10.
The Apollonian gasket is a fractal constructed from a triple of circles, where each circle is tangent to the other two. Each level continues this pattern, adding 2·3n more circles on the nth level of the gasket, for a total of 3n+1 + 2 circles after n stages. Repeating this process and taking the limit gives an object like the gasket pictured above on the left.
The Apollonian gasket is also closely related to the undirected graph known as the Apollonian network. The network can be created by first taking three tangent circles, inscribing a circle in the gap created by the three circles, and continuing this process, and then giving each circle a vertex and each pair of tangent circles an edge. This process is seen in the second picture above which shows how it is related to the gasket, and the construction leads to the object pictured on the right.
Pretty interesting relation between the continuous fractal and the discrete graph!
A haiku about prime numbers
Any number that
is divisible by one
and itself is prime.