An Archimedian spiral has the polar equation r = b θ1/n This post will look at the case n = 1. I may look at more general values of n in a future post. The case n = 1 is the simplest case, and it’s the case I needed for the client project that motivated this post. In this case the spacing between points where the spiral crosses an axis is constant. Call this constant h. Then h = 2πb. For example, when rolling up a carpet, h corresponds to the thickness of the carpet. Suppose θ runs from 0 to 2πm, wrapping around the origin m times. We could approximate the spiral by m concentric ..read more

It’s occasionally necessary to evaluate a hypergeometric function at a large negative argument. I was working on a project today that involved evaluating F(a, b; c; z) where z is a large negative number. The hypergeometric function F(a, b; c; z) is defined by a power series in z whose coefficients are functions of a, b, and c. However, this power series has radius of convergence 1. This means you can’t use the series to evaluate F(a, b; c; z) for z < −1. It’s important to keep in mind the difference between a function and its power series representation. The former may exist where the latte ..read more

I recently ran across a tweet from Allen Downey saying So much of 20th century statistics was just a waste of time, computing precise answers to useless questions. He’s right. I taught mathematical statistics at GSBS [1, 2] several times, and each time I taught it I became more aware of how pointless some of the material was. I do believe mathematical statistics is useful, even some parts whose usefulness isn’t immediately obvious, but there were other parts of the curriculum I couldn’t justify spending time on [3]. Fun and profit I’ll say this in partial defense of computing precise answers t ..read more

An article by Y. L. Cheung [1] gives reasons why poker is usually played with five cards. The author gives several reasons, but here I’ll just look at one reason: pairs don’t act like you might expect if you have more than five cards. In five-card poker, the more pairs the better. Better here means less likely. One pair is better than no pair, and two pairs is better than one pair. But in six-card or seven-card poker, a hand with no pair is less likely than a hand with one pair. For a five-card hand, the probabilities of 0, 1, or 2 pair are 0.5012, 0.4226, and 0.0475 respectively. For a six-ca ..read more

Yesterday I stumbled on the fact that the size of Jupiter is roughly the geometric mean between the sizes of Earth and the Sun. That’s not too surprising: in some sense (i.e. on a logarithmic scale) Jupiter is the middle sized object in our solar system. What I find more surprising is that a systematic search finds mean relationships that are far more accurate. The radius of Jupiter is within 5% of the geometric mean of the radii of the Earth and Sun. But all the mean relations below have an error less than 1%. The radius of Mercury equals the geometric mean of the radii of the Moon and Mars ..read more

The size of Jupiter is approximately the geometric mean of the sizes of Sun and Earth. In terms of radii, The ratio on the left equals 9.95 and the ratio on the left equals 10.98. The subscripts are the astronomical symbols for the Sun (☉, U+2609), Jupiter (♃, U+2643), and Earth (, U+1F728). I produced them in LaTeX using the mathabx package and the commands \Sun, Jupiter, and Earth. The the mathabx symbol for Jupiter is a little unusual. It looks italicized, but that’s not because the symbol is being used in math mode. Notice that the vertical bar in the symbol for Earth is vertical, i.e. no ..read more

I was listening to the latest episode of the Space Rocket History podcast. The show includes some audio from a documentary on Pioneer 11 that mentioned that a man would weigh 500 pounds on Jupiter. My immediate thought was “Is that all?! Is this ‘man’ a 100 pound boy?” The documentary was correct and my intuition was wrong. And the implied mass of the man in the documentary is 190 pounds. Jupiter has more than 300 times more mass than the earth. Why is its surface gravity only 2.6 times that of the earth? Although Jupiter is very massive, it is also very large. Gravitational attraction is pro ..read more

Are guidance documents laws? Strictly speaking, no. The people who generate such documents are not legislators. Legislators delegate to agencies to make rules, and agencies delegate to other organizations to make guidelines. For example [1], Even HHS, which has express cybersecurity rulemaking authority under the Health Insurance Portability and Accountability Act (HIPAA), has put a lot of the details of what it considers adequate cybersecurity into non-binding guidelines. I’m not a lawyer, so nothing I can should be considered legal advice. However, the authors of [1] are lawyers. The legal ..read more

A week or two ago I wrote about Laguerre’s root-finding method and made some associated images. This post gives a couple more examples. Laguerre’s method is very robust in the sense that it is likely to converge to a root, regardless of the starting point. However, it may be difficult to predict which root the method will end up at. To visualize this, we color points according to which root they converge to. First, let’s look at the polynomial (x − 2)(x − 4)(x − 24) which clearly has roots at 2, 4, and 24. We’ll generate random starting points and color them blue, orange, or green depending on ..read more

I set up a GitHub account for a new employee this morning and spent a ridiculous amount of time proving that I’m human. The captcha was to listen to three audio clips at a time and say which one contains bird sounds. This is a really clever test, because humans can tell the difference between real bird sounds and synthesized bird-like sounds. And we’re generally good at recognizing bird sounds even against a background of competing sounds. But some of these were ambiguous, and I had real birds chirping outside my window while I was doing the captcha. You have to do 20 of these tests, and appar ..read more