Blog Profile / The Endeavour

Filed Under:Academics
Posts on Regator:1469
Posts / Week:4.6
Archived Since:April 26, 2011

Blog Post Archive

Student’s future, teacher’s past

“Teachers should prepare the student for the student’s future, not for the teacher’s past.” — Richard Hamming I ran across the above quote from Hamming this morning. It made me wonder whether I tried to prepare students for my past when I used to teach college students. How do you prepare a student for the […]

Ideal background for algebraic geometry

From Foundations of Algebraic Geometry: … in an ideal world, people would learn this material over many years, after having background courses in commutative algebra, algebraic topology, differential geometry, complex analysis, homological algebra, number theory, and French literature.

Changing your mind

From Dorothy Sayers’ essay Why Work? It is always strange and painful to have to change a habit of mind; though, when we have made the effort, we may find a great relief, even a sense of adventure and delight, in getting rid of the false and returning to the true.

Volume of a rose-shaped torus

Start with a rose, as described in the previous post: Now spin that rose around a vertical line a distance R from the center of the rose. This makes a torus (doughnut) shape whose cross sections look like the rose above. You could think of having a cutout shaped like the rose above and extruding Play-Doh […]

Length of a rose

The polar graph of r = cos(k?) is called a rose. If k is even, the curve will trace out 2k petals as ? runs between 0 and 2?. If k is odd, it will trace out k petals, tracing each one twice. For example, here’s a rose with k = 5. (I rotated the […]

When length equals area

The graph of hyperbolic cosine is called a catenary. A catenary has the following curious property: the length of a catenary between two points equals the area under the catenary between those two points. The proof is surprisingly simple. Start with the following: Now integrate the first and last expressions between two points a and […]

Solving systems of polynomial equations

In a high school algebra class, you learn how to solve polynomial equations in one variable, and systems of linear equations. You might reasonably ask “So when do we combine these and learn to solve systems of polynomial equations?” The answer would be “Maybe years from now, but most likely never.” There are systematic ways to […]

Generating pink noise

Different colors of noise are named by analogy with colors of light. Pink noise is between white noise and red noise. White noise has equal power at all frequencies. The power in red noise drops off like 1/f2 where f is frequency. The power in pink noise drops off like 1/f. Generating pink noise is […]

Building software the right way

Yesterday a friend told me about a software project whose owners said “We’re going to do this the right way.” I told him I have two opposite reactions when I hear that: Ooh, that sounds like fun! Run away! I’ve been on several projects where the sponsors have identified some aspect of the status quo […]

The 3n+1 problem and Benford’s law

This is the third, and last, of a series of posts on Benford’s law, this time looking at a famous open problem in computer science, the 3n + 1 problem, also known as the Collatz conjecture. Start with a positive integer n. Compute 3n + 1 and divide by 2 repeatedly until you get an odd […]

Cauchy, Benford, and a problem with NHST

Introduction Samples from a Cauchy distribution nearly follow Benford’s law. I’ll demonstrate this below. The more data you see, the more confident you should be of this. But with a typical statistical approach, crudely applied NHST (null hypothesis significance testing), the more data you see, the less convinced you are. This post assumes you’ve read the […]

Weibull distribution and Benford’s law

Introduction to Benford’s law In 1881, Simon Newcomb noticed that the edges of the first pages in a book of logarithms were dirty while the edges of the later pages were clean. From this he concluded that people were far more likely to look up the logarithms of numbers with leading digit 1 than of […]

Golden angle

The golden angle is related to the golden ratio, but it is not as well known. And the relationship is not quite what you might think at first. The golden ratio ? is (1 + ?5)/2. A golden rectangle is one in which the ratio of the longer side to the shorter side is ?. […]

What personality classifications have in common

There are many ways to divide people into four personality types, from the classical—sanguine, choleric, melancholic, and phlegmatic—to contemporary systems such as the DISC profile. The Myers-Briggs system divides people into sixteen personality types. Show More Summary

Denver airport, Weierstrass, and A&S

Last night I was driving toward the Denver airport and the airport reminded me of the cover of Abramowitz and Stegun’s Handbook of Mathematical Functions. Here’s the airport: And here’s the book cover: I’ve written about the image on book cover before. Someone asked me what function it graphed and I decided it was probably […]

Resisting simplicity

As much as we admire simplicity and strive for simplicity, something in us isn’t happy when we achieve it. Sometimes we’re disappointed with a simple solution because, although we don’t realize it yet, we didn’t properly frame the problem it solves. I’ve been in numerous conversations where someone says effectively, “I understand that 2+3 = […]

Flying through a 3D fractal

A Menger sponge is created by starting with a cube a recursively removing chunks of it. Draw a 3×3 grid on one face of the cube, then remove the middle square, all the way through the cube. Then do the same for each of the eight remaining squares. Repeat this process over and over, and do it […]

Computing harmonic numbers

The harmonic numbers are defined by Harmonic numbers are sort of a discrete analog of logarithms since As n goes to infinity, the difference between Hn and log n is Euler’s constant ? = 0.57721… [1] How would you compute Hn? For small n, simply use the definition. But if n is very large, there’s a way […]

Technical notes and other relatively hidden conent

I’ve written quite a few pages that are separate from the timeline of the blog. These are a little hidden, not because I want to hide them, but because you can’t make everything equally easy to find. These notes cover a variety of topics: Math diagrams Numerical computing Probability and approximations Differential equations Python Regular expressions […]

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