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In the novel Carry On, Mr. Bowditch, a sailor asked Nathaniel Bowditch to teach him how to do navigational calculations, but the man only knows how to add and subtract by counting on his fingers. He had not heard of multiplication. Bowditch is surprised, but realizes if he made tables of logs of trig functions, […] The
Nothing grows exponentially forever. What appears to be an exponential curve often turns out to be some sort of S curve, such as a logistic curve. Suppose you’re collecting data on the left side of the curve. If there’s even a small amount of error in your data, you won’t be able to predict the […] The post Why fitting
There’s a common saying in discussion of fractals that the length of a coastline depends on how small a device you use to measure it. I thought this was a hypothetical, say as applied to the steps in the construction of the Koch snowflake. But the saying has its roots in actually surveying. Lewis Fry […] The post Empir
Suppose you want to solve a triangle. You know two sides and the angle between them. Then you can solve for the third side using the law of cosines. Now suppose you want to solve a big triangle, a triangle on the surface of the earth so large that the curvature of the earth matters. You […] The post Haversine law first
I recently bought a copy of Los Alamos Rolodex, a book displaying business cards from Los Alamos Nation Labs from 1967 to 1978. You can find some examples of the cards here. One of the cards in the book is for Eugene Frank, President of B & F Instruments. His card lists his phone number […] The post Phone words first a
A couple days ago a friend told me about the book Carry On, Mr. Bowditch, a fictional account of the life of Nathaniel Bowditch (1773–1838). I’ve been listening to the book on Audible, and apparently it’s only lightly fictionalized. Bowditch was a self-educated mathematician and astronomer, best known for his book The
A rising tide may lift all boats, but the AI tide lifts some boats much more than others. By all accounts, the best programmers have had the biggest productivity boost from AI. And top tier mathematicians are using AI to settle long-standing mathematical conjectures. AI is a powerful tool, but tools don’t come to life
I’ve written three posts on cosine similarity lately. The first looked at interpreting cosine similarity. The second looked at an approximation related to the first. The third looked at how ranking according to cosine similarity works better than cosine similarity itself. Normalized word vectors are points on a high di
The product of four consecutive Fibonacci numbers equals the product of two consecutive integers. For example, 3 × 5 × 8 × 13 = 39 × 40. I ran across this theorem in a note [1] that says “The product of any four consecutive Fibonacci numbers is twice a triangular number.” Since triangular numbers have […] The post Fibo
Yesterday I wrote about the canonical example of how vector embeddings of words add: “king” − “man” + “woman” ≈ “queen” This should be interpreted as saying that the word vector for king, minus the word vector for man, plus the word vector for woman, is in some sense close to the word vector for queen. This post will [
The previous post looked at how to interpret cosine similarity, or equivalently angles between word vectors. In a high-dimensional space, randomly chosen vectors are likely nearly perpendicular, and so relatively large angles, such as 50°, indicate very closely related words. Another way to look at this, as explained i
Machine learning represents words as vectors and measures the similarity of words by the angles between the vectors. For vectors x and y, where θ is the angle between the vectors, and so This is the cosine similarity between the words represented by x and y. Small angles have large cosines, and so words with larger cos
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