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Tag: Math

Antenna length: Another rule of 72
The famous Rule of 72 says that to find out how many years it takes an investment to double in value, divide 72 by the annual percentage rate. I’ll come back to that in a little bit. This morning I read a really good article, Fifty Things you can do …
Hallucinations of AI Science Models
AlphaFold 2, FourCastNet and CorrDiff are exciting. AI-driven autonomous labs are going to be a big deal [1]. Science codes now use AI and machine learning to make scientific discoveries on the world’s most powerful computers [2]. It’s common practice for scientists to ask questions about the validity, reliability and …
Double super factorial
I saw someone point out recently that 10! = 7! × 5! × 3! × 1! Are there more examples like this? What would you call the pattern on the right? I don’t think there’s a standard name, but here’s why I think it should be called double super factorial …
Laguerre’s root finding method
Edmond Laguerre (1834–1886) came up with a method for finding zeros of polynomials. Unlike Newton’s method for finding zeros of general functions, Laguerre’s method is specialized for polynomials. Laguerre’s method converges an order of magnitude faster than Newton’s method, i.e. the error is cubed on each step rather than squared. …
Distance from a point to a line
Eric Lengyel’s new book Projective Geometric Algebra Illuminated arrived yesterday and I’m enjoying reading it. Imagine if someone started with ideas like dot products, cross products, and determinants that you might see in your first year of college, then thought deeply about those things for years. That’s kinda what the …
Accelerating Archimedes
One way to approximate π is to find the areas of polygons inscribed inside a circle and polygons circumscribed outside the circle. The approximation improves as the number of sides in the polygons increases. This idea goes back at least as far as Archimedes (287–212 BC). Maybe you’ve tried this. …
How much will a cable sag? A simple approximation
Suppose you have a cable of length 2s suspended from two poles of equal height a distance 2x apart. Assuming the cable hangs in the shape of a catenary, how much does it sag in the middle? If the cable were pulled perfectly taught, we would have s = x …
A surprising result about surprise index
Surprise index Warren Weaver [1] introduced what he called the surprise index to quantify how surprising an event is. At first it might seem that the probability of an event is enough for this purpose: the lower the probability of an event, the more surprise when it occurs. But Weaver’s …
Blow up in finite time
A few years ago I wrote a post about approximating the solution to a differential equation even though the solution did not exist. You can ask a numerical method for a solution at a point past where the solution blows up to infinity, and it will dutifully give you a …
Finite differences and Pascal’s triangle
The key to solving a lot of elementary what-number-comes-next puzzles is to take first or second differences. For example, if asked what the next item in the series 14, 29, 50, 77, 110, … the answer (or at lest the answer the person posing the question is most likely looking …
Bounding the perimeter of a triangle between circles
Suppose you have a triangle and you know the size of the largest circle that can fit inside (the incircle) and the size of the smallest circle that can fit outside (the circumcircle). How would you estimate the perimeter of the triangle? In terms of the figure below, if you …
Area of quadrilateral as a determinant
I’ve written several posts about how determinants come up in geometry. These determinants often look similar, having columns related to coordinates and a column of ones. You can find several examples here along with an explanation for this pattern. If you have three points z1, z2, and z3 in the …
Approximate roots of fractions
This post will discuss a curious approximation with a curious history. Approximation Let x be a number near 1, written as a fraction x = p / q. Then define s and d as the sum and difference of the numerator and denominator. s = p + q d = …
A knight’s tour of an infinite chessboard
Let ℤ² be the lattice of points in the plane with integer coordinates. You could think of these points as being the centers of the squares in a chessboard extending to infinity in every direction. Cantor tells us that the points in ℤ² are countable. What’s more surprising is that …
Frequency analysis
Suppose you have a list of encrypted surnames names of US citizens. If the list is long enough, the encrypted name that occurs most often probably corresponds to Smith. The second most common encrypted name probably corresponds to Johnson, and so forth. This kind of inference is analogous to solving …
Advanced questions about a basic diagram
I saw a hand-drawn version of the diagram above yesterday and noticed that the points were too evenly distributed. That got me to thinking: is there any objective way to say that this famous diagram is in some sense complete? If you were to make a diagram with more points, …
The Borwein integrals
The Borwein integrals introduced in [1] are a famous example of how proof-by-example can go wrong. Define sinc(x) as sin(x)/x. Then the following equations hold. However where δ ≈ 2.3 × 10−11. This is where many presentations end, concluding with the moral that a pattern can hold for a while …
This-way-up and Knuth arrows
I was looking today at a cardboard box that had the “this way up” symbol on it and wondered whether there is a Unicode value for it. Apparently not. But there is an ISO code for it: ISO 7000 symbol 0623. It’s an international standard symbol for indicating how to …