John D. Cook

Let be the upper half plane, the set of complex real numbers with positive imaginary part. When we measure distances the way we've discussed in the last couple posts, the geometry of is hyperbolic. What is a circle of radius r in ? The same as a circle in any geometry: it's the [...]The post A circle in the hyperbolic plane first appeared on John D. Cook.

The previous post described a metric for the Poincare upper half plane. The development is geometrical rather than analytical. There are also analytical formulas for the metric, at least four that I've seen. It's not at all obvious that the four equations are equivalent, or that any of them matches the expression in the previous [...]The post Equal things that don't look equal first appeared on John D. Cook.

One common model of the hyperbolic plane is the Poincare upper half plane . This is the set of points in the complex plane with positive imaginary part. Straight lines are either vertical, a set of points with constant imaginary part, or arcs of circles centered on the real axis. The real axis is not [...]The post Hyperbolic metric first appeared on John D. Cook.

The opening line of William Gibson's novel Neuromancer is famous: The sky above the port was the color of a television, tuned to a dead channel. When I read this line, I knew immediately what he meant, and thought it was a brilliant line. Later I learned that younger readers didn't know what he was [...]The post TV tuned to a dead channel first appeared on John D. Cook.

Suppose Alice runs a confidential restaurant. Alice doesn't want there to be any record of who visited her restaurant but does want to get paid for her food. She accepts Monero, and instead of a cash register there are two QR codes on display, one corresponding to her public view keyA and the other corresponding [...]The post How stealth addresses work in Monero first appeared on John D. Cook.

I was reading about Shackleton's incredible expedition to Antarctica, and the Weddell Sea features prominently. That name sounded familiar, and I was trying to remember where I'd heard of Weddell in math. I figured out that it wasn't Weddell exactly but Weddle I was thinking of. The Weddell Sea is named after James Weddell (1787-1834). [...]The post Weddle integration rule first appeared on John D. Cook.

The previous post includes code for solving the equation Hn = m i.e. finding the value ofn for which thenth harmonic number is the closest tom. It works well for small values ofm. It works for largem in the sense that the solution is very close tom, but it's not necessarily the best solution. For [...]The post Solving H_n = 100 first appeared on John D. Cook.

I mentioned in the previous post that the harmonic numbers Hn are never integers for n > 1. In the spirit of that post, I'd like to find the value of n such that Hn is closest to a given integer m. We have two problems to solve. First, how do we accurately and efficiently [...]The post Closest harmonic number to an integer first appeared on John D. Cook.

Jozsef Kurschak proved in 1908 that the function is never an integer for 0 < m < n. In particular, the harmonic numbers are never integers for n > 1. The function f(m,n) can get arbitrarily close to any integer value by taking m andn large enough, but it can never exactly equal an integer. [...]The post Closest consecutive reciprocal sum to an integer first appeared on John D. Cook.

Five posts on Pythagorean triangles and Pythagorean triples Primitive Pythagorean triangles with the same area Sparse binary Pythagorean triples Matrix Pythagorean triples Approximation by Pythagorean triangles Fibonacci meets PythagorasThe post Pythagorean triples first appeared on John D. Cook.

The last couple posts have been about group pairings, specifically Tate pairings as they're used in cryptography. This post will show that RSA encryption can be seen as a special case of pairing-based cryptography. The idea comes from Ben Lynn's 2007 dissertation. Lynn is the L" in BLS signatures-one of the topics in his dissertations-and [...]The post RSA as a pairing first appeared on John D. Cook.

Elliptic curve Diffie-Hellman Given a point P on an elliptic curve E, and a random number a, aP means to add P to itself a times, using the addition on E. The pointaP can be computed efficiently, even ifa is a very large number [1]. However, if E has a large number of points, and [...]The post Three-party Diffie-Hellman in one shot first appeared on John D. Cook.

Pairings can mean a variety of related things in group theory, but for our purposes a pairing is a bilinear mapping from two groups to a third group. e: G1 * G2 GT Typically the group operation on G1 and G2 is written addititvely and the group operation on GT is written multiplicatively. In [...]The post Elliptic curve pairings in cryptography first appeared on John D. Cook.

Let p be a prime number. Then the integers mod p form a finite field. The number of elements in a finite field must be a power of a prime, i.e. the order q = pn for some n. When n > 1, we can take the elements of our field to be polynomials of [...]The post Adding an imaginary unit to a finite field first appeared on John D. Cook.

Here are four theorems that generalize the Pythagorean theorem. Follow the links for more details regarding each equation. 1. Theorem by Apollonius for general triangles. 2. Edsgar Dijkstra's extension of the Pythagorean theorem for general triangles. 3. A generalization of the Pythagorean theorem to tetrahedra. 4. A unified Pythagorean theorem that covers spherical, plane, and [...]The post Four generalizations of the Pythagorean theorem first appeared on John D. Cook.

Themth elementary symmetric polynomial of degreen is the sum of all terms containing a product of m variables. So, for example, These polynomials came up in the previous post. The problem was choosing weights to minimize the variance of a weighted sum of random variables can be solved using elementary symmetric polynomials. To state the [...]The post Elementary symmetric polynomials and optimization first appeared on John D. Cook.

Suppose you have $100 to invest in two independent assets, A andB, and you want to minimize volatility. SupposeA is more volatile thanB. Then putting all your money onA would be the worst thing to do, but putting all your money onB would not be the best thing to do. The optimal allocation would be [...]The post Weighting an average to minimize variance first appeared on John D. Cook.

Excellent video by Almost Sure: What does Riemann Zeta have to do with Brownian Motion? Connects several things that I've written about here including Brownian motion, the Riemann zeta function, and the Kolmogorov-Smirnov test.The post Brownian motion and Riemann zeta first appeared on John D. Cook.

Suppose you have data on the closing prices of two stocks over 1,000 days and you want to look at the correlation between the two asset prices over time in rolling 30 day windows. It seems that the rolling correlation is periodic. peaking about every 50 days. But this is an artifact of the rolling [...]The post Rolling correlation first appeared on John D. Cook.

The area of a triangle can be computed directly from the lengths of its sides via Heron's formula. Heres is the semiperimeter,s = (a +b +c)/2. Is there an analogous formula for spherical triangles? It's not obvious there should be, but there is a formula by Simon Antoine Jean L'Huilier (1750-1840). Here we denote area [...]The post Analog of Heron's formula on a sphere first appeared on John D. Cook.

There's increasing talk of gigawatt data centers. Currently the largest data center, Switch's Citadel Campus in Nevada, uses 850 megawatts of power. OpenAI's Stargate data center, under construction, is supposed to use 1.2 gigawatts. Gigawatt An average French nuclear reactor produces about a gigawatt of power. If the US were allowed build nuclear reactors, we [...]The post How much is a gigawatt? first appeared on John D. Cook.

Here's an interesting theorem that leads to some aesthetically pleasing images. It's known as the Japanese cyclic polygon theorem. For all triangulations of a cyclic polygon, the sum of inradii of the triangles is constant. Conversely, if the sum of inradii is independent of the triangulation, then the polygon is cyclic. The image above shows [...]The post Japanese polygon theorem first appeared on John D. Cook.

A tetrahedron has four triangular faces. Suppose three of those faces come together like the corner of a cube, each perpendicular to the other. Let A1, A2, and A3 be the areas of the three triangles that meet at this corner and let A0 be the area of remaining face, the one opposite the right [...]The post Tetrahedral analog of the Pythagorean theorem first appeared on John D. Cook.

As I've written about several times lately, the Smith chart is the image of a rectangular grid in the right half-plane under the function f(z) = (z- 1)/(z+ 1). What would the image of a grid in the left half-plane look like? For starters, sincef maps the right half-plane to the interior of the unit [...]The post The anti-Smith chart first appeared on John D. Cook.

A few days ago I wrote two posts about how to create a Smith chart, a graphical device used for impedance calculations. Then someone emailed me to point out the connection between the Smith chart and triangular numbers. The Smith chart is the image of a rectangular grid in the right half-plane under the function [...]The post Impedance and Triangular Numbers first appeared on John D. Cook.

The cross ratio of four pointsA,B,C,D is defined by whereXY denotes the length of the line segment fromX toY. The idea of a cross ratio goes back at least as far as Pappus of Alexandria (c.290 - c.350 AD). Numerous theorems from geometry are stated in terms of the cross ratio. For example, the cross [...]The post Cross ratio first appeared on John D. Cook.

Let's make a QR code out of a sentence two ways: mixed case and upper case. We'll use Python with the qrcode library. >>> import qrcode >>> s = "The quick brown fox jumps over the lazy dog." >>> qrcode.make(s).save("mixed.png") >>> qrcode.make(s.upper()).save("upper.png") Here are the mixed case and upper case QR codes. The QR code [...]The post Text case changes the size of QR codes first appeared on John D. Cook.

Apollonius of Perga (c. 262 BC - c. 190 BC) discovered a theorem that generalizes the Pythagorean theorem but isn't nearly as well known. Let ABC be a general triangle, and let D be the midpoint of the segment AB. Leta be the length of the side opposite A andb the length of the side [...]The post An ancient generalization of the Pythagorean theorem first appeared on John D. Cook.

The previous post required computing After writing the post, I thought about how you would mentally approximate log2 5. The most crude approximation would round 5 down to 4 and use log2 4 = 2 to approximate log2 5.That would be good enough for an order of magnitude guess, but we can do much better [...]The post Mentally compute logs base 2 first appeared on John D. Cook.

A physical key, such as a house key, is a piece of metal with cuts of differing depths. Typically there may be around 6 cuts, with five different possible depths for each cut. This allows 56 = 15,625 possible keys. Encryption keys, such as AES keys, are a string of bits, often 128 bits, for [...]The post Physical Keys and Encryption Keys first appeared on John D. Cook.

The Freshman's dream" is the statement (x+ y)p = xp + yp It's not true in general, but itis true modp if p is a prime. It's a cute result, but it's also useful in applications, such as finite field computations in cryptography. Here's a demonstration of the Freshman's dream in Python. >>> p = [...]The post Freshman's dream first appeared on John D. Cook.

Yesterday's post looked at a recently mined Bitcoin block, the 920,994th block in the blockchain, and verified that it contains the hash of the previous block. This post will look at the same block and verify its Merkle tree root. Before getting down to the bytes, we'll back up and say what a Merkle tree [...]The post Why and how Bitcoin uses Merkle trees first appeared on John D. Cook.

The high-level explanation of a blockchain says that each block contains a cryptographic hash of the previous block. That's how the blocks are chained together. That's not exactly true, and it leaves out a lot of detail. This post will look in full detail at how Bitcoin blocks are chained together by inspecting the bits [...]The post How blocks are chained in a blockchain first appeared on John D. Cook.

I recently saw someone post [1] that 987654321/123456789 is very nearly 8, specifically 8.0000000729. I wondered whether there's anything distinct about base 10 in this. For example, would the ratio of 54321six and 12345six be close to an integer? The ratio is 4.00268, which is pretty close to 4. What about a larger base? Let's [...]The post 987654321 / 123456789 first appeared on John D. Cook.

The previous post looked at the basics of how to create a Smith chart. The Smith chart is the image of a Cartesian grid in the right half-plane under the function f(z) = (z- 1)/(z+ 1). At the end of the post I noted that evenly distributed grid lines in thez plane result in very [...]The post Spacing the circles on the Smith chart first appeared on John D. Cook.

The Smith chart from electrical engineering is the image of a Cartesian grid under the function f(z) = (z - 1)/(z + 1). More specifically, it's the image of a grid in the right half-plane. This post will derive the basic mathematical properties of this graph but will not go into the applications. Said another [...]The post How to make a Smith chart first appeared on John D. Cook.

The previous post looked at how to generate random points on a sphere, generating spherical coordinates directly. I wanted to include a demonstration that this generates points with the same distribution as the more customary way of generating points on a sphere, and then decided the demonstration should be its own post. I'll generate random [...]The post Generating random points in Colorado first appeared on John D. Cook.

The topic of generating random points on a unit sphere has come up several times here. The standard method using normal random samples generates points (x, y, z) in Cartesian coordinates. If you wanted points in spherical coordinates, you could first generate points in Cartesian coordinates, then convert the points to spherical coordinates. But it [...]The post Random spherical coordinates first appeared on John D. Cook.

One of the more subtle ideas to convey in an introductory statistics class is that statistics have distributions. Students implicitly think that when you calculate a statistic on a data set, say the mean, that then you have THE mean. But if your data are (modeled as) samples from a random variable, then anything you [...]The post Distribution of correlation first appeared on John D. Cook.

Almost Sure posted an interesting fact on X: If a point (x, y, z) is chosen at random uniformly on the unit sphere, then x, y, and z each have the uniform distribution on [-1, 1] and zero correlations (but not independent!) This follows from Archimedes' On the Sphere and Cylinder" published in 225BC. Archimedes [...]The post Distribution of coordinates on a sphere first appeared on John D. Cook.

I was calculating a correlation coefficient this afternoon and ran into something interesting. Suppose you have two uncorrelated random variablesX andY. If you draw, say, a thousand samples each from X andY and compute Pearson's correlation coefficient, you almost certainly will not get 0, though you very likely will get a small number. How do [...]The post ODE to Fisher's transform first appeared on John D. Cook.

Sometimes you'll hear that a process has so many nines of reliability or that the error rate is so many sigmas. A few years ago I wrote a post on converting between nines and sigmas. See that post for details, approximations, etc. Here I'd like to post a new graph that I believe is an [...]The post Quality metrics first appeared on John D. Cook.

In 2013, John Conway and Alex Ryba published a brief note [1] on how to convert identities involving sine and cosine into identities involving Fibonacci and Lucas numbers. Fibonacci and Lucas The Fibonacci numbers Fn are defined by F0 = 0, F1 = 1, and Fn+2 = Fn + Fn+1 for n > 1. Similarly, [...]The post Turning trig identities into Fibonacci identities first appeared on John D. Cook.

OpenAI Codex is now generally available (see here, here). I'm using the Codex extension in the Cursor code editor with my OpenAI account. Codex is very helpful for some tasks, such as complex code refactoring, implementing skeleton code for an operation, or writing a single small self-contained piece of code. Models have come a long [...]The post Experiences with GPT-5-Codex first appeared on John D. Cook.

I've written before about Bitcoin's elliptic curve and Monero's elliptic curve. In the Monero post I wrote Bitcoin and Ethereum use the elliptic curvesecp256k1." That's true, but it's incomplete. Ethereum does use the elliptic curve secp256k1 for digital signatures, as does Bitcoin, but Ethereum also uses a different elliptic curve for its consensus layer. Ethereum's [...]The post Ethereum's consensus layer elliptic curve first appeared on John D. Cook.

The inverse of the matrix is the matrix assumingad - bc 0. Also, the inverse of the bilinear function (a.k.a. Mobius transformation) is the function again assumingad - bc 0. The elementary takeaway is that here are two useful equations that are similar in appearance, so memorizing one makes it easy to memorize [...]The post Inverting matrices and bilinear functions first appeared on John D. Cook.

For reasons I don't understand, it's more common, at least in my experience, to see references on how to generate pointson a sphere thanin a sphere. But the latter is easily obtained from the former. The customary way to generate points on a sphere inn dimensions is to generate points from ann-dimensional normal (Gaussian) distribution [...]The post Generate random points inside a sphere first appeared on John D. Cook.

Exactly one month ago I wrote about sampling points from a triangle. This post will look at the three dimensional analog, sampling from a tetrahedron. The generalization to higher dimensions works as well. Sampling from a triangle In the triangle post, I showed that a naive approach doesn't work but a variation does. If the [...]The post Random samples from a tetrahedron first appeared on John D. Cook.

Many hope that AI will be smart enough" to make breakthrough scientific discoveries in the near future, such as find a cure for cancer. Some research efforts have sought to create an AI Scientist" that can make discoveries in an automated or semi-automated way; for a recent example, see [1]. Others [2] have called out [...]The post GPT-5 for AI-assisted discovery first appeared on John D. Cook.

This morning I took an old blog post and turned it into an X thread. I think the thread is easier to read. More expository and less rigorous. The post and thread look at generalizations of the fact that every integer and its fifth power end in the same digit. The conclusion is thatn andnk [...]The post More on Carmichael first appeared on John D. Cook.