# hadamard (blogs) — RSS Amplifier Recent posts from the 3 feeds in the RSS Amplifier directory that cover hadamard. Page: Feed: --- ## [How would you know whether an ancient culture had zero?](https://www.johndcook.com/blog/2026/08/21/ancient-number-system/) _2026-08-21 · John · John D. Cook_ A few weeks ago I wrote about the number system used in labeling spreadsheet columns. Labels run from A through Z, then AA through AZ, etc. This looks a lot like base 26, but it’s not quite the same. It has no analog of zero. If Z were like zero, Y would be followed by \[…\] The post How would you know whether an ancient culture had zero? first appeared on John D. Cook . ## [AI-generated ASCII diagrams](https://www.johndcook.com/blog/2026/08/20/ai-generated-ascii-diagrams/) _2026-08-20 · John · John D. Cook_ I like AI-generated ASCII diagrams. Because nobody would ask AI to generate ASCII diagrams, and so, it’s congruous. I like incongruity \[1\]. Aside from the incongruity of using a gazillion-parameter neural network to make 1970’s style ASCII art, ASCII diagrams have some uses. They’re absolutely tiny compared to image files. But more importantly they can \[…\] The post AI-generated ASCII diagrams… ## [Big little hexagon](https://www.johndcook.com/blog/2026/08/18/big-little-hexagon/) _2026-08-19 · John · John D. Cook_ A new paper just came out, The Maximum-Area Small Polygon Problem. The paper solves the problem of finding, for each n, the n-gon with diameter 1 and maximum area. For odd n, the solution is what you might expect: a regular n-gon. I would expect this to be the solution for even n as well, \[…\] The post Big little hexagon first appeared on John D. Cook . ## [Big little hexagon](https://www.johndcook.com/blog/2026/08/18/big-little-hexagon/) _2026-08-19 · John · John D. Cook_ A new paper just came out, The Maximum-Area Small Polygon Problem. The paper solves the problem of finding, for each n, the n-gon with diameter 1 and maximum area. For odd n, the solution is what you might expect: a regular n-gon. I would expect this to be the solution for even n as well, \[…\] The post Big little hexagon first appeared on John D. Cook . ## [Mean distance to the sun and a new graph theorem](https://johndcookconsulting.substack.com/p/mean-distance-to-the-sun-and-a-new) _2026-08-18 · John Cook · John D. Cook Consulting Substack_ The mean distance, over time, between a planet and its star is approximately the semimajor axis of its orbit. ## [The imbalance theorem](https://www.johndcook.com/blog/2026/08/18/the-imbalance-theorem/) _2026-08-18 · John · John D. Cook_ The imbalance conjecture is now a theorem. James Alexander Schreib and Yousof Yavari posted a proof last week. What does the conjecture theorem say? Start with a graph G with no edge between two nodes of the same degree. Then for every edge, calculate the absolute value of the difference of the degree of each end. \[…\] The post The imbalance theorem first appeared on John D. Cook . ## [The imbalance theorem](https://www.johndcook.com/blog/2026/08/18/the-imbalance-theorem/) _2026-08-18 · John · John D. Cook_ The imbalance conjecture is now a theorem. James Alexander Schreib and Yousof Yavari posted a proof last week. What does the conjecture theorem say? Start with a graph G with no edge between two nodes of the same degree. Then for every edge, calculate the absolute value of the difference of the degree of each end. \[…\] The post The imbalance theorem first appeared on John D. Cook . ## [Mean distance to the sun](https://www.johndcook.com/blog/2026/08/18/mean-distance-to-the-sun/) _2026-08-18 · John · John D. Cook_ Suppose you have a planet in an elliptical orbit around a star. The math is identical for any light object orbiting a heavy object, such as a moon or satellite orbiting a planet, but we’ll call the heavy object a star and the light object a planet. The center of the star is not quite \[…\] The post Mean distance to the sun first appeared on John D. Cook . ## [Mean distance to the sun](https://www.johndcook.com/blog/2026/08/18/mean-distance-to-the-sun/) _2026-08-18 · John · John D. Cook_ Suppose you have a planet in an elliptical orbit around a star. The math is identical for any light object orbiting a heavy object, such as a moon or satellite orbiting a planet, but we’ll call the heavy object a star and the light object a planet. The center of the star is not quite \[…\] The post Mean distance to the sun first appeared on John D. Cook . ## [Proportion of 1s in a Hadamard matrix](https://www.johndcook.com/blog/2026/08/16/proportion-of-1s-in-a-hadamard-matrix/) _2026-08-17 · John · John D. Cook_ The first post in the recent series of posts on Hadamard matrices describes a way of constructing new Hadamard matrices from two other Hadamard matrices by taking their Kronecker product. Starting with a Hadamard matrix H0 and a Hadamard matrix G, you can construct a sequence of Hadamard matrices by Hn+1 = G ⊗ Hn for \[…\] The post Proportion of 1s in a Hadamard matrix first appeared on John D. Cook… ## [Ship software peacefully (Sponsored)](https://crawlproof.com/a/rIynZdtcv9Cy) _2026-08-17 · **Sponsored**_ Deploy apps from your repo with auto-config, instant previews, scaling, and monitoring. ## [Compressing Matrices and Correcting Codes](https://johndcookconsulting.substack.com/p/compressing-matrices-and-correcting) _2026-08-15 · John Cook · John D. Cook Consulting Substack_ Hadamard matrices are very special matrices, so you’d expect that they could be determined by less data than the full matrix. ## [Probability of correcting errors](https://www.johndcook.com/blog/2026/08/15/probability-of-correcting-errors/) _2026-08-15 · John · John D. Cook_ Error correcting codes are most simply described in terms of the errors they can certainly correct. For example, the Hadamard code used for the Mariner 9 probe to Mars encoded each 6-bit pixel to a 32-bit codeword in such a way that the original pixel could be recovered if no more than 7 bits were \[ \] The post Probability of correcting errors first appeared on John D. Cook . ## [Probability of correcting errors](https://www.johndcook.com/blog/2026/08/15/probability-of-correcting-errors/) _2026-08-15 · John · John D. Cook_ Error correcting codes are most simply described in terms of the errors they can certainly correct. For example, the Hadamard code used for the Mariner 9 probe to Mars encoded each 6-bit pixel to a 32-bit codeword in such a way that the original pixel could be recovered if no more than 7 bits were \[ \] The post Probability of correcting errors first appeared on John D. Cook . ## [Compressing a Hadamard matrix](https://www.johndcook.com/blog/2026/08/15/compressing-a-hadamard-matrix/) _2026-08-15 · John · John D. Cook_ Hadamard matrices are in the news following the recent announcement of a newly discovered Hadamard matrix. I ve written three posts on Hadamard matrices recently, one as a sort of introduction and two on applications: the error correcting code used in the Mariner 9 probe and constructing sphere packings. A Hadamard matrix is an orthogonal matrix \[ \] The post Compressing a Hadamard matrix first… ## [Compressing a Hadamard matrix](https://www.johndcook.com/blog/2026/08/15/compressing-a-hadamard-matrix/) _2026-08-15 · John · John D. Cook_ Hadamard matrices are in the news following the recent announcement of a newly discovered Hadamard matrix. I ve written three posts on Hadamard matrices recently, one as a sort of introduction and two on applications: the error correcting code used in the Mariner 9 probe and constructing sphere packings. A Hadamard matrix is an orthogonal matrix \[ \] The post Compressing a Hadamard matrix first… ## [Applications of Hadamard Matrices](https://johndcookconsulting.substack.com/p/applications-of-hadamard-matrices) _2026-08-14 · John Cook · John D. Cook Consulting Substack_ This morning I wrote a post, mentioned in the previous letter, about constructing Hadamard matrices. ## [Hadamard Codes and Sphere Packing](https://www.johndcook.com/blog/2026/08/13/hadamard-sphere-packing/) _2026-08-14 · John · John D. Cook_ Yesterday Levent Alpöge announced that he and his colleagues had discovered a new Hadamard matrix using Claude AI. That motivated a post I wrote this morning on how to construct Hadamard matrices. I mentioned in that post that these matrices arise in applications. This evening I gave an example, describing how NASA used a Hadamard \[ \] The post Hadamard Codes and Sphere Packing first appeared on… ## [How NASA’s Mariner 9 probe encoded images](https://www.johndcook.com/blog/2026/08/13/mariner-hadamard/) _2026-08-13 · John · John D. Cook_ NASA set Mariner 9 to photograph Mars in 1971. The images had to be encoded for transmission using an error-correcting code, otherwise they would be significantly corrupted when they were received on Earth. The images were encoded for transmission using a code based on Hadamard matrices, specifically a (32, 6, 16) Hadamard code. This means \[ \] The post How NASA’s Mariner 9 probe encoded images… ## [Dogs, Shells, Retro Crypto, etc.](https://johndcookconsulting.substack.com/p/dogs-shells-retro-crypto-etc) _2026-08-13 · John Cook · John D. Cook Consulting Substack_ I’ve written several unrelated posts since the previous newsletter. ## [Constructing Hadamard matrices](https://www.johndcook.com/blog/2026/08/13/constructing-hadamard-matrices/) _2026-08-13 · John · John D. Cook_ A Hadamard matrix is an orthogonal matrix whose entries are all either 1 or − 1. For example is a Hadamard matrix of order 2. True to Stigler s law of eponymy, James Joseph Sylvester investigated Hadamard matrices before Jacques Hadamard. Sylvester saw how to bootstrap the example above into more examples. If H is a \[ \] The post Constructing Hadamard matrices first appeared on John D. Cook . ## [Unlimited Mobile Internet, One V-SIM (Sponsored)](https://crawlproof.com/a/8Kc2v00KiYID) _2026-08-13 · **Sponsored**_ Automatically switches across T-Mobile, AT&T and Verizon; no throttling, no contract. ## [Constructing Hadamard matrices](https://www.johndcook.com/blog/2026/08/13/constructing-hadamard-matrices/) _2026-08-13 · John · John D. Cook_ A Hadamard matrix is an orthogonal matrix whose entries are all either 1 or − 1. For example is a Hadamard matrix of order 2. True to Stigler s law of eponymy, James Joseph Sylvester investigated Hadamard matrices before Jacques Hadamard. Sylvester saw how to bootstrap the example above into more examples. If H is a \[ \] The post Constructing Hadamard matrices first appeared on John D. Cook . ## [Cryptic but consistent](https://www.johndcook.com/blog/2026/08/12/cryptic-but-consistent/) _2026-08-12 · John · John D. Cook_ Suppose you ve never worked at the command line and you re reading a book about the bash shell. You read that !$ is a shortcut to refer to the last word of the previous command. That little fact will almost certainly not stick in your head for a couple reasons. First, you probably see no need \[ \] The post Cryptic but consistent first appeared on John D. Cook . ## [Cryptic but consistent](https://www.johndcook.com/blog/2026/08/12/cryptic-but-consistent/) _2026-08-12 · John · John D. Cook_ Suppose you ve never worked at the command line and you re reading a book about the bash shell. You read that !$ is a shortcut to refer to the last word of the previous command. That little fact will almost certainly not stick in your head for a couple reasons. First, you probably see no need \[ \] The post Cryptic but consistent first appeared on John D. Cook . ## [Dogs and fat tails](https://www.johndcook.com/blog/2026/08/11/dogs-and-fat-tails/) _2026-08-11 · John · John D. Cook_ I was reading a blog post on boat names because it was on Hacker News this morning. The post contained a link to a data set on dog names in NYC and I poked around the data a little. The top names were not at all what I expected, but then again this is limited \[ \] The post Dogs and fat tails first appeared on John D. Cook . ## [Dogs and fat tails](https://www.johndcook.com/blog/2026/08/11/dogs-and-fat-tails/) _2026-08-11 · John · John D. Cook_ I was reading a blog post on boat names because it was on Hacker News this morning. The post contained a link to a data set on dog names in NYC and I poked around the data a little. The top names were not at all what I expected, but then again this is limited \[ \] The post Dogs and fat tails first appeared on John D. Cook . ## [Manually unbreakable cryptography](https://www.johndcook.com/blog/2026/08/11/manually-unbreakable-cryptography/) _2026-08-11 · John · John D. Cook_ Suppose you were able to go back in time, to an era before computers, and give someone contemporary cryptography. Encryption methods that are essentially unbreakable now would certainly be unbreakable then. But there s a catch: not only do attackers not have computers, neither do users. Manual cryptography If you told someone about RSA encryption, for \[ \] The post Manually unbreakable cryptography… ## [Manually unbreakable cryptography](https://www.johndcook.com/blog/2026/08/11/manually-unbreakable-cryptography/) _2026-08-11 · John · John D. Cook_ Suppose you were able to go back in time, to an era before computers, and give someone contemporary cryptography. Encryption methods that are essentially unbreakable now would certainly be unbreakable then. But there s a catch: not only do attackers not have computers, neither do users. Manual cryptography If you told someone about RSA encryption, for \[ \] The post Manually unbreakable cryptography… ## [Learning from historical mistakes](https://www.johndcook.com/blog/2026/08/10/learning-from-historical-mistakes/) _2026-08-10 · John · John D. Cook_ The following extraordinary paragraph comes from Knuth s TAOCP Volume 4A, right before the last set of exercises. Many of the exercises below ask a modern reader to find and/or to correct errors in the literature of bygone days. The point is not to gloat over how smart we are in the 21st century; the point \[ \] The post Learning from historical mistakes first appeared on John D. Cook . ## [Learning from historical mistakes](https://www.johndcook.com/blog/2026/08/10/learning-from-historical-mistakes/) _2026-08-10 · John · John D. Cook_ The following extraordinary paragraph comes from Knuth s TAOCP Volume 4A, right before the last set of exercises. Many of the exercises below ask a modern reader to find and/or to correct errors in the literature of bygone days. The point is not to gloat over how smart we are in the 21st century; the point \[ \] The post Learning from historical mistakes first appeared on John D. Cook . ## [Inverse differential equations](https://www.johndcook.com/blog/2026/08/10/inverse-differential-equations/) _2026-08-10 · John · John D. Cook_ In science and engineering classes, you might describe a system using Newton s laws and end up with a differential equation. You then solve the differential equation, analytically or numerically, to see how the solutions behave. You might also do the opposite, especially in a mathematics class: look at what differential equation a set of functions \[ \] The post Inverse differential equations first… ## [Collect Open Source Legends (Sponsored)](https://crawlproof.com/a/WEXtgt3ELwxU) _2026-08-10 · **Sponsored**_ Open-licensed art, limited foil trading packs, and optional on-chain mints. ## [Inverse differential equations](https://www.johndcook.com/blog/2026/08/10/inverse-differential-equations/) _2026-08-10 · John · John D. Cook_ In science and engineering classes, you might describe a system using Newton s laws and end up with a differential equation. You then solve the differential equation, analytically or numerically, to see how the solutions behave. You might also do the opposite, especially in a mathematics class: look at what differential equation a set of functions \[ \] The post Inverse differential equations first… ## [DNA, Range Reduction, and Apostrophes](https://johndcookconsulting.substack.com/p/dna-range-reduction-and-apostrophes) _2026-08-09 · John Cook · John D. Cook Consulting Substack_ The latest post is a brief note about how Bessel functions where used in discovering the structure of DNA. ## [DNA and Bessel functions](https://www.johndcook.com/blog/2026/08/09/dna-and-bessel-functions/) _2026-08-09 · John · John D. Cook_ I was reading a book on the history of the discovery of the structure of DNA \[1\] and was surprised by a few passing references to Bessel functions. According to Claude, When X-rays are diffracted by a helical structure, the resulting diffraction pattern breaks into a series of horizontal layer lines. Cochran, Crick, and Vand \[ \] The post DNA and Bessel functions first appeared on John D. Cook . ## [DNA and Bessel functions](https://www.johndcook.com/blog/2026/08/09/dna-and-bessel-functions/) _2026-08-09 · John · John D. Cook_ I was reading a book on the history of the discovery of the structure of DNA \[1\] and was surprised by a few passing references to Bessel functions. According to Claude, When X-rays are diffracted by a helical structure, the resulting diffraction pattern breaks into a series of horizontal layer lines. Cochran, Crick, and Vand \[ \] The post DNA and Bessel functions first appeared on John D. Cook . ## [A simple range reduction method](https://www.johndcook.com/blog/2026/08/09/simple-range-reduction/) _2026-08-09 · John · John D. Cook_ At the end of my post on how not to calculate cosine I said that the first step in calculating cosine, particularly cosine of a large number, would be to do range reduction. This post will present a simple range reduction method by Cody and Waite that is adequate for moderately large arguments. If you \[ \] The post A simple range reduction method first appeared on John D. Cook . ## [Corrupted apostrophes](https://www.johndcook.com/blog/2026/08/07/corrupted-apostrophes/) _2026-08-08 · John · John D. Cook_ I have a program that shares files between my laptop and my phone. It works well, except for apostrophes. When I type an apostrophe ' on my laptop, it becomes ’ on my phone. And when I type 's on my phone, it becomes 痴 on my laptop. Apparently the phone turns the apostrophe (U+0027) \[ \] The post Corrupted apostrophes first appeared on John D. Cook . ## [Corrupted apostrophes](https://www.johndcook.com/blog/2026/08/07/corrupted-apostrophes/) _2026-08-08 · John · John D. Cook_ I have a program that shares files between my laptop and my phone. It works well, except for apostrophes. When I type an apostrophe ' on my laptop, it becomes ’ on my phone. And when I type 's on my phone, it becomes 痴 on my laptop. Apparently the phone turns the apostrophe (U+0027) \[ \] The post Corrupted apostrophes first appeared on John D. Cook . ## [How not to calculate cosine](https://www.johndcook.com/blog/2026/08/07/how-not-to-calculate-cos/) _2026-08-07 · John · John D. Cook_ Calculus professors with no experience in numerical computing will tell students that computers calculate trig functions with power series. They don t. I worked on the implementation of trig functions in hardware, and I can assure you we didn t just use power series. Power series are an excellent way to calculate functions near the center of \[ \] The post How not to calculate cosine first appeared… ## [How not to calculate cosine](https://www.johndcook.com/blog/2026/08/07/how-not-to-calculate-cos/) _2026-08-07 · John · John D. Cook_ Calculus professors with no experience in numerical computing will tell students that computers calculate trig functions with power series. They don t. I worked on the implementation of trig functions in hardware, and I can assure you we didn t just use power series. Power series are an excellent way to calculate functions near the center of \[ \] The post How not to calculate cosine first appeared… ## [Math with big numbers](https://johndcookconsulting.substack.com/p/math-with-big-numbers) _2026-08-07 · John Cook · John D. Cook Consulting Substack_ My three latest post all deal with doing math on numbers too large to represent as a standard floating point number. ## [cos(200!)](https://www.johndcook.com/blog/2026/08/07/cos200/) _2026-08-07 · John · John D. Cook_ In a footnote to the previous post, I said that Python s math library can calculate the logarithm of extremely large numbers but not the cosine. This post will expand on that comment. In this post I ll use n = 200! as my example rather than 1000! because this value of N is larger than the \[ \] The post cos(200!) first appeared on John D. Cook . ## [cos(200!)](https://www.johndcook.com/blog/2026/08/07/cos200/) _2026-08-07 · John · John D. Cook_ In a footnote to the previous post, I said that Python s math library can calculate the logarithm of extremely large numbers but not the cosine. This post will expand on that comment. In this post I ll use n = 200! as my example rather than 1000! because this value of N is larger than the \[ \] The post cos(200!) first appeared on John D. Cook . ## [Calculating log(1000!)](https://www.johndcook.com/blog/2026/08/06/log1000/) _2026-08-06 · John · John D. Cook_ The previous post pointed out that the following code such as the following unexpectedly works. from math import log, factorial log(factorial(1000)) 5912.128178488163 If you don t find this unexpected, note that if you replace math.log with numpy.log the code will fail \[1\]. Functions like natural logarithm operate on real numbers. Real numbers are represented \[ \] The post Calculating log(1000!)… ## [Calculating log(1000!)](https://www.johndcook.com/blog/2026/08/06/log1000/) _2026-08-06 · John · John D. Cook_ The previous post pointed out that the following code such as the following unexpectedly works. from math import log, factorial log(factorial(1000)) 5912.128178488163 If you don t find this unexpected, note that if you replace math.log with numpy.log the code will fail \[1\]. Functions like natural logarithm operate on real numbers. Real numbers are represented \[ \] The post Calculating log(1000!)… ## [The code that didn’t break](https://www.johndcook.com/blog/2026/08/05/math-log/) _2026-08-05 · John · John D. Cook_ Last week I wrote a post on hiding cryptographic keys in decks of cards. I wrote some code for that post that shouldn t work, but before fixing I noticed that it in fact did work. The code computes logarithms for integers larger than the largest representable float. For example, the largest float is on the \[ \] The post The code that didn’t break first appeared on John D. Cook . ## [Enumerating trees and circles](https://www.johndcook.com/blog/2026/08/05/enumerating-trees-and-circles/) _2026-08-05 · John · John D. Cook_ A few days ago I wrote a post on counting rooted trees. That post looked at the sequence c(n) which counts the number of rooted trees with n nodes. Here one node is distinguished as the root, but the nodes below the root are not distinguished from each other; all that matters is how the \[ \] The post Enumerating trees and circles first appeared on John D. Cook . ## [Enumerating trees and circles](https://www.johndcook.com/blog/2026/08/05/enumerating-trees-and-circles/) _2026-08-05 · John · John D. Cook_ A few days ago I wrote a post on counting rooted trees. That post looked at the sequence c(n) which counts the number of rooted trees with n nodes. Here one node is distinguished as the root, but the nodes below the root are not distinguished from each other; all that matters is how the \[ \] The post Enumerating trees and circles first appeared on John D. Cook . ## [Mathematical Alchemy](https://johndcookconsulting.substack.com/p/mathematical-alchemy) _2026-08-04 · John Cook · John D. Cook Consulting Substack_ My three latest posts have been about alchemy, sorta. ## [Mathematical alchemy](https://www.johndcook.com/blog/2026/08/04/metallic-alchemy/) _2026-08-04 · John · John D. Cook_ After writing the previous post about metallic ratios, I thought about the analogy to alchemy and the attempt to make precious metals out of base metals. When can you make one metallic ratio out of another? Can you make the golden ratio out of the lead ratio? Before we can make gold out of lead, \[ \] The post Mathematical alchemy first appeared on John D. Cook . ## [Mathematical alchemy](https://www.johndcook.com/blog/2026/08/04/metallic-alchemy/) _2026-08-04 · John · John D. Cook_ After writing the previous post about metallic ratios, I thought about the analogy to alchemy and the attempt to make precious metals out of base metals. When can you make one metallic ratio out of another? Can you make the golden ratio out of the lead ratio? Before we can make gold out of lead, \[ \] The post Mathematical alchemy first appeared on John D. Cook .