You can represent octonions as 2 by 2 matrices with quaternion entries. But you have to change the definition of matrix multiplication slightly to make it work.

John D. Cook Consulting Blog

26.05.2025

Looking for keys under the lamppost

What drunks and technical consultants have in common.

John D. Cook Consulting Blog

25.05.2025

My favorite paper: H = W

The best paper tile, in my opinion, was H = W. This post unpacks what the title means.

John D. Cook Consulting Blog

25.05.2025

Structured frameworks for complex systems

Sometimes an LLM makes creative connections. Maybe not entirely sensical, but interesting nevertheless.

John D. Cook Consulting Blog

25.05.2025

Dungeons, Dragons, and Numbers

Riffing on a joke by Dan Piponi giving examples of numbers in the D&D alignment matrix.

John D. Cook Consulting Blog

23.05.2025

Wilkinson's polynomial | Dependence of roots on coefficients

Wilkinson's polynomial demonstrates that a tiny change to a coefficient in a polynomial might result in a huge change to the roots.

John D. Cook Consulting Blog

22.05.2025

Effective graph resistance

Computing effective graph resistance is an application of the Moore-Penrose pseudoinverse.

John D. Cook Consulting Blog

22.05.2025

Interpolation instability

When interpolating a function at evenly spaced points, adding more interpolation points can make the accuracy worse, in practice even when not in theory.

John D. Cook Consulting Blog

22.05.2025

Drazin pseudoinverse of a matrix

The Drazin pseudoinverse is not as well known as the Moore-Penrose pseudoinverse. The Drazin pseudoinverse behaves nicely with exponents.

John D. Cook Consulting Blog

21.05.2025

Multiplying a matrix by its transpose

Partitioning a matrix to make multiplication by its transpose more efficient. Link to a new algorithm.

John D. Cook Consulting Blog

21.05.2025

A bit-twiddling marvel

A very clever, very cryptic hack.

John D. Cook Consulting Blog

20.05.2025

Probability of rolling a Yahtzee

Probability of rolling a Yahtzee in no more than n rolls.

John D. Cook Consulting Blog

20.05.2025

Partitioned matrices | What works and what doesn't

You can partition a matrix and think of it as a matrix whose entries are matrices. You can manipulate these matrices almost as if the blocks were numbers.

John D. Cook Consulting Blog

16.05.2025

Trinomial Coefficients and Kings

There are two meanings of the term trinomial coefficients. One of these is related to the movement of a king on a chessboard.

John D. Cook Consulting Blog

16.05.2025

Riff on an integration bee integral

Using an integral from a competition quiz as a springboard to discuss related ideas.

John D. Cook Consulting Blog

15.05.2025

Multiplying by quaternions on the left and right

Matrix representations for multiplying by a quaternion on the left and on the right.

John D. Cook Consulting Blog

15.05.2025

Another little chess puzzle

Another chess puzzle by Martin Gardner. Place two rooks, two bishops, and two knights on a 4 by 4 board so that no piece attacks any other.

John D. Cook Consulting Blog

15.05.2025

Embeddings, projections, adjoints, and pseduoinverses

Embeddings and projections are not inverses of each other, but they're like inverses. Three generalizations of inverse: left and right, adjoint, & pseudoinverse

John D. Cook Consulting Blog

14.05.2025

Alternative exp and log notation

Alternative notation for exponents and logs using up and down arrows has several advantages. Mathematical symmetries come out as visual symmetries.

John D. Cook Consulting Blog

14.05.2025

Decimal Separator and Internationalization

Avoiding problems related to different conventions for representing decimals and separating thousands.

John D. Cook Consulting Blog

13.05.2025

A crowded little chess puzzle

A chess puzzle by Martin Garnder and links to similar articles.

John D. Cook Consulting Blog

12.05.2025

Special solutions to the eight queens problem

There are 12 equivalence classes of solutions to the 8 queens problem. One class is highly symmetric. One is a solution to the no-three-in-line problem.

John D. Cook Consulting Blog

12.05.2025

Formulating eight queens as a SAT problem

How to turn the eight queens problem into a Boolean satisfiability (SAT) problem.

John D. Cook Consulting Blog

10.05.2025

The non-attacking bishops problem

How many bishops can you place on a chessboard so that no bishop is attacking any other bishop?

John D. Cook Consulting Blog

10.05.2025

Sorting Roman numerals

Alphabetical order and numerical order match surprisingly well for moderately small numbers.

John D. Cook Consulting Blog

10.05.2025

Frequency of names of English monarchs

Frequency of names of English monarchs, from the most common (Edward) to the least (John).

John D. Cook Consulting Blog

09.05.2025

Why do LLMs have emergent properties?

Large language models show unexpected "emergent" behaviors as they are scaled up. This should not at all be surprising. Here we give a possible explanation.

John D. Cook Consulting Blog

09.05.2025

Square root of a small number

Why might it be a problem if a numerical algorithm takes the square root of a small number? Why the most obvious fix is insufficient.

John D. Cook Consulting Blog

09.05.2025

Frequency of papal names

Frequency of names for popes. The most popular has been John, used 21 times, even though there was a John XXIII.

John D. Cook Consulting Blog

08.05.2025

Composing rotations using quaternions

Rotations can be represented by matrices or by quaternions. An advantage of quaternions is that the composition of rotations is very simple.

John D. Cook Consulting Blog

08.05.2025

Converting between quaternions and rotation matrices

Equations and Python code for going back and forth between quaternion and matrix representations of rotations.

John D. Cook Consulting Blog

07.05.2025

A simple way to generate random points on a sphere

A way of generating points on a sphere that is intuitively clear and efficient. Comparison to the more common algorithm.

home›John D. Cook Consulting Blog›Science

John D. Cook Consulting Blog - Science

Fitting a parabola to an ellipse and vice versa

Gardener's ellipse | nails and string method of drawing

The ellipse hidden inside Pascal’s triangle

Stacking positive and negative cannonballs

How Mathematica Draws a Dragonfly

How often to the hands of a clock line up?

Naive version of the Lucas-Lehmer test for Mersenne primes

Octonions sometimes associate

Representing octonions as matrices, sorta

Looking for keys under the lamppost

My favorite paper: H = W

Structured frameworks for complex systems

Dungeons, Dragons, and Numbers

Wilkinson's polynomial | Dependence of roots on coefficients

Effective graph resistance

Interpolation instability

Drazin pseudoinverse of a matrix

Multiplying a matrix by its transpose

A bit-twiddling marvel

Probability of rolling a Yahtzee

Partitioned matrices | What works and what doesn't

Trinomial Coefficients and Kings

Riff on an integration bee integral

Multiplying by quaternions on the left and right

Another little chess puzzle

Embeddings, projections, adjoints, and pseduoinverses

Alternative exp and log notation

Decimal Separator and Internationalization

A crowded little chess puzzle

Special solutions to the eight queens problem

Formulating eight queens as a SAT problem

The non-attacking bishops problem

Sorting Roman numerals

Frequency of names of English monarchs

Why do LLMs have emergent properties?

Square root of a small number

Frequency of papal names

Composing rotations using quaternions

Converting between quaternions and rotation matrices

A simple way to generate random points on a sphere

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