iiinova — where your digital self keeps evolving

Justin Lanier @jlanier.bsky.social · 19/06/2026

Hi! In July I'm once again running a free online program for K-12 math teachers called "21st century mathematics". Its goal is to help teachers to connect math they teach with some math that has been discovered in the 21st century. Pass it along! Info+register: justinlanier.org/21st-century...

A 4x4 puzzle grid, indicating an example of an alternating sign matrix; an example of a configuration of 21 points that maximized the number of pairs that are a unit distance apart. These are the two mathematical themes of the 2026 program.

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Justin Lanier @jlanier.bsky.social · 17/06/2026

Something that is then fun is that these same ideas will extend to larger grids!

4 by 4, 5 by 5, and 6 by 6 grids of the counting numbers from 1 to n squared.

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Justin Lanier @jlanier.bsky.social · 19/07/2025

Independent of thinking about AI, Terry Tao himself has an interesting and encouraging perspective regarding that first question:

from “Does one have to be a genius to do maths?” by Terry Tao:
"The answer is an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”.
...
The number of interesting mathematical research areas and problems to work on is vast – far more than can be covered in detail just by the “best” mathematicians, and sometimes the set of tools or ideas that you have will find something that other good mathematicians have overlooked, especially given that even the greatest mathematicians still have weaknesses in some aspects of mathematical research."

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Justin Lanier @jlanier.bsky.social · 27/06/2025

Simon Stevin, what a hero. mathshistory.st-andrews.ac.uk/HistTopics/R...

Stevin made a number of other important advances in the study of the real numbers. He argued strongly in L'Arithmetique (1585) that all numbers such as square roots, irrational numbers, surds, negative numbers etc should all be treated as numbers and not distinguished as being different in nature. He wrote:-

"Thesis 1: That unity is a number.

Thesis 2: That any given numbers can be square, cubes, fourth powers etc.

Thesis 3: That any given root is a number.

Thesis 4: That there are no absurd, irrational, irregular, inexplicable or surd numbers.

It is a very common thing amongst authors of arithmetics to treat numbers like √8 and similar ones, which they call absurd, irrational, irregular, inexplicable or surds etc and which we deny to be the case for number which turns up."

His first thesis was to argue against the Greek idea that 1 is not a number but a unit and the numbers 2, 3, 4, ... were composed of units. The other three theses were encouraging people to treat different types of numbers, which were at that time treated separately, as a single entity - namely a number.

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Justin Lanier @jlanier.bsky.social · 27/06/2025

TIL that we narrowly missing lugging around not only the monikers "imaginary", "negative", "complex", and "irrational" for numbers, but also "fictitious limits". Real glad for that. 😅 from MacTutor's "The real numbers: Stevin to Hilbert"

Two years after the publication of Hankel's monograph, Méray published "Remarques sur la nature des quantités" in which he considered Cauchy sequences of rational numbers which, if they did not converge to a rational limit, had what he called a "fictitious limit". He then considered the real numbers to consist of the rational numbers and his fictitious limits.

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Justin Lanier @jlanier.bsky.social · 23/06/2025

There are lots of chances for teachers to work on problems collaboratively, encounter new math ideas, and grow as teachers and mathematicians in community. We also have a great lineup of guest mathematicians who will visit us and share about their work.

Photographs of eight mathematicians: Dr. Frank Morgan, Eve Parrott, Dr. Maia Karpovich, Dr. Emanuel Milman, Dr. Joe Neeman, Dr. Parker Duncan, Rory O'Dwyer, Dr. Eviatar Procaccia, Harun Khan, and Dr. Bernadette Faye.

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Justin Lanier @jlanier.bsky.social · 23/06/2025

We are two weeks out from the start of "21st century mathematics", a program I run for K-12 math teachers. (Register now!) The program helps teachers connect the math they teach with some math that has been discovered in the 21st century. The program is flexible, free, & online.

A picture of a double bubble. A picture of the first 18 Fibonacci numbers, with the ones that are perfect powers highlighted.

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Justin Lanier @jlanier.bsky.social · 20/06/2025

I didn't know The Matrix was filmed in Sydney! But I saw this marker on Google Maps when I was investigating your "Where's Nalini?" prompt.

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Justin Lanier @jlanier.bsky.social · 06/05/2025

www.nytimes.com/2025/05/05/u...

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Justin Lanier @jlanier.bsky.social · 03/05/2025

No voting in Australian elections for me, but I still showed up for the democracy sausage. And bake sale!

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Justin Lanier @jlanier.bsky.social · 28/04/2025

Ten squares. The first example has perimeter 14. The second one has perimeter 22. Can anyone find ones with smaller or bigger perimeters? Or other examples that match these? Intrigued? Check out "21st century mathematics", an enrichment program for teachers: justinlanier.org/21st-century...

Ten squares arranged in two different ways: a 2x3 rectangle with a 2x2 square stuck to its side, and one that goes like this:
00X00
0XXX0
XX0XX
X000X

where the X's are the squares and the 0'x are empty space.

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Justin Lanier @jlanier.bsky.social · 23/04/2025

The program webpage has all the details (and the full video with captions): bit.ly/21math2025 Video part 2:

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Justin Lanier @jlanier.bsky.social · 23/04/2025

“21st century mathematics”: a free month-long online enrichment program for math teachers. Info + register: bit.ly/21math2025 Video part 1:

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Justin Lanier @jlanier.bsky.social · 08/03/2025

Find the derivative of g(x) = x^(1 / ln x) in two different ways.

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Justin Lanier @jlanier.bsky.social · 10/01/2025

As a happy coincidence, I ran across a kindred sentiment while reading Felix Klein's “Erlanger Antrittsrede” (1872) today: www.sciencedirect.com/science/arti...

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Justin Lanier @jlanier.bsky.social · 09/01/2025

"If there are at least 7 people in a room, can they always break up into groups of 3 or 4? Explain why they can, or give an example of a number of people where this is not possible."

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