Leon P Smith

https://ioc.exchange/@leon_p_smith

Communications engineer and mathematician. Longtime functional programming and Haskell enthusiast, occasional Schemer. Inventor of corecursive queues, postgresql-simple, an aggregate theory of concrete mathematics, and self-documenting cryptography. Currently aspiring to become an Epistemic Frame Engineer.

Pronouns: he/him

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Last Notes

2026-08-23 11:47:28 UTC

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As Tad Stoermer explains, in this context, for an external threat, think of an unpopular foreign invader. The resistance movement is "simple" because the population has no difficulty identifying the abuser as a legitimate threat that needs a response.
Threats that develop internally, such as a politician who grows into a dictator, an abusive family member, abusive boss, or abusive coworker, attempting to point out the facts to other involved parties generally is not going to go well. The resistance is "complex" because it will be difficult to impossible to generate any kind of coordinated action against the abuse, because everybody will be debating whether or not it's really happening or not, or if you are just being overly dramatic.
Personally, my estranged biological brother and Donald Trump remind me a lot of each other. My dad reminds me a lot of the current Democratic leadership: he refuses to recognize that the issue is his problem too. It's easier and more convenient for him to blame me for the way my brother has treated me, than to live in reality.
It's related to "insider threats" in the IT sense: actually, in an act of physical mathematics, I came up with a scheme to tweak the password hashing process to force certain bad actors including internal threats to be slightly more "honest", for some value of "honest".
If mathstodon were to deploy this idea, then all passwords would be automatically suffixed with a phrase akin to "this password is for mathstodon.xyz, please report abuse to ..." or whatever.
Then when the password database is stolen, the password cracker must incorporate that phrase as part of their guess, otherwise their guess is wrong (even if the cracker guessed the correct password)
If on the other hand, the password cracker is guessing the magic incantation acknowledging the source of the password hash, then the cracker has that phrase in its possession, somehow, which is evidence that might lead to a report of the password leak back to the abuse reporting line.
I originally came to this methodology by thinking of it like a transmission medium. Of course this virtual "transmission medium" is purely game theoretic and has no underlying physical mechanism: rather it's attempting to impose a topological constraint on physical messaging networks, a theory of mind viruses if you will.
But of course the culture that IT has around crypto is beyond dysfunctional, so getting anything actually fixed is usually impossible, and difficult in the best of circumstances.

2026-08-22 11:53:58 UTC

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That reminds me of Tad Stoermer's recent explanation of simple resistance against external abuse, versus complex resistance against internal abuse.
Personally, I seem to be have cursed with a cornucopia of internal threats throughout my life. Ugh.
https://www.youtube.com/watch?v=vqZXRzFBcz4

2026-06-01 15:11:13 UTC

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That's a good point; I mean, I don't want to oversell the difference, it's just that I personally think that making the Stern-Brocot tree the frontispiece makes somewhat more sense.
Both your paper and "Recounting the Rationals" have been very memorable, for sure, though I suspect I should take the time to do a closer re-read of them.
Basically, I think a high-level overview of an ideal elementary curriculum looks something like this:
1. In addition to more typical introductory topics, which should include boolean logic and computer programming, introduce students to Pascal's Triangle, the symmetry group of the square, and the Stern-Brocot trees. (The Calkin-Wilf tree could also be included here, but should be emphasized that it's related but different)
2. The first lessons on the Symmetry Group of the square should emphasize a purely intuitive understanding using the mechanical calculator in my constructive symmetry repo. Early lessons should also emphasize that this arithmetic is associative but not commutative, and connect the commutativity of multiplication to the fact that rotating a rectangle doesn't change its area.
https://github.com/constructive-symmetry/constructive-symmetry/tree/master/D002_Book_of_Algebra
3. Followup lessons should include a comprehensive overview of how to calculate the arithmetic of the symmetry group of the square via pencil and paper.
This includes at least two if not all three interesting variants of the semidirect product. The first leads to modular arithmetic and generalizes naturally into the symmetries of the regular polygons, aka the dihedral groups.
The second semidirect product connected to matrix arithmetic and naturally generalizes into the symmetry group of tesseracts and other higher-dimensional cube-like shapes.
The third semidirect product is very convenient for software bitsliced implementations: it's very easy to be able to add two vectors of 10 elements of D4 together in less than 10 instructions on any 32-bit CPU that supports standard shift and bitwise-logical operations, typically achieving a throughput of greater than one operation per clock cycle on nearly any modern CPU, even modern embedded microcontrollers!
(Of course, modern application processors typically support these bitwise operations on special registers that are 128 bits or longer, which support 42-element vector addition very efficiently. I'm not sure this is practically useful, but the concept of software bitslicing is interesting and undoubtedly useful in some practical contexts)
But I digress, other pencil-and-paper methods for calculating D4 include an introduction to permutations of four elements, and by the multiplication of 2x2 integer matrices.
It's interesting that the second semidirect product actually captures the essence of the implementation of D4 via 2x2 integer matrices.
4. Once matrix multiplication is on the table, the Stern-Brocot tree needs to be revisited, and then use matrices here too, to reveal SL(2,N). And yeah, it would seem a little weird to not have already
revealed the Calkin-Wilf tree by now, but if not it's obviously becoming a much higher priority.
5. Of course, at this point starting to discuss the modular group becomes a very appealing target.
6. Also, if one is interested in understanding the semidirect product as a general concept, this naturally leads to the concept of an automorphism of groups. I honestly think this can reasonably be introduced to older elementary and junior high kids by role playing being on a Star Trek-like science team who's goal is to decode alien heiroglyphs. (Aka, "solving" simple substitution ciphers for various finite groups, especially D4, Z4, and K4)
7. Once the concepts from both 5 and 6 are in play, it makes sense to point out that the general modular group GL(2,Z) is isomorphic to Aut(Z²), a.k.a.the symmetry group of the addition of pairs of integers.
8. Once modular arithmetic is in play, it's worth pointing out that modding SL(2,N) by 7 and then by {-1,1) gets you PSL(2,7), which is isomorphic to the automorphisms of the Fano plane.
I don't quite understand this yet: I understand why PSL(3,2) gets you the automorphisms of the Fano plane, and while I find PSL(2,7) an easy-enough group to count, I don't (yet!) understand the exceptional isomorphism between these groups.
9. Introducing a little bit of projective geometry and homogeneous coordinates once we start discussing direct linear relations of the form y=mx seems like a no-brainer, as that starts to explain why the Stern-Brocot tree is related to Diophantine approximation.
Of course, these remarks is basically from the point of view of the symmetry group of the square. I have a somewhat reasonable understanding of what I'd like to do with the Stern-Brocot tree, but only a very broad outline of what I'd like to do with Pascal's Triangle.
This seems to have a lot of the earlier material, though via the principles of iterative deepening, it's not like this sequence of major curriculum events should be focused on to the exclusion of other important threads of thought.
I hope you'd agree I've started to expose a rich interplay between the symmetry group of the square and the Stern-Brocot and Calkin-Wilf trees.
@npub1nf4…nqe4

2026-05-29 22:13:38 UTC

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I rediscovered the unified treatment of the two trees for myself in late 2021 or early 2022; honestly I'm not in the slightest bit surprised that somebody else had noticed.
The topograph formulation is new to me. I'm reminded a bit of "From Christoffel Words to Markoff Numbers" by Christophe Reutenauer, though I really don't understand very much from that book (yet?).
I was introduced to the Stern-Brocot and Calkin-Wilf trees via [@jer_gib](https://functional.cafe/@jer_gib ) et al's "Functional Pearl: Enumerating the Rationals" circa 2008. I wish I had the foresight to see a connection to linear algebra, but I got the connection between linear algebra and the Stern-Brocot tree from "Concrete Mathematics" in 2021.
I'm sure my appreciation for the Calkin-Wilf tree could be much better, but at least for the purposes of elementary education, I think the Stern-Brocot tree is a bit more naturally motivated, as it is a binary search tree.
Though I wasn't familiar with the paper you posted, Dr. Stange has had a big impact on my thinking! I am especially pleased with her video "Rethinking the Rationals". I was immediately amused by the easter egg of the Apollonian Gasket that's on the whiteboard on that video.
Incidentally, when I "discovered" the connection between the modular group and SL(2,N), I started learning about PSL(2,Z) and friends and found a number of intriguing comments you've made over the years, which guided some of my endeavors.
I also found your lecture "This Weeks Finds 14: The Three Strand Braid Group" extraordinarily fascinating. The first time I watched it, most of the lecture went over my head. I think I have a grasp on the broad outlines now, even if I don't (yet?) understand many of the details.
@npub1euc…fkvk

2026-05-28 21:40:34 UTC

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But yes, I have long loved the fact that the Stern-Brocot tree is a incredible, wonderful playground for discussing tree and graph traversals. That's part of the reason I wrote "Lloyd Allison's Corecursive Queue" nearly twenty years ago now.
I mean, at the time, I didn't understand many of the connections I listed in my previous post, but I still was enamored with it as an example.
https://themonadreader.wordpress.com/wp-content/uploads/2009/07/issue142.pdf

2026-05-28 21:36:34 UTC

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I have a philosophy of math education that believes that every child should be introduced to the symmetry group of the square, the Stern-Brocot Tree, and Pascal's Triangle as soon as vaguely practical.
I honestly believe that I could have taken on all of these concepts (in a _shallow_ way) as a five-year old, but I realize I'm not exactly normal in this regard.
Anyway, using these to discuss tree and graph traversals is definitely among the reasons I think introducing the Stern-Brocot Tree early is important!
In fact, applying breadth-first search to the Stern-Brocot tree provides an alternate proof of Cantor's equivalence between the natural numbers and the positive rationals.
A breadth-first search applied to a complete binary tree can be optimized away, and once you've done this, the alternative proof becomes much more efficient than the standard proof. For example, figuring out what the octillionth enumerated fraction is on the Stern-Brocot proof is fast and easy, whereas figuring out the octillionth enumerated fraction using the standard proof is doable, but is much slower and involves some complicated algorithms involving heavy-duty number theory.
The classic Farey Sequences are inorder traversals of finite subgraphs of the Stern-Brocot tree. As a depth-first search, inorder traversals don't work on infinite subgraphs: for example, a classic observation in analysis is that the smallest positive fraction does not exist, thus one cannot perform an inorder traversal of the full Stern-Brocot tree.
A family of Farey Sequences of increasing order is an example of an iteratively deepened depth-first search applied to the Stern-Brocot tree. The Franel-Landau theorem states that certain asymptotic behaviors of these sequences are equivalent to the Riemann Hypothesis.
Iterative deeping is actually the less-obvious but fourth new ingredient critical to my philosophy of math education: nearly anybody would learn more math if they employ iterative deepening as a learning and study strategy, than most any other heuristic.
Iterative deepening is almost always preferable to breadth-first search with a queue, as it requires much much less memory for any graph with an exponential number of nodes reachable after N hops.
Also, in theory iterative deepening is never much slower than breadth-first search in these sorts of contexts, but in practice iterative deepening is almost always much faster than breadth-first search due to the reduced memory requirements.

2025-08-06 02:47:38 UTC

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I was taught a little tiny bit of the tail end of new math, mostly set theory, which I did find useful.
But I've also been long interested in how to take new math and make it so much more.
I would love to see integrated computer programming and math, so that you use computer programming to learn about math, and use math to learn about computer programming.
I would love to see more coverage of heuristics, especially the importance of iterative deepening as a learning and study strategy, and to develop problem solving skills via practice.
Add to that the Stern-Brocot Tree, the Symmetry Group of the Square, and Pascal's Triangle, and I call the result "an Aggregate Theory of Concrete Mathematics".
https://github.com/constructive-symmetry/constructive-symmetry

2025-06-16 17:58:47 UTC

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I did recently come to appreciate at least one practical advantage of the point-free style, namely in the context of partial evaluation:
https://hackage-content.haskell.org/package/sha256-0.1.0.3/docs/Crypto-Sha256-Hkdf.html
And, I have to hand it to the Squiggolists for coming up with their lazy breadth-first numbering algorithm. That is a truly impressive bit of work.
But for more workaday applications, yeah I tend to agree. I mean, above I ended up
writing those cryptographic functions
in a relatively direct way, then manually
transformed them to the point-free style because it matters.
Maybe I don't appreciate other potential benefits of the point-free style well enough, but for more typical code I don't like it.
I prefer to write in a _relatively_ point-free style, mostly sticking to point-free idioms I commonly use because I find them exceptionally useful.

2025-06-05 00:11:32 UTC

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Fun fact, Sir Mix-a-lot likes to build high-power radios:
https://www.youtube.com/watch?v=6VpZOvADLW0

2025-03-12 02:22:12 UTC

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He's not particularly wealthy, though he's done better than I have. Men who act and behave like Trump is actually shockingly common, but they tend to limit their future at least somewhat more effectively than Trump has. My brother has undoubtedly gotten much futher than most who try this route, though he'll never be nearly as "successful" as Trump.
It's a matter of public record actually: I wrote a bunch of software at my brother's request that was used to answer E911 calls at Parke County Indiana from 2013 to 2018. He never paid me, then sued me claiming me as his "employee" that he admitted in court to never paying.
I consider that blackmail, and I refuse to have anything to do with him again unless he obtains a felony criminal conviction and serves multiple years in prison.
One of the subtexts of "Kevin Bacon and the Stern Brocot Tree" and my Aggregate Theory of Concrete Mathematics is that it was in part motivated by the existence of my estranged nephews, and in part calling out this pattern of behavior.
One of the complications is that my parents are Anabaptist and take an unnecessarily dim view of legal proceedings when they are necessary.
I actually grew up listening to Joan Baez, and her recordings continue to be one of my favorite parts of my parents music collection!

2025-03-12 01:51:14 UTC

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Well, there's the idea that depressions are where the biggest fortunes are made, even if it won't be realized until the rebound. The billionaires will be fine, it's probably the moderately wealthy retirees (like my parents) that'll be hardest hit.
... I am estranged from my biological brother. When Trump started running in 2015, I read so many articles analyzing his psychology. Nearly all of them you could have substituted Trump's name with my brothers and been left with a shockingly accurate article.
It really feels like my personal life foretold the national story, in the way it both parallels and preceded it. Unfortunately I have much less faith in even the vaguest sense of rational self-preservation on the count of a substantial subset of the human population, or the willingness of witnesses to bad behavior to remember long enough to hold anybody accountable.

2025-03-10 02:14:00 UTC

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admittedly I'm not an expert in graphics, X11, or Wayland, but my general impression is that Wayland's architecture has largely failed to significantly improve upon X11 in most meaningful ways, and has regressed in others.

2025-03-06 22:10:39 UTC

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No, the amount of visible light continues to rise as temperature goes up, even if _most_ of the emissions at the highest temperatures are not visible to the naked eye.
It is possible to verify that if you hold the frequency constant and increase the temperature of Planck's Law, power continues to increase:
https://en.wikipedia.org/wiki/Black-body_radiation

2025-01-25 07:44:28 UTC

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Ever considered working on radio stuff? There should be a lot of potential in building a new high-speed digital-mode radio protocol aimed at amateur radio and emergency response.
(Amateur radio tends to be very slow, partly by laws and regulation which have been relaxed. So high-speed is still very slow compared to wifi, though at much longer ranges)
My idea is something primarily aimed at 2m and 70cm, a modern APRS replacement with error correcting codes and cryptographic authentication. The aim should be to build a virtual bulletin board that is capable of running in a mesh mode, and that is capable of maintaining consensus as robustly as possible with minimal fuss.
So... lots of challenging distributed systems problems to think about, among other things.

2024-12-23 20:02:40 UTC

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I've been browsing "Squigonometry: The Study of Imperfect Circles" by Poodiack and Wood this weekend, which talks a lot about this. I know this is all intimately connected to the modular group and my Aggregate Theory of Concrete Mathematics, but I've not figured out many of the details yet.
...It's been a long time since I've really thought about calculus.
https://link.springer.com/book/10.1007/978-3-031-13783-9

2024-12-09 15:34:22 UTC

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I'm going to wait until NVME is available on the Pi 500

2024-11-18 19:21:00 UTC

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Incidentally I first noticed the group-like structure that bcrypt's key expansion exhibits, then @npub1nf4…nqe4 posted something where he actually used a specific noun for a grouplike structure (loop? don't exactly remember which one...), and that reminded me I should figure out which noun described bcrypt's key expansion.
Turns out it's a quasigroup, so later when [@jer_gib](https://functional.cafe/@jer_gib ) pointed out that averaging two numbers is an example of something commutative but not associative, I recalled the other binary operation I thought of when I first got deeply interested in applying the Symmetry Group of the Square to math pedagogy nearly 20 years ago.
Namely, the binary operation over a domain of 3 elements that, when provided two distinct elements, returns the element that is not among the inputs, and when provided two identical elements, returns that element, I recognized that as a quasigroup thanks to my experience with bcrypt and Jeremy's remark.
At Jeremy's suggestion, I decided to try to give it the name the "tetheral quasigroup", which also gave me enough breadcrumbs to piece together a "justification" for using it as an example in more basic, general-interest math courses:
https://github.com/constructive-symmetry/constructive-symmetry/commit/fecc52a59db54ba8913e0a6389ab4800a7177e38

2024-11-18 19:01:01 UTC

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I've been thinking about something that can be accurately described as adversarial literate programming for a few years now, specifically in the context of password hashing.
Alice gets to specify an algorithm with some associated documentation. Eve wants to allow Jack to run Alice's algorithm himself on his own hardware while denying Jack access to Alice's documentation.
The security goal is to either force Eve to give Jack a sporting chance of recovering Alice's documentation from Eve's implementation, or give up on allowing Jack to run Alice's algorithm for himself.
My idea is to include that documentation as a (often non-secret) part of the inputs to a cryptographic hash function. This game-theoretic transmission medium is purely mathematical in nature, thus much like a virus is dependent on other forms of life to complete its lifecycle, this is ultimately dependent on other transmission media backed by physical processes to actually deliver its message.
Thus this is a cryptographic mind virus. As these mind viruses cannot possibly be adversely relevant to one's interests unless one is doing something less than proper to Alice and others, that presumably makes this mind virus "woke". Thus adversarial literate programming employs a transmission medium of woke mind viruses.
The main design principle that in retrospect I followed was to take PBKDF2 and bcrypt, apply a continuation-passing-style transform on it, and then tweak things to maximize the number of repetitions of Alice's documentation, and maximize the utility of every continuation and every possible partial evaluation.
It's way more than a "small program whose purpose is mainly to explain an idea", but it is also that. So sorry @npub1yp3…un2n for hijacking your thread a bit.
https://github.com/auth-global/self-documenting-cryptography

2024-11-07 19:44:43 UTC

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Indeed, I agree completely. But part of Elon's argument is we have to colonize to Mars immediately, because 0.5/4.7 is uncomfortably close to the end, which is absurd nonsense.

2024-11-07 17:17:00 UTC

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I've been thinking a lot about something like this lately.
I've long wondered about the formation rate of technological civilizations within the universe... and how long they typically survive.
Often I've wondered, what if the formation rate is so low, and the self-extinction rate kinda awful, that on average there's only say, 0.1 technological civilizations active in any large galaxy that's reasonably hospitable to life like our Milky way? And many of them don't get all that far with technology?
I mean, that's kinda been my gut instinct for a while, though I've not delved into it very carefully. Still, it does suggest that we really should try our damnedest to stay alive as long as we possibly can.
Elon's rush to Mars pisses me off, because he shits on the fact that we have 100s of millions of years left on this biosphere, and will shit on the biosphere to colonize Mars quickly. My guess is that if we manage to colonize Mars within the next 3000-5000 years, we'll be doing extremely well in terms of overall survivability.

2024-08-11 22:44:50 UTC

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I'd love to have a bunch of "smart" appliances that follow the principles of API-first design. Switches that I can control from my computer easily and effectively without relying on third-party servers. A furnace I can monitor and instruct via ethernet and USB. But these attempts to insert yourself into my business so you can sell my switch-using or furnace-using habits to anybody with a bit of money, nope.