https://robertandrewspencer.com/ Recent content on Robert Andrew Spencer Hugo -- gohugo.io en Sun, 04 Oct 2026 10:00:19 +0100 https://robertandrewspencer.com/amateur/ Sun, 04 Oct 2026 10:00:19 +0100 https://robertandrewspencer.com/amateur/ Call me an Amateur Mathematician. Not because my work is amateurish — that is, not very good — but rather in the original sense of the word: I do mathematics for the love of it, rather than for financial compensation. After completing my PhD in 2022 I have kept the lights on at home by writing software of various forms. The realities of academia, particularly as someone from the Global South, made it difficult to justify doing mathematics for money. https://robertandrewspencer.com/research/cell-modules-for-a-n-webs/ Sat, 28 Mar 2026 00:00:00 +0000 https://robertandrewspencer.com/research/cell-modules-for-a-n-webs/ https://robertandrewspencer.com/research/cell-modules-for-tln/ Wed, 28 Jan 2026 00:00:00 +0000 https://robertandrewspencer.com/research/cell-modules-for-tln/ https://robertandrewspencer.com/dont-try-this-at-home/ Wed, 28 Jan 2026 00:00:00 +0100 https://robertandrewspencer.com/dont-try-this-at-home/ A few months ago I found myself with a fun little problem and decided to solve it in the most insane way I knew how. Background I was maintaining PyBloqs, an open source framework for creating data-heavy reports from python. The specifics are not important to this story, but I’d recently added a dynamic server to the project. In a nutshell, users could define dynamic dashboards similar to streamlit or dash, but much simpler. https://robertandrewspencer.com/native_dot_py/ Mon, 08 Dec 2025 15:22:46 +0000 https://robertandrewspencer.com/native_dot_py/ Here’s something you can do but definitely shouldn’t. To be clear, there is no trickery here. All these commands were executed as shown above1. The magic trick is actually in the codec header. In a move shamelessly stolen from dict-unpacking-at-home we introduce a .pth file that installs our decoder. The decoder lives in the package native-dot-py=0.1.0 and simply copies the file to a temporary directory, runs cargo build and loads the dynamic library as any other native extension would. https://robertandrewspencer.com/wizard/ Thu, 08 May 2025 07:11:21 +0100 https://robertandrewspencer.com/wizard/ A strange thing keeps happening in my professional life: people keep calling me a wizard1. Its not because of my large beard or pointy hat, I’m sure, but my skill with the computer. As the famous aphoroism goes: Any sufficiently advanced technology is indistinguishable from magic. - Clarke's third law However, I’d posit: Any sufficiently proficient practitioner of technology is indistinguishable from a magician. - A logical conclusion While technology as magic is not a new concept (though I think there is a lot more to be said about it), I think likening me to a wizard is more correct than either of my colleagues realised. https://robertandrewspencer.com/inline_typehints/ Mon, 21 Apr 2025 00:00:01 +0000 https://robertandrewspencer.com/inline_typehints/ Which of these identically implemented Python functions has the best signature?1 def add_new_item_to_shopping_list( list_: tuple[str, ...], item: str ) -> tuple[str, ...]: return tuple((new_item, *shopping_list)) def anitsl(l: tuple[str, ...], i: str) -> tuple[str, ...]: return tuple((i, *l)) def add_item(shopping_list: tuple[str, ...], new_item: str) -> tuple[str, ...]: return tuple((shopping_list, *new_item)) Well, the second option is obviously terrible (though some style guides would claim that the only problem is that there are five too many letters in the function name). https://robertandrewspencer.com/rainbow/ Wed, 25 Sep 2024 07:04:00 +0000 https://robertandrewspencer.com/rainbow/ A long time ago I had a professor who was into photographing atmospheric optical phenomena like sun dogs. This inspired me to try work out exactly how these rainbow-like oddities occur. Any physics class will give you the basics: light reflects off water droplets and spreads out into a spectrum which we see as a rainbow. But how does it reflect off the water droplets? Why does it split into a specturm? https://robertandrewspencer.com/static_html_generator/ Sun, 25 Aug 2024 21:54:33 +0100 https://robertandrewspencer.com/static_html_generator/ This bank holiday weekend I found myself writing a small amount of documentation for project I was doing. I wanted it to be simple and easy to edit but still look good and feel professional. I started out writing some pure HTML and used simple.css to style the semantic elements. It was looking very neat, and I was enjoying getting back to the roots of web development by writing out tags manually instead of using a markdown parser. https://robertandrewspencer.com/matt_levine_securities_fraud/ Tue, 06 Aug 2024 06:26:56 +0000 https://robertandrewspencer.com/matt_levine_securities_fraud/ Matt Levine is a columnist at Bloomberg where he has a column called Money Stuff. One of the running themes in his pieces is that everything — according to him — is securities fraud . One would expect him to cover a fair amount of fraud as a financial columnist with legal training, but the list of things he’s claimed to be securities fraud over the years is… extensive. However, I couldn’t find a definitive list of Things Matt Levine Thinks are Securities Fraud, so I wrote a quick script to scrape the archive on Bloomberg and, well… https://robertandrewspencer.com/twenty_one/ Sat, 09 Mar 2024 12:00:00 +0000 https://robertandrewspencer.com/twenty_one/ https://robertandrewspencer.com/paku_waka/ Sun, 11 Feb 2024 12:00:00 +0000 https://robertandrewspencer.com/paku_waka/ They say that imitation is the sincerest form of flattery. This was my first foray into small game development. I decided to make some things easier, and add restrictions to other things. These were the rules when I started. Paku paku is a lot of fun. I am copying recreating it, and the game should have no more scope than that. The game should be written in five one-hour sprints. For the first four hours, I am only allowed to compile the code during the last ten minutes. https://robertandrewspencer.com/pick-a-number/ Mon, 14 Aug 2023 15:55:19 +0100 https://robertandrewspencer.com/pick-a-number/ … any number. You’ll probably want this number to have three or four digits. Fewer might be boring and more may get a bit complicated. If you don’t feel like working through this trick open the example boxes. It’s really better if you do it yourself (the first time) though. Example Let’s choose $5701$. Let’s say the number has $n$ digits. We are going to make $n$ “layers” of numbers. The first layer only has one number in it: the number you chose. https://robertandrewspencer.com/python_ellipses/ Sun, 23 Apr 2023 11:08:37 +0100 https://robertandrewspencer.com/python_ellipses/ The following is valid python code: def some_function(): ... That’s not a placeholder, that’s a constant called Ellipsis: >>> ... Ellipsis >>> Ellipsis Ellipsis >>> ... is Ellipsis True The result is that the function some_function runs, it just returns nothing (since the only expression evaluates a constant and does nothing with it). Its like the todo! in rust except it doesn’t crash. So what’s the point? Well just to name a few, docs where you don’t want to fill in the function body, type hints like Tuple[int, . https://robertandrewspencer.com/nan_nah/ Fri, 17 Mar 2023 19:24:07 +0000 https://robertandrewspencer.com/nan_nah/ Let’s talk about floats. Floating point numbers are complicated. They are meant to represent “any number” by which we probably mean any real number. However this is of course impossible, since there are only $2^{64}$ possible values that can be stored in a 64-bit float, and the number of reals is uncountably infinite. To solve this, we represent a subset of rational numbers and declare the job done. For good measure, we throw in $\pm \infty$ and the concept of “not-a-number” or NaN. https://robertandrewspencer.com/research/phd/ Sun, 22 Jan 2023 00:00:00 +0000 https://robertandrewspencer.com/research/phd/ This is my thesis for my Doctor of Philosophy degree. https://robertandrewspencer.com/git_lg_with_links/ Sat, 07 Jan 2023 16:57:42 +0000 https://robertandrewspencer.com/git_lg_with_links/ Click on commit hashes in git log --graph in order to check out that commit. I’ve recently started using the terminal kitty, and something that’s impressed me is the custom handling of terminal hyperlinks. While clicking on files to open them in your chosen editor, or cd to the directory output of ls --hyperlink=auto is neat, something I’ve been doing a lot of recently is jumping between commits in a complicated git history. https://robertandrewspencer.com/aoc_2022_1/ Mon, 12 Dec 2022 00:00:01 +0000 https://robertandrewspencer.com/aoc_2022_1/ This year, I’m doing the Advent of Code and for career reasons I’m doing it in q. In many ways q is a terrifying language1 and this seemed like a good opportunity to get more experience with it. On the first day of AoC, the problem came to be: Find the maximum sum of a given set of lists of positive integers. The input will be a flat array of integers with None separating the lists. https://robertandrewspencer.com/sort-by-hue/ Wed, 09 Nov 2022 18:54:28 +0000 https://robertandrewspencer.com/sort-by-hue/ My childhood bookshelf was sorted according to the Dewey Decimal System. It’s an imperfect system concocted by an imperfect man1 and it showed. Fiction 823, the bulk of my teenage library, sat awkwardly in the middle between general maths 511 and world history 909. Algorithm design 005.1 sat uncomfortably close to unexplained mysteries 001.94. As a result, my collection didn’t stay that way for long. I kept the good bits of the system (fiction at eye level and sorted by author surname) and discarded the less sensible — putting maths and computer science together. https://robertandrewspencer.com/quantum-numbers-definitions/ Mon, 01 Aug 2022 00:00:00 +0000 https://robertandrewspencer.com/quantum-numbers-definitions/ In the course of my PhD, I was often working with quantum numbers. These are quite mysterious entities and though we know a fair amount about them, its often a lot of folklore. Nonetheless, they are a fascinating generalisation of “normal” numbers, so I’m going write a series about them. In this first installment, we’ll define them, and play with them, and then ask a question I honestly don’t know how to answer. https://robertandrewspencer.com/i-hate-paintings/ Sat, 04 Jun 2022 15:16:36 +0000 https://robertandrewspencer.com/i-hate-paintings/ An excerpt from “The Pocket Guide to Artistry in the 20th Century” by Sir A. R. Tiste, with a post-note by the author. Chapter 4: Dealing with Friends, Family, and Acquaintances My candle burns at both ends; It will not last the night; But ah, my foes, and oh, my friends— It gives a lovely light! — Edna St. Vincent Millay The established artist is appreciated by her peers, accepted by her friends and revered by the public. https://robertandrewspencer.com/antennapod-rewind/ Wed, 16 Mar 2022 15:05:14 +0000 https://robertandrewspencer.com/antennapod-rewind/ I use AntennaPod to listen to podcasts. Its a great little piece of software: one of the few open source mobile applications that feels polished. Onboarding is great, its easy to find and subscribe to new podcasts, and it has all the gratuitous settings I expect by now of open source software. The interface is mostly intuitive and it JustWorks™. One of the podcasts I listen to is 99% Invisible. Its a show about design in all its many shapes and forms and I honestly didn’t expect to be the kind of person who cares much for “Design”. https://robertandrewspencer.com/naked_science/naked-science-8/ Mon, 14 Mar 2022 14:25:24 +0000 https://robertandrewspencer.com/naked_science/naked-science-8/ Producer Robert Spencer Presenter Robert Spencer Guest voices John James and Viv James Adrian Glover Mensun Bound Lead-in Julia Ravey https://robertandrewspencer.com/naked_science/naked-science-7/ Sun, 06 Mar 2022 14:25:24 +0000 https://robertandrewspencer.com/naked_science/naked-science-7/ Producer Robert Spencer Presenters Robert Spencer and Chris Smith Guests Solomiya (From Kyiv) James Lyne (Cyber in context) Chris Folkerd and Stephen Crow (Attack vs Defense) Heather Harrison-Dinnis (Legal) 🇺🇦 Slava Ukraini 🇺🇦 https://robertandrewspencer.com/naked_science/naked-science-6/ Fri, 04 Mar 2022 14:25:24 +0000 https://robertandrewspencer.com/naked_science/naked-science-6/ Producer Robert Spencer Presenter Robert Spencer Guests Charlotte Birkmanis Lead-in Chris Smith https://robertandrewspencer.com/met-statistics/ Fri, 04 Mar 2022 10:22:46 +0000 https://robertandrewspencer.com/met-statistics/ There are three types of lies. Lies, damned lies, and statistics. — Unknown The Metropolitan police in London uses Live Facial Recognition (LFR) to find offenders on watch-lists in public places. The practice is controversial, to say the least. It has been challenged in UK courts and Amazon has indefinitely halted providing its service, Rekogition, to law enforcement agencies. In the spirit of accountability, the Met keeps records of when it deploys LFR. https://robertandrewspencer.com/naked_science/naked-science-5/ Tue, 22 Feb 2022 14:25:24 +0000 https://robertandrewspencer.com/naked_science/naked-science-5/ Producer Robert Spencer Presenter Robert Spencer Guests Jun Ye Lead-in and Lead-out Julia Ravey https://robertandrewspencer.com/bring-me-back/ Fri, 11 Feb 2022 16:26:56 +0000 https://robertandrewspencer.com/bring-me-back/ I am too young to reminisce. My parents spent more time on IRC1s than I did, but there is one aspect of the system I sorely miss. tulip: turns out Einstein was a real guy tulip: im shook tulip: i thought he was just some theoretical physicist * rspencer sighs into his beard That last line was triggered by a /me command. Instead of sending a normal message, I’d have typed /me sighs into his beard. https://robertandrewspencer.com/naked_science/naked-science-4/ Fri, 11 Feb 2022 10:25:24 +0000 https://robertandrewspencer.com/naked_science/naked-science-4/ Producer Robert Spencer Presenter Robert Spencer Guests Gareth Mitchell, Stephanie Hare, and Lord Clement-Jones https://robertandrewspencer.com/naked_science/naked-science-3/ Wed, 09 Feb 2022 14:25:24 +0000 https://robertandrewspencer.com/naked_science/naked-science-3/ Producer Robert Spencer Presenter Robert Spencer Guest Zachary Schutte Lead-in Otis Kingsman https://robertandrewspencer.com/naked_science/naked-science-2/ Mon, 07 Feb 2022 10:00:24 +0000 https://robertandrewspencer.com/naked_science/naked-science-2/ Producers Anoushka Handa and Robert Spencer Presenters Anoushka Handa and Robert Spencer Guests Richard Buggs (Ash Dieback) David Coomes (Alder and Frankia Alni) Richard Head (Longbows) Chantal Helm (Cambridge Botanic Gardens) James Littlwood (Cambridge PPF) https://robertandrewspencer.com/naked_science/naked-science-1/ Fri, 04 Feb 2022 14:25:24 +0000 https://robertandrewspencer.com/naked_science/naked-science-1/ Producer Robert Spencer Presenter Robert Spencer Guests Ville Salo and Ilkka Törmä Lead-in Anoushka Handa https://robertandrewspencer.com/wherebeen/ Sat, 18 Dec 2021 20:49:21 +0000 https://robertandrewspencer.com/wherebeen/ Here is an interesting picture: Earlier this year, my partner and I holidayed in Dubrovnik, Croatia. This is a picture of that trip. More exactly, its a picture of where I was during that holiday. I had set my phone to record its position every fifteen seconds or so. Then, when I was back home, I translated that data into the above image. Every black pixel represents a point reported by my phone and we can pick out some pretty interesting features. https://robertandrewspencer.com/research/non-induced-modular-representations-of-cyclic-groups/ Mon, 01 Nov 2021 00:00:00 +0000 https://robertandrewspencer.com/research/non-induced-modular-representations-of-cyclic-groups/ https://robertandrewspencer.com/microjson/ Sun, 24 Oct 2021 18:06:54 +0100 https://robertandrewspencer.com/microjson/ I published microjson today. It’s a tiny crate, designed to do exactly one thing — read JSON. It’s not very fast, doesn’t let you serialize objects, doesn’t unpack JSON into custom structs1 and doesn’t expose multiple APIs. But there is one other thing it doesn’t do which I think is important: it doesn’t allocate any memory. It also doesn’t have any dependencies. No allocations: no_std The use case for microjson is tiny applications (embedded) that need to extract a small amount of data from some JSON once. https://robertandrewspencer.com/eggs/ Fri, 10 Sep 2021 17:03:24 +0100 https://robertandrewspencer.com/eggs/ How should you eat your eggs? I don’t mean sunny-side up or scrambled or as an omelette. I’m asking which order you should remove them from the carton. See, a while back the internet exploded over whether left-to-right or right-to-left was the “correct choice”1. Eventually some people started to insist that from the centre out was the only option to “preserve balance”. But is it? A colleague and I were discussing this today2 and realised there is a nice little puzzle here. https://robertandrewspencer.com/wfc1/ Fri, 27 Aug 2021 14:08:59 +0100 https://robertandrewspencer.com/wfc1/ Here’s a picture from a project I’ve been tinkering with for a few weeks. Scroll to the bottom for a moving version. An Introduction to the Algorithm The idea of this algorithm is to use Wave Function Collapse to construct a natural feeling castle shape in 3D. I’ve tinkered with WFC a few times in the past but this is my first attempt at implementing it on a 3D grid (as opposed to a 2D image). https://robertandrewspencer.com/rust_static_analysis/ Tue, 03 Aug 2021 18:46:52 +0100 https://robertandrewspencer.com/rust_static_analysis/ I recently was browsing some code on GitHub when I came across a strange source file. From what I could read, there was a glaring bug in the code. It wouldn’t mean that the code didn’t work, merely that it wouldn’t work as expected (or documented). What was strange though was how small the file was (barely more than a screen) and how clearly the bug stood out. Surely it should have been caught, I thought. https://robertandrewspencer.com/research/modular-valenced-temperley-lieb/ Sun, 01 Aug 2021 00:00:00 +0000 https://robertandrewspencer.com/research/modular-valenced-temperley-lieb/ https://robertandrewspencer.com/writing-proofs-clearly-stated/ Fri, 28 May 2021 11:28:58 +0200 https://robertandrewspencer.com/writing-proofs-clearly-stated/ This one is specifically for the students writing exams. If you are asked to clearly state a result in your proof, do so. I don’t like to give full proofs in this series, as I believe that these are general tips not tied to any particular subject. However, I think this one is most easily demonstrated with an example. Consider this problem1: Problem: Show that the minimum weight a BCH code with design distance \(\delta\) is at least \(\delta\). https://robertandrewspencer.com/writing-proofs-exercise/ Mon, 24 May 2021 18:20:43 +0200 https://robertandrewspencer.com/writing-proofs-exercise/ This one is easy: Never explicitly leave part of the proof “up to the reader”. If you are writing a proof, you must convince the reader that the result holds. Asking the reader to show intermediate results or perform calculations just goes towards convincing them that you don’t have a complete solution1. There are some easy phrases you should avoid: It can be shown that… A quick calculation yields… It isn’t hard to see that… Showing this is a quick exercise. https://robertandrewspencer.com/writing-proofs-induction/ Thu, 20 May 2021 10:41:28 +0200 https://robertandrewspencer.com/writing-proofs-induction/ Of all the types of proof, proof by induction is the one I see botched the most often. Induction proofs are a very specific type of proof with some very specific requirements. Failing to cover all of the requirements, at best is not a complete proof, and at worst will result in fatal flaws in the argument. Anatomy of an Induction Proof All induction proofs need to have the same basic steps: https://robertandrewspencer.com/writing-proofs-notation/ Mon, 17 May 2021 13:10:35 +0200 https://robertandrewspencer.com/writing-proofs-notation/ When writing a proof we often have a choice of notation. Perhaps we are naming variables or deciding if functions act from the left \( \theta(x) \), or right \( (x)\vartheta\). We may be referring to existing theorems or constructions (is it the HOMFLY polynomial or the HOMFLYPT?) or may need to introduce our own. In all cases there is a single golden rule: be consistent. As long as you set out your notation early and stick to it, you are being clear. https://robertandrewspencer.com/writing-proofs-recap/ Fri, 14 May 2021 08:41:21 +0200 https://robertandrewspencer.com/writing-proofs-recap/ Proof can rapidly get complicated or involve multiple cases and edge conditions. Once you have finished tying up all the loose ends, take the time to give a very quick recap of what you have actually done. Using the Going Up theorem we can extend … and so \(\dim R \ge \dim S\). On the other hand, if we have … and thus \(\dim S \ge \dim R\). We have thus shown that \(\dim R \ge \dim S\) and \(\dim R\le \dim S\) and so these must in fact be equal. https://robertandrewspencer.com/latex_2/ Wed, 12 May 2021 17:16:15 +0200 https://robertandrewspencer.com/latex_2/ You may recall my rant about LaTeX in 2020. In short, trying to “update” an installation of TeXLive had me running down internet rabbit holes, fuming about poor versioning, and generally tearing out my hair. It was not pretty. Thankfully that story has a happy ending. Shortly after writing that post, I discovered TinyTeX. TinyTeX is one of those rare pieces of software that does what it says on the tin, correctly, first time. https://robertandrewspencer.com/writing-proofs-intro/ Mon, 12 Apr 2021 16:34:38 +0200 https://robertandrewspencer.com/writing-proofs-intro/ I’ve done a fair share of reading mathematical proofs. It comes with the territory. As an aspiring professional mathematician I expect to spend a substantial chunk of my remaining lifetime reading (and hopefully writing) proofs. During my PhD, I’ve taught scores of undergraduate students and marked hundreds of problem sets. I’ve seen it all, from rote copies of the results from lectures (complete with errors verbatim), to proofs that were unintelligible, to submissions that made me want to read them again because they were just that good. https://robertandrewspencer.com/research/l-p-jones-wenzl/ Tue, 16 Feb 2021 00:00:00 +0000 https://robertandrewspencer.com/research/l-p-jones-wenzl/ https://robertandrewspencer.com/research/modular-temperley-lieb/ Mon, 02 Nov 2020 00:00:00 +0000 https://robertandrewspencer.com/research/modular-temperley-lieb/ https://robertandrewspencer.com/latex/ Sun, 09 Aug 2020 19:03:04 +0100 https://robertandrewspencer.com/latex/ My PhD is in Pure Mathematics. Every day I read beautiful documents typeset in the wondrous system of LaTeX. It is neat, elegant, easy on the eyes and designed with mathematicians in mind. I love LaTeX documents, and so should you. But it is not only in academic writing that the TeX’ed document appeals. The archery club I run has its minutes taken (not by me) in LaTeX. This blog, when it has formulae, will use some form of TeX. https://robertandrewspencer.com/circles/ Sat, 08 Aug 2020 12:13:10 +0100 https://robertandrewspencer.com/circles/ https://robertandrewspencer.com/research/jacobsthal/ Mon, 01 Jan 0001 00:00:00 +0000 https://robertandrewspencer.com/research/jacobsthal/ The Jacobsthal elements, \(\mathcal{J}_n\) of \(\text{TL}_n\) are defined by Boyd and Hepworth12. They are so named as the number of diagrams in their support is given by the Jacobsthal numbers. There is an exact sequence \[0 \xrightarrow{} \mathcal{F}_n \xrightarrow{} \text{TL}_n\xrightarrow{-\cdot\mathcal{J}_n} \text{TL}_n\] where \(\mathcal{F}_n\) is the mysterious “Fineberg” module. We present some of the Jacobsthal elements and some information about them below. In these calculations, \(X = \mu/ \lambda = -v\) and the loop resolves to a factor of \(v + v^{-1} = -X - X^{-1}\). https://robertandrewspencer.com/research/jones-wenzl/ Mon, 01 Jan 0001 00:00:00 +0000 https://robertandrewspencer.com/research/jones-wenzl/ Here are some of the Jones-Wenzl idempotents in the Temperley-Lieb algebras. The coefficients are computed using Scott Morrison’s formula and my library temperley_lieb_cat. The data files can be deserialised, though calculation through jw(n) should be fast enough for all the values below. The pdf files use the convention that a closed loop resolves to \(X\). \(\text{JW}_1\) pdf data \(\text{JW}_2\) pdf data \(\text{JW}_3\) pdf data \(\text{JW}_4\) pdf data \(\text{JW}_5\) pdf data \(\text{JW}_6\) pdf data \(\text{JW}_7\) data \(\text{JW}_8\) data \(\text{JW}_9\) data From \(\text{JW}_7\) onwards, \(\LaTeX\) refuses to compile the document and from around 10 onwards, the coefficients overflow signed 128-bit integers and I haven’t yet implemented larger number values.