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The sceptics were right about everything. The physics didn't work, the error rates were catastrophic, the whole thing was theoretically cute and practically useless. Engineers received a spec. I wrote the history of what happened next. https://historyofquantum.com

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Original post on mathstodon.xyz

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13. BSD Unix, Part 3 - The Beginning of BSD Unix

From Pascal and ex tapes to a VAX system: Bill Joy handles BSD distribution and debugging, then joins Ozalp Babaoglu to bring virtual memory to Unix.

fosscomics.com

Original post on mathstodon.xyz

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AI model gets a 32-bit RISC-V CPU compiled directly into its weights, then runs DOOM on it — experimental AI model can run Doom and C code using its own weights as RAM Qwenputer is an experimental blend of the stock Qwen 3.5 4B LLM with a 32-bit RISC-V CPU compiled and inserted directly into its weights. #hardware https://www.tomshardware.com/tech-industry/artificial-intelligence/ai-engineer-successfully-compiles-a-32-bit-risc-v-cpu-in-qwen-3-5-4b-llm-inserted-directly-into-its-weights-built-in-cpu-can-run-processor-arithmetic-operations-code-and-of-course-doom

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Wubi reimagined? Here's a new open source project that lets you dual boot Linux from Windows without needing any USB or DVD. https://itsfoss.com/libertix/

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Making peace with the projective plane

In my newsletter I give an introduction to the latest few posts and occasionally give some additional commentary. In my latest newsletter I wrote > Projective space and homogeneous coordinates seemed like arbitrary abstractions when I was a student because they were presented without any motivation. Much later I came to realize that they’re natural, almost inevitable. Exceptional cases and hand-wavy references to objects “at infinity” go away when moving to projective space. My mathematics education was very top-down: abstractions first, specifics later (if ever). I was OK with that for a while, until I hit my abstraction tolerance sometime in grad school. Now I would advocate more of a bottom-up approach, motivating abstractions first rather than presenting them as a _fait acompli_. The first time I saw the projective plane was probably in a topology class where someone said you could construct the plane by starting with a square and gluing opposite edges in a physically impossible way. This seemed like a gimmick, a way to construct a non-orientable manifold but not anything practical. Another construction is to say the projective plane is an ordinary sphere with antipodal points identified. I found that a little more satisfying, but it still unmotivated. What I would have appreciated was someone saying that the projective plane was invented to make geometry more elegant by eliminating exceptional cases and making rigorous the idea of things “at infinity.” For example, geometric calculations often have to treat vertical lines specially, but vertical lines are not special in projective geometry. You could start by saying we’re going to represent points in the plane not simply as (_x_ , _y_) but as (_x_ , _y_ , 1) and use (_x_ , _y_ , 0) for the point at infinity in the direction (_x_ , _y_). That may seem like a hack, but it’s certainly motivated. It’s exactly the sort of thing a graphics programmer would do. In fact, I imagine many programmers have done just that, independently bumping into projective geometry. After having introduced this extra coordinate, it’s natural to ask what would it mean if the last coordinate were something other than 0 or 1. You’ll quickly run into this when you start doing calculations on these extended coordinates. You could have a rule that you always normalize by dividing by the last coordinate, if it’s not zero, and then you have essentially discovered homogeneous coordinates. You’ll see something like the approach above in elliptic curve cryptography. Programmers wanting to implement encryption algorithms are thinking concretely in terms of (_x_ , _y_) points, until they have to worry about a mysterious “point at infinity” and handle it correctly. Then homogenous coordinates appear as a tool for getting out of a bind. *** I thought about titling this post something like “Dr. Stragelove, or: How I Learned to Stop Worrying and and Love the Projective Plane” but I wasn’t sure how well that would go over. I doubt many readers have seen the 1964 movie this is alluding to, but the trope “How I Learned to Stop Worrying and Love …” is more widely known than the movie itself.

johndcook.com

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A unified view of poles and polars

I wrote several posts on poles and polars a few days ago. First, I wrote about the dual of a spherical triangle, called the polar triangle. Then I wrote about poles and polars in plane geometry and briefly mentioned the extension to projective geometry. Poles and polars are a little awkward in the plane because you have to pay attention to special cases. Poles and polars are relative to a conic section, and are defined/computed differently depending on whether you’re taking the dual with respect to an ellipse, parabola, or hyperbola. Also, the polar of the center of a conic section isn’t defined. ## Projective space If you move from the Euclidean plane to the projective plane, the exceptions go away, though it may not seem so if you’re talking about points at infinity or lines at infinity. If you fully commit to working in projective space, there aren’t any hand-wavy points or lines “at infinity.” There’s not even a distinction between ellipses, parabolas, and hyperbolas. There’s no such thing as a free lunch, and the cost of this elegant unification is moving to a bigger geometric space with a different coordinate system. Instead of working in the Euclidean spaces _R_ ² or _R_ _n_ , we work in _P_ _n_ over a field _F_ which could be _R_ but doesn’t need to be [1]. The coordinates on _P_ _n_ are homogeneous coordinates, (_n_ + 1)-tuples of field elements, not all equal to zero, with two (_n_ + 1)-tuples representing the same point if they are proportional, i.e. one is a non-zero multiple of the other. We can embed _R_ _n_ in _P_ _n_ by adding an extra coordinate on the end, set equal to 1. You could think of this extra coordinate as a **switch** , set to 1 for finite points and set to 0 for infinite points. But this distinction between finite and infinite points is an artifact of the embedding, not a property of _P_ _n_ : there are no special points in _P_ _n_. A point that is “at infinity” in one embedding of _R_ _n_ in _P_ _n_ need not be in another embedding. ## Poles and polars Once you accept the need to work in homogeneous coordinates, everything becomes simple and consistent. A quadric (a conic section when _n_ = 2) is the set of points {_x_ : _x_ ᵀ _A_ _x_ = 0} where _A_ is a non-singular (_n_ + 1) by (_n_ + 1) symmetric matrix over _F_. The **polar** of a point _p_ is the hyperplane {_x_ : _p_ ᵀ _A_ _x_ = 0} and the **pole** of a hyperplane _H_ = {_x_ : _u_ ᵀ _x_ = 0} is the point _p_ = _A_ −1 _u_. Notice there’s no mention of “infinity” in this section. All the calculations take place in the field _F_ with no points at infinity added. [1] The only requirement is that the field _F_ cannot have characteristic 2. So _F_ could be the reals, the rationals, the complex numbers, etc. _F_ could even be a finite field as long as the number of elements in _F_ is not a power of 2.

johndcook.com

Brazilian lawyers busted for attempting to manipulate court's AI with prompt injection hidden in filing The lawyers of a plaintiff in a labor case attempted to use prompt injection to influence the court's AI but were caught. They won the case anyway, as the defendant did not respond, making the move totally unnecessary. #hardware https://www.tomshardware.com/tech-industry/artificial-intelligence/brazilian-lawyers-busted-for-attempting-to-manipulate-courts-ai-with-prompt-injection-hidden-in-filing-courts-ai-flags-hidden-text-that-demanded-superficial-legal-scrutiny-lawyers-fined-usd16-000

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