> For graphics rendering Euler equation doesnt matter.
Huh?? "Euler angles" are one of the most popular representations in computer graphics! The only other good alternative is quaternions, where as you say this also matters!
Yes, but do you really need to scale all your values to make the rotation amount fit into a sin() call, only to have to scale them back again when you're done?
I think what they're getting at is "why are we scaling everything so one full turn is sin(6.28ish) when we can just say sin(1)?"
You could easily try it out by defining a function that replaces sin(), cos(), and tan() with their "turn" equivalents, and seeing how you get on.
If you look at the implementation of sqrt for a computer, it's usually implemented with Newton's algorithm, which is an iterative numerical method with high speed convergence. It is computationally efficient and looks approximately zero how √ looks.
Sometimes a person will tell a story which is untrue, with the purpose of bringing levity to a conversation. This may be termed a joke. These stories may also be used to illustrate a point.
It's important to note that such cases are not always clearly signaled as being humor or untrue. It is a part of the joke's effect that the reader or listener will not at first know it is a joke, but will realize it after noticing an absurdity.
I’ve been writing code for 46 years. Not once have I had to code a derivative.
And for all the people who are concerned about how sin' 2πx = 2π cos 2πx, in actual code, it doesn’t matter. Let’s say that I’m writing a basic graphing function and I want to be able to display the slope of the sin curve at any point.
I am not going to expose the turn-based units to the user. Caring about slopes implies that I’m doing calculus and thus assuming radians. So even though my internal values are [0,1], I will label them as [0, 2π] (and the actual numeric values on the display may actually be something like [50,450] which is yet another numeric value we don’t display). So to get the slope at π/4, I’ll calculate cos_t 0.125 and display that value.
We do all kinds of unit translations in computing without worrying about it. This is just another case of that which observes that numerically speaking, using turns is better aligned with the underlying numerical algorithm for calculating trig values.
> I’ve been writing code for 46 years. Not once have I had to code a derivative.
Haha !
I have been coding for much shorter time but having done some ML on orientations and on spheres in my time, I have had to take their derivatives all the time.
It will be interesting to consider folks who do machine learning on robot trajectories or analysing dynamics of robotic arms.
> But ask them to enumerate all the intermediate steps required to create a formal direct proof, and it will loose attention and forget important details as they go out of their input window size.
It's interesting how people will comment on LLM capabilities despite clearly not having engaged with frontier models in any meaningful way in a long time
Having models write Lean proofs of mathematical claims is standard operating procedure for any LLM math discovery!
Yeah but the LLM can only handle proofs that hold inside its context window. Proofs for novel theories requiring thousands of pages with dozen millions of steps will need support from external tools to organize the full structure of the formal document; it cannot be done by the LLM inference process alone, which was my point. It would be like asking a mathematician to proof theorems without pen and paper; external tooling is a must, the statistical essential nature of generating content from weights is 1) error prone and 2) not suitable for chains of systematic reasoning that are longer than the attention span.
The proofs will be only as good as the framework for linking successive instances of reasoning.
Not sure how true this restrictions is once you have the agents hammering on at a big code base of formalised proofs.
Proofs stretching thousands of pages are split into lemmas, grouped into sub theories.
What I haven’t seen agents do yet is to develop new ideas for entire such theories. I have usually seen them bite into some existing idea and grinding out related results. But I am less sure than ever that they won’t!
When you try to maximize ad revenue, you add multiple advertising SDKs to your website, each of which can often do live bidding with hundreds of ad/data brokers
You can usually check the ads.txt file on a website to see which companies are allowed to bid for ad space on there. For example, for dict.cc, the website in question:
When we launch a site it is seldom more than perhaps Hotjar, Google Analytics, and two-three other services connected.
And then through the years product managers and other stakeholders gets sold on adding LinkedIn, Instagram, Meta, and so on. So we add those.
Next a specific service ”to better track the sales funnel from in-store salespeople to the web” gets added. Then another ”analyse the data quality versus bounce rate” tracker gets added. And so on.
Before long the developers have streamlined the process of adding new scripts/analytics/trackers that editors can add them on their own, and that is when the floodgates open.
analytics: A/B testing, "if x does user click y"?, unique page visits, etc.
ads: integrating with an ad provider comes with hundreds of trackers, because they want to
- know if you bought a product after clicking on an ad
- show you targeted ads for shoes after you googled shoes
- build a profile of you (age, gender, location, profession) to show relevant ads across different websites
Likely has little relationship to what is actually in the page. They had to do GDPR, didn't or couldn't spend a lot of time on it -- or had an especially conservative corporate counsel -- and ended up just getting a list of every company they've ever worked with, for any reason, "to be safe".
For most companies this can easily be thousands of partners, and going through that list and figuring out exactly who might get data in reality, through every possible permutation of workflow, is a horrendously expensive proposition.
You might be surprised how many well-meaning regulations leave even the best-intentioned implementers in an impossible situation.
Oh yeah, that combination of fear and lack of knowledge probably plays a big part. I was once involved with creating a privacy policy for a B2B(!) web application. What a farce. In the end, the process was cut short (counsel too expensive and not nearly familiar enough with tech). The resulting document was at least 50 % stuff the app simply does not do.
Ultimately that is what they are having to do though, it's just costing them twice as much by pretending that being conservative and not actually looking at the problem saved them.
I remember my parents needing to buy very expensive map DVD disks in order to update the car’s navigation system to update their car’s navigation computer.
The disks were very expensive and if you didn’t update them it wouldn’t know about new roads (slower trip, missed turns, etc).
That's why we have Android Auto: move the updates to a thing that already has frequent Internet connection, and if the update goes wrong it's clearly the fault of this other company, plus the car itself still works fine.
Jeep already had an OTA the broke the ability for the car to be driven.
Yes, boondoggle subsidies allow you to un-economically bring fiber to a subset of random places. I say this as the beneficiary of one such boondoggle. It doesn't scale well
It's going to turn into slop, exponentially fast, any moment now.
No but seriously it could imply grinding to a halt, turning into a laughing stock of slop, or, imho much less likely given the current state of AI (barring huge technological breakthrough and a couple of other assumptions) you were probably implying an intelligence explosion.
Not if you use "turn"-trigonometric functions, as the author suggests.
turn-sin `t sin(x) = sin(2pi * x)` has `d/dx tsin(x) = 2pi * tcos(x)`.
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