#quaternions — Public Fediverse posts

"Papers I Have Loved" by Casey Muratori (2016)

https://www.youtube.com/watch?v=SDS5gLSiLg0

#pwlconf #marchingcubes #quaternions

But wait, there's more! I also wrote a C++ program and Lua scripts to help you make animations of these higher-dimensional fractals. You can find these at https://codeberg.org/caten/RealAlgebraPlot

Not only can you see animations of the rock-paper-scissors fractal of central interest in my paper, you can also take slices of even higher-dimensional fractals coming from other real algebras, such as the octonions.

I wrote everything here from scratch as much as possible, so it has basically zero dependencies. I even opted to use Netpbm (https://en.wikipedia.org/wiki/Netpbm) so that I'd have an easy time writing image files directly without using a library. I am using #GraphicsMagick to produce animations. The same utility can be used to convert the output image files to a more familiar format like png.

#math #mathematics #algebra #analysis #ComplexAnalysis #fractals #Mandelbrot #MandelbrotSet #UniversalAlgebra #octonions #quaternions #cpp #Lua #LuaLang

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Euclid Update – Revised Timeline

In a post last week I hinted that I hoped soon to be able to provide an update on progress with the European Space Agency’s Euclid Mission. People within the Euclid Consortium have known for some time, but it has now been officially announced by ESA, that the timeline for the first full Data Release (DR1; originally scheduled for October 2026) has now been revised.

The plan now is for DR1 to happen in stages, with a first tranche to occur in November 2026 (precise date yet to be announced). The complete data release will take place in mid-2027 (probably in June). The sky area covered by DR1 will be about covering a large sky area of about 1900 deg².

To give a little more background, the data products from the Euclid survey are divided into three levels of data processing function:

• LE1 – the “raw” data frames, prior to calibration, generated at the Science Operations Centre from the time-ordered data and telemetry received from the spacecraft
• LE2 – the calibrated and corrected data for the two instruments (VIS and NISP) – images, spectra, catalogues of point sources, etc
• LE3 – the high-level data products (galaxy catalogues, cosmic shear maps, etc) designed for cosmological analysis

The first release will comprise LE1 and LE2 only. This will be called DR-Foundation. The LE3 data will be added next year to make the full DR1. Since LE3 is required for the cosmological analysis that is the prime motivation for the Euclid mission, it follows that there will be no official cosmology results from Euclid DR1 until mid-2027 at the earliest. Other results based on “Foundation” data may of course emerge before then.

I hope this clarifies the situation.

P.S. Irish physicists and astronomers will be particularly interested to know that the LE1 data includes information about the spacecraft and instrument pointing orientations, which is stored in the form of quaternions…

What a romantic, though! #WilliamRowanHamilton #Quaternions #RomanticGestures

Two sculptures beside the beautiful canal bridge at Broombridge, in memory of the day in 1843 that 38-year-old William Rowan Hamilton, walking with his wife Helen, scratched the recipe for quaternions into the stone of the bridge in case he'd forget it later
#Quaternions #RowanHamilton #Broombridge

Reworked the 24-cell sketch, much easier to understand now.

Every one of the 12 visible #spinors has a button that brings the yellow spinor (the 1 state) to the clicked one because in 3D spinors double as rotations. (both are #quaternions so there exists a quotient between spinors, which is a rotation).

I think what's clear is that thinking in terms of a spinorial/rotational lattice makes things much easier to grasp than in terms of a vectorial/translational lattice

https://editor.p5js.org/aap/sketches/-N-Bnk_jn

This article by Simon Altman on #quaternions and rotations may be the most lucid i've seen on the topic: https://worrydream.com/refs/Altmann_1989_-_Hamilton,_Rodrigues,_and_the_Quaternion_Scandal.pdf

My beloved #spinors are also mentioned. It is rarely explicitly stressed that the quaternionic imaginaries are 180° rotations, not 90° rotations. The same *can* be said of the complex numbers too of course, but commutativity makes things special there.

P.S.
I couldn't work out in my head what would be an appropriate order in which to enumerate the vertices of a four-dimensional hypercube given e.g. by
α+βi+γj+δk
where each of the Greek letters is 0 or 1.

#FourDimensional
#Quaternions

And he sketched a crooked tesseract.

#FourDimensional

#AndHeBuiltACrookedHouse
#RobertHeinlein

#Canal
#Hamilton
#Ireland
#Quaternions

Project complete!

https://quaternion.cafe/ is a "vim-style" interactive tutorial that teaches sensor fusion with quaternions. It's mostly an aggregation of stuff I learned over the past few years while messing with inertial sensors.

#hardware #math #quaternions

I was up until 2am last night working on this. Whoops! I implemented a basic real-time code-editor and the visualizations. It uses the Intersection Observer API to only play the canvas when the visualization is in the browser viewport (kinda nifty!). #javascript #quaternions #math

Ah, another riveting #GitHub repository, because who wouldn't want to perform "linalg operations" on vectors and #quaternions when they could be doing literally anything else? 🤦‍♂️ It's perfect for those who dream of spending their nights #debugging math libraries instead of having a social life. 💤
https://github.com/David-OConnor/lin-alg #linalg #mathlibraries #programminghumor #HackerNews #ngated

The brilliant story behind quaternions:
#mathematics #vector #analysis #quaternions #science
🌐

https://theconversation.com/three-letters-one-number-a-knife-and-a-stone-bridge-how-a-graffitied-equation-changed-mathematical-history-241034

#GodotEngine #Quaternions #PleaseHelp

Added another simpler version to the forum post:

https://forum.godotengine.org/t/quaternion-rotates-but-then-reverses/73642/2

🦇

#GodotEngine #Quaternions #help

I have posted a question on the forums. Anyone feeling brain enough to help?

https://forum.godotengine.org/t/quaternion-rotates-but-then-reverses/73642

🦇

#GodotEngine #Quaternions

Hey! I'm getting the hang of this Quaternion thang! Look at it rotate!

0 degrees.
90 degrees. WhoooO!
180 degrees! Man, the excite!

90 degrees? Huh?

It's now oscillating. I don't even oscillate any more. I'm too old!
Please stop.

<Imaginary Error: Quaternions are allergic to 2/PI+PI. Please report this to the Lovecraft hotline. Good luck in hell>

🦇

I watched this excellent Swift Introduction to Geometric Algebra, https://youtu.be/60z_hpEAtD8?si=Gv_rHCafbNXj3WXr . I enjoyed it! Geometric algebra allows you to multiply and divide scalars, vectors, and other geometric objects. It's fascinating and helps make sense of concepts like imaginary numbers and quaternions.
#geometricAlgebra
#quaternions
#vectors

This is a fantastic article on rotations and #quaternions by @TheNumbat based on lecture notes by @keenancrane

http://thenumb.at/Exponential-Rotations/

My son saw the test model there "Spot the cow", and we just had to #3dprint one and color it in :)

Get yours over at Keenan's model repository:
https://www.cs.cmu.edu/~kmcrane/Projects/ModelRepository/

#GeometryProcessing

@acegikmo excellent lecture! My frustration is that I still do not have an intuitive grasp of #quaternions the way I do with complex numbers and the complex plane.

have you ever used #quaternions?

Quaternions are one of those things I see a lot of people in CG being confused about (doesn't help that a lot of people leave out "normalized"). If you are only planning to use them for rotations, their meaning is simple: A quaternion is a rotation about an angle around an axis. That's it. They are NOT three angles. I wrote a small derivation of the maybe confusing qpq* formula. Only really requires trigonometry and vectors.
https://liascript.github.io/course/?https://sibaku.github.io/quaternions/quaternions.md
#math #cg #quaternions