Demystifying Sir William Rowan Hamilton's Quaternions

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FractalWoman

Күн бұрын

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@alanthayer8797 9 ай бұрын

Fractal women is about PRECISION and Simplicity! I’m all n for that Movement!

@FractalWoman 9 ай бұрын

Thanks. That is EXACTLY what I am all about.

@pixbee3759 9 ай бұрын

Thanks to you and this video I just attempted ‘to code’ for the first time in my life and I felt really cool... now tempted to look for a beginners course …😂

@FractalWoman 9 ай бұрын

Fantastic. Did you download my code?

@AstraGroupDF 9 ай бұрын

The square root is a tricky one in complex analysis because you need to define where is the image of your function otherwise you're gonna get one function mapping a point to two points. Same goes to the logarithm where you would have a infinite number of images of a single point.

@FractalWoman 9 ай бұрын

When you represent SQUARE roots as SQUARE matrices, then all of the confusion goes away. The square root of [+1] is: [+1 0] [ 0 +1] and [-1 0] [ 0 -1] The square root of [-1] is: [ 0 +1] [ -1 0] and [ 0 -1] [ +1 0] Easy as PI. Pun intended.

@ed_dev 8 ай бұрын

Have you looked into geometric algebra (clifford algebras)? I recently read some of Eric Lengyel's first volume for his game engine books and he recently released a new book called Projective Geometric Algebra illuminated. He gives a lot of insights on quats and other concepts as well.

@FractalWoman 8 ай бұрын

Yes, I have looked into this. But on my quest for nature's math, I find geometric algebra too complicated for my liking. Fractal geometry is closer to the truth what nature does (in my opinion) and that is why I think it is so important to demystify complex numbers. The most famous fractal, the Mandelbrot Set ONLY needs complex numbers. Have a look at my paper, The Mandelbrot Set as a Quasi-Black Hole and you will understand my line of thinking.

@Kraflyn 7 ай бұрын

aren't quaternions spinor matrices? Not rotation matrices?

@CalibrationProject 9 ай бұрын

First of all thank you for your well thought out and clearly presented videos. You are my favorite KZbinr and an inspiration. I have a couple of questions/comments on this video: (1) When the whole angle is used (around minute 49) it wasn't clear to me why the points squish together when viewed from the side , and why the two-step multiplication is required with the conjugate. My trigonometry is very rusty but the use of half angles rings a bell. Could it be that the half angle identities relating sine and cosine are hiding some additional information in the i, j, and k matrices (as 0 and 1 were hiding sine and cosine of 90 degrees)? (2) I attempted to replicate the squishing effect by fiddling with your code, trying to comment out the half-angle with conjugate portion but was unsuccessful ... so I decided that building my own from scratch would be the best way to understand (both the concepts and how to fiddle with it), which leads me to my next observation/question: (3) I successfully implemented matrix multiplication for 4x4 matrices, but in testing it with your i, j, and k (minute 31) I noticed that i^2 = j^2 = k^2 = -1 works correctly. But ij = not as described by Hamilton's expressions (minute 21). Same with ji = , (not ). The signs are opposite. I pulled out pen and paper to verify that it's not a bug or artifact of my code but I'm coming up with the same. For example the bottom-left cell (Row4, Column1) of your k matrix is +1, but multiplying and summing the 4th row of i with the 1st column of j yields -1. Signs are reversed from Hamilton's equations. Is this important? (4) You have mentioned previously that although the signs in the backward diagonal for the imaginary component must be opposite, either bottom left or top right could get the negative or the positive. Therefore switching the signs in *one* of your i, j, or k would bring your matrices into sync with Hamilton's equations. (5) Suggestion for your JavaScript code: I find strict mode to be incredibly helpful in troubleshooting bugs and errors, and it works in p5.js with the standard "use strict" as the first line in the code. The extra effort to fix silent-but-innocuous errors that it catches is more than compensated for in the numerous silently-buggy errors (especially variable name typos/capitalization mistakes) that I inevitably make.

@FractalWoman 9 ай бұрын

@CalibrationProject: Thank you for your kind words about my channel. It's a lot of work and sometimes, I wonder why I am doing this. Then I get a message like yours. 🙂 1) According my my research, it seems that lengths aren't conserved when doing quaternion multiplication. This doesn't happen when using complex numbers for rotations. I believe this has something to do with the non-commutativity of quaternions. 2) Yes, better to write your own code. I understand things better when I write my own code. 3) 4) With quaternions, there are three things that affect the direction of rotation. a) Row convention vs column convention when implementing matrices on the computer. b) Pre multiplying the point with the matrix vs post multiplication. c) Using the conjugate matrix vs the original matrix. I am not sure what would cause the direction to be correct but the rules to be backwards. I would have to see your code. Truthfully, it doesn't really matter in my opinion. When you are writing the code, you get to decide what convention you want to use. 5) Thanks for the suggestions. I will give that a try. Good luck with your code. 6) I like your KZbin name. What inspired it?

@FractalWoman 9 ай бұрын

There is one more thing that affects the direction of rotation and that is left handed coordinate system vs right handed. That is why your spin is the right way but your rules are backwards. P5JS is a Left handed coordinate system. In my code, I had to switch to right handed. See if you can find out how I did this.

@CalibrationProject 9 ай бұрын

The inspiration for my KZbin name is that in order to accurately understand reality, our tools must be accurately recording information, and our programs must be accurately interpreting data. To calibrate is to bring into conformance with a standard, such that it records and conveys accurate information. Most critically, our minds need to be calibrated so that we think accurately and understand reality. The standard of truth that I seek to calibrate my thinking is the Bible (God’s revealed truth to man). My particular interest is science and history of the beginnings, from Creation through approximately Exodus in the Bible. My dream is to start my own channel with the aim of recalibrating our thoughts to accurately understand the world (reality, the way things are/were/began), based on the standard of truth. When I said that you are an inspiration I really admire your pursuit of accurate understanding and “re-writing the program”, and your precision in defining terms. I admire the clarity with which you explain concepts, and how you respectfully present arguments and evidence where you disagree with the standard model. Your re-interpretation of the missing time in Plank’s equation is compelling. Your switch from radians to cycles and explicitly including cycles in the units is compelling. Your focus on unit analysis is spot on (I’m an engineer and one of my catch phrases has always been “the units are your roadmap”, when solving any problem).

@FractalWoman 9 ай бұрын

@@CalibrationProject Wow. Thanks so much. As you may know, I work in the field of medical physics. We build medical devices for various applications including biopsy and surgical procedures. Since I am the one developing the software, I am in charge of calibration. I am "the calibrator" in our lab. For our medical devices, I HAVE to get the calibration right or the doctor is going to make a mistake in the operating room. That is why I take calibration so seriously. Regarding unit analysis, several years ago I wrote a paper called, "Calibrating the Universe and why we need to do it" which was published in Physics Essays. This was my first attempt as Modified Unit Analysis. Here is a link to the pre-print. www.researchgate.net/publication/308094064_Calibrating_the_universe_and_why_we_need_to_do_it You said: "My dream is to start my own channel with the aim of recalibrating our thoughts to accurately understand the world" This is a great dream and I think you should continue perusing it. If you start making videos, I will watch them.

@CalibrationProject 9 ай бұрын

@@FractalWoman @FractalWoman I downloaded your paper 2 years ago shortly after finding your channel but have to admit that I got caught up in catching up on all your past videos and never actually read it. I will do so. I work in the drinking water and wastewater treatment industry where we have strict water quality standards, and customers that rely on safe drinking water. Almost everything is automated based on analyzer-recorded data which gets passed for example from probe (electrical signal) > analyzer (mg/L) > wire (4-20 mA) > PLC input card (0-20,000 raw input signal) > programed calculations (mg/L, converted as needed in formulas). And then the signal chain reverses in the opposite direction to control the speed of the pump that ultimately injects a specific quantity of actual chlorine solution to the actual water being treated. At every point in the chain, the signal conversion has to be precisely calibrated from one device to the next or "the number" loses its physical meaning. One of my favorite aspects is troubleshooting when things aren't working which often involves identifying what part of the chain is mis-calibrated, and/or adjusting to accommodate real-world limitations (time-delays, precision limitations along the signal chain, adjusting for natural decay of chemicals, etc.). In a couple of your videos you point out that a unitless constant in an equation often results from a calibration error ... I have used that principle in reverse to adjust to real-world fluctuations of parameters that we can't control, or that are infrequently measured and then inferred/estimated between (e.g. chlorine spontaneously decays over time, which is a "known unknown" that we can apply a unitless calibration factor to compensate for). If the analyzer is calibrated and the pump is calibrated, but the result is drifting over time (within an expected range of course) we can apply a constant multiplier to compensate and maintain the end-product water quality.

@martinnjoroge6006 9 ай бұрын

The procedure for deriving the Quaternion matrix is well understood (by me). It gives the matrix forms of the i, j, k Quaternion symbols. If I now apply that method of deriving the matrix form of symbols to the imaginary number i, I would form two expressions : Q = w + xi and Q' = w' + x'i. I then multiply them : QQ' = (w + xi)(w' + x'i) =ww' + wx'i + xw'i - xx' = w''' + x''i where w'' = ww' - xx' and x'' = wx' + xw'. This then gives the "imaginary matrix" [ [+w -x], [+x +w] ]. x is the variable associated with the imaginary number i in this process so we get a matrix [ [0 -1], [+1 0] ] which would seem to be the matrix form of the symbol [+i] going with how the Quaternion symbols had their respective matrices derived. However, your matrix representation of [+i] is [ [0 +1], [-1 0] ] which is the negative of my derivation here. Why would that be the case?

@FractalWoman 9 ай бұрын

Good question. I use row convention when writing my matrices and you are using column convention. I like row convention better because of the way my compiler displays matrices in the debugger.

@FractalWoman 9 ай бұрын

To clarify my answer, in this video, I switch to column convention for Hamilton's quaternion because that appears to be the convention he is using. In all of my other videos about complex numbers and quaternions, I use row convention. That is why I show my "i" matrix as [0 +1], [-1 0] in this video. I do mention this in the video but I guess you missed it.

@martinnjoroge6006 9 ай бұрын

@@FractalWoman I guess I hadn't "well understood" it as I thought. Thank you for the correction.

@kathleenstewart44 9 ай бұрын

Why do you not put cos(90), so cos(theta) in your code, for the off diagonal zeroes? Is it because they’re imaginary units where the diagonal is real?

@FractalWoman 9 ай бұрын

That is a very good question. When you look at the matrix at the 45:00 mark, you will see that I multiply each matrix entry a normalized multiplication factor nW, nX, nY or nZ. So technically, all those zeros matrix entries are still sine terms but with the normalized factor set equal to zero. For example, Iin the "i" matrix at the 34:21 mark, nW and nX are set to 1.0, and nY and nZ are set to zero. This zeros out the Y and Z components creating a rotation about the X axis only. You will also notice that the cos(90) is zero so technically zero but I still write it for completeness since nW is equal to one. I write the terms where n? is 1 and I zero out the terms where n? is zero. This is for teaching purposes only. In my code, all the cos and sin terms are "written" even when they are zero or when n? is zero and I let the computer sort all that out. Hope this makes sense.

@mjfk872 9 ай бұрын

Thanks for the video. I have a book recommendation: "What is light? wave theory of light and origins of ether in science". It would be great if you give a feedback on this book.

@FractalWoman 9 ай бұрын

Sounds like an interesting book. I have a feeling you are the author. Am I correct? If so, is there any way of you sending me a pdf of the book? I don't have a lot of time for reading these days, but this summer I may have some time.