Applications Of Conformal Geometric Algebra To Transmission Line Theory

[@liammccreary2941]

I cannot stress how unbelievably invaluable this video is! I study nuclear magnetic resonance and everything I do is typically modeled using quantum mechanics and the algebra gets a bit dicey and counterintuitive at times. This video alone is enough to develop a powerful intuition for pulse sequence manipulations and I’m convinced that CGA will soon take over the field of NMR and quantum studies

[@kentgauen]

This is my favorite Geometric Algebra explanation I have seen. Thank you for making these concepts accessible!

[@810labs]

thanks glad it was helpful to you.

[@AndreaCalaon73]

Beautiful! Well done! Eventually we get out if the vector spaces pits!

[@jackkelty1057]

how tf did I end up here still goated vid tho ngl

[@timelsen2236]

Wow, Clifford algebra grown up!

[@mathunt1130]

You have no idea. Read the book entitled, "Geometric algebra for physicists".

[@tanvach]

The link between conformal space time algebra and circuit theory is very surprising! (Though probably not that surprising in hindsight since spacetime is kind of embedded in Maxwell’s equations). Mind blowing stuff, lots to unpack and think about.

[@810labs]

true. would be nice to work out details.

[@pseudolullus]

Extremely interesting, especially how you bypassed the need for differential equations by using groups and GA!

[@810labs]

i also find this aspect really interesting.

[@sari54754]

Most engineers will not be familiar with GA. We are drawn with Gibbs vector calculus and do not see the beauty of GA. Great work, indeed.

[@BCarli1395]

Thank you. This is a very helpful approach to the topic.

[@ARBB1]

Great work Alex, more people should be aware of GA.

[@810labs]

thanks, !

[@pianavela]

Very nice explanation. The paper on IEEE access is also very interesting, although is long and contains a lot of material. I would like to point out that in the slide Paths generated by discrete bivectors the role of X and R are probably exchanged. Nice work! make more videos with more details. Thank you!

[@thomasolson7447]

Cool, you're investigating the same stuff I'm investigating. The Smith Chart, conformal maps (sqrt(x^2+1)+x, -sqrt(x^2+1)+x, sqrt(x^2+x+1)+x, -sqrt(x^2+x+1)+x), distribution of points on double angle sphere [x^2-y^2-z^2, 2*x*y, 2*x*z], spinnors, and I imagine you noticed gamma by now. So, two complex numbers behind all of this. Possibly two quaternions behind each for conformal mappings. I don't plan on going down this road though. I'm doing DSML in collage. Best I can ever hope for from a physics professor is a giggle and maybe a comment about my pretty graphs.

[@danstiurca7963]

Very nice and very useful. Will need to watch this 4-5 times to let it sink in, but very well done! Thank you.

[@810labs]

thanks, the paper might be helpful for you also.

[@danstiurca7963]

@@810labs thank you

[@SteveAcomb]

oh my god I can’t believe I found this channel. amazing work

[@810labs]

Glad you enjoy it!

[@ramkitty]

fascinating, the impedance mismatch theta seems like a torsional stress in the skin effect creating a helical wave.

[@henrikr8183]

It's pronounced like Ree-muhn

[@danielmilyutin9914]

8:43 misprints must be: (a+b)^2 = a^2 +b^2 + ab + ba I suppose definition is: a dot b = (ab + ba)/2

[@berg0002]

Thanks a lot, very well done and inspiring. Have you done some work to see how it describes nature when we upgrade Minkowsy spacetime to 3 complex orthogonal planes? Consider a quite significant flaw in the interpretation of Einstein’s discovery of the constant of speed of light; what if c isn’t a *speed* (a vector pointing in the direction of motion) but just a fixed scaling factor, manifesting the ratio between distance and time, as 2 properties of the same, only differing locally due a phase shift, which can be represented by a complex number at each position in the grid. This would imply time to be a 3D vector as well, as it scales to distance with a constant scaling factor c. A wave observed from a fixed position and time is a phase shift of position in 3D space vs a moment in 3D time. I am pretty sure we can succesfully detune our observations from Minkowsky spacetime manifold to a static grid. As a next step in the transformation of our observations we should then model and observe (measure) cosmic as well as minute expressions of the universe in GA language (G3+3 Space, a space spanned by 3 orthogonal basis vectors in real 3D space and 3 orthogonal basis vectors in imaginary 3D time). Each basis vector of space pairs with its imaginary Counterpart basis vector of time. Dynamics come into existence through phase-shifting the grid positions In the 3D field of moments of time with the grid positions in the 3D field of position in space. Entanglement is a phase-lock relationship of 2 separate positions in 3D space; superposition is a phase-lock situation of two separate Positions in The 3D time field at a single position in 3D space. It would be great fun to demodulate the forms constructed over space in the 3D field of time experienced from a fixed position in space, which is a superposition of phase shifts, hence waves at a certain fixed position. Who volunteers to colaborate with me to work this out. The project should render 3D animation models demonstrating properties of fundamental particles, superposition, entanglement and should demystify observations such as in the double slit experiment. It would explain the measurement problem and dequalify the Copenhagen interpretation of the collapse of the wavefunction, which is nothing more than opening ourselves to 2 new time dimensions that operating at slight phase shifts around moments of observations in linear time.

[@810labs]

Geometric Algebra emphasizes real algebra. complex algebra is ambiguous geometrically.

[@berg0002]

What do you think of the suggested interpretation of c (scalar vs. vector)? With time being a 3D vector too? If we could visualize this in 2 x 3D. Time being the 3D phase modulation of each position in 3D space?

[@frankdimeglio8216]

@@810labs THE ABSOLUTE, BALANCED, EXTENSIVE, AND CLEAR MATHEMATICAL PROOF REGARDING E=MC2 AS F=MA: Gravity IS ELECTROMAGNETISM/energy, AS E=MC2 IS F=MA. The Earth (A PLANET) is a MIDDLE DISTANCE form that is in BALANCED relation to the Sun AND the speed of light (c), AS the stars AND PLANETS are POINTS in the night sky; AS E=mc2 IS F=ma !!! This NECESSARILY represents, INVOLVES, AND DESCRIBES what is possible/potential AND actual IN BALANCE, AS ELECTROMAGNETISM/energy is gravity. Very importantly, A PHOTON may be placed at the center of WHAT IS THE SUN (as A POINT, of course); AS the reduction of SPACE is offset by (or BALANCED with) the speed of light (c); AS E=mc2 is F=ma; AS ELECTROMAGNETISM/ENERGY IS GRAVITY. The stars AND PLANETS are POINTS in the night sky. The sky is blue, AND the Earth is ALSO BLUE. (OVERLAY what is THE EYE in BALANCED RELATION to/WITH what is THE EARTH.) E=MC2 IS F=ma !!! TIME DILATION ULTIMATELY proves (ON BALANCE) that ELECTROMAGNETISM/energy is gravity, AS E=mc2 is F=ma. INDEED, TIME is NECESSARILY possible/potential AND actual IN BALANCE; AS E=mc2 is F=ma; AS ELECTROMAGNETISM/energy is gravity. The stars AND PLANETS are POINTS in the night sky. SO, THE EARTH is E=mc2 AS F=ma IN BALANCE. Objects fall at the SAME RATE (neglecting air resistance, of course), AS ELECTROMAGNETISM/energy is gravity. Gravity IS ELECTROMAGNETISM/energy. It ALL CLEARLY makes perfect sense. BALANCE and completeness go hand in hand. So, get a very good LOOK at what is THE EYE. (Very importantly, outer "space" involves full inertia; AND it is fully invisible AND black.) "Mass"/ENERGY IS GRAVITY. ELECTROMAGNETISM/energy is gravity. E=MC2 IS F=ma. Gravity AND ELECTROMAGNETISM/energy are linked AND BALANCED opposites, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity !!! Gravity IS ELECTROMAGNETISM/energy. BALANCE and completeness go hand in hand. Consider what is the speed of light (c) as well. Carefully consider the man who IS standing on what is THE EARTH/ground. Touch AND feeling BLEND, AS ELECTROMAGNETISM/energy IS gravity. Energy has/involves GRAVITY, AND ENERGY has/involves inertia/INERTIAL RESISTANCE. The ultimate mathematical unification of physics/physical experience combines, BALANCES, AND INCLUDES opposites, AS E=MC2 IS F=ma (ON BALANCE !!!); AS ELECTROMAGNETISM/energy is gravity !!! "Mass"/ENERGY involves BALANCED inertia/INERTIAL RESISTANCE consistent with/as what is BALANCED electromagnetic/gravitational force/ENERGY, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/ENERGY IS GRAVITY. Gravitational force/ENERGY IS proportional to (or BALANCED with/as) inertia/INERTIAL RESISTANCE, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. Gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE, AS E=mc2 is F=ma IN BALANCE; AS ELECTROMAGNETISM/ENERGY IS GRAVITY !!! Therefore, objects (AND the falling men) fall at the SAME RATE (neglecting air resistance, of course); AND the rotation of WHAT IS THE MOON matches it's revolution. MOREOVER, a given PLANET (INCLUDING WHAT IS THE EARTH) sweeps out EQUAL AREAS in equal times consistent WITH/AS F=ma, E=mc2, AND what is perpetual motion; AS ELECTROMAGNETISM/energy is gravity; AS E=MC2 IS F=ma IN BALANCE !!!! Gravity IS ELECTROMAGNETISM/energy, AS E=MC2 IS F=ma. This NECESSARILY represents, INVOLVES, AND DESCRIBES what is possible/potential AND actual IN BALANCE !!! Great. By Frank DiMeglio

[@angeldude101]

When you put it like that, I get the impression that this might be better described as time being a bivector space (or pseudovector space). The bivectors in G(3) behave very much like complex numbers to the point that you can argue that e12 simply is i, with the other basis bivectors forming the quaternions j and k. Since bivectors generally work very well for rotations, it seems like you're suggesting that we could rotate a position in space by a position in time, which is an interesting idea. This would also equate time the the quaternions if you include the scalar part. The alternative to that would be a G(3,3) space where time has its own separate basis vectors that square to -1 rather than acting as bivectors of the spacial basis vectors. This would give an absolutely massive 64-dimensional multivector space which I personally would rather not think about. Honestly, if you think there's merit to the idea, try it out! If it appears to accurately describe observations in real life and matches the existing math, then it could very well end up changing our fundamental understanding of the universe. If it doesn't work out, so what? How often to you get to play around with a 6 dimensional vector space‽

[@dean532]

Wow this is applied to TLA theory as well.

[@AndreaCalaon73]

Reeman

[@Ottmar555]

Rayman sphere

[@SphereofTime]

14:07 geometric algebra for physics

[@ixion2001kx76]

Please provide a link to the paper

[@810labs]

ieeexplore.ieee.org/document/7982607

[@joepike1972]

I know this should apply to human vocal tract analogy and resonance for vowel sounds. I see bits and parts I recognize, but I am still at a loss. I know how to take vocal tract cross sectional areas and with cascading operation I can get the formants, F1, F2, F3,... And I would like more direct methods, such as rotations. But I don't see the connection.

[@chimetimepaprika]

Duuuuuude, this is sweet as hell

[@bocckoka]

If I ever do this, I will say okkay, I promise.

[@jask320]

Cool

[@ixion2001kx76]

Could you please write down the titles and author names of the other Geometric Algebra books?

[@mathunt1130]

Chris Dorst and Anthony Lasenby.

[@810labs]

@@mathunt1130 Chris Doran . Leo Dorst also has a book , GA for computer graphics.

[@mathunt1130]

@@810labs I have the book by Leo Dorst and I found it very hard to read in certain parts.

[@JosBergervoet]

Can you extend it to N-port impedance matrices with N>1?

[@810labs]

i have another video about how to model 2-ports. i worked on the n-port problem for a while but never figured out a good model. i think its possible.

[@JosBergervoet]

​@@810labs Especially interesting would be the calibration of an N-port network analyzer with a 2N-port "error network" in the connection. Leading of course to a "calibration group", isomorpic to Sp(2N,C) if I'm correct. (Since a reciprocal S-matrix gives a symplectic T-matrix. Lots of interesting things!)

[@810labs]

​@@JosBergervoet it would be really neat. feel free to email me.

[@deltalima6703]

This is hilarious! If you know all the math ahead of time it makes perfect sense, if not you will learn nothing here. Interesting stuff.

[@insouciantFox]

There is the time before a scientist knows about GA and the time after.

[@mathunt1130]

REE-man not Rayman.

[@810labs]

tomato tamato potato patato

[@yamiyugi8123]

My question is two part and concerns the video at 8:07. Part A) to my question begins: you place the square of two added vectors aka their expanded dot product as a second supposedly-equivalent statement a^2 + b^2 + 2(ab+ba). I must check from you that these are intended to be the nonorthogonal case because otherwise our dot product would be 0. And if this nonorthogonality stands, then this looks to me exactly as the law of cosines should look which would equal out to some scalar quantity c^2= a^2 + b^2 + 2a•b. If so, then 2(ab+ba) looks confusing and my first question is can you clarify why we are using this 2(ab+ba)? Part B) Still concerns 8:07, where I would like to know why this a•b would also be the same as (ab+ba). This is below your first formula. We are taught in the GA that the dot product is 1/2(ab+ba), and it also says so in the following slide which is quite different compared to your equation of (a+b)^2 which when stated as the law of cosines still retains a great symmetrical and deceptively simplified structure. My second question is could you please explain how the statement a•b is the same as (ab+ba) when I believed that 2a•b= (ab+ba) and hence the 1/2 compensation for a singular nonorthogonal and symmetrical dot product a•b ? 2a•b is still a scalar in the absolute value sense. Thank you for your assistance in advance!

[@810labs]

no sure i understand. there may be missing factors or 2, but you get the idea. can find more about GA in those books or on wikipedia.

[@yamiyugi8123]

Sure. Got it! Thanks for the quick response!

[@csnyc1]

REEEEEEman