Demystifying Euler's Formula

In this video, I delve deeper into Euler's formula and explain in more detail why I think the 2x2 matrix version of Euler's formula is sooooo important.

[@ThisSideUp12]

Hey, I jut found your videos and I wanted to say you are a wonderful teacher! Thank you so much for sharing all of this with us!

[@FractalWoman]

You are very welcome.

[@carlospaiva9083]

Brilliant once again, thanks! :D

[@romado59]

It always good to set these ideals to you memory or mind.

[@rustedcrab]

Have you red the great little book "God's secret formula" by Peter Plichta? Tells the story of his lifelong search for the numerical generative principles of the universe that he saw manifesting in his studies in chemistry, physics, biology, etc. and he forms together a compelling picture starting with the basic numbers 1, 2 and 3, their powers and reciprocals, the cycles of primes, the euler number, the natural logarithm, the pascal triangle, binomial coefficients, the sierpinsky fractal... In a beautiful perspective of interrelated phenomena.

[@pepe6666]

oh man i am SO LOOKING FORWARD TO THIS. before i watch im going to get some food. im endlessly struggling with oilers identity for doing fourier transforms and such.

[@joepike1972]

This idea is not related to this question but I have been trying to think about an electron as a current flowing in the ether with circumference the Compton wave length. I am envisioning toroid vorticies along this current flow. What I realized is these can't be placed too close to each other since their points of contacts would be in opposite directions. I am trying to think about the math for the spacing of these magnetic sheer induce vorticies.

[@rustedcrab]

Using tau for the full cycle renders the formula even more obviously simple.

[@FractalWoman]

For sure. I agree.

[@dwinantosaputra6679]

very brave, and I must say very informative. I think today physics an energy content must be able to describe epsilon, and not only a particle movement / particulate in micros motion. angular is always a heavy topic, gratz 😍

[@reynoldsbeng3756]

See Dr V Lewe, 1915. Website at ACE Consultancy UK has all about his physics dissertation, regarding matrix calculus and elastic membrane theory, to Engineering. The Nuclear Force is revealed to us. The energy of bending light, m^4

[@TheMemesofDestruction]

Very Helpful! Thank you! ^.^

[@8thsinner]

Also, whilst on the topic, why did the pyramid builders use it 11 times in the pyramids? I personally suspect this relates to the birkeland currents, because you have 5 sections and 6 intersections. Also, if you happen to know the angles of momentum at each of those 11 placements please let me know, I don't think I know the correct language to find those details, maybe you do.

[@rogerconnolly3688]

In Vastu architecture they teach that creation begins with a swastika which spins, the gaps then join and this turn expands out to an 8 x 8 grid from which 3D life begins.

[@8thsinner]

@@rogerconnolly3688 Is there a name for that composite shape, not sure I follow on the grid size when it meets itself, where does one grid end and another start?

[@Penrose707]

Can you please expand on the idea of equivalency of the Cartesian and Argand planes? Thank you

[@FractalWoman]

When you implement complex numbers as 2x2 matrices (including Euler's formula), there is no difference between the cartesian coordinate system and the complex plane. There is literally no need for the language of real and imaginary numbers when you do math this way. The forward diagonal of the 2x2 matrix holds the X coordinate of the X-Y plane and the backward diagonal of the matrix holds the Y coordinate. Technically, the forward diagonal is "real" and the backward diagonal is "imaginary" there is really no need for these terms. We could just as easily have called them "forward" numbers and "backward" numbers for the forward and backward diagonals.

[@alexanderjenkins7929]

Eidomorphism by neven knezvic explores an ontology based around eulers formula

[@FractalWoman]

Found the book. Got the book. "Thus, eidomorphism is a theory of mathematical energy imbued with mind: in other words, all that exists is minded mathematical energy following the structure of the Euler formula. We call the dimensionless point acting as the origin of the formula the monad, as it is the net resultant of the real, imaginary, positive and negative axes of the formula and is an undivided whole. The monad is the most basic eidomorph. It is the noumenon, and we can tease from of it every ontological and phenomenological consequence imaginable. The theory of these monads is a monadology of sorts, and the monadology is likened to the description of a program that generates the Universe". Thanks for the tip.

[@alex29661]

How do you distinguish between "man's language" and "God's language" in math?

[@FractalWoman]

God's math is used in the creation process. Man's math is used after the fact. Man's math give us a description. God's math gives us an explanation. And, as Ken Wheeler always says, "descriptions are not explanations".

[@alex29661]

@@FractalWomanBut can you please explain? Perhaps I have trouble understanding how to distinguish between them.

[@FractalWoman]

@@alex29661 Let's use the Mandelbrot Fractal as an example. When I use the equation, Z = Z squared + C to generate the Mandelbrot set, that is a creation process. No calculus required. Calculus could be used to analyze the Mandelbrot Fractal after it is generated. But the calculus didn't generate the Mandelbrot Set. The only math I need is the 2x2 matrix version of complex numbers, addition and multiplication. God's math is simple. Mans' math is complicated. God's math is beautiful (like the Mandelbrot fractal). Man's math is kind of ugly, like a black board full of equations.

[@alex29661]

@@FractalWoman It seems to me that due to your subjective and limited perspective of what part of math is "beautiful and simple," you think that you found out what kind of mathematics my Lord used in the creation of the universe. In reality, you are just using the Fourier series to gain any desired output you want, it should be noted that the Euler formula should be used for studying objects only of a repetitive nature. The function you are using belongs to the so-called "Hilbert Space", which is why you can actually use it as any other function with the same property to output any kind of solution as you wish. Fourier analysis is based on Euler's formula, and we use it to figure out what kind of frequency spectrum the measurements have. You can also use it to find out what the rest of the unknown function would look like if it has a repetitive nature and you have enough data points. That is the reason why you can make any fractals using the Euler number. While Euler's number is a useful mathematical constant in many ways, it's not necessary for creating fractals. Fractals can be generated using a wide range of mathematical techniques which can be equally beautiful and simple. Now I will try to explain why Euler's formula is not distinguished from other parts of math as "God's math". It is because you can't describe all observations we have in nature with Euler's number. Let us take the simplest example of the spring-mass oscillations. The differential equation that describes our oscillation is mx'' = -kx. This is where you can use the Eulers formula to find a simple solution which is x=Ce^(it). Please note that it is not we couldn't use any other function that will output the same solutions. But e^it makes perfect sense because of its oscillatory simple nature. But what if we now add friction? Then the solution is not pure oscillation anymore, and C*e^(it) is not the correct solution. We have friction, which will decrease the energy of the system, and now the solution will look more like x=C*e^(-a*t)*e^(it), where "a" is the real number. This new solution is simple and intuitive; we understand every element with the highest intuition and reasoning. Where e^(-a*t) explains the decay of energy of the system (transferred to heat by friction), and e^(it) explains oscillatory motion. Or what if we have some oscillatory force pushing the spring making it resonate? Then the solution would look like x=t*e^(it). This is just like in bridge resonance, the frequency of oscillations that resonate with wind or walking will stay the same, but the amplitude will increase over time. Now, if you want to complicate things, you can. We know the property (not so unique) of the Euler formula to output every possible number in every possible order, just like, for example, power series. It is proven mathematically that everything can be described with a power series because of that special property which I won't bother explaining. That means that we can modify the Eulers function to fit our solution without exponential decay e^(-at) or amplitude growth t. For example, in the case of friction, in order for the Euler formula to fit the solution, you can add a negative sine wave with an infinitely long period, and that will output the same numbers of positions as the solution (C*e^(-at)*e^(it)) we got before. In the case of resonance, fitting the Euler function would be more complex, but it must be possible. One of the key differences is that the modified e^it solution is not intuitive and reasonable in this examples, thus it implies that after infinitely many days have passed, the system will gain energy again (amplitude will increase), which is not possible. There are many more examples. I'm studying controls in mechanical engineering, and if I want, I could write a whole book where Euler's formula would provide nonsensical solutions. The same case would be with any other function with the same property and there are many of them. Your thinking reminds me of a time when physicists couldn't find a reasonable and intuitive explanation for the experiment results from particle collider, but instead of investigating it more to gain more intuitive and reasonable answers, they fit their result in power series (note that they could use Eulers or any other function) and they have claimed that by that formula the universe works with that type of atom collisions. The formula looked terrible, it had over 50 coefficients. I noticed that I'm maybe using too many power series examples, but there are so many more functions that are found with the exact same property that the Euler formula does. Simple research can provide many more wonderful functions. The power series has the same property in the sense that it can be configured to have any desired output, but it doesn't even belong to Hilbert space as the Euler function does. Hilbert space is worth looking up, has very interesting properties, if you like Euler's formula you would love Hilbert space it has functions of a similar and even better nature depending on application. Any function in Hilbert space could be studied like you study Euler's formula. Your theory might work well for the imaginary repetitive world you have shown us. But in reality, it is far from the truth that it will provide us with reasonable and logical solutions that make sense in real-world physical behavior. If you wanted to understand what kind of logic our Lord used, it wouldn't fit your head. Our most advanced math and physics can't grasp it. But imagine if everything was described with Euler's formula, besides all things mentioned. It would imply that our world is repetitive, that God is limited, and that everything we have seen we will see again and again for infinite times without stopping. And that spring from example that slowed down due to friction will, after infinitely many days, magically (due to Euler's formula) start gaining speed again, which is not possible.

[@reynoldsbeng3756]

God's maths MUST exist. If nothing else existed what mathematical truth would remain? The simplest complex number; so which is really the imaginary part?

[@8thsinner]

I'd like to know how to use these measuring certain formulas relative to certain numbers but the math language side of things always limits me, how they were used in the pyramids is what I want to be able to extrapolate into other designs. This is only one stabilization part of the universal equation though, I mean, thats why it was used to design the stock market, and other banking procedures. The thieves who stole it from the pyramid builders knew how powerful this number is.

[@MS-od7je]

Can’t wait. Btw I assume ( you know what that means) that the Julia sets are incorporated herein also. Mandelbulbs etc.

[@FractalWoman]

For sure the Julia sets are incorporated here. The Mandelbulbs do not use complex numbers and therefore, are not directly related. Only things that use complex numbers are related. This includes quantum mechanics which I forgot to mention in this video.

[@ibaha411]

First 2 mins of the video and two mistakes already; phi not theta, it’s identity if you see it as -e^i pi=1.0 🤦🏻‍♂️ Still watching….

[@FractalWoman]

pi is theta, there is no "phi" in my video. What most people don't realize is, as soon as you SEE the letter "i", we are in the domain of complex numbers and complex numbers are 2x2 matrices under the hood. So, lets have a look at this from the perspective of the 2x2 matrix. Let's deal with this part first e^(ipi) e^(ipi) = cos(pi) + i sin(pi) but pi = 180 degrees, so I am going to switch to degrees for clarity. e^(ipi) = cos(180) + i sin(180) In matrix form, looks like this: [ cos(180) 0 ] [ 0 sin(180) ] [ 0 cos(180) ] + [ -sin(180) 0 ] = [cos(180) sin(180) ] [-sin(180) cos(180)] = [ -1 0 ] [ 0 -1 ] When you multiply this by -1, you get [1 0] [0 1] You see, there is nothing wrong with what I said. PI (180 degrees) is THETA. In case you didn't know, theta refers to an angle and pi radians is equal to 180 degrees which is an angle. I think you made a mistake. 🙂

[@romado59]

Maybe add one more grid called "phase"?

[@ChrisAthanas]

Helpful

[@sistajoseph]

I think you have got it, you just got to translate into language that "man" can understand. The ontology is as simple as it looks. That part about the identity is fantastic. The whole thing should hang together properly. If one is the identity, which I accept, what is -1?

[@FractalWoman]

In the 2x2 matrix version of Euler's formula, [1] is equal to the identity matrix: [1 0] [0 1] and [-1] is equal to the matrix: [-1 0] [0 -1] I call this the negation matrix. When you multiply another 2x2 matrix by this matrix, it negates all the value. Yes, I agree that we need to make the language accessible to "man". That is what I am trying to do. Baby steps.

[@jgarris0577]

Really want to see someone put thsi to use in blender.

[@FractalWoman]

I wish I was more savvy with Blender. I prefer to write my own programs and not depend on an external platform. That said, Blender is AMAZING and extremely powerful. I highly recommend it.

[@stridedeck]

Following, using mathematical terms, ie. vectors, etc, is using man's language and conditioning. You are trying to get to the creation of the Source of the universe. Basically, to me, this is what you are doing: alphabet and language are structures. Nature, the world, the universe is unstructured. Man's language is imposing their conditioning structure onto an unstructured universe. Structures distorts, misleads, and always fall short! Keep doing what you are doing!

[@FractalWoman]

Yes. You are right. Or as I like to say, "the problem is with the language". It always falls short.

[@stridedeck]

@@FractalWoman yes, very mystical. The concept I have is this: at every point in space and at every moment in time, there is a force expanding at the speed of light in all directions that moves everything, even our neurons. (ie. a 4D hypersphere). This concept uses all scientific laws, experiments, but has this simple interpretation. It explains the dynamic fabric of space (gravity), and other quantum features (time, distance, entanglement, etc.). Also, consciousness (we have 4D mind), yet our 3D neurons can not access this 4D force. Also, how we got something (matter) from nothing. I could go on, but I will be sounding more and more like a mad, crazy man!

[@MapSpawn]

I really strongly disagree on Euler's identity, because your interpretation is incorrectly mathematical. It symbolizes a perfect rotation back to the original start. This is to be interpreted that entropy does not need to affect everything. For example, culture doesn't have to disappear to entropic forces. If humanity was + or -, that is to say stretched thin or over exuberant, both lead to a downfall. This is a physics equation, not a mathematical one. It can only be substantiated using mathematics, but what it reveals in mathematics is that there absolutely is a smallest divisible space, because otherwise you'd have a rad function with an infinite floating point value. It is extremely beautiful that you can rotate something and absolutely know it is back to its original position; if the universe had no size limit this would not be possible.

[@FractalWoman]

Contrary to what you are suggesting, Euler's formula IS purely mathematical, although it can be APPLIED to physics as a rotation. Euler's formula has nothing to do with entropic forces or the divisibility of space. You may be thinking of Euler's number but I do not talk about that in this video. In this video, I am reinterpreting Euler's formula in terms of the correct interpretation of complex numbers as a 2x2 matrix. Matrices are mathematical objects, not physics. The sine and cosine terms are mathematical, but can be applied to physics. Just because something can be applies to physics, doesn't make it purely physics. I can apply this formula to other things that have nothing to do with physics like rotating an object around a point in a computer graphics program. '"you'd have a rad function with an infinite floating point value." What is a "rad function"? If you are talking about sine and cosine, then you are wrong because the outputs for sine and cosine functions are always between -1 and +1 and can ever be infinite no matter what "rad" value you put into the function.

[@MapSpawn]

@@FractalWoman If you can only be more precise and more precise, then there is always a margin of error. Euler's demonstrates the matrices which have xyz coordinates to express where a point is in space, these coordinates can only be expressed to a limited decimal. It doesn't go for infinity, it has a precise end. This goes against Quantum mechanics (which I reject) because in Quantum you would be never reach the same position again because there is no smallest position.

[@FractalWoman]

@@MapSpawn In computer science, there is a thing called the "limit to the digits of precision". Using even the best computer you can find, there is always going to be a limit to the digits of precision that you can use for calculations. In quantum mechanics, there is a similar limit called the Planck limit. There is no such thing as infinity. Not in the computer science and not in nature. So I'm not really sure what point you are trying to make.