A sphere is a two dimensional surface embedded in a three dimensional space that obeys Euclidean Geometry. This can be misleading, as the curved four-dimensional space-time of our Universe does not necessarily need to be embedded in a space of more than four dimensions in order for General Relativity to work. It could just be that the Universe we live in is simply not governed by Euclidean Geometry.

Can Python, MATLAB, Mathematica, or even TikZ be used for this?

I make my animations with the software Poser. Poser does not have built in functions for complex variables. I had to create these myself. I explain how I make my 3D animations in my video at kzbin.info/www/bejne/bHnPZpesdp1ri9E

@@EugeneKhutoryansky There was a question asked on Zhihu: "I still don’t understand the difference between real-valued functions and complex-valued functions. One studies real numbers, the other studies complex numbers? The graphs seem the same to me..." Someone responded with the following answer: "The essence of a function is mapping. A real-valued function is a mapping from R to R, while a complex-valued function is a mapping from C to C. As the algebraically closed and most complete field of numbers, studying mappings from the complex number field to itself is of significant importance. Our number system has undergone two expansions: the first from the rational numbers to the real numbers, which gave the number system completeness; the second from the real numbers to the complex numbers, which extended the number system to a two-dimensional plane. As a mapping from C to C, the graph of a complex function is four-dimensional, and is impossible to visualize by intuition. Therefore, if we really want to draw the graph of a complex function, one way to do so is by coloring it (since there is no concept of color in mathematics itself, coloring is used to help us visualize the graph of a complex function). So I don’t know where the original poster saw the graphs being the same. Just as the expansion from the real numbers to the complex numbers brought the number system to its ultimate form, complex analysis takes classical analysis to its extreme. Meanwhile, real analysis, built on set theory and measure theory, opened the door to modern analysis. A typical example is in real analysis, where we treat two functions as the same if they are equal almost everywhere. This led to the redefinition of concepts like integration, which have a broader scope and greater power." Do you think the first half of this person's answer makes sense?

@@EugeneKhutoryansky There was a question asked on Zhihu: "I still don’t understand the difference between real-valued functions and complex-valued functions. One studies real numbers, the other studies complex numbers? The graphs seem the same to me..." Someone responded with the following answer: "The essence of a function is mapping. A real-valued function is a mapping from R to R, while a complex-valued function is a mapping from C to C. As the algebraically closed and most complete field of numbers, studying mappings from the complex number field to itself is of significant importance. Our number system has undergone two expansions: the first from the rational numbers to the real numbers, which gave the number system completeness; the second from the real numbers to the complex numbers, which extended the number system to a two-dimensional plane. As a mapping from C to C, the graph of a complex function is four-dimensional, and is impossible to visualize by intuition. Therefore, if we really want to draw the graph of a complex function, one way to do so is by coloring it (since there is no concept of color in mathematics itself, coloring is used to help us visualize the graph of a complex function). So I don’t know where the original poster saw the graphs being the same. Just as the expansion from the real numbers to the complex numbers brought the number system to its ultimate form, complex analysis takes classical analysis to its extreme. Meanwhile, real analysis, built on set theory and measure theory, opened the door to modern analysis. A typical example is in real analysis, where we treat two functions as the same if they are equal almost everywhere. This led to the redefinition of concepts like integration, which have a broader scope and greater power." Do you think the first half of this person's answer makes sense?

Thanks.

The whole point of General Relativity and the equivalence principle is that an observer can believe that they are at rest, even when they are in a non-inertial reference frame. This is not the case in Special Relativity.

Thanks.

Thanks for the compliment about my video and my animations.

I don't work in a University. And yes, I plan to continue making videos after retirement.

Thanks for the compliments. I am glad that my animations are helpful.

Thanks.

I sort of did that with my video on the Block Universe at kzbin.info/www/bejne/pKLJapJ6oZlgjrs Thanks.

Each different set of possible observation is counted as one state. Without a detector, we only have two possible sets of observations. With a detector, we have four sets of possible observations.

C1 and C2 can be any value. They are scalars that are multiplying the matrix [1,0]: (c1+c2) [1,0] = [ (c1+c2) , 0 ]

@@EugeneKhutoryansky ah i see. The brackets are round and not rectangular. Well now it makes sense. However, than there is no correlation between the first example and the second example. We compare two different situations which each other. The math doesnt explain why there is a difference between both scenarios or am i wrong? It looks more like a mathematical description of both scenarios

In the first example, it is explained that the two values of the two waves are added together, while in the second example we only consider the result that we can observe. In this case, the electron would have the characteristics of a particle

Each different set of possible observation is counted as one state. Without a detector, we only have two possible sets of observations. With a detector, we have four sets of possible observations.

10:20

Thanks. I am glad my animations were helpful.

You are welcome and thanks.

No

String Theory is on my list of topics for future videos.

@@EugeneKhutoryansky Yeah I've study some topology Such as open and closed intervals Cartesian products And I've also heard they have something to do with string theory

@@EugeneKhutoryansky quite ambitious lol

Thanks.

I COULD USE A LITTLE FUEL MYSELF AND WE COULD ALL USE A LITTLE.. CHANGGGEEE

My man it’s an experiment you can do at home using technology derived from understanding about quantum physics (lasers)

This is a criticism that could be made of most explanations. Do the equations explaining our physical laws really explain phenomena, or do they just describe them?

​@@EugeneKhutoryansky Good point

I am glad my video was helpful. Good luck with your exams.

Thanks.

Thanks for the compliments.

I have another video where I discuss the various different philosophical interpretations of Quantum Mechanics. It is at kzbin.info/www/bejne/joKVZnhvnNpnp6s

I am glad you enjoyed my video. Thanks for the compliments.

The video is correct. Everything in this video is DC. I cover AC and transformers in many of my other videos. I also cover EM waves in many of my other videos.

@EugeneKhutoryansky sorry I realized that. Should have made it more clear in my comment. I have just seen these types of videos explaining electricity in general terms.