Preserving structure is a symmetry -- syntax. Syntax (structure) is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force. Symmetry is dual to conservation -- the duality of Noether's theorem. Duality is being conserved -- the 5th law of thermodynamics! "Always two there are" -- Yoda. Force carriers are dual -- attraction is dual to repulsion. Action is dual to reaction -- Sir Isaac Newton. Photons and Gluons are force carriers hence they are dual. Bosons are dual to Fermions -- atomic duality. Subgroups are dual to subfields -- the Galois correspondence!
Preserving structure is a symmetry -- syntax. Syntax (structure) is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force. Symmetry is dual to conservation -- the duality of Noether's theorem. Duality is being conserved -- the 5th law of thermodynamics! "Always two there are" -- Yoda. Force carriers are dual -- attraction is dual to repulsion. Action is dual to reaction -- Sir Isaac Newton. Photons and Gluons are force carriers hence they are dual. Bosons are dual to Fermions -- atomic duality. Subgroups are dual to subfields -- the Galois correspondence!
You were right! I totally see the application of group theory here - seems like it's very important to the more abstract theories of physics. (Selfishly) I'm not sure to what extent i'll be able to apply it to games programming, but it's definitely given me a deeper insight.
Preserving structure is a symmetry -- syntax. Syntax (structure) is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force. Symmetry is dual to conservation -- the duality of Noether's theorem. Duality is being conserved -- the 5th law of thermodynamics! "Always two there are" -- Yoda. Force carriers are dual -- attraction is dual to repulsion. Action is dual to reaction -- Sir Isaac Newton. Photons and Gluons are force carriers hence they are dual. Bosons are dual to Fermions -- atomic duality. Subgroups are dual to subfields -- the Galois correspondence!
Keep going, this is one of the rare channels that has a proper grasp of how to explain the material, from the visualizations to the curriculum itself (excellent choice starting from group theory and only _then_ go to linear algebra)
@@AllAnglesMath Preserving structure is a symmetry -- syntax. Syntax (structure) is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force. Symmetry is dual to conservation -- the duality of Noether's theorem. Duality is being conserved -- the 5th law of thermodynamics! "Always two there are" -- Yoda. Force carriers are dual -- attraction is dual to repulsion. Action is dual to reaction -- Sir Isaac Newton. Photons and Gluons are force carriers hence they are dual. Bosons are dual to Fermions -- atomic duality.
It's funny how cathegory theory schematics sneak in as well. How the operation, that takes place instead of transposition (the knify T one), for unitary matrices called?
That's a really good question. For a finite group it's easy: you just multiply the generators together to obtain the other group elements. So when you have a finite group of matrices, you just multiply the generator matrices together to obtain all the other matrices. But for an infinite group (and in particular a continuous one), you can't just "multiply" an infinite number of factors together. Instead, you use exponentiation. The easiest example is the circle group. We explain it in this video: th-cam.com/video/3geVAJvJM8c/w-d-xo.html . The key is that you need an additional parameter such as theta. As you continuously vary the value of theta, the exponential e^(i*theta) will continuously walk across the circle, generating all the rotation matrices. So we still have only a single generator i, but we also have an extra parameter. I hope this answers your question. By the way, think of exponentiation as repeated multiplication, where the factors are all infinitely close to 1. This is how the finite and infinite concepts of generators are linked to each other.
@@AllAnglesMaththank you very much for the long exaustive response! I was wandering in the case of more general/abstract continuous groups how would you define exponentiation and why out of all operation is this particular one at play, Is it always traceable back to matrix exponentiation by finding a matrix representation of the group? This topic is very interesting! Hope my questions arent too much to answer, in case i will be happy to wait for the next videos to satisfy my curiosity
@@fedebonons8453 This is a very big and interesting topic called Lie groups, and I hope to make a series of videos about it in the future. For now, the informal way I think about it is that exponentiation is just repeated multiplication. We take a tiny step, and we repeat it many times. That's how you slowly walk around a circle, or around a higher-dimensional blob: By starting off in a certain direction, and then repeating that little step as often as needed. This is exactly what exponentiation *is*. The video about Euler's formula already makes this a little more formal, by showing you what happens in the limit (when the steps become infinitely small, and there are infinitely many of them).
@@AllAnglesMath actually the more i think about it the more makes sense i saw the video about euler formula but its still a bit hard to think of it in a more general sense once again thank you!
No, hemos utilizado nuestra propia biblioteca para la renderización de animaciones. Está escrito en Python. Usamos OpenCV para producir el video final.
17:18 but SL(2) is more than just SO(2), no? We get SL as a kernel. Edit: for example, SL(2) also contains SO(1, 1). I don’t know any cookie recipes sorry. 😑
That's a good point. When you start from GL(2) and take the determinant, its kernel is indeed SL(2), the group of all matrices with a determinant of 1. But in the video I explicitly say that we limit the determinant to O(2) first. In that case, the kernel is only SO(2), which contains only ortho matrices with determinant 1.
Preserving structure is a symmetry -- syntax. Syntax (structure) is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force. Symmetry is dual to conservation -- the duality of Noether's theorem. Duality is being conserved -- the 5th law of thermodynamics! "Always two there are" -- Yoda. Force carriers are dual -- attraction is dual to repulsion. Action is dual to reaction -- Sir Isaac Newton. Photons and Gluons are force carriers hence they are dual. Bosons are dual to Fermions -- atomic duality. Subgroups are dual to subfields -- the Galois correspondence!
Preserving structure is a symmetry -- syntax. Syntax (structure) is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force. Symmetry is dual to conservation -- the duality of Noether's theorem. Duality is being conserved -- the 5th law of thermodynamics! "Always two there are" -- Yoda. Force carriers are dual -- attraction is dual to repulsion. Action is dual to reaction -- Sir Isaac Newton. Photons and Gluons are force carriers hence they are dual. Bosons are dual to Fermions -- atomic duality.
Preserving structure is a symmetry -- syntax. Syntax (structure) is dual to semantics -- languages, communication or information. If mathematics is a language then it is dual. Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force. Symmetry is dual to conservation -- the duality of Noether's theorem. Duality is being conserved -- the 5th law of thermodynamics! "Always two there are" -- Yoda. Force carriers are dual -- attraction is dual to repulsion. Action is dual to reaction -- Sir Isaac Newton. Photons and Gluons are force carriers hence they are dual. Bosons are dual to Fermions -- atomic duality. Subgroups are dual to subfields -- the Galois correspondence!
Your cookie recipes mean the universe to me
Somebody watched all the way to the end of the video ;-)
Preserving structure is a symmetry -- syntax.
Syntax (structure) is dual to semantics -- languages, communication or information.
If mathematics is a language then it is dual.
Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force.
Symmetry is dual to conservation -- the duality of Noether's theorem.
Duality is being conserved -- the 5th law of thermodynamics!
"Always two there are" -- Yoda.
Force carriers are dual -- attraction is dual to repulsion.
Action is dual to reaction -- Sir Isaac Newton.
Photons and Gluons are force carriers hence they are dual.
Bosons are dual to Fermions -- atomic duality.
Subgroups are dual to subfields -- the Galois correspondence!
Absolutly fascinating , you cover so much material in such an easy and lucid manner, incredible
Quick, new angles just dropped!
Beautiful! I always tell that mathematics is the only real natural science, the most profound one. It investigates the very fabric of the universe.
it is study of connections and pattrens but physics is real science object which exist in reality
Preserving structure is a symmetry -- syntax.
Syntax (structure) is dual to semantics -- languages, communication or information.
If mathematics is a language then it is dual.
Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force.
Symmetry is dual to conservation -- the duality of Noether's theorem.
Duality is being conserved -- the 5th law of thermodynamics!
"Always two there are" -- Yoda.
Force carriers are dual -- attraction is dual to repulsion.
Action is dual to reaction -- Sir Isaac Newton.
Photons and Gluons are force carriers hence they are dual.
Bosons are dual to Fermions -- atomic duality.
Subgroups are dual to subfields -- the Galois correspondence!
i mean its easy brothers: SU(4) with 15 kinds of gravitons. theory of everything solved
Or it's described by SU(1) with 1^2-1=0 kinds of gravitrons because they don't exist :(
Good one 🤣
@@greenbear1561 That’s why SU(1) doesn’t happen in physics, yep. They’ve no need for it 😐😞
I don't know any cookie recipes but I've subscribed.
Thanks 😉
You were right! I totally see the application of group theory here - seems like it's very important to the more abstract theories of physics.
(Selfishly) I'm not sure to what extent i'll be able to apply it to games programming, but it's definitely given me a deeper insight.
Preserving structure is a symmetry -- syntax.
Syntax (structure) is dual to semantics -- languages, communication or information.
If mathematics is a language then it is dual.
Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force.
Symmetry is dual to conservation -- the duality of Noether's theorem.
Duality is being conserved -- the 5th law of thermodynamics!
"Always two there are" -- Yoda.
Force carriers are dual -- attraction is dual to repulsion.
Action is dual to reaction -- Sir Isaac Newton.
Photons and Gluons are force carriers hence they are dual.
Bosons are dual to Fermions -- atomic duality.
Subgroups are dual to subfields -- the Galois correspondence!
Keep going, this is one of the rare channels that has a proper grasp of how to explain the material, from the visualizations to the curriculum itself (excellent choice starting from group theory and only _then_ go to linear algebra)
Thank you for the encouragement!
@@AllAnglesMath Preserving structure is a symmetry -- syntax.
Syntax (structure) is dual to semantics -- languages, communication or information.
If mathematics is a language then it is dual.
Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force.
Symmetry is dual to conservation -- the duality of Noether's theorem.
Duality is being conserved -- the 5th law of thermodynamics!
"Always two there are" -- Yoda.
Force carriers are dual -- attraction is dual to repulsion.
Action is dual to reaction -- Sir Isaac Newton.
Photons and Gluons are force carriers hence they are dual.
Bosons are dual to Fermions -- atomic duality.
Thank you!
mind blown!
subscribed! (:
If you wish to make a chocolate chip cookie from scratch, you must first invent the universe
Ah, I see, your recipe will begin with the big bang 😆
It's funny how cathegory theory schematics sneak in as well.
How the operation, that takes place instead of transposition (the knify T one), for unitary matrices called?
It's called the Conjugate Transpose or the Hermitian Transpose: en.wikipedia.org/wiki/Conjugate_transpose
@@AllAnglesMath all right, simple enough - transpose and flip sign for all imaginary parts.
How do the generators generate those "uncountable"/"continuous" groups?
Love the videos keep up! ❤
Exponentiation helps a lot I think
That's a really good question.
For a finite group it's easy: you just multiply the generators together to obtain the other group elements. So when you have a finite group of matrices, you just multiply the generator matrices together to obtain all the other matrices.
But for an infinite group (and in particular a continuous one), you can't just "multiply" an infinite number of factors together. Instead, you use exponentiation. The easiest example is the circle group. We explain it in this video: th-cam.com/video/3geVAJvJM8c/w-d-xo.html . The key is that you need an additional parameter such as theta. As you continuously vary the value of theta, the exponential e^(i*theta) will continuously walk across the circle, generating all the rotation matrices. So we still have only a single generator i, but we also have an extra parameter. I hope this answers your question.
By the way, think of exponentiation as repeated multiplication, where the factors are all infinitely close to 1. This is how the finite and infinite concepts of generators are linked to each other.
@@AllAnglesMaththank you very much for the long exaustive response!
I was wandering in the case of more general/abstract continuous groups how would you define exponentiation and why out of all operation is this particular one at play,
Is it always traceable back to matrix exponentiation by finding a matrix representation of the group?
This topic is very interesting!
Hope my questions arent too much to answer, in case i will be happy to wait for the next videos to satisfy my curiosity
@@fedebonons8453 This is a very big and interesting topic called Lie groups, and I hope to make a series of videos about it in the future.
For now, the informal way I think about it is that exponentiation is just repeated multiplication. We take a tiny step, and we repeat it many times. That's how you slowly walk around a circle, or around a higher-dimensional blob: By starting off in a certain direction, and then repeating that little step as often as needed. This is exactly what exponentiation *is*.
The video about Euler's formula already makes this a little more formal, by showing you what happens in the limit (when the steps become infinitely small, and there are infinitely many of them).
@@AllAnglesMath actually the more i think about it the more makes sense
i saw the video about euler formula but its still a bit hard to think of it in a more general sense
once again thank you!
Gracias! Muy buen material. Estas usando manim en las presentaciones, o es otra aplicacion?
No, hemos utilizado nuestra propia biblioteca para la renderización de animaciones. Está escrito en Python. Usamos OpenCV para producir el video final.
17:18 but SL(2) is more than just SO(2), no? We get SL as a kernel.
Edit: for example, SL(2) also contains SO(1, 1).
I don’t know any cookie recipes sorry. 😑
That's a good point. When you start from GL(2) and take the determinant, its kernel is indeed SL(2), the group of all matrices with a determinant of 1.
But in the video I explicitly say that we limit the determinant to O(2) first. In that case, the kernel is only SO(2), which contains only ortho matrices with determinant 1.
@@AllAnglesMath Oh, sorry, I missed that part by going back and forth 😅 uh
@@05degrees No problem, it was still a good point to make.
Preserving structure is a symmetry -- syntax.
Syntax (structure) is dual to semantics -- languages, communication or information.
If mathematics is a language then it is dual.
Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force.
Symmetry is dual to conservation -- the duality of Noether's theorem.
Duality is being conserved -- the 5th law of thermodynamics!
"Always two there are" -- Yoda.
Force carriers are dual -- attraction is dual to repulsion.
Action is dual to reaction -- Sir Isaac Newton.
Photons and Gluons are force carriers hence they are dual.
Bosons are dual to Fermions -- atomic duality.
Subgroups are dual to subfields -- the Galois correspondence!
...would selling cookies and donating the proceeds work?
Absolutely 😋
Preserving structure is a symmetry -- syntax.
Syntax (structure) is dual to semantics -- languages, communication or information.
If mathematics is a language then it is dual.
Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force.
Symmetry is dual to conservation -- the duality of Noether's theorem.
Duality is being conserved -- the 5th law of thermodynamics!
"Always two there are" -- Yoda.
Force carriers are dual -- attraction is dual to repulsion.
Action is dual to reaction -- Sir Isaac Newton.
Photons and Gluons are force carriers hence they are dual.
Bosons are dual to Fermions -- atomic duality.
WHERES da Geometric Algebra & Geometric Calculus aka NON Newtonian mechanics
Be patient, young one. Good things will come to those who wait.
Preserving structure is a symmetry -- syntax.
Syntax (structure) is dual to semantics -- languages, communication or information.
If mathematics is a language then it is dual.
Beta plus decay is dual to beta minus decay or the W+ Boson is dual to the W- Boson -- the electro-weak force.
Symmetry is dual to conservation -- the duality of Noether's theorem.
Duality is being conserved -- the 5th law of thermodynamics!
"Always two there are" -- Yoda.
Force carriers are dual -- attraction is dual to repulsion.
Action is dual to reaction -- Sir Isaac Newton.
Photons and Gluons are force carriers hence they are dual.
Bosons are dual to Fermions -- atomic duality.
Subgroups are dual to subfields -- the Galois correspondence!