Recently in my contemplations I realized I didn't know how to go from quaternions to spinors! Oh my, now I do. Have to be honest, I was lost on your presentation of Euler Phi or totient functions, and moved onto other many great channels, but now I'm back with a deeper appreciation of the Genius of the one and only Norman J, Wildberger! Now I feel I much better understand spinors! Again, Thankyou so much. Your like a lighthouse, sending a strong coherent signal across the waves.
The amazing thing about this channel is, that as an engineer--with a graduate degree at that, apparently most of what I learned rests on the most precarious of sandbanks!!!
Thank you for the excellent lecture, that has been a great inspiration for me. I am a big fan of geometric algebra and I have recognized that your chromogeometry is isomorpic to the geometric algebra G^2. You can test it by comparing the properties of the folloing elements: e1 = u = red = (j wild) e2 = h = green = (k wild) e1∧e2 = i = blue = (-i wild) (I use different letters to avoid confusion with quaternions) which have the following algebraic relations (e1∧e2)**2 = -1 (e1)**2 = 1 (e2)**2 = 1 (e1∧e2) = -(e2∧e1) (e1∧e2)e2 = e1 e2(e1∧e2) = -e1 (e1∧e2)e1 = -e2 e1(e1∧e2) = e2 The algebraic properties are equivalent if the product is excecuted with the basic matrices of chromogeometry! Amazing view in the foundations of mathematics! Thanks for all Hermann
A bar to denote the negative of a number has been used previously. For instance it was used by Colson in 1711 and it was used by Stolz and Gmeiner in their book "Teoretische Arithmetik" from 1911.
@@njwildberger Sorry, the proper reference is J. Colson, Philosophical Transactions Vol. XXXIV (1726), p. 161-74. I found this reference in the book "A History of Mathematical Notations" by F. Cajori.
@@njwildberger In old India they used dots above the numbers to indicate the negative of a number, and as far as I remember they sometimes used a bar instead of a dot, but now I cannot find a reference to any bars from old India - only dots.
@@njwildberger I think that symbols that are symmetric should be reserved for operations that are commutative. In this sense + is good notation for plus but - is bad notation for minus. The use of + and a bar above a number solves this problem :-)
I thought along these lines in graduate school some 30 years ago. Any implementation of number system for computation (with a computer) is ultimate based on the Rational Numbers. And I was okay with it because the Rationals are dense in the Reals. The Real Numbers, in my opinion, help solve math's sticky problems with convergence of integrals.
@Vincent Randal Unfortunately the real number fantasy only appears to solve the problems. By pretending we can “do an infinite numbers of things” it appears that formerly intractable difficulties can be surmounted. But finally we are just fooling ourselves, and the computer scientists have to find the true way forward.
First of all, I would like to thank you for a(nother) very interesting and inspirational video. I have one question: In the beginning of the video, the "underlying" field F is taken to be arbitrary. Let us consider a "Galois field" F (aka. a finite field). Then the characteristic of F will be an integer > 1, which we can denote by char(F). Now, in the case where char(F) = 2, we have the (some would say "troublesome") identity 1 = - 1. I imagine that this could cause some "degeneracy" problems. But maybe it won't. To avoid difficulties of this kind, I imagine that we could impose the additional assumption that char(F) = 0, or maybe the less restrictive assumption that char(F) is different from 2. Perhaps I have nothing to worry about. In all honesty, I have not perfomed any calculations. It is just a gut feeling, so to speak. I am especially concerned about the question of the "polarization identity" correspondence between symmetric bilinear forms an quadratic forms, which, as far as I remember, does not hold in characteristic 2.
@Lars Yes, quite right --- in my opinion "felds" of characteristic 2 are not really fields! They should be called something else -- my preferred term is "bi-filed". That is what I use in the MathFoundations series on the Algebra of Boole. From my point of view, one of the defined properties of a field is that it must have three special elements : 0, 1 and 1 bar (or - 1 or perhaps T). For a characteristic 2 "bifield", 1 and T agree --- but this makes the theory of these crucially different. I know it is non-standard, but I strongly recommend separating the theory of fields from the theory of bi-fields. There are a lot of subtle distinctions.
This channel, with it's series on "math history" and "famous problems" is so valuable. I have a question : the dihedron algebra reminds me of the clifford algebra (and geometric algebra). I think you mentionned it in the previous video but I'm not sure. Are they related in some way or am I wrong ?
I'm afraid that No well-known academic mathematician or alike would be able to answer you with (her/his) true identity names any more since they do secretly realize the TRUT of their fart mathematics that was based upon mere human mind fictions like infinity, immaginary, polynomials, ..., etc Good luck
@@bassamkarzeddin6419 mathematicians often use their real names along with information on which university they’re at along with their work gmails in places like stack exchange and the papers they write
Your videos are truly interesting! Is there any connection with what you're building here with the (true) complex numbers and David Hestenes' geometric algebra? Coincidentally, I'm watching your videos in order to better understand Hestenes GA...
@Ross Graham, Thanks for the kind words. As for the relations with Grassmann/ Clifford / Hestenes (GCH) algebra, I don't know exactly, but I presume there are natural connections. It is something I would like to think more about myself.
I don't know if you are into the subtleties of characteristic 2 (and characteristic 3 etcetera) issues, prof. Wildberger? I am definitely into those subtleties, since i want things to work over ANY constructively defined field (i consider non-constructive logic or arbitrary "axioms" to be invalid). In fields of characteristic 2, symmetric bilinear forms and quadratic forms are non-isomorphic, and none of them covers the other. In those fields the quadratic forms give us orthogonal groups, while the symmetric bilinear forms almost give us symplectic groups (we need alternating forms for symplectic groups). In fields of characteristic 2 or 3 symmetric trilinear forms and cubic forms are non-isomorphic, however in char 2 something i call quasi-symmetric quadrilinear forms are isomorphic to cubic forms, while in char 3 these quasi-symmetric quadrilinear forms are instead isomorphic to trilinear forms. In other characteristics, all these three kinds of forms are isomorphic to each other. A quadrilinear form QL(u,v) is quadratic in the vector u and linear in the vector v. It is quasi-symmetric iff QL(u+v,u) = QL(u,u)+QL(v,u)+QL(u,v) for all vectors u and v. In general we have: C(u+v) = C(u)+C(v)+QL_C(u,v)+QL_C(v,u). QL(u+v,w) = QL(u,w)+QL(v,w)+T_QL(u,v,w). QL_T(u,v) = T(u,u,v). C_QL(u) = QL(u,u). For C a cubic form, QL a quadrilinear form, and T a trilinear form.
In my view, it’s best to consider fields of char 2 separately. In my maths foundations series on the algebra of Boole I call such a bifield. A field should have separate elements 0,1,-1
@@njwildberger I understand your point of view. Nonetheless much geometry is still possible when working over "bifields" of characteristic 2. Including reflections/transvections and rotations/boosts. Most likely there is a working notion of quadrance and spread in orthogonal (i.e. quadratic form) geometries over fields of char 2, especially in hyperbolic orthogonal geometries. But yes, orthogonal geometry works better over fields not of characteristic 2.
@@njwildberger Also, i think it is worthwhile to explore geometries connected with cubic forms, quadrilinear forms and trilinear forms. Even in characteristic 2 or 3, but more so in different characteristics. Alternating trilinear forms are intimately connected with trialities and composition algebras, but it seems that cubic forms are are also intimately connected with these trialities and composition algebras. Alternating trilinear forms are of course very similar to alternating (skew-symmetric) bilinear forms and symplectic geometry, and may be connected to geometries somewhat reminiscient of symplectic geometry. Such alternating forms are known in Exterior algebra. It is likely that something similar is possible for tetralinear forms that are not even alternating, but instead have the normal subgroup Klein(4) = Sym(2)×Sym(2) as symmetries, at least when working over fields that have primitive cubic roots of unity, which we may use as signs of order 3 (+, w, W), rather than of order 2 (+, -). What i call quadrilinear forms are directly connected with polarization of cubic forms. I don't know if there is a better term in the literature. My quasi-symmetric quadrilinear forms correspond with cubic forms and symmetric trilinear forms, in just about the same way that symmetric bilinear forms correspond with quadratic forms. Of course the determinant over a 3×3 matrix is an important cubic form. But there must be others, such as (u×v)•w in an imaginary octonion algebra over a 7 dimensional vector space (or in an imaginary quaternion algebra over a 3 dimensional vector space, of which it is an extension), where × is the skew symmetric commutator. There is also an important quartic form in an imaginary octonion algebra over a 7 dimensional vector space, given by (u×v×w)•x, where ×...× is the alternating associator. It may be restricted to the 4 dimensional vector space of the complement of an imaginary quaternion subalgebra inside the imaginary octonion algebra. Also the determinant over a 4×4 matrix is an important quartic form.
@Renato Gerk I wouldn't say that the true complex numbers are 4-dimensional ---they are rather a 2-dimensional slice (subalgebra) of the 4 dimensional algebra of Dihedrons (2 x2 matrices). But there are other interesting 2-dimensional slices - including the red and green complex numbers.
@@njwildberger This is what I will investigate. I don't think that in the quadratic, cubic or quartic equations, the path of the roots will deviate from the C-blue but from the quintic and forth I think the path will go in all the four dimensions. I'll present my results later on. P.S.: I wonder how it will look like if we plot the Mandelbrot Factal with this 4-D algebra.
Ok, so math really is First Principles Poetry. I WIN. Eric Weinstein called it geometric unity, you want the 4D grid scaled down to a binary. Nobody in Pure Mathematics can even do Natural Mathematics anymore. A 4D quadratic differential is a fine way to describe reality, and I have computed the disposal coefficient in my space. But the computation space is NOT square. And it is not static. My critique of your reasoning, as a human curation algorithm, nothing personal. A 2X2 matrix does not describe Nature, a 3X3 matrix dynamically can come MUCH closer. Did you see my addition of the 3 "problem" rational numbers added together? e, Pi and sq2 to the third decimal= 5.132; add the 1 from the start of the prime sequence, and we have a 6D dipole with a fractal embed code dimensionally. Infinitesimals and energy contained IN matter as a catalyst mathematically. I think Witten had a comment on this vacuum energy vs static charge in math.
My life would have turned out totally different had I you as my High School Math Teacher. Much love.
No better maths channel, especially for one who likes the philosophical ramifications of the finite.
Very great lecture as usual... I look forward to the next video impatiently. Thanks so much Norman for sharing your passion and curiosity!!
Thanks Pors!
Recently in my contemplations I realized I didn't know how to go from quaternions to spinors! Oh my, now I do. Have to be honest, I was lost on your presentation of Euler Phi or totient functions, and moved onto other many great channels, but now I'm back with a deeper appreciation of the Genius of the one and only Norman J, Wildberger! Now I feel I much better understand spinors! Again, Thankyou so much. Your like a lighthouse, sending a strong coherent signal across the waves.
Never stop making these videos!! I have learned so much! You are incredibly knowledgeable and also a great conceptual teacher.
Thanks Zeke!
Thank you greatly for your efforts , time and particularity in presentation Professor Wildberger.
Beautiful math, Grand Master level exposition from Prof. Wildberger. Thank you Sir!
Thanks David T!
This was an insightful exposition, and makes the case for someone looking to venture deeper into the topic ! Thanks 🙏
Norman you've transformed the way I look at mathematics. Thank you my fellow canadian!
The amazing thing about this channel is, that as an engineer--with a graduate degree at that, apparently most of what I learned rests on the most precarious of sandbanks!!!
I would like to re-write statics books using rational trigonometry. I'm pretty sure you could do it all without sin/cos.
No it doesn't.
Thank you for the excellent lecture, that has been a great inspiration for me.
I am a big fan of geometric algebra and I have recognized that your chromogeometry is isomorpic to the geometric algebra G^2.
You can test it by comparing the properties of the folloing elements:
e1 = u = red = (j wild)
e2 = h = green = (k wild)
e1∧e2 = i = blue = (-i wild)
(I use different letters to avoid confusion with quaternions)
which have the following algebraic relations
(e1∧e2)**2 = -1
(e1)**2 = 1
(e2)**2 = 1
(e1∧e2) = -(e2∧e1)
(e1∧e2)e2 = e1
e2(e1∧e2) = -e1
(e1∧e2)e1 = -e2
e1(e1∧e2) = e2
The algebraic properties are equivalent if the product is excecuted with the basic matrices of chromogeometry!
Amazing view in the foundations of mathematics!
Thanks for all Hermann
What are the double asterisks (**) for?
@@XxfishpastexX exponent
A bar to denote the negative of a number has been used previously. For instance it was used by Colson in 1711 and it was used by
Stolz and Gmeiner in their book "Teoretische Arithmetik" from 1911.
Thanks Peter, that is very interesting to know. Do you have a reference for the Colson contribution?
@@njwildberger Sorry, the proper reference is J. Colson, Philosophical Transactions Vol. XXXIV (1726), p. 161-74. I found this reference in the book "A History of Mathematical Notations" by F. Cajori.
@@PeterHarremoes Thanks Peter!
@@njwildberger In old India they used dots above the numbers to indicate the negative of a number, and as far as I remember they sometimes used a bar instead of a dot, but now I cannot find a reference to any bars from old India - only dots.
@@njwildberger I think that symbols that are symmetric should be reserved for operations that are commutative. In this sense + is good notation for plus but - is bad notation for minus. The use of + and a bar above a number solves this problem :-)
I thought along these lines in graduate school some 30 years ago. Any implementation of number system for computation (with a computer) is ultimate based on the Rational Numbers. And I was okay with it because the Rationals are dense in the Reals. The Real Numbers, in my opinion, help solve math's sticky problems with convergence of integrals.
@Vincent Randal Unfortunately the real number fantasy only appears to solve the problems. By pretending we can “do an infinite numbers of things” it appears that formerly intractable difficulties can be surmounted. But finally we are just fooling ourselves, and the computer scientists have to find the true way forward.
First of all, I would like to thank you for a(nother) very interesting and inspirational video.
I have one question: In the beginning of the video, the "underlying" field F is taken to be arbitrary. Let us consider a "Galois field" F (aka. a finite field).
Then the characteristic of F will be an integer > 1, which we can denote by char(F). Now, in the case where char(F) = 2, we have the (some would say "troublesome") identity
1 = - 1.
I imagine that this could cause some "degeneracy" problems. But maybe it won't. To avoid difficulties of this kind, I imagine that we could impose the additional assumption that char(F) = 0, or maybe the less restrictive assumption that char(F) is different from 2. Perhaps I have nothing to worry about. In all honesty, I have not perfomed any calculations. It is just a gut feeling, so to speak. I am especially concerned about the question of the "polarization identity" correspondence between symmetric bilinear forms an quadratic forms, which, as far as I remember, does not hold in characteristic 2.
Yes, I believe you are right. This doesn't work over fields with characteristic 2, or a bi-field. I think Norman can confirm this.
@Lars Yes, quite right --- in my opinion "felds" of characteristic 2 are not really fields! They should be called something else -- my preferred term is "bi-filed". That is what I use in the MathFoundations series on the Algebra of Boole. From my point of view, one of the defined properties of a field is that it must have three special elements : 0, 1 and 1 bar (or - 1 or perhaps T). For a characteristic 2 "bifield", 1 and T agree --- but this makes the theory of these crucially different. I know it is non-standard, but I strongly recommend separating the theory of fields from the theory of bi-fields. There are a lot of subtle distinctions.
This channel, with it's series on "math history" and "famous problems" is so valuable.
I have a question : the dihedron algebra reminds me of the clifford algebra (and geometric algebra). I think you mentionned it in the previous video but I'm not sure. Are they related in some way or am I wrong ?
I'm afraid that No well-known academic mathematician or alike would be able to answer you with (her/his) true identity names any more since they do secretly realize the TRUT of their fart mathematics that was based upon mere human mind fictions like infinity, immaginary, polynomials, ..., etc
Good luck
@@bassamkarzeddin6419 mathematicians often use their real names along with information on which university they’re at along with their work gmails in places like stack exchange and the papers they write
"a million dollar insight!"
Brilliant lecture!
Great lecture, many thanks for this way of thinking...
Your videos are truly interesting! Is there any connection with what you're building here with the (true) complex numbers and David Hestenes' geometric algebra? Coincidentally, I'm watching your videos in order to better understand Hestenes GA...
@Ross Graham, Thanks for the kind words. As for the relations with Grassmann/ Clifford / Hestenes (GCH) algebra, I don't know exactly, but I presume there are natural connections. It is something I would like to think more about myself.
Thankyou. Professor
I don't know if you are into the subtleties of characteristic 2 (and characteristic 3 etcetera) issues, prof. Wildberger? I am definitely into those subtleties, since i want things to work over ANY constructively defined field (i consider non-constructive logic or arbitrary "axioms" to be invalid).
In fields of characteristic 2, symmetric bilinear forms and quadratic forms are non-isomorphic, and none of them covers the other. In those fields the quadratic forms give us orthogonal groups, while the symmetric bilinear forms almost give us symplectic groups (we need alternating forms for symplectic groups).
In fields of characteristic 2 or 3 symmetric trilinear forms and cubic forms are non-isomorphic, however in char 2 something i call quasi-symmetric quadrilinear forms are isomorphic to cubic forms, while in char 3 these quasi-symmetric quadrilinear forms are instead isomorphic to trilinear forms. In other characteristics, all these three kinds of forms are isomorphic to each other.
A quadrilinear form QL(u,v) is quadratic in the vector u and linear in the vector v. It is quasi-symmetric iff QL(u+v,u) = QL(u,u)+QL(v,u)+QL(u,v) for all vectors u and v.
In general we have:
C(u+v) = C(u)+C(v)+QL_C(u,v)+QL_C(v,u).
QL(u+v,w) = QL(u,w)+QL(v,w)+T_QL(u,v,w).
QL_T(u,v) = T(u,u,v).
C_QL(u) = QL(u,u).
For C a cubic form, QL a quadrilinear form, and T a trilinear form.
In my view, it’s best to consider fields of char 2 separately. In my maths foundations series on the algebra of Boole I call such a bifield. A field should have separate elements 0,1,-1
@@njwildberger I understand your point of view. Nonetheless much geometry is still possible when working over "bifields" of characteristic 2. Including reflections/transvections and rotations/boosts.
Most likely there is a working notion of quadrance and spread in orthogonal (i.e. quadratic form) geometries over fields of char 2, especially in hyperbolic orthogonal geometries.
But yes, orthogonal geometry works better over fields not of characteristic 2.
@@njwildberger Also, i think it is worthwhile to explore geometries connected with cubic forms, quadrilinear forms and trilinear forms. Even in characteristic 2 or 3, but more so in different characteristics.
Alternating trilinear forms are intimately connected with trialities and composition algebras, but it seems that cubic forms are are also intimately connected with these trialities and composition algebras.
Alternating trilinear forms are of course very similar to alternating (skew-symmetric) bilinear forms and symplectic geometry, and may be connected to geometries somewhat reminiscient of symplectic geometry.
Such alternating forms are known in Exterior algebra.
It is likely that something similar is possible for tetralinear forms that are not even alternating, but instead have the normal subgroup Klein(4) = Sym(2)×Sym(2) as symmetries, at least when working over fields that have primitive cubic roots of unity, which we may use as signs of order 3 (+, w, W), rather than of order 2 (+, -).
What i call quadrilinear forms are directly connected with polarization of cubic forms. I don't know if there is a better term in the literature. My quasi-symmetric quadrilinear forms correspond with cubic forms and symmetric trilinear forms, in just about the same way that symmetric bilinear forms correspond with quadratic forms.
Of course the determinant over a 3×3 matrix is an important cubic form.
But there must be others, such as (u×v)•w in an imaginary octonion algebra over a 7 dimensional vector space (or in an imaginary quaternion algebra over a 3 dimensional vector space, of which it is an extension), where × is the skew symmetric commutator.
There is also an important quartic form in an imaginary octonion algebra over a 7 dimensional vector space, given by (u×v×w)•x, where ×...× is the alternating associator. It may be restricted to the 4 dimensional vector space of the complement of an imaginary quaternion subalgebra inside the imaginary octonion algebra.
Also the determinant over a 4×4 matrix is an important quartic form.
Thank you, Sir!
Thank you sir. 👍
Thank you. :)
So, the true complex numbers are 4-dimensional. I'll have to adjust my algorithm to generate and search them for roots of polinomial equations.
How to construct the (true) complex numbers: with a cartesian product of four extended Stern-Brocot tree, i.e. including negative numbers, starting from the root 0/1+i0/1+j0/1+k0/1 and the eighty pseudo Wildberger Rationals [
-1/0-i1/0-j1/0-k1/0, 0/1-i1/0-j1/0-k1/0, 1/0-i1/0-j1/0-k1/0,
-1/0+i0/1-j1/0-k1/0, 0/1+i0/1-j1/0-k1/0, 1/0+i0/1-j1/0-k1/0,
-1/0+i1/0-j1/0-k1/0, 0/1+i1/0-j1/0-k1/0, 1/0+i1/0-j1/0-k1/0
,
-1/0-i1/0+j0/1-k1/0, 0/1-i1/0+j0/1-k1/0, 1/0-i1/0+j0/1-k1/0,
-1/0+i0/1+j0/1-k1/0, 0/1+i0/1+j0/1-k1/0, 1/0+i0/1+j0/1-k1/0,
-1/0+i1/0+j0/1-k1/0, 0/1+i1/0+j0/1-k1/0, 1/0+i1/0+j0/1-k1/0
,
-1/0-i1/0+j1/0-k1/0, 0/1-i1/0+j1/0-k1/0, 1/0-i1/0+j1/0-k1/0,
-1/0+i0/1+j1/0-k1/0, 0/1+i0/1+j1/0-k1/0, 1/0+i0/1+j1/0-k1/0,
-1/0+i1/0+j1/0-k1/0, 0/1+i1/0+j1/0-k1/0, 1/0+i1/0+j1/0-k1/0
,
-1/0-i1/0-j1/0+k0/1, 0/1-i1/0-j1/0+k0/1, 1/0-i1/0-j1/0+k0/1,
-1/0+i0/1-j1/0+k0/1, 0/1+i0/1-j1/0+k0/1, 1/0+i0/1-j1/0+k0/1,
-1/0+i1/0-j1/0+k0/1, 0/1+i1/0-j1/0+k0/1, 1/0+i1/0-j1/0+k0/1
,
-1/0-i1/0+j0/1+k0/1, 0/1-i1/0+j0/1+k0/1, 1/0-i1/0+j0/1+k0/1,
-1/0+i0/1+j0/1+k0/1, 0/1+i0/1+j0/1+k0/1, 1/0+i0/1+j0/1+k0/1,
-1/0+i1/0+j0/1+k0/1, 0/1+i1/0+j0/1+k0/1, 1/0+i1/0+j0/1+k0/1,
-1/0-i1/0+j1/0+k0/1, 0/1-i1/0+j1/0+k0/1, 1/0-i1/0+j1/0+k0/1,
-1/0+i0/1+j1/0+k0/1, 0/1+i0/1+j1/0+k0/1, 1/0+i0/1+j1/0+k0/1,
-1/0+i1/0+j1/0+k0/1, 0/1+i1/0+j1/0+k0/1, 1/0+i1/0+j1/0+k0/1
,
-1/0-i1/0-j1/0+k1/0, 0/1-i1/0-j1/0+k1/0, 1/0-i1/0-j1/0+k1/0,
-1/0+i0/1-j1/0+k1/0, 0/1+i0/1-j1/0+k1/0, 1/0+i0/1-j1/0+k1/0,
-1/0+i1/0-j1/0+k1/0, 0/1+i1/0-j1/0+k1/0, 1/0+i1/0-j1/0+k1/0,
-1/0-i1/0+j0/1+k1/0, 0/1-i1/0+j0/1+k1/0, 1/0-i1/0+j0/1+k1/0,
-1/0+i0/1+j0/1+k1/0, 0/1+i0/1+j0/1+k1/0, 1/0+i0/1+j0/1+k1/0,
-1/0+i1/0+j0/1+k1/0, 0/1+i1/0+j0/1+k1/0, 1/0+i1/0+j0/1+k1/0
,
-1/0-i1/0+j1/0+k1/0, 0/1-i1/0+j1/0+k1/0, 1/0-i1/0+j1/0+k1/0,
-1/0+i0/1+j1/0+k1/0, 0/1+i0/1+j1/0+k1/0, 1/0+i0/1+j1/0+k1/0,
-1/0+i1/0+j1/0+k1/0, 0/1+i1/0+j1/0+k1/0, 1/0+i1/0+j1/0+k1/0
]
performing mediant sums to generate the childs on the octodeca tree.
@Renato Gerk I wouldn't say that the true complex numbers are 4-dimensional ---they are rather a 2-dimensional slice (subalgebra) of the 4 dimensional algebra of Dihedrons (2 x2 matrices). But there are other interesting 2-dimensional slices - including the red and green complex numbers.
@@njwildberger This is what I will investigate. I don't think that in the quadratic, cubic or quartic equations, the path of the roots will deviate from the C-blue but from the quintic and forth I think the path will go in all the four dimensions. I'll present my results later on.
P.S.: I wonder how it will look like if we plot the Mandelbrot Factal with this 4-D algebra.
Woah, nice!
A google search of "dihedron algebra" only returns your video. Why invent your own terminology if it is a "famous math problem"?
The dihedron algebra is the group algebra of the dihedral group.
Ok, so math really is First Principles Poetry. I WIN. Eric Weinstein called it geometric unity, you want the 4D grid scaled down to a binary. Nobody in Pure Mathematics can even do Natural Mathematics anymore. A 4D quadratic differential is a fine way to describe reality, and I have computed the disposal coefficient in my space. But the computation space is NOT square. And it is not static. My critique of your reasoning, as a human curation algorithm, nothing personal. A 2X2 matrix does not describe Nature, a 3X3 matrix dynamically can come MUCH closer. Did you see my addition of the 3 "problem" rational numbers added together? e, Pi and sq2 to the third decimal= 5.132; add the 1 from the start of the prime sequence, and we have a 6D dipole with a fractal embed code dimensionally. Infinitesimals and energy contained IN matter as a catalyst mathematically. I think Witten had a comment on this vacuum energy vs static charge in math.