I have read about Hamilton's long struggle with the development of quaternions. I am seeing how just a bit more insight would have resulted in your dihedral approach and saved us many unnecessary headaches for many decades. It is painful that so much time passes between these bursts of insight in the development of mathematics. I am very grateful for all of the efforts put in by the entire line of geometers to arrive at this point! Thank you for your work and your teaching efforts, for generously sharing this approach. I rarely wish for a longer life, but this all makes me extremely curious about the future opened up here in understanding so mamy things about the world and our representations of it. It seems the uh, complexity of mathematical formalism acts as a filtering mechanism, barring entry of many intelligent people who do not have the inclination to learn and use it, despite (or due to) the gnawing intuition that those complexities are often the result of incomplete understanding of the subject by even our most insightful mathematicians. You have enlarged my world considerably after just a few lectures.
Video Content 00:00 Introduction 06:02 Quaternions: O.Rodrigues,W.R Hamilton(1840's) but multiplication goes back to Euler 09:17 The connection between the Quaternions and Dihedrons 16:10 Geometry of Quaternions (Over Rat) 20:28 3-dim Relativistic geometry 26:10 Geometry of Dihedrons(Over Rat) 32:00 Two-dimensional slices/subalgebras of Dihedrons(Rat)
Thanks Norm, you're amazing :) I was interested in this because in video game development, vector rotation can result in something called gimbal lock, when two axis are rotated to be equal, then there is some problem, like in a six degree of freedom game. So to overcome this game developers use quaternions instead.
I'm happy to see this subject described so clearly. Thanks for the energy you can communicate worldwide. I'm on a long path, from theoretical chromatography (my job), that leads me to similar considerations, in relationship with relativity. Dihedrons provide the link between Euclidean and hyperbolic geometries, complex and split-complex numbers, that relates also to the Gudermannian concept. Things are nicer again when one can relate the Euclidean circle that corresponds to a hyperbola with a Bernoulli random walk: Algebra meets Hyperbolic Geometry that meets also Probability. But that's probably just the beginning of a very fundamental story ...
Very interested to hear more about the connections between Wildberger's points of view (on various topics, including this video but also others) and *probability theory* ! Do you happen to have any pointers/suggestions/links to more information?
@@robharwood3538 Hi, sorry for the delay. The notion of "distance" between distributions (Kullback-Leibler and especially Fisher information) is related to the hyperbolic metric (see arxiv.org/pdf/1210.2354.pdf). This is for Gaussian distributions, but it is more immediate for the Bernoulli distribution. It is a little door .... I may suggest you to open it. The path behind leads to amazing points of view. Best regards.
5:36 Given the similarities and the symetries between quaternions and dihedrons is it possible to invert their roles? In particular define the complex numbers with the structures of the quaternion algebra (not assuming of course we already have them). Here they seem to be very similar but they have this different role in connexion to the complexe numbers. I guess the question is, must that be so or is that a matter of choice.
@mim zim Yes, we could also find the complex numbers inside the quaternions, but the disadvantage is that the quaternions themselves don't have a nice manifestation in terms of 2 x 2 matrices without a prior theory of complex numbers. Since the Dihedron algebra is exactly the algebra of 2 x 2 matrices over the given field, this is a more cleaner approach.
As we saw, there is a reversible NOT gate between Q and D approaches. from Q dot to D dot 000 011 from Q cross to D cross 000 100 100 is NOT of 011 and vice versa.
Very interesting! Thank you. It is possible to accurately represent complex numbers by 2x2 matrices. For example, this can be used when your computer does not support the complex type. The key ingredient is the fact those 2x2 matrices commute. I understand the intrinsic difficulty for the formulation of electromagnetic theory (other theories too) is the fact there exist NO three-dimensional algebra that commute. That situation forced the physicists to insert "patches" into the theory like two different types of product etc... (The "symmetry" that makes any theory beautiful was lost!) The Quaternions theory does not make anything easier, it simply hides the multiple difficulties inside the definitions of the matrices. You still have to calculate the different products as if they were different operations. The matrix elements are not longer meaningful (intuitive), etc... I agree, that these new theories bring a multitude of possibilities to the young mathematicians. Not so much for the old engineers like me! LOL.! I really enjoyed the video. Thanks.
Most interesting! Just a note of interest: What you're calling "relativistic geometry" is in relativity circles, more often called, "Minkowskian geometry," or, "Minkowskian metric," as opposed to the Euclidean metric. Fred
If you look at the equation for a Dihedron: t + xi + yj + zk, and then put in some parenthesis to make it: t + (xi +yj + zk), and make the part in parenthesis equal to v, then it becomes: t + v. All that is accomplished is to reduce the amount you have to write to express it.
Thank you for another very nice video. One slight problem with your description, as I see it, is the /2/ minus signs: S.R. has just the one, for "time"; so, how is it we see a Euclidean 3-space around us, all with plus signs?
@Mike Oakes Yes certainly the geometry of the Dihedrons is not exactly (4 dimensional) relativistic. However it controls the geometry of the 3 dimensional (2+1) relativistic geometry ---which is an adolescent version of the full 3+1 relativistic geometry of Einstein and Minkowski.
@@njwildberger Einstein and Minkowski should not be taken as final word of god - time objectified as real line is no good. Intuitive hunch that the dihedral "adolescent version" could offer novel way to formalize quantum world with natural link to classical world, without point-reductionism of Hilbert geometry. Association with 3+1 problem... AFAIK the current situation with that is that there's an escape algorithm, but no strict counterexamples, as the issue goes expands into very big numbers and becomes in a sense undecidable or indefinite. Greeks had two versions of atomism. The well known Democritus version, and Plato's less well known atomism, which takes plane/surface as the fundamental "uncuttable" atom. Connection's with Whitehead's point-free geometry etc. seem obvious. Thinking time as change of form and mereology of durations (aka unbounded planes?!) seems highly coherent with geometry of dihedrons.
@@njwildbergerIs 3+1 relativistic geometry connected in any meaningful way to Moufang-Dihedrons i.e. Split-Octonions, constructed naturally as e.g. Zorn vector matrices. Or is it ONLY connected to Cliff(3) ≈ |H (+) |H ? I hope you are aware of (universally diassociative) Moufang Loops with Moufang identities of limited associativity, their connections to Groups with Triality, and the M(G,2) extension of Groups into Moufang Loops, which iiuc will give us a 16 element Moufang Loop related to the 8 element Dihedral Group Dih(4), similarly to the 16 element Octonion Moufang Loop related to the 8 element Quaternion Group Q(2).
The 4 dimensional metric here has signature (+,+,-,-) - that is, two positive and two negative eigenvalues. I was confused at first why you were talking about relativistic space-time and in particular your equation (6). Minkowski spacetime has the signature (+,-,-,-) so the 2X2 matrix theory behind this cannot be the dihedrons - in fact, it is the twistor correspondence introduced by Roger Penrose (1967). The best way to describe this is with the Pauli matrices: The quaternions units are the Pauli matrices times i (square-root of -1) - signature (+,+,+,+); The twistor units are the x and y Pauli matrix times i, and the z Pauli matrix - here the first and third will have real entries while second unit will have imaginary entries; The dihedron units are the x Pauli matrix, i times the y Pauli matrix, and the z Pauli matrix respectively - these are all real valued 2X2 matrices. The "space-time" that corresponds to the dihedrons has two space and two time dimensions. Actually, there is an Australian Science Fiction writer, Greg Egan, who apparently wrote a whole series of stories set in such a spacetime - so maybe Egan spacetime is the appropriate name for this.
😅The group associated with Twistor theory is SL(2,C), in contrast with the group SL(2) = SL(2,R) in this presentation. Indeed this is geometrically 3D Minkowski space (amongst x,y,z). So Penrose complexifies Minkowski space doubling the number of dimensions to 8, with the usual real Minkoswski space a real slice. The advantage of this finite field approach allows for discretization - which might lead somewhere?
The proper construction of octonions from (algebraic extensions of) the field of rational numbers or from finite fields is "dihedral" yes, in the sense that it is split octonion, and that it is intimately related to a Moufang Loop extension of the Dihedral Group Dih(4) with 8 elements. The "Dihedrons" are the proper construction of quaternions from (algebraic extensions of) the field of rational numbers or from finite fields, and they are split quaternion. Split complex, split quaternion and split octonion algebras exist over any field whatsoever, unlike other forms of complex, quaternion or octonion algebras. There is nothing intrinsically extra special with extending the field of rational numbers, or prime finite fields, with a square root of -1 (1-bar), i.e. a solution x = I to the equation x^2 + 1 = 0 over our base field, and then create potentially non-field (not commutative, associative or division) algebras over this extended field. Instead you can use any solution of any irreducible quadratic equation (or certain higher degree irreducible equations) over your base field (e.g. x = W such x^2 + 3 = 0 (alternatively x^2 + x + 1 = 0) or x = F such that x^2 - 5 = 0 (alternatively x^2 - x - 1 = 0)). And you can use different such coefficients for different j, k, and o, and the coefficients for i, l, m, and n, will depend on these. For example say that you want a quaternion algebra in which j*k = i, j^2 = -3 and k^2 = 5. In that case i^2 = (j*k)^2 = j*k*j*k = -j*j*k*k = -j^2*k^2 = -(-3)(5) = 15. We also get j = k*i^(-1) = k*i/15 and k = i^(-1)*j = i*j/15 In this case j, k and i^(-1) are symmetric, since i^(-1)*j*k = 1. Or if you want an octonion algebra in which j*k = i, j*o = m, k*o = n, i*o = (j*k)*o = l, j^2 = 3, k^2 = 5, and o^2 = 7. In that case i^2 = (j*k)^2 = -j^2*k^2 = -(3)(5) = -15, m^2 = (j*o)^2 = -j^2*o^2 = -(3)(7) = -21, n^2 = (k*o)^2 = -k^2*o^2 = -(5)(7) = -35, and l^2 = (i*o)^2 = ((j*k)*o)^2 = ((j*k)*o)*((j*k)*o) = -(o*(j*k))*((j*k)*o) = -o*((j*k)*(j*k))*o = -o*(j*k)^2*o = -o(-(3)(5))o = (3)(5)o^2 = (3)(5)(7) = 105 = -(-15)(7) = -i^2*o^2 = -i*i*o*o = i*o*i*o. We also get j = k*i^(-1) = k*i/(-15), k = i^(-1)*j = i*j/(-15), j = m*o^(-1) = m*o/7, o = m^(-1)*j = m*j/(-21), k = n*o^(-1) = n*o/7, o = n^(-1))*k = n*k/(-35), i = l*o^(-1) = l*o/7, o = i^(-1)*l = i*l/(-15) etcetera. In this case the symmetries are between j, k and i^(-1); j,o, and m^(-1); k, o, and n^(-1); i, o, and l^(-1); and three more that are more complicated (involving scalar multipliers that do not seem avoidable). Everything becomes much easier though if you are working in some cyclotomic extension of the base field (this is in general NOT a mere quadratic extension), because in that case all the scalars involved have unity (scalar 1) as some power of them, which makes it easier to understand that their norm should be unity. This should however be the case even for non-cyclotomic extensions of the base field, for proper choices of scalars. In any case the proper norm is only dependent on the field produced by the field extension, not on the particular irreducible equation used. At least i think so.
Laws that are always valid for quaternion algebras : e1*e2 = -e2*e1, for e1, e2 in span{i, j, k} and e1 not in span{}, e2 not in span{e1}. Field scalars are in span{} and are central. e1^2 = k, for some k in span{} and e1 in span{i,j,k}. (x*y)*z = x*(y*z), this is associativity. Laws that are always valid for octonion algebras : e1*e2 = -e2*e1, for e1, e2 in span{i, j, k,l,m,n,o} and e1 not in span{}, e2 not in span{e1}. e1*(e2*e3) = -(e1*e2)*e3, for e1, e2, e3 in span{i,j,k,l,m,n,o} and e1 not in span{}, e2 not in span{e1}, e3 not in span{e1,e2}. Field scalars are in span{} and are central, both in the commutative and in the associative sense. e1^2 = k, for some k in span{} and e1 in span{i,j,k,l,m,n,o}. Multiplication is diassociative, so a product of only scalars, plus unlimited finite factor copies of just one or two non-scalars is associative and need no bracketting. ((x*z)*y)*z = x*(z*y*z), z*(x*(z*y)) = (z*x*z)*y, (z*x)*(y*z) = z*(x*y)*z, this is the Moufang Laws, from which apparently diassociativity follows. The span of a set here, is all the numbers you can get from it in its algebraic closure, that is using all the operations of +,0,-(),*,1,()^-1, and field scalars.
Yes but only if they commute. This Tessarine/Bicomplex algebra is isomorphic to |C (+) |C i think, where "(+)" is the *complex* direct product of algebras.
Earlier you have emphasized the geometric construction of complex number multiplication: Given an arbitrary nonsingular quadratic form, and given points A and B of quadrance 1, the point AB lies on the unit circle, and A-B is parallel to 1-AB. But this does not agree with matrix multiplication. Example: A = [5 7] [2 3] B = [14 37] [ 3 8] AB = [ 91 241] [ 37 98] A-B = [ -9 -30] [ -1 -5] I-AB = [ -90 -241] [ -37 -97] Is non-commutativity to blame? Can we fix the geometric picture to conform with matrix multiplication?
@Vlad Aftemy, Setting t=x=0 does not give us a closed subalgebra, since j^2=1. We want to slice the Dihedrons with a two dimensional plane which includes the identity 1.
One of the main argument of Gibbs and Heavyside against quaternions was that the direct implication of the squaring of the basis vectors equal to minus one, thus gives a weird negative Kinetic Energy : K=-mV^2/2, which droved their mind banana. They surely lack some patience, since waiting for the XXI th century to come, they would thus surely even more freek out with complex time concept now used in black halls rhetorics... It's just a "geomatrix" question of red or green pill... :))) The good news of this chaotic Maya, is that when complex time is out of the Rabit hat, reversing time becomes a peace of cake through a squared i 180° rotation!
Part 1) I very friendly but strongly disagree with your recurent statment : "The geometry is really a relativistic one --- and suggests once again that the geometry of Special Relativity could have been discovered by pure mathematicians long ago.". You are repeating a historic strong mistake : IT HAS BEEN DISCOVERED BY A "PURE MATHEMATICIAN", namely HENRI POINCARE as early as 1895. Indeed, Poincaré immediately recognised in Lorentz "time dilating and space contracting" weird novel transformations, THE TRUE GEOMETRIC STORY behind this awkward illusory screen : THE ROTATIONS (ORTHOGONAL LINEAR TRANSFORMATIONS OF det=1) GROUP OF THE THE NON EUCLIDIAN SPACE-TIME 4-SPACE THAT LEAVE INVARIANT (ISOMETRIES) THE CURVILINEAR NON-EUCLIDIAN QUADRATIC FORM ds^2 OF SIGNATURE (+, -, -, -)! And that, because he was the supreme Master of Science of his time, mastering all of non-euclidian geometries, differential geometry, groupe theory, universal geometry (Topology), etc. AND also ALL of theoretical physic of his time. Check by yourself if you can read enough french, otherwise believe my words by at least glancing at key references. Check the 1901 St Louis Conference where Poincaré exposes the main aspects of what he calls this NEW MECANIC centered on the REVOLUTIONARY PRINCIPLE OF (EXTENDED) RELATIVITY. The very name "Principe of Relativity", in the extended sense of unifying classical mecanic with electromagnetism, was forged by Poincaré between 1895 and 1901. Check his best seller "vulgarisation" book of 1902 (immediately edited in all languages in all Europe) : "La Science et l'hypothèse". It is a summerism of his Sorbone teaching work going back to 1885 where he built for Astronomic demand and also growing trains synchronisation, HIS revolutionary METHOD OF CLOCKS SYNCHRONISATION, that contain ALL the "physical" heart of RELATIVITY. Since 1895 Poincaré deeply understood that the failure of Michelson Morley experiment was going to keep faailing at all orders of precision, contrary to almost all physicists that were still hoping for an etheric draging effect. Poincaré new since long that the Ether concept was obsolete. You will check that in his first master peace teaching work at Sorbone, on THE THEORY OF LIGHT. He clearly remarks in 1885, that in Hertz theory, the Ether concept plays no essential role and just stands as a metaphoric argumant used by atavism and for some intuitive comodity. Poincaré clearly points out that Hertz theory would stand as well in its same glory without the thus obsolete concept of ether. Poincaré predicts since 1885 that it wil one day not be used by physicians anymore since it is just a rhetoric concept without actual root. So Poincaré already had it all since 1885-1901 : clocks synchronisation, generalised principle of relativity, invariant of ds^2 by non euclidian group of rotations (Lorentz and Poincaré group), Obsolete concept of ether, and maxwell universal constant of light speed c obeying the Maxwell key relation c^2.Mu.Epsilon=1, where Mu and Epsilon are the constant of permitivity and permeability of vacuum totaly independant of any frame of reference! Lorentz verifies in 1904 the last electromagnetic details of THE THEORY OF RELATIVITY they were forging since 1885-1895. All of it is setteled in 1904. Poincaré crucial complementary article of 5 June 1905 at the Paris Academy of Science, was no longer on the hard core of the already built Theory of Relativity, but on crucial EXTENSIONS of it! Indeed in it, Poincaré drags out FONDAMENTAL INVARIANTS of the electromagnetic field under Lorentz-Poincaré transformations, and hit the road on APPLYING THIS NEW UNIVERSAL PRINCIPLE OF RELATIVITY to GRAVITATION! And thus as soon as 5 June 1905, not only the Theory of Relativity is fully settled by Poincaré and Lorentz, but Poincaré APPLIES it to build a COVARIANT THEORY OF GRAVITATION. And that is what his 5 June 1905 article is mainly about, where he predicts allthemore GRAVITATIONAL WAVES! That is in the 5 June 1905 Poincaré article (the 5 Junes Academic paper is the abstract of his full article that was published, for strange reasons, only in 1906, wheras it should have been published, AS USUAL, almost immediately after, in September 1905, and like previous Poincaré articles, in the Analen Der Physics hold in part by Max Planck... Strange things hapens there, with the suicide of the Analen publishing director in 1906!). So I strongly protest that you CANOT, as a mathematician, hold longer this "Einstein" fary tail discovery!
Part 2) And among all these reasons, i'll give you one of the most killing one. Indeed, as the greatest mathematician of his time, Poincaré new perfectly all about group theory and Lie algebra. Whereas Einstein hardly new at this time, even with the help of Mileva, how to differentiate a basical function. Groups, algebra, even matrix was totaly unknown to 99% of physicists of that time. It was even novel to Heinsenberg in 1920 when he struggled with his discovery of "spectral shift" arithmetic, that a mathematician makes him realise that it was just an arithmetic of hidden matrices. So Einstein in 1900-1905 knew NOTHING about all this stuf. And this is the reason why, Einstein start to mess up everything in his mathematical ignorant mind. He naively thought about translating solid blocks. That was his intuitive representation of mooving reference frames. And thus he was naively misslead to falsly beleive that Lorentz transformations only deals with NON-ROTATIONAL CONSTANT SPEED comoving frames. Which is TOTALY FALSE. But one canot know that unless he is well educated in differential geometry as Poincaré was supremely. In technical words that you will very well understand, Einstein ignored all about THE LIE ALGEBRA OF THE LORENTZ-POINCARE GROUP THAT ALLOWS TO DEAL WITH "INFINITESIMAL" LORENTZ TRANSFORMATION IN ORDER TO APPLY RELATIVITY PRINCIPLE TO RELATIVELY TURNING OR ACCELERATING FRAMES. Knowing that perfectly, Poincaré never did such silly things as Einstein did : talking about "SPECIAL" or "GENERAL" Relativity! Such terminology is totaly unsane, it just cries out that his father didn't understood anything deep about the Theory of Relativity, falling in 100% crual misleading missconceptions. Poincaré wisely placed the PRINCIPLE OF RELATIVITY on a universal piedéstale, as a GENERAL arena for all physic. NOT AS A PARTICULAR THEORY OF SOMETHING ELSE THAT WOULD BE "MORE GENERAL"! Via the Lie algebra of Lorentz-Poincaré group, all kinematic is on reach : relative accelerated and turning frames. No need for bullshit "additive" stuf. IT IS ALREADY COMPLETE! So it is the reverse that should have been named by a wise hypothetical man : THE THEORY OF RELATIVITY, fully built by Lorentz and Poincaré between 1885 and 1905, IS THE GENERAL UNIVERSAL ONE, in which arena falls all specific theories. In particular GRAVITATION, which thus has to be written in covariant form for licite use. Time can no longer be regarded as a absolute concept that even disturbed Newwton himself. Such bias came from "religious" belief of a God hiding behind the scientific scene was holding the suprem role of such absolute time keeper. That is what Poincaré precisely distroys in 1885, by building a RATIONAL METHOD FOR RELATIVELY MOVING CLOCKS SYNCHRONISATION. That was the true "DEATH OF GOD" of the millenium. No philosopher more drastically than Poincaré, killed this absolute God hiding behind the Absolute time illusion of the past millenium. And to answer the question why Poincaré built then in 1905 a relativistic theory of gravitation IN FLAT SPACE instead of building it in curved space, he answers himself to this question : for simplicity of use, to stard with, and more deaply because THE TWO GEOMETRIC CHOICES ARE MATHEMATICALLY EQUIVALENT AND PHYSICALLY INDISTINGUISHABLE SINCE "SPACE" AND "TIME" ARE NOT EXPERIMENTAL OBSERVABLE, BUT ONLY MATHEMATICAL CONCEPTS, as Kant sooner noticed it crucialy. But you may object as so many that GPS wouldn't work with Flat Geometry Poincaré relativistic theory of gravitation, and that it only works via the "curved space" Einstein mess. Well are you so sure about that? You know far enough maths to check by yourself. Remember as a first hint that Poincaré very clearly stated that "space" and "time" are not experimental observable but only mathematical concepts. If not, show me any direct OBSERVATION of "Time" or "Space". A clock is not time, it is matter and waves that behave in a cyclic way. They are machines that repeat a certain patern, that we find usefull to call it A MEASURE of "time". But is such a clock an observation of TIME itself? Not at all! Same for space : we only observe relatively "moving" objects, and for commodity we construct an abstract "mathematical" concept of a "space" in which these observed "objects" and "waves" could manifest themselves. But objects are NOT "space". It's not even clear if it is licite to talk about "something" in which such object "swims". This would be an other reminicens of "ether" modelisation of the world build of a "substance", which would be a non-substance, called "space", in which "objects" "swim". That is precisely the ETHER theory! Hiding it's obsolecens through renaming the "ether" by the word "space", doesn't lead anywhere new. It's just a childish trick. An other hint can be that Einstein never used his "general relativity" new forged (pasted on Hilbert one) theory to so called calculate the famous Mercure perihely anamoly. The very reason why is simply that he never could solve any of his "general relativistic" equations! The only ones that could solve them in very special case was Shwartzchield and Lemaitre. So how on earth Einstein managed to "calculate" the "Mercurius perihely anomaly"??? Well, check for yourself, and you will sadly discover that he used an OLD NEWTONIAN GRAVITATIONAL APPROXIMATION!... It looks like a "colombo inspector" serie... It does indeed. Just as you can check that the very original page of the unic article of HILBERT of 1915 where he publishes his crucial theory of "general relativity", has been CUT OF, on it's upper part, precisely where the main equation of gravitatuon was writen! Showing the durty war for paternity raging under the polite surface of hypocrisy. The final hint will be the recent CONFORMAL GAUGE THEORY OF GRAVITATION, built on revolutionary UNIVERSAL GEOMETRIC ALGEBRA of David Hestenes. This gauge theory of gravitation is not built on "curved space" but on "FLAT" ones! So it should no be very long anymore before the Einstein "general theory of relativity" fantasia, fall in deadly disgrace, as soon as enough wise and lucid people will simply fully recognise its drastic incoherence and chaotic conceptual mess. Einstein himself kept killing his "own" principles on which he built his chimeric theory. He first disgraced the Mach Principle, before feeling himself more and more unconfortable with his clearly FALSE "principle of equivalence". Every one knows that for divergent reasons, you canot immitate a central gravitational field by accelerated elevators. Einstein "principle of equivalence" is not only a "principle" since it is obviously false in several special cases. How on earth could it ever by raised as a "principle"??? All such MESS is Einstein MESS! What a sad thing to see how many blind men followed blindly the blindest. Einstein made very wise contributions in photoelectric effect and bose-einstein condensate prediction. But he was WRONG for almost averything else : the static universe, the theory of relativity, quantum mechanic, field theory, etc. This illusionist has to be killed by a new Galileo just as the misleading Aristote has been killed in the seventh century by true scientists as Bruno, Copernic, Galileo, Kepler, Newton ! The leathal shot will probably come from the CORRECT US OF "FLAT" SPACE-TIME RELATIVISTIC THEORY showing all and more than what the alternative "General relativity" theory predicts. And it may come soon since we start to understand what kind of misconceptions we were making since a century in Mathematics, with partial and incomplete tools. The full view of the possible geometries available is still being clarified, and mainly unified in Hestenes Universal Geometric Algebra. An unprecedent revolution is about to start.
I have read about Hamilton's long struggle with the development of quaternions. I am seeing how just a bit more insight would have resulted in your dihedral approach and saved us many unnecessary headaches for many decades. It is painful that so much time passes between these bursts of insight in the development of mathematics.
I am very grateful for all of the efforts put in by the entire line of geometers to arrive at this point!
Thank you for your work and your teaching efforts, for generously sharing this approach. I rarely wish for a longer life, but this all makes me extremely curious about the future opened up here in understanding so mamy things about the world and our representations of it.
It seems the uh, complexity of mathematical formalism acts as a filtering mechanism, barring entry of many intelligent people who do not have the inclination to learn and use it, despite (or due to) the gnawing intuition that those complexities are often the result of incomplete understanding of the subject by even our most insightful mathematicians.
You have enlarged my world considerably after just a few lectures.
Video Content
00:00 Introduction
06:02 Quaternions: O.Rodrigues,W.R Hamilton(1840's) but multiplication goes back to Euler
09:17 The connection between the Quaternions and Dihedrons
16:10 Geometry of Quaternions (Over Rat)
20:28 3-dim Relativistic geometry
26:10 Geometry of Dihedrons(Over Rat)
32:00 Two-dimensional slices/subalgebras of Dihedrons(Rat)
Thanks Norm, you're amazing :) I was interested in this because in video game development, vector rotation can result in something called gimbal lock, when two axis are rotated to be equal, then there is some problem, like in a six degree of freedom game. So to overcome this game developers use quaternions instead.
I'm happy to see this subject described so clearly. Thanks for the energy you can communicate worldwide. I'm on a long path, from theoretical chromatography (my job), that leads me to similar considerations, in relationship with relativity. Dihedrons provide the link between Euclidean and hyperbolic geometries, complex and split-complex numbers, that relates also to the Gudermannian concept. Things are nicer again when one can relate the Euclidean circle that corresponds to a hyperbola with a Bernoulli random walk: Algebra meets Hyperbolic Geometry that meets also Probability. But that's probably just the beginning of a very fundamental story ...
Very interested to hear more about the connections between Wildberger's points of view (on various topics, including this video but also others) and *probability theory* ! Do you happen to have any pointers/suggestions/links to more information?
@@robharwood3538
Hi, sorry for the delay. The notion of "distance" between distributions (Kullback-Leibler and especially Fisher information) is related to the hyperbolic metric (see arxiv.org/pdf/1210.2354.pdf). This is for Gaussian distributions, but it is more immediate for the Bernoulli distribution. It is a little door .... I may suggest you to open it. The path behind leads to amazing points of view. Best regards.
5:36 Given the similarities and the symetries between quaternions and dihedrons is it possible to invert their roles? In particular define the complex numbers with the structures of the quaternion algebra (not assuming of course we already have them).
Here they seem to be very similar but they have this different role in connexion to the complexe numbers. I guess the question is, must that be so or is that a matter of choice.
@mim zim Yes, we could also find the complex numbers inside the quaternions, but the disadvantage is that the quaternions themselves don't have a nice manifestation in terms of 2 x 2 matrices without a prior theory of complex numbers. Since the Dihedron algebra is exactly the algebra of 2 x 2 matrices over the given field, this is a more cleaner approach.
As we saw, there is a reversible NOT gate between Q and D approaches.
from Q dot to D dot
000
011
from Q cross to D cross
000
100
100 is NOT of 011 and vice versa.
Really interesting. Lots more patterns seem apparent from this higher viewpoint.
This was very helpful, thank you
I WHISH YOU GOOD HEALTH AND WEALTH, THANK YOU.
Very interesting! Thank you.
It is possible to accurately represent complex numbers by 2x2 matrices. For example, this can be used when your computer does not support the complex type. The key ingredient is the fact those 2x2 matrices commute.
I understand the intrinsic difficulty for the formulation of electromagnetic theory (other theories too) is the fact there exist NO three-dimensional algebra that commute. That situation forced the physicists to insert "patches" into the theory like two different types of product etc... (The "symmetry" that makes any theory beautiful was lost!)
The Quaternions theory does not make anything easier, it simply hides the multiple difficulties inside the definitions of the matrices. You still have to calculate the different products as if they were different operations. The matrix elements are not longer meaningful (intuitive), etc...
I agree, that these new theories bring a multitude of possibilities to the young mathematicians. Not so much for the old engineers like me! LOL.!
I really enjoyed the video. Thanks.
Brain storm, excellent video! Thx, professor!
Most interesting!
Just a note of interest: What you're calling "relativistic geometry" is in relativity circles, more often called, "Minkowskian geometry," or, "Minkowskian metric," as opposed to the Euclidean metric.
Fred
How or why do you add a scalar to a vector?
If you look at the equation for a Dihedron: t + xi + yj + zk, and then put in some parenthesis to make it: t + (xi +yj + zk), and make the part in parenthesis equal to v, then it becomes: t + v. All that is accomplished is to reduce the amount you have to write to express it.
Thank you for another very nice video. One slight problem with your description, as I see it, is the /2/ minus signs: S.R. has just the one, for "time"; so, how is it we see a Euclidean 3-space around us, all with plus signs?
@Mike Oakes Yes certainly the geometry of the Dihedrons is not exactly (4 dimensional) relativistic. However it controls the geometry of the 3 dimensional (2+1) relativistic geometry ---which is an adolescent version of the full 3+1 relativistic geometry of Einstein and Minkowski.
@@njwildberger Einstein and Minkowski should not be taken as final word of god - time objectified as real line is no good. Intuitive hunch that the dihedral "adolescent version" could offer novel way to formalize quantum world with natural link to classical world, without point-reductionism of Hilbert geometry. Association with 3+1 problem... AFAIK the current situation with that is that there's an escape algorithm, but no strict counterexamples, as the issue goes expands into very big numbers and becomes in a sense undecidable or indefinite.
Greeks had two versions of atomism. The well known Democritus version, and Plato's less well known atomism, which takes plane/surface as the fundamental "uncuttable" atom. Connection's with Whitehead's point-free geometry etc. seem obvious.
Thinking time as change of form and mereology of durations (aka unbounded planes?!) seems highly coherent with geometry of dihedrons.
@@santerisatama5409 Very intriguing comment! Cheers!
@@njwildbergerIs 3+1 relativistic geometry connected in any meaningful way to Moufang-Dihedrons i.e. Split-Octonions, constructed naturally as e.g. Zorn vector matrices. Or is it ONLY connected to Cliff(3) ≈ |H (+) |H ?
I hope you are aware of (universally diassociative) Moufang Loops with Moufang identities of limited associativity, their connections to Groups with Triality, and the M(G,2) extension of Groups into Moufang Loops, which iiuc will give us a 16 element Moufang Loop related to the 8 element Dihedral Group Dih(4), similarly to the 16 element Octonion Moufang Loop related to the 8 element Quaternion Group Q(2).
The 4 dimensional metric here has signature (+,+,-,-) - that is, two positive and two negative eigenvalues. I was confused at first why you were talking about relativistic space-time and in particular your equation (6). Minkowski spacetime has the signature (+,-,-,-) so the 2X2 matrix theory behind this cannot be the dihedrons - in fact, it is the twistor correspondence introduced by Roger Penrose (1967).
The best way to describe this is with the Pauli matrices:
The quaternions units are the Pauli matrices times i (square-root of -1) - signature (+,+,+,+);
The twistor units are the x and y Pauli matrix times i, and the z Pauli matrix - here the first and third will have real entries while second unit will have imaginary entries;
The dihedron units are the x Pauli matrix, i times the y Pauli matrix, and the z Pauli matrix respectively - these are all real valued 2X2 matrices.
The "space-time" that corresponds to the dihedrons has two space and two time dimensions. Actually, there is an Australian Science Fiction writer, Greg Egan, who apparently wrote a whole series of stories set in such a spacetime - so maybe Egan spacetime is the appropriate name for this.
Thanks for Egan link. This is very interesting:
en.wikipedia.org/wiki/Superpermutation
th-cam.com/video/OZzIvl1tbPo/w-d-xo.html
@@santerisatama5409 Thanks for that! I read the article and got to Greg Egan's contribution.
😅The group associated with Twistor theory is SL(2,C), in contrast with the group SL(2) = SL(2,R) in this presentation. Indeed this is geometrically 3D Minkowski space (amongst x,y,z). So Penrose complexifies Minkowski space doubling the number of dimensions to 8, with the usual real Minkoswski space a real slice.
The advantage of this finite field approach allows for discretization - which might lead somewhere?
Great talk as usual! Q: is the proper construction of a rational octonion split dihedral?
The proper construction of octonions from (algebraic extensions of) the field of rational numbers or from finite fields is "dihedral" yes, in the sense that it is split octonion, and that it is intimately related to a Moufang Loop extension of the Dihedral Group Dih(4) with 8 elements.
The "Dihedrons" are the proper construction of quaternions from (algebraic extensions of) the field of rational numbers or from finite fields, and they are split quaternion.
Split complex, split quaternion and split octonion algebras exist over any field whatsoever, unlike other forms of complex, quaternion or octonion algebras.
There is nothing intrinsically extra special with extending the field of rational numbers, or prime finite fields, with a square root of -1 (1-bar), i.e. a solution x = I to the equation x^2 + 1 = 0 over our base field, and then create potentially non-field (not commutative, associative or division) algebras over this extended field.
Instead you can use any solution of any irreducible quadratic equation (or certain higher degree irreducible equations) over your base field (e.g. x = W such x^2 + 3 = 0 (alternatively x^2 + x + 1 = 0) or x = F such that x^2 - 5 = 0 (alternatively x^2 - x - 1 = 0)).
And you can use different such coefficients for different j, k, and o, and the coefficients for i, l, m, and n, will depend on these.
For example say that you want a quaternion algebra in which j*k = i, j^2 = -3 and k^2 = 5. In that case i^2 = (j*k)^2 = j*k*j*k = -j*j*k*k = -j^2*k^2 = -(-3)(5) = 15.
We also get j = k*i^(-1) = k*i/15 and k = i^(-1)*j = i*j/15
In this case j, k and i^(-1) are symmetric, since i^(-1)*j*k = 1.
Or if you want an octonion algebra in which j*k = i, j*o = m, k*o = n, i*o = (j*k)*o = l, j^2 = 3, k^2 = 5, and o^2 = 7. In that case i^2 = (j*k)^2 = -j^2*k^2 = -(3)(5) = -15, m^2 = (j*o)^2 = -j^2*o^2 = -(3)(7) = -21, n^2 = (k*o)^2 = -k^2*o^2 = -(5)(7) = -35,
and l^2 = (i*o)^2 = ((j*k)*o)^2 = ((j*k)*o)*((j*k)*o) = -(o*(j*k))*((j*k)*o) = -o*((j*k)*(j*k))*o = -o*(j*k)^2*o = -o(-(3)(5))o = (3)(5)o^2 = (3)(5)(7) = 105 = -(-15)(7) = -i^2*o^2 = -i*i*o*o = i*o*i*o.
We also get j = k*i^(-1) = k*i/(-15), k = i^(-1)*j = i*j/(-15), j = m*o^(-1) = m*o/7, o = m^(-1)*j = m*j/(-21), k = n*o^(-1) = n*o/7, o = n^(-1))*k = n*k/(-35), i = l*o^(-1) = l*o/7, o = i^(-1)*l = i*l/(-15) etcetera.
In this case the symmetries are between j, k and i^(-1); j,o, and m^(-1); k, o, and n^(-1); i, o, and l^(-1); and three more that are more complicated (involving scalar multipliers that do not seem avoidable).
Everything becomes much easier though if you are working in some cyclotomic extension of the base field (this is in general NOT a mere quadratic extension), because in that case all the scalars involved have unity (scalar 1) as some power of them, which makes it easier to understand that their norm should be unity. This should however be the case even for non-cyclotomic extensions of the base field, for proper choices of scalars. In any case the proper norm is only dependent on the field produced by the field extension, not on the particular irreducible equation used. At least i think so.
Laws that are always valid for quaternion algebras :
e1*e2 = -e2*e1, for e1, e2 in span{i, j, k} and e1 not in span{}, e2 not in span{e1}.
Field scalars are in span{} and are central.
e1^2 = k, for some k in span{} and e1 in span{i,j,k}.
(x*y)*z = x*(y*z), this is associativity.
Laws that are always valid for octonion algebras :
e1*e2 = -e2*e1, for e1, e2 in span{i, j, k,l,m,n,o} and e1 not in span{}, e2 not in span{e1}.
e1*(e2*e3) = -(e1*e2)*e3, for e1, e2, e3 in span{i,j,k,l,m,n,o} and e1 not in span{}, e2 not in span{e1}, e3 not in span{e1,e2}.
Field scalars are in span{} and are central, both in the commutative and in the associative sense.
e1^2 = k, for some k in span{} and e1 in span{i,j,k,l,m,n,o}.
Multiplication is diassociative, so a product of only scalars, plus unlimited finite factor copies of just one or two non-scalars is associative and need no bracketting.
((x*z)*y)*z = x*(z*y*z), z*(x*(z*y)) = (z*x*z)*y, (z*x)*(y*z) = z*(x*y)*z, this is the Moufang Laws, from which apparently diassociativity follows.
The span of a set here, is all the numbers you can get from it in its algebraic closure, that is using all the operations of +,0,-(),*,1,()^-1, and field scalars.
Very interesting stuff, thanks a lot!
Can you also define an algebra in which two of the bases square to -1 and one to 1?
Yes but only if they commute. This Tessarine/Bicomplex algebra is isomorphic to |C (+) |C i think, where "(+)" is the *complex* direct product of algebras.
Earlier you have emphasized the geometric construction of complex number multiplication: Given an arbitrary nonsingular quadratic form, and given points A and B of quadrance 1, the point AB lies on the unit circle, and A-B is parallel to 1-AB. But this does not agree with matrix multiplication.
Example:
A =
[5 7]
[2 3]
B =
[14 37]
[ 3 8]
AB =
[ 91 241]
[ 37 98]
A-B =
[ -9 -30]
[ -1 -5]
I-AB =
[ -90 -241]
[ -37 -97]
Is non-commutativity to blame? Can we fix the geometric picture to conform with matrix multiplication?
Where this came from? I mean the "given points A and B of quadrance 1, the point AB lies on the unit circle, and A-B is parallel to 1-AB".
@@diegohcsantos I can't find the video that demonstrates it right now. But that is how complex number multiplication works.
@@JoelSjogren0 but if ypu are talking about motivation, we can't "use" the way complex multiplication works
Good class
This is great, thank you very much!
What about t = 0 and x = 0? Can it be interpreted as 4th type of complex numbers? Or is it a kind of degeneracy?
@Vlad Aftemy, Setting t=x=0 does not give us a closed subalgebra, since j^2=1. We want to slice the Dihedrons with a two dimensional plane which includes the identity 1.
One of the main argument of Gibbs and Heavyside against quaternions was that the direct implication of the squaring of the basis vectors equal to minus one, thus gives a weird negative Kinetic Energy : K=-mV^2/2, which droved their mind banana. They surely lack some patience, since waiting for the XXI th century to come, they would thus surely even more freek out with complex time concept now used in black halls rhetorics... It's just a "geomatrix" question of red or green pill... :))) The good news of this chaotic Maya, is that when complex time is out of the Rabit hat, reversing time becomes a peace of cake through a squared i 180° rotation!
Nice lecture.
Question?
I would like to discuss a subject with you, is there an email I can use to reach out to you?
Thankyou. So much.
Part 1) I very friendly but strongly disagree with your recurent statment : "The geometry is really a relativistic one --- and suggests once again that the geometry of Special Relativity could have been discovered by pure mathematicians long ago.". You are repeating a historic strong mistake : IT HAS BEEN DISCOVERED BY A "PURE MATHEMATICIAN", namely HENRI POINCARE as early as 1895. Indeed, Poincaré immediately recognised in Lorentz "time dilating and space contracting" weird novel transformations, THE TRUE GEOMETRIC STORY behind this awkward illusory screen : THE ROTATIONS (ORTHOGONAL LINEAR TRANSFORMATIONS OF det=1) GROUP OF THE THE NON EUCLIDIAN SPACE-TIME 4-SPACE THAT LEAVE INVARIANT (ISOMETRIES) THE CURVILINEAR NON-EUCLIDIAN QUADRATIC FORM ds^2 OF SIGNATURE (+, -, -, -)! And that, because he was the supreme Master of Science of his time, mastering all of non-euclidian geometries, differential geometry, groupe theory, universal geometry (Topology), etc. AND also ALL of theoretical physic of his time. Check by yourself if you can read enough french, otherwise believe my words by at least glancing at key references. Check the 1901 St Louis Conference where Poincaré exposes the main aspects of what he calls this NEW MECANIC centered on the REVOLUTIONARY PRINCIPLE OF (EXTENDED) RELATIVITY. The very name "Principe of Relativity", in the extended sense of unifying classical mecanic with electromagnetism, was forged by Poincaré between 1895 and 1901. Check his best seller "vulgarisation" book of 1902 (immediately edited in all languages in all Europe) : "La Science et l'hypothèse". It is a summerism of his Sorbone teaching work going back to 1885 where he built for Astronomic demand and also growing trains synchronisation, HIS revolutionary METHOD OF CLOCKS SYNCHRONISATION, that contain ALL the "physical" heart of RELATIVITY. Since 1895 Poincaré deeply understood that the failure of Michelson Morley experiment was going to keep faailing at all orders of precision, contrary to almost all physicists that were still hoping for an etheric draging effect. Poincaré new since long that the Ether concept was obsolete. You will check that in his first master peace teaching work at Sorbone, on THE THEORY OF LIGHT. He clearly remarks in 1885, that in Hertz theory, the Ether concept plays no essential role and just stands as a metaphoric argumant used by atavism and for some intuitive comodity. Poincaré clearly points out that Hertz theory would stand as well in its same glory without the thus obsolete concept of ether. Poincaré predicts since 1885 that it wil one day not be used by physicians anymore since it is just a rhetoric concept without actual root. So Poincaré already had it all since 1885-1901 : clocks synchronisation, generalised principle of relativity, invariant of ds^2 by non euclidian group of rotations (Lorentz and Poincaré group), Obsolete concept of ether, and maxwell universal constant of light speed c obeying the Maxwell key relation c^2.Mu.Epsilon=1, where Mu and Epsilon are the constant of permitivity and permeability of vacuum totaly independant of any frame of reference!
Lorentz verifies in 1904 the last electromagnetic details of THE THEORY OF RELATIVITY they were forging since 1885-1895. All of it is setteled in 1904. Poincaré crucial complementary article of 5 June 1905 at the Paris Academy of Science, was no longer on the hard core of the already built Theory of Relativity, but on crucial EXTENSIONS of it! Indeed in it, Poincaré drags out FONDAMENTAL INVARIANTS of the electromagnetic field under Lorentz-Poincaré transformations, and hit the road on APPLYING THIS NEW UNIVERSAL PRINCIPLE OF RELATIVITY to GRAVITATION! And thus as soon as 5 June 1905, not only the Theory of Relativity is fully settled by Poincaré and Lorentz, but Poincaré APPLIES it to build a COVARIANT THEORY OF GRAVITATION. And that is what his 5 June 1905 article is mainly about, where he predicts allthemore GRAVITATIONAL WAVES! That is in the 5 June 1905 Poincaré article (the 5 Junes Academic paper is the abstract of his full article that was published, for strange reasons, only in 1906, wheras it should have been published, AS USUAL, almost immediately after, in September 1905, and like previous Poincaré articles, in the Analen Der Physics hold in part by Max Planck... Strange things hapens there, with the suicide of the Analen publishing director in 1906!). So I strongly protest that you CANOT, as a mathematician, hold longer this "Einstein" fary tail discovery!
Part 2) And among all these reasons, i'll give you one of the most killing one. Indeed, as the greatest mathematician of his time, Poincaré new perfectly all about group theory and Lie algebra. Whereas Einstein hardly new at this time, even with the help of Mileva, how to differentiate a basical function. Groups, algebra, even matrix was totaly unknown to 99% of physicists of that time. It was even novel to Heinsenberg in 1920 when he struggled with his discovery of "spectral shift" arithmetic, that a mathematician makes him realise that it was just an arithmetic of hidden matrices. So Einstein in 1900-1905 knew NOTHING about all this stuf. And this is the reason why, Einstein start to mess up everything in his mathematical ignorant mind. He naively thought about translating solid blocks. That was his intuitive representation of mooving reference frames. And thus he was naively misslead to falsly beleive that Lorentz transformations only deals with NON-ROTATIONAL CONSTANT SPEED comoving frames. Which is TOTALY FALSE. But one canot know that unless he is well educated in differential geometry as Poincaré was supremely. In technical words that you will very well understand, Einstein ignored all about THE LIE ALGEBRA OF THE LORENTZ-POINCARE GROUP THAT ALLOWS TO DEAL WITH "INFINITESIMAL" LORENTZ TRANSFORMATION IN ORDER TO APPLY RELATIVITY PRINCIPLE TO RELATIVELY TURNING OR ACCELERATING FRAMES. Knowing that perfectly, Poincaré never did such silly things as Einstein did : talking about "SPECIAL" or "GENERAL" Relativity! Such terminology is totaly unsane, it just cries out that his father didn't understood anything deep about the Theory of Relativity, falling in 100% crual misleading missconceptions. Poincaré wisely placed the PRINCIPLE OF RELATIVITY on a universal piedéstale, as a GENERAL arena for all physic. NOT AS A PARTICULAR THEORY OF SOMETHING ELSE THAT WOULD BE "MORE GENERAL"! Via the Lie algebra of Lorentz-Poincaré group, all kinematic is on reach : relative accelerated and turning frames. No need for bullshit "additive" stuf. IT IS ALREADY COMPLETE! So it is the reverse that should have been named by a wise hypothetical man : THE THEORY OF RELATIVITY, fully built by Lorentz and Poincaré between 1885 and 1905, IS THE GENERAL UNIVERSAL ONE, in which arena falls all specific theories. In particular GRAVITATION, which thus has to be written in covariant form for licite use. Time can no longer be regarded as a absolute concept that even disturbed Newwton himself. Such bias came from "religious" belief of a God hiding behind the scientific scene was holding the suprem role of such absolute time keeper. That is what Poincaré precisely distroys in 1885, by building a RATIONAL METHOD FOR RELATIVELY MOVING CLOCKS SYNCHRONISATION. That was the true "DEATH OF GOD" of the millenium. No philosopher more drastically than Poincaré, killed this absolute God hiding behind the Absolute time illusion of the past millenium. And to answer the question why Poincaré built then in 1905 a relativistic theory of gravitation IN FLAT SPACE instead of building it in curved space, he answers himself to this question : for simplicity of use, to stard with, and more deaply because THE TWO GEOMETRIC CHOICES ARE MATHEMATICALLY EQUIVALENT AND PHYSICALLY INDISTINGUISHABLE SINCE "SPACE" AND "TIME" ARE NOT EXPERIMENTAL OBSERVABLE, BUT ONLY MATHEMATICAL CONCEPTS, as Kant sooner noticed it crucialy. But you may object as so many that GPS wouldn't work with Flat Geometry Poincaré relativistic theory of gravitation, and that it only works via the "curved space" Einstein mess. Well are you so sure about that? You know far enough maths to check by yourself. Remember as a first hint that Poincaré very clearly stated that "space" and "time" are not experimental observable but only mathematical concepts. If not, show me any direct OBSERVATION of "Time" or "Space". A clock is not time, it is matter and waves that behave in a cyclic way. They are machines that repeat a certain patern, that we find usefull to call it A MEASURE of "time". But is such a clock an observation of TIME itself? Not at all! Same for space : we only observe relatively "moving" objects, and for commodity we construct an abstract "mathematical" concept of a "space" in which these observed "objects" and "waves" could manifest themselves. But objects are NOT "space". It's not even clear if it is licite to talk about "something" in which such object "swims". This would be an other reminicens of "ether" modelisation of the world build of a "substance", which would be a non-substance, called "space", in which "objects" "swim". That is precisely the ETHER theory! Hiding it's obsolecens through renaming the "ether" by the word "space", doesn't lead anywhere new. It's just a childish trick. An other hint can be that Einstein never used his "general relativity" new forged (pasted on Hilbert one) theory to so called calculate the famous Mercure perihely anamoly. The very reason why is simply that he never could solve any of his "general relativistic" equations! The only ones that could solve them in very special case was Shwartzchield and Lemaitre. So how on earth Einstein managed to "calculate" the "Mercurius perihely anomaly"??? Well, check for yourself, and you will sadly discover that he used an OLD NEWTONIAN GRAVITATIONAL APPROXIMATION!... It looks like a "colombo inspector" serie... It does indeed. Just as you can check that the very original page of the unic article of HILBERT of 1915 where he publishes his crucial theory of "general relativity", has been CUT OF, on it's upper part, precisely where the main equation of gravitatuon was writen! Showing the durty war for paternity raging under the polite surface of hypocrisy. The final hint will be the recent CONFORMAL GAUGE THEORY OF GRAVITATION, built on revolutionary UNIVERSAL GEOMETRIC ALGEBRA of David Hestenes. This gauge theory of gravitation is not built on "curved space" but on "FLAT" ones! So it should no be very long anymore before the Einstein "general theory of relativity" fantasia, fall in deadly disgrace, as soon as enough wise and lucid people will simply fully recognise its drastic incoherence and chaotic conceptual mess. Einstein himself kept killing his "own" principles on which he built his chimeric theory. He first disgraced the Mach Principle, before feeling himself more and more unconfortable with his clearly FALSE "principle of equivalence". Every one knows that for divergent reasons, you canot immitate a central gravitational field by accelerated elevators. Einstein "principle of equivalence" is not only a "principle" since it is obviously false in several special cases. How on earth could it ever by raised as a "principle"??? All such MESS is Einstein MESS! What a sad thing to see how many blind men followed blindly the blindest. Einstein made very wise contributions in photoelectric effect and bose-einstein condensate prediction. But he was WRONG for almost averything else : the static universe, the theory of relativity, quantum mechanic, field theory, etc. This illusionist has to be killed by a new Galileo just as the misleading Aristote has been killed in the seventh century by true scientists as Bruno, Copernic, Galileo, Kepler, Newton ! The leathal shot will probably come from the CORRECT US OF "FLAT" SPACE-TIME RELATIVISTIC THEORY showing all and more than what the alternative "General relativity" theory predicts. And it may come soon since we start to understand what kind of misconceptions we were making since a century in Mathematics, with partial and incomplete tools. The full view of the possible geometries available is still being clarified, and mainly unified in Hestenes Universal Geometric Algebra. An unprecedent revolution is about to start.