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.

@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.

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.

@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).

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.

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.

Thanks for Egan link. This is very interesting: en.wikipedia.org/wiki/Superpermutation kzfaq.info/get/bejne/hcCqfNmdltrGgKM.html

@@santerisatama5409 Thanks for that! I read the article and got to Greg Egan's contribution.

@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.

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