Rick,

Hmmm. Very interested. You'll recall I've supported your view in principle before but I'm no mathematician. I've invoked Maxwells 4 states mechanistically without ever understanding quaternions. You conclude; "Nature's choice will surely be Octonion Algebra." I think I've found it already has done! Are you familiar with Dirac's QM twin stacked orthogonal inverse pair equation? I'd like you to check out a very important finding for me (with non-linear Chirality) advise, and maybe sign up to help with the algebra!

It's all in my essay (with matching computer code and plot in Declan Traill's) amazingly appearing to reproduce QM's predictions classically, shocking enough but all barriers to compatibility with 'SR' are then also lifted!!

Thanks for yours, very hopeful and helpful I hope and pencilled in for a possible 10. I don't think it was surprising physicists; "had difficulty working full four dimensional algebraic elements into physical theories. But I'm now convinced the inverse of that will work! So Octonians may indeed be prove a critical reality.

Excellent job. Please do ask any questions of mine if not clear.

Very Best

Peter

    I remember years ago contact, but not context. Anyhum, I'll try to prove I've read the essay (I am somewhat of a slacker in this regard):

    "Most noticeable was J. C. Maxwell some 30 years later. He saw within individual Quaternion differentiation sub-forms all of the differentiation product forms he needed for his famous 4 equations,but could not work it out with full 4 dimensional algebraic elements."

    Part of the reason I was awarded a PhD was that at the end of two years of not very stellar grad school there was an oral qualifier exam. I was asked a question about Maxwell's equations. I responded that to answer it I'd have to write the equations in a form the professors had never seen before. They were skeptical, but I had all 4 written as one using quaternions. They had in fact never seen anything like it. I passed the qualifier. :) I should add that it is not clear to me what "Quaternion differentiation sub-forms" means. That kind of thing makes reading difficult for me.

    Speaking of quaternions: "It has three separate Complex sub-algebras." Well, ok, if we just look at the basis units, but really there are infinitely many sub-algebras isomorphic to C, one for every element of norm 1 with no real part, so topologically, S2. This is just a quibble.

    "One should expect Octonion Algebra to fundamentally speak to us louder and perhaps with more authority than this Geometric Algebra." Expect? Authority? The problem with this in my view is that in having an expectation of this sort you are in a sense instructing the maths where you want it to go. It may not listen. (On the other hand, you're not wrong, IMO.)

    "For some, the fact that the above Geometric Algebra is associative for multiplication makes it more attractive. This reluctance towards the use of Octonion Algebra is a fundamentally false criticism."

    Well, yes. But you're glossing over the fact that the algebra of actions of O on itself is an associative geometric algebra for a 6-D space. Anyway.

    "If division must be a characteristic of Nature's Algebra, we can stop at Octonion Algebra, since it is the end of the line."

    I of course agree that O is the end of the line, but rather because there are only 4 parallelizable spheres, and only 4 sequences of classical Lie groups. Any higher dimensional algebra you may concoct will have none of the properties and associations that make R,C,H and O exceptional, generative, and resonant. If you start with spheres, no further arguments are needed.

    "We have useful three dimensional items like the magnetic, electric and gravitational fields that we must find homes for in the eight dimensional structure of the Octonions."

    Let me say that I heartily approve of pursuing this line of thinking, for whether or not it is physically correct, it can generate mathematical insight that can lead to better physical applications. If, that is, one listens to the mathematics. I question whether trying to find homes for gravity and EM within the relatively small structure of O you are allowing your ideas of what the mathematics ought to do lead you. It may have other ideas that your preconceptions blind you to seeing. Maybe.

    I am intrigued my your thoughts on the applicability of left-right versions of O. They are, of course, isomorphic, but ...

    Anyway, in conclusion, you have real ideas here, and, as suggested, "so little time" to develop them all. But what better way to use that time than by listening to the whispers from the universe.

    Hmm. Am I "all about applying group theory"? Groups arise as secondary structures from the mathematics inherent in tensored division algebras. This was first noted in conjunction with color SU(3) 45 years ago at Yale by Feza Gürsey and company. His starting point, as is mine, is the division algebras: specifically, CâŠ--O. Clearly his group had SU(3) as a goal, and were initially very pleased with how naturally it arose from the mathematics. But they tried to shoehorn this elegant connection into QM. Their failure in this regard made them turn rather vehemently away from trying for any further connections of O to physics. They had reputations to protect.

    I chose to ignore QM and just pay attention to the algebraic structure. You say: "I do not want to tell the math what to do, I want it to tell me how it must be." We are on the same page here. Yes, I did assume that the mathematics would lead from CâŠ--HâŠ--O to U(1)xSU(2)xSU(3). And in fact, even in the absence of any knowledge of physics it does. So, cool. But in my paper "Seeable Matter; Unseeable Antimatter" I take the mathematical structure well beyond what we all know and love. I shan't be taking those ideas any further, but to get to that point, if the mathematics had ever not presented an easily followed path I would have given up on the whole thing. In particular, the mathematics never had anything to say about gravitation that I could see, so I never tried to incorporate it. And if that above mentioned paper is in fact correct (which of course it is ... ahem), then gravity has a whole new playground - one I freely admit I am not competent to enter.

    Having the table of ordered permutation triplet basis product rules for Right Octonion Algebra in the body of my essay, it can be used to visualize my algebraic variance/invariance sieve, and demonstrate that all product terms for any number of Octonion algebraic element products will fall into an algebraically invariant set, or one of 14 algebraically variant sets.

    To make sure everyone gets the drift on the ordered permutation triplet basis product rule, going cyclically left to right, the consecutive product of two basis elements is + the third element, and commuting the product order going cyclically right to left, the consecutive product of two basis elements is - the third element. For any three unlike non-scalar basis elements there are two possible definitions, starting with one order then forming another rule exchanging any two elements which is fully equivalent to ordering them in the opposite direction. This singular rule change has the effect of changing the sign on all 6 basis product pairs. Very important that it is just a sign change.

    It will not be of any importance that I use a table for Right Octonions and not Left Octonions, nor that the table is relative to algebra R0. Any other choice will sieve the same product terms into the same sets, the only difference will be the relative signs within the variant sets.

    I will do a proof by example using a sequence of basis element products describing the product history of the final resultant basis element. The product e1 * e2 is ruled by the ordered permutation triplet including the set {e1 e2 e3}, using curly braces to not imply a sign rule quite yet. For R0, R1, R2 and R3 the sign rule is (e1 e2 e3) and for R4, R5, R6 and R7 the rule is the opposite: (e3 e2 e1). The correspondence between algebra enumerations and triplet enumerations is no accident, they were purposefully enumerated to make it such. Anyway, for the former 4 the result is +e3, and for the latter 4 it is -e3. This would be a simple example of an algebraically variant product term; it can change signs when a change of algebra definition is made.

    Now take the e3 result and multiply on the left by e2, effectively doing e2 * (e1 * e2). This is based on the same triplet rule {e1 e2 e3}, so any algebra change induced negation will be done twice, meaning for every possible Octonion Algebra choice, the result will always be +e1. This is an example of a non-trivial (not singularly defined e0 * en, ej * ej etc.) algebraically invariant product term.

    If instead we did e5 * (e1 * e2), the second product would be using the rule for {e6 e5 e3} so the sign on the final result basis element e6 would be dependent on both {e1 e2 e3} and {e6 e5 e3} rules, and they can change in different ways for specific changes in algebra. Relative to R0, which has all +1 values in its column and results in +e6, the e6 sign for some other Right algebra will be determined by the product of the row {e1 e2 e3} and {e6 e5 e3} values for that algebra's column. If the row product is +1, the result in that algebra will be the same as in R0: +e6, and if the row product is -1, the result for that algebra will have opposite sign as R0 indicates, or -e6. These column products are precisely the compositions mentioned in my essay. {e1 e2 e3} and {e6 e5 e3} have e3 in common, and the only other triplet including e3 is {e7 e4 e3} which indeed is the resultant row from the composition operation. So we can look at this row and see for R0, R3, R4 and R7 the final result for e5 * (e1 * e2) will be +e6, and for R1, R2, R5 and R6 the result will be -e6. Doing the same row composition on our first example above, the composition ends up on the all +1 row, which is where every algebraic invariant product term will end up, independent of the number of products. If along the way we needed to multiply by the scalar e0 or the very same current result basis, these rules are singularly defined, so one must stay on the current row.

    You can see now 7 algebraically variant rows to land on, and might be questioning how I get to my claim of twice this number. We must bring in the Left Octonion Algebras into the discussion, or more precisely the anti-automorphism Right to Left morph which negates all seven triplet rules. If you ended up on a particular variant row through the application of an even number of variant products, changing every one would not change the final result. However, if you did an odd number of variant products, the anti-automorphism would change the result sign. Thus the variant count parity doubles the 7 to 14. My V+{abc} is even count, and V-{abc} is odd count. Since the composition rule is closed, any number of basis products may be done with comparable results.

    Now, what happens in each of these variant sets is that for any legitimate Octonion Algebra definition change, every product term in a set will either change sign, or not change sign. My "Law of Octonion Algebraic Invariance" states any observable described by Octonion Algebra must be an algebraically invariant form. The corollary to this is any algebraically variant form is not observable. If we individually add/subtract per sign all product terms in a variant set and force a zero result for each set, a mixed bag of variant and invariant product terms now becomes fully invariant since +0 = -0. These are my "Homogeneous Equations of Algebraic Constraint". It is easy to believe experimentation will not show everything that needs to be seen. I am thinking these equations of constraint are extremely important.

    Octonion Algebra is not talking softly here, it is shouting!

      Very hard to grasp this without a blackboard. I think we have likely both experienced the frustration of an audience failing to immediately grasp ideas with which we are so familiar that we can no longer imagine everyone does not see what we see.

      Why do you use Porteous's (e1,e2,e3) quaternionic triple? In 1993 at the first Octoshop (which I organized in hopes of finding meterial to finish my first book), everyone was using tables for which the set of quaternion index triples was invariant wrt index doubling and cycling. At the time mine was based on the index triple, (1,2,6), and Martin Cederwall (inventor of the octonion X-product, and whose university hosted the Octoshop (Porteous was there, too)) started with the triple (1,2,4). Conway and Sloane also used (1,2,4), and I eventually succumbed to peer pressure and now also start with (1,2,4). Anyway, if you are not already familiar with it, you should get familiar with the X-product, and my extension of it, the XY-product). It's very cool stuff.

      However, I have not played with this stuff in a number of years, so even to follow my own work would now require some effort.

      I need to see your stuff with fewer words, and more symbols.

      • [deleted]

      • Post #21

      I think it will be worth your time to get through this. Doing an equivalent to the Octonion 8-force-work expressions that have an outside differentiation that are analogous to the divergence of the Electrodynamics stress-energy-momentum tensor allows the conservation of energy and momentum equations to be formed in the Octonion Algebra framework. The 8-force-work is 9 pages of differential equations and the outside differentiation equivalent is 24 pages, tough to pull out of the 9 pages without some help. The help is looking at all 9 algebraically invariant terms of the form below for for d_j representing partial differentiation, A_j the Octonion 8-potential coefficients and bases e_j

      d_i e_i * {( d_j e_j * A_k e_k) * (d_l e_l * A_m e_m)}

      which of course is determined by basis products e_i * { (e_j * e_k) * (e_l * e_m) }

      The Electrodynamics portion is exactly as it is with 4D tensor representations but the whole includes more fields and forces, including Gravitation and a number of not EM or Gravity related rotational fields. Of course this can be written much simpler inside a half page by representing things in terms of time rate of change and gradients of rotational and irrotational energy densities, time rate of change and divergence of the 7D Octonion Poynting vector, rotational and irrotational dyadics, etc just like the Electrodynamics approach but in more dimensions, necessary to span more stuff.

      When I first derived this with the aid of my home grown symbolic algebra software, I was blown away that this many terms actually balanced out as an equality. THIS WAS NOT A SIMPLE COINCIDENCE. The power and truth of Octonion Algebraic Invariance if formidable.

      As for my choice of triplet enumeration technically I do not think it matters. I already mentioned the "exclusive or" logic operation bit wise on binary values 1 through 7 naturally provides 7 closed sets perfect for the triplets:

      1^2^3 = 7^6^1 = 5^7^2 = 6^5^3 = 5^4^1 = 6^4^2 = 7^4^3 = 0

      So 1^2 = 3, 2^3 = 1 and 3^1 = 2 fully commutative closed set, same for other triplets.

      I think this is more "cool" not to mention advantageous within my symbolic algebra software since all processors to the exclusive or as a native uP instruction.

      Spend sometime with the Invariance/variance issue, it is all there in the table and my thread note. It is critical to mathematical physics within the bounds of Octonion Algebra.

      Rick

      What a piece of something, last was me. By time I got though entry and several attempts at the I am not a robot, it signed me out.

      I must cnofess that you are too frank a person who can criticise one's own writings. Truth is quite illusive though it can also be termed as simple in nature! Mahatama Gandhi often said that his is not that much a struggle against British rule over India as it is a search for truth! That gave rise to Peaceful Non-cooperation Movement for India's freedom. Easy said than done, today if we look critically as to how India got freedom, several factors come on the scene, like rebellion among the soldiers of native Indians in British army, Netaji Subhash Chandra Bose movement with the help of Japanese, etc. British found it difficult to continue their rule. Also, they thought dividing India into two through creation of Pakistan will be enough for them to let the continent difficult to manage! British were not all wrong in their suppositions as Pakistan is faced with several unsurmoutable problems and India continued to remain poor and insufficiently developed. Only recently, leadership of current PM Modi that hopes have arisen for India to progress towards a status due to it on account of population and talent it posesses intinsically.The wisdom behind lies both in friendly foreign relations and effective internale fforts towards development and wealth generation. May i say that just Truth is not enough, its practice in actions and character building of individuals and community together are needed during implementation....

        Kindly also elaborate on the wrod ' Octonian '. in your words as language used by others have not helped me comprehend this word adequately!

        Mr. Rick Lockyer,

        Very nicely written, thank you for some insights.

        Do you see any similarities between these 3 essays? {link: fqxi.org/community/forum/topic/2951]1[/link], 2, 4

        Maybe you read them or maybe not, but it seems to me that all of them are pointing in the same direction, with different approaches. Don't take me too serious though, I'm just playing with "images leftovers" from the essays.

        Silviu

          *no relation other than this contest between me and any of the above authors

          My earlier post on my Algebraic Invariance and Variance Sieve seems to have missed the mark of being presented in an understandable way since Geoffrey Dixon did not follow it, telling me I did a poor job. Let me try to improve things. First, what the heck is this all about?

          In my essay I state the different ways Octonion Algebra can be defined is fully covered by select orientation changes within the seven Quaternion subalgebra triplets, a significant simplification over changes to 64 element multiplication tables. There is no reason to believe one of the 16 is preferred over the others, but the outcome of changes between definitions is the possibility of a result sign change. So if the result is our attempt to describe something we can observe to be one sign and never the other sign, the algebra change would be problematic if the result might change sign.

          Fortunately there are Octonion product terms that maintain consistent signage for all algebra definitions. These should be considered invariant symmetries of Octonion Algebra. If we can't assign a preference to one algebra definition over another we must insure the results of any theoretical application of Octonion Algebra to physical reality must fully reside inside the set of invariant product terms. But how can we simply determine whether or not a product term is an Algebraic Invariant?

          Product terms that are not invariant to Octonion Algebra definition changes will have one sign for eight of the definitions and the opposite sign for the other 8. These should be considered anti-symmetric variants of Octonion Algebra. All Algebraically Variant product terms can be sorted into one of 14 sets, where set members all change signs in exactly the same way for all Octonion Algebra definition changes.

          Every product term for any possible series of Octonion algebraic element products can be classified as a member of the invariant set or one of the variant sets. Which set is fully determined by the product history from initial native algebraic element definitions through the final results after some number of multiplications have been performed. Product order matters, and at each step of the product history there is a multiplication between two basis elements with an algebra specific rule determining the result. After the first multiplication in a sequence, one of these elements is a composite result of all earlier products and the other is the next up in the history. The algebraic rule is going to be determined by one variable definition Quaternion triplet rule, or the consistently defined products including the scalar basis or between like basis elements.

          For the sieve algorithm applied to a product history, at each basis element pair product, a move is prescribed from the current row in the Hadamard Matrix table in my essay to either itself or a new row. Which move is entirely determined by the next basis pair product. If the rule this product is governed by is not a variant triplet rule, remain at the current row. If not, the new row is found by doing the row composition described in my essay between the Hadamard elements of the current row and the Hadamard elements for the row labeled by the triplet called out by the basis element product. One could consider the stay on the same row as a composition move specified between the current row and the "none" row which is the composition identity element.

          Procedurally you start a product history on the identity "none" row. The first basis element product rule composition is applied, and you change rows or don't as described above. Next determine the rule between the result basis element from the last product with the next basis element up in the product history sequence, use it to make the next row change or stay. Repeat until the product history sequence completes. If you end up on the "none" row, the product sequence is an Algebraic Invariant. Otherwise you will be on a row with a triplet label and the product sequence is an Algebraic Variant partially described by the label, let's call it {abc}. If the sequence involved an even number of triplet compositions, the variant set will be what I call V{abc}, if odd count, it will be in the set V-{abc}.

          Hopefully this clears things up.

            Did you read my essay or are you simply resting to the title?

            My concept of truth here has nothing to do with one's opinion, and certainly not one's behavior.

            Consult Wikipedia but spell Octonion correctly for answers to your next post.

            Rick

            My inability to understand something frequently has little to do with how well an idea has been presented. You may have done a great job presenting the ideas; and I may have been (and likely was) too lazy to dig into it adequately. I grasp ideas through a kind of osmosis, which takes time.

            I put a comment in my essay section trying to prove something that is maybe irrelevant to your discussion, viz., all representations of O are isomorphic. I really have no idea if that is pertinent. Quite possibly not. It doesn't change your notion - which I share - that left and right representations are in some ways distinct. In my work, as I recall, the distinction arises from the X-product. Using our ea, a=1,...,7, type of representation of O, there are two distinct categories. Two representations are in the same category if one can get from one to the other via an X-product revision. Using this method of mapping one rep to another, it becomes clear that the collection of reps is divided in two, all reps in each half can be linked by an X-product variation, but there is no way to do this from one half to the other. I am hoping that this means we are talking about the same thing, at least mathematically. I'm not yet close to figuring out your connection to physics. Again, likely my fault.

            I need to reread your essay. Soon. For that matter, I also need to reread my own material on all this stuff. As a wise man once said (you), so little time.

            Excellent essay Rick!

            What can I tell you? You hit all the bases. You speak directly to the question of what is fundamental. You present a compelling argument for your choices; though I might add 'and this also'... I am in general agreement with your premise and I look forward to working with your octonion calculation software, possibly for creating some higher-d fractals. I will have to re-access some of your prior work too, since I see hooks into what I am now working on that didn't pop out before.

            I will read this again, but offer my rating now while I am still impressed.

            All the Best,

            Jonathan

              I should add this...

              My brief conversation with Tevian Dray at GR21 affirms what you are saying about the fact that non-associativity is a necessity and not something that should be avoided. I like the way Geoffrey said it on another thread; non-associativity is not a bug, it is an amazing feature that comes into play in the perfect way. P.C. Kainen also shares in the conviction that non-associative algebra and geometry must play a part in Physics.

              And your statement that the octonions need to drive or drive the process is spot on. They are the Big Daddy in terms of evolutive processes, and made so by their non-associativity, so that is most certainly not a defect - and instead it is quite possibly the most powerful attribute an algebra can have. So I applaud your efforts to make the octonions more accessible for people working in Physics and Cosmology.

              All the Best,

              Jonathan

              Rick,

              I went through the below and was doing fine but am no mathematician, my skills are elsewhere, so I faltered when I reached "ordered permutation triplet basis product rules."

              My question is; will octonians help in a mathematical description from twin pairs of handed (complementary) momenta with inverse Cos distributions? so finding the QM Hamiltonian from the Lagrangian?

              As an insight, I replied to your post about barmaids on my string as follows.;;

              Rick,

              You're right,ish, Classic QM was trickier for barmaids than logical SR, but I've shown it possible with my rotating sphere. Viz; Get her to shut her eyes, spin it on a vertical axis, then;

              1. Touch her finger on a pole and ask 'Is it going left or right?' (= 0)

              2. Do so on the equator & ask 'Is that clockwise? or anticlockwise? (= 0)

              3. Touch it on the N pole & ask 'Is it going clockwise or anti..? (=-1)

              4. Touch it on the S pole & ask 'Is it going clockwise or anti..? (=+1)

              5. Do so on the equator her side & ask 'Is it going left or right?' (=-1)

              6. Same on the other side (or flip the poles) & ask 'left or right?'(=+1)

              7. Finally at latitude 45supo & ask is it moving or rotating? (=both)

              Now we KNOW the spin AND linear speed both change NON-linearly, by Cos latitude. Rotate the polar axis in any plane and that doesn't change. Three out of five barmaids understand.

              Now ALSO tell them each sphere re-emits at 'c' with respect to itself whatever the original 'closing speed', and there are millions on the surface of a lens, and her understanding of SR allows complete unification with QM. There are a number of barmaids around who now understand that (more) logical analysis! Some were impressed enough to... well you'll need to use imagination.

              Can you find logical or epistemological fault?

              Very Best

              Peter

              The ordered triplet rule is simple when explicitly stated, which I did not. For a set of three basis elements, say {e1 e2 e3}, they can specify two cyclic product rules based on their order. The ordered rule (e1 e2 e3) is a simple mnemonic for 6 separate products where there are three possible products going through the order shown cyclically left to right and three more going right to left. The basis element at one of the 3 positions multiplied by the next one to its right is the + the next in progression. '"Cyclic" means when you get to the end, the next in progression is on the other end, you wrap around. If you go right to left instead, this is equivalent to flipping (commuting) the order of the multiplication and this instead specifies the result is - the third basis element.

              So the ordered permutation triplet basis product rule (e1 e2 e3) is a shorthand expression for all 6 possible basis pair products between two of three elements, implying also the operation is closed for the set, meaning of course the product of two set members is another set member. The explicit results implied by (e1 e2 e3) are

              e1 * e2 = +e3, e2 * e3 = +e1, e3 * e1 = +e2 cyclic left to right

              e2 * e1 = -e3, e3 * e2 = -e1, e1 * e3 = -e2 cyclic right to left

              If we instead did the same rule application on (e3 e2 e1), we would find opposite signs on all 6 products above. This is the only alternate definition. Every possible permutation of the three basis elements will fall into one of these 2 rules.

              You could look at (x y z) describing a right hand vector cross product and (z y x) describing a left hand vector cross product. The triplet rule also expresses all products of different non-scalar Quaternion basis elements and the 2 separate ways it can be defined.

              On Bell's "theorem", I once had a collegial relationship with someone who thought he had something to say on the subject using essentially the two ways to define Quaternion Algebra as fair coin statistical choices. He made the mistake of singularly enumerating the basis elements but using both rules in the same expression. This "spooky algebra at a distance" gave the desired minus cosine response because simultaneously applying both rules to the same basis names changed the algebra from Quaternion to an algebra with only a scalar product. I tried my level best to help him understand he made what on the surface looked like a simple sign error. A number of other people also tried to make him understand his mistake. Instead of listening and admitting he made a mistake, he and his supporters set out on a years long campaign of personal attacks. My last words on the subject was in Retraction Watch, where he was whining about his published paper with the very same math errors being retracted. After the personal attacks I have zero respect for him and all of his supporters, and will have nothing to do with any of them going forward. I am a little touchy on the subject.

              Bell did not touch the metrology of the experimental equipment nor the causal effects on it by the particles. I do not think this can be divorced from the discussion. You seem to be attempting to take a pass with your spherical analogy getting to the requisite +1 and -1 detector clicks. I think the difficulty is going from a sequence of +1 and -1 data values to the minus cosine angular effect. If you take axiomatically the existence of a probability distribution behind each +1 or -1 result, it certainly can be recovered from a large number of samples. This is what QM does, but classical non-stochastic methods can't. QM also seems to force on the subject its credo nothing is known on either side until one side makes a measurement and this forces the distant measurement results to be what they become, but with "spooky" non-communication that has no possible classical description. This pushes the probability analysis in favor of the QM position. The problem of insuring the measurements actually are on "entangled " pairs can easily provide a mechanism to cherry pick the data removing data that may refute the QM position. In short, the game seems rigged. As far as my tackling this, it is so far down the list I will die before it comes up.

              Rick

              Hello yet again Rick,

              I'd like to be kept in the loop once your calculational software is available for use. I've kept in touch with Louis Kauffman in Chicago, who was one of the authors of a book called "Hypercomplex Iterations" about higher dimensional fractals, and I just mentioned your software package in an e-mail to him. So don't be surprised if there are some people interested to see what it can do. I hope you can get a community of people working with the octonions, as a result of your software development efforts.

              All the Best,

              Jonathan

              Jonathan,

              I have read your essay a couple of times since it first came out, really do not know how to react to it so haven't. My ideas about Gravitation are a bit more pedestrian, so I have no feel for how the Mandelbrot set might fit in. My ignorance, not my neglect of your efforts.

              Thanks for your kind words and support over the years.

              On the symbolic algebra software, it was developed to run on freeware Nodejs online downloadable. I have not developed a nice user interface above this since have no time and ok with opening a Windows old style dos command box, typing in command line info to launch node.js pointed at my specific script file that is in a directory with a subdirectory holding my classes and methods that do the work. The command box is the console out, good for errors but not much else, I only look at log file text data generated by the show function output to the console optionally logging to a text file.

              I will be putting up a boilerplate script file that has comments saying your stuff starts here and ends here so users can use a text editor to put in calls to the classes to do what they want. Also will be a finished and complete script verifying my conservation equations. It cranks out a ~260k text file with the details.

              Maybe some kind person with the time and skills would write something to bring in, edit, save and run user scripts with some contextual type-ahead once you start entering a method giving instant parameter documentation to ease use. Open source would be nice.

              I am partially through documentation of classes and methods, someone would have to dig it out of the source code without it but what I put out will have it all fully visible, no library.

              Rick