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Introduction: Enon Harris

For the past thirteen years, I have been interested in Geometric Algebra (GA, real-valued Clifford Algebras with geometric interpretation), which gives a standard formalism for all sorts of problems, including a more comprehensible QM, Gauge Theory Gravity, as well as practical applications in graphics and path planning for robots and machine tools. What I most like about it is that nearly every paper is written to be understood, starting from scratch and assuming little prior knowledge. It is also coordinate-free, terse, visualizable and just general enough to represent anything physical. Clifford spectra of tree structures is a current application that I find particularly interesting.

Another of my interests is a computer language called "Frink", created by Alan Eliasen. The most radical thing about it is that it is physically-typed, so that every quantity can have physical units, and it will recognize virtually any sort of units, no matter how obscure. (For instance if you want to calculate orbital energy as "roods / microfortnight^2", it can do it.)

Frink runs on anything with a JVM, has both simple terminal-like interactive desk-calculator and full programming modes, and has many other advanced and convenient features for all sorts of practical purposes, including arbitrarily large integers and precise rational numbers, interval arithmetic, exact time computations in various time systems, currency conversion and inflation-adjusted value calculations, natural language translations, regexp, self-evaluation, fast number theory functions, convenient built-in data structures, and some modest symbolic math capabilities. I use Frink all the time and hope some of you will try it out. It also has some of the best documentation I have ever seen.

Working with Frink led me to try to figure out the patterns in the factors of the various unit classes (a unit class would be something such as "area" rather than a specific unit such as "hectare"). It turns out that there is a pattern that lets the vast majority of the different unit classes be mapped out in a coherent arrangement that fits on a single page, and gives new insight into how physical quantities relate to each other. I'll post more on this soon.

More relevant to Azimuth is the idea of "seed factories", minimal toolsets sufficient to produce the tools that make the tools ... that make as large a fraction as possible of products in the economy. The software used for materials, tooling and product routing, scheduling and accounting, and the co-design of products together with all the processes needed to make them is the biggest challenge. Similar software will be needed for off-planet industrial ecologies and for product-producing molecular nanotechnology, so this software will be some of the most critical infrastructure of the future.

Great gains in efficiency and sustainability are possible with seed factories, and the lend themselves to decentralized ownership and organization that allows closely coupling demand to production and widely distributing the economic rents of productive machines.

I hope we will together think up and build things that have a big positive effect in the real world.

Comments

  • 1.

    Hi Enon, welcome! You have some interesting interests. Your post got me curious about Clifford and Geometric algebras...this could be a major distraction if I am not careful :)

    Comment Source:Hi Enon, welcome! You have some interesting interests. Your post got me curious about Clifford and Geometric algebras...this could be a major distraction if I am not careful :)
  • 2.

    Hi and welcome,

    You might enjoy John Denker's very clear intro to GA: https://www.av8n.com/physics/clifford-intro.htm to go with along with Hestenes spacetime algebra: http://geocalc.clas.asu.edu/pdf/SpacetimePhysics.pdf

    Comment Source:Hi and welcome, You might enjoy John Denker's very clear intro to GA: https://www.av8n.com/physics/clifford-intro.htm to go with along with Hestenes spacetime algebra: http://geocalc.clas.asu.edu/pdf/SpacetimePhysics.pdf
  • 3.

    Those are good references. Denker also has a follow-up article on using GA for electromagnetics which is good. Slehar's GA intro has the prettiest pictures. Jaap Suter's Geometric Algebra Primer is also a good place to start for an intro with math. Papers and books with David Hestenes, Leo Dorst, or Chis Doran among the authors are all great.

    For software to help with visualization, particularly in the 3D + 2 conformal model Dorst's GAviewer and tutorials are indispensable.

    For extra mathy math see Lundholm and Svenson's Clifford algebra, geometric algebra, and applications lecture notes.

    Comment Source:Those are good references. Denker also has a follow-up article on using GA for electromagnetics which is good. Slehar's [GA intro](https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/) has the prettiest pictures. Jaap Suter's [Geometric Algebra Primer](http://www.jaapsuter.com/geometric-algebra.pdf) is also a good place to start for an intro with math. Papers and books with [David Hestenes](http://geocalc.clas.asu.edu/), [Leo Dorst](http://www.science.uva.nl/research/ias/ga/index.html), or [Chis Doran](http://geometry.mrao.cam.ac.uk/) among the authors are all great. For software to help with visualization, particularly in the 3D + 2 conformal model Dorst's [GAviewer and tutorials](http://www.science.uva.nl/research/ias/ga/viewer/content_viewer.html) are indispensable. For extra mathy math see Lundholm and Svenson's [Clifford algebra, geometric algebra, and applications](http://arxiv.org/abs/0907.5356) lecture notes.
  • 4.

    Hi Enon and thank you for the links. I'm curious what you mean by Clifford spectra of tree stuctures.

    Comment Source:Hi Enon and thank you for the links. I'm curious what you mean by Clifford spectra of tree stuctures.
  • 5.

    Sorry, should have replied to you a while back.

    What I mean by a Clifford spectrum of a tree structure can be visualized with a sapling on a shaker table such as is used to test the vibration tolerance of equipment. The table can shake at any frequency or combination of frequencies up to some maximum force exerted (which may vary depending on frequency etc.) The table can move not only linearly in z,y,z (e_1,e_2,e_3 in the usual Clifford / CA Geometric Algebra / GA notation) but can also rotate in all three perpendicular planes (e_12,e_13,e_23). In a GA "multivector" in a space of n dimensions (n orthogonal basis vectors) there are a weighted sum of 2^n parts, or (orthogonal unit basis) "blades", each which are formed by taking the outer "^" product of all combinations of the n basis vectors: each algebra has a scalar, which is 0 basis vectors multiplied together each algebra has a pseudoscalar with all basis vectors multiplied together, (e.g. e_123 for 3D) which is "dual" to the scalar. Similarly the three basis vectors in 3D are dual to the three planes of rotation. In general, each basis "blade" of the algebra which is formed from the outer product of some set of basis vectors is dual to another blade formed from the remaining basis vectors of the algebra. In 3D e_1 is dual to e_23, e_2 is dual to e_13, e_3 is dual to e_12. ( In 4D, e_12 is dual to e_34, there is no unique vector perpendicular to a given plane of rotation but rather a plane perpendicular to the plane. The number of blades of each degree in an n-D algebra goes as the nth row of Pascal's triangle. The geometric product of different basis blades (=outer product for basis blades since they are orthogonal) can be found by noting that the square of a basis vector is 1 (or -1 if that is its signature), and that swapping any two basis vectors in the basis blade results in a sign change - so you don't need a multiplication table.)

    In any dimension, rotating in a given plane can be accomplished by multiplying the thing being rotated by e (2.718...) raised to a bivector (for simplicity, formed from the product of two different basis vectors which define a plane of rotation, and by their order define a direction of rotation (AB=-BA, non-orthogonal cases more complicated), and by their size (1) mean a rotation by 90 degrees). Actually, the thing, X, being rotated is usually multiplied from both sides like this: e^(-B/2) X e^(B/2), where B is the bivector of rotation. (The exponential of a bivector is defined with Taylor series, odd and even term sums are sin and cos, outer and inner products whose sum is the "geometric product", e^B. Unit bivectors square to -1, just like i, assuming the underlying vectors have the same square, 1 or -1.) Chains of rotations can be formed by sandwiching that whole thing in another pair of exponentials, e^(-C/2) e^(-B/2) X e^(B/2) e^(C/2), which can be chained as many times as you like and works in any dimension.

    So back to the sapling on the shaker table, it has at least six different mechanical spectra, one for each direction and plane of rotation (if it were a "tree" made of balloons and put inside a vibrating piston, it would have a pseudoscalar spectrum as well.)

    Imagine shaking the sapling in a gradually increasing frequency along an axis. At certain resonance frequencies determined by the length and stiffness of the different levels of branches in the sapling there will be large amplitude movements of the parts of the tree that are in resonance. These six or seven spectra are distinct, but can be coupled in regular ways. The sum of all the different basis blade spectra is a Clifford spectrum. Each structure has a mechanical Clifford spectrum as well as electromagnetic Clifford spectrum. (EM is very clean in CA/ GA, but that's another topic.)

    These multivectors formed from the sum of up to 2^n parts act like "hypercomplex" numbers and form various algebras depending on the dimension and the signature (signs of the squares of the basis vectors of the algebra.) These algebras generally allow solving for the inverse of a multivector in an equation which has many applications as mentioned in my intro post.

    Comment Source:Sorry, should have replied to you a while back. What I mean by a Clifford spectrum of a tree structure can be visualized with a sapling on a shaker table such as is used to test the vibration tolerance of equipment. The table can shake at any frequency or combination of frequencies up to some maximum force exerted (which may vary depending on frequency etc.) The table can move not only linearly in z,y,z (e_1,e_2,e_3 in the usual Clifford / CA Geometric Algebra / GA notation) but can also rotate in all three perpendicular planes (e_12,e_13,e_23). In a GA "multivector" in a space of n dimensions (n orthogonal basis vectors) there are a weighted sum of 2^n parts, or (orthogonal unit basis) "blades", each which are formed by taking the outer "^" product of all combinations of the n basis vectors: each algebra has a scalar, which is 0 basis vectors multiplied together each algebra has a pseudoscalar with all basis vectors multiplied together, (e.g. e_123 for 3D) which is "dual" to the scalar. Similarly the three basis vectors in 3D are dual to the three planes of rotation. In general, each basis "blade" of the algebra which is formed from the outer product of some set of basis vectors is dual to another blade formed from the remaining basis vectors of the algebra. In 3D e_1 is dual to e_23, e_2 is dual to e_13, e_3 is dual to e_12. ( In 4D, e_12 is dual to e_34, there is no unique vector perpendicular to a given plane of rotation but rather a plane perpendicular to the plane. The number of blades of each degree in an n-D algebra goes as the nth row of Pascal's triangle. The geometric product of different basis blades (=outer product for basis blades since they are orthogonal) can be found by noting that the square of a basis vector is 1 (or -1 if that is its signature), and that swapping any two basis vectors in the basis blade results in a sign change - so you don't need a multiplication table.) In any dimension, rotating in a given plane can be accomplished by multiplying the thing being rotated by e (2.718...) raised to a bivector (for simplicity, formed from the product of two different basis vectors which define a plane of rotation, and by their order define a direction of rotation (AB=-BA, non-orthogonal cases more complicated), and by their size (1) mean a rotation by 90 degrees). Actually, the thing, X, being rotated is usually multiplied from both sides like this: e^(-B/2) X e^(B/2), where B is the bivector of rotation. (The exponential of a bivector is defined with Taylor series, odd and even term sums are sin and cos, outer and inner products whose sum is the "geometric product", e^B. Unit bivectors square to -1, just like i, assuming the underlying vectors have the same square, 1 or -1.) Chains of rotations can be formed by sandwiching that whole thing in another pair of exponentials, e^(-C/2) e^(-B/2) X e^(B/2) e^(C/2), which can be chained as many times as you like and works in any dimension. So back to the sapling on the shaker table, it has at least six different mechanical spectra, one for each direction and plane of rotation (if it were a "tree" made of balloons and put inside a vibrating piston, it would have a pseudoscalar spectrum as well.) Imagine shaking the sapling in a gradually increasing frequency along an axis. At certain resonance frequencies determined by the length and stiffness of the different levels of branches in the sapling there will be large amplitude movements of the parts of the tree that are in resonance. These six or seven spectra are distinct, but can be coupled in regular ways. The sum of all the different basis blade spectra is a Clifford spectrum. Each structure has a mechanical Clifford spectrum as well as electromagnetic Clifford spectrum. (EM is very clean in CA/ GA, but that's another topic.) These multivectors formed from the sum of up to 2^n parts act like "hypercomplex" numbers and form various algebras depending on the dimension and the signature (signs of the squares of the basis vectors of the algebra.) These algebras generally allow solving for the inverse of a multivector in an equation which has many applications as mentioned in my intro post.
  • 6.
    edited March 12

    Enon, Thank you for explaining, and for being here.

    Comment Source:Enon, Thank you for explaining, and for being here.
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