Basic Twistor Theory, Bi-twistors, and Split-octonions - Roger Penrose

Abstract: Twistor theory was introduced in the mid 1960s as an approach to
combining quantum theory with space-time structure. A driving
force behind the introduction of Twistor Theory was to combine the
quantum-field theoretic requirement of positive frequency with the
structure of space-time. In order to achieve this, the notion of
twistor space was introduced to codify the structure of space-time in
a way which related it to the splitting of the twistor space into two
halves, one representing positive frequency, and the other
representing negative frequency. Standard twistor theory involves a
complex projective 3-space PT which naturally divides into two
halves PT + and PT - , joined by their common 5-real-dimensional
boundary PN. The points of the space PN represent light rays in
Minkowski space-time. However, this splitting has two quite
different basic physical interpretations, namely positive/negative
helicity and positive/negative frequency, which ought not to be
confused in the formalism, and the notion of “bi-twistors” is
introduced to resolve this issue. It is found that quantized bi-twistors
have a previously unnoticed G 2 * structure, which enables the split-
octonion algebra to be directly formulated in terms of quantized bi-
twistors, once the appropriate complex structure is incorporated.
-------------------------------------------------------------------------------------------------------------------------
Join us at sites.google.c...

[@proximitygaming8253]

I feel like this is one of those scientists/mathematicians that go down as famous for fundamental discoveries, except that Penrose is alive in our era.

[@skhotzim_bacon]

Oh yeah. This man is one of the greatest of our generation. He's on the same level as Einstein and Hawking. An immortal unforgettable great man for sure.

[@SpotterVideo]

What do the Twistors of Roger Penrose and the Hopf Fibrations of Eric Weinstein and the "Belt Trick" of Paul Dirac have in common? It takes two complete turns to get down the "rabbit hole" (Alpha Funnel 3D--->4D) to produce one twist cycle (Quantum unit). Can both Matter and Energy be described as "Quanta" of Spatial Curvature? (A string is revealed to be a twisted cord when viewed up close.) Mass= 1/Length

[@proximitygaming8253]

@@SpotterVideo I don't quite understand most of what you said, but it seems like peculiar and quirky phenomena! I'm planning to study differential geometry sometime soon, so I might get it later.

[@AbsoluteGameBoys]

couldn’t agree more

[@hyperduality2838]

Twistor space is an equivalent or dual description of space/time. Space is dual to time -- Einstein. Space/time symmetries are dual to mobius maps -- stereographic projection (conformal invariance). Points are dual to lines -- the principle of duality in geometry. Null points (twistor space) are dual to null lines (light rays in space/time). Clockwise is dual to anti-clockwise -- helicity. Spinors (matter, Fermions) are dual to vectors (waves, Bosons) -- wave/particle duality. Duality means that there is a 4th law of thermodynamics:- Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! All observers make predictions to track targets, goals and objectives -- a syntropic process, teleological. Teleological physics (syntropy) is dual to non teleological physics (entropy). Thesis (action) is dual to anti-thesis (reaction) creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic. Action is dual to reaction -- Sir Isaac Newton or the duality of force. Attraction (sympathy) is dual to repulsion (antipathy), stretch is dual to squeeze, push is dual to pull -- forces are dual. All forces are dual in physics hence energy must be dual:- Energy = force * distance -- simple physics. Energy is duality, duality is energy. Potential energy is dual to kinetic energy -- gravitational energy is dual. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual. Curvature or gravitation is dual. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). The point duality theorem is dual to the line duality theorem. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching.

[@giovannifontanetto9604]

A very young researcher

[@hyperduality2838]

Twistor space is an equivalent or dual description of space/time. Space is dual to time -- Einstein. Space/time symmetries are dual to mobius maps -- stereographic projection (conformal invariance). Points are dual to lines -- the principle of duality in geometry. Null points (twistor space) are dual to null lines (light rays in space/time). Clockwise is dual to anti-clockwise -- helicity. Spinors (matter, Fermions) are dual to vectors (waves, Bosons) -- wave/particle duality. Duality means that there is a 4th law of thermodynamics:- Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! All observers make predictions to track targets, goals and objectives -- a syntropic process, teleological. Teleological physics (syntropy) is dual to non teleological physics (entropy). Thesis (action) is dual to anti-thesis (reaction) creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic. Action is dual to reaction -- Sir Isaac Newton or the duality of force. Attraction (sympathy) is dual to repulsion (antipathy), stretch is dual to squeeze, push is dual to pull -- forces are dual. All forces are dual in physics hence energy must be dual:- Energy = force * distance -- simple physics. Energy is duality, duality is energy. Potential energy is dual to kinetic energy -- gravitational energy is dual. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual. Curvature or gravitation is dual. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). The point duality theorem is dual to the line duality theorem. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. Classical reality is dual to quantum reality synthesizes true reality -- Roger Penrose using the Hegelian dialectic.

[@brendawilliams8062]

Amazing and beloved human. Thankyou

[@postiemania]

At 8:38 this is the field geometry of an actual engine I am building. Sir Roger Penrose and his work is very inspiring to me.

[@gilleslalancette7933]

Thank you Sir Penrose!

[@hyperduality2838]

Twistor space is an equivalent or dual description of space/time. Space is dual to time -- Einstein. Space/time symmetries are dual to mobius maps -- stereographic projection (conformal invariance). Points are dual to lines -- the principle of duality in geometry. Null points (twistor space) are dual to null lines (light rays in space/time). Clockwise is dual to anti-clockwise -- helicity. Spinors (matter, Fermions) are dual to vectors (waves, Bosons) -- wave/particle duality. Duality means that there is a 4th law of thermodynamics:- Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! All observers make predictions to track targets, goals and objectives -- a syntropic process, teleological. Teleological physics (syntropy) is dual to non teleological physics (entropy). Thesis (action) is dual to anti-thesis (reaction) creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic. Action is dual to reaction -- Sir Isaac Newton or the duality of force. Attraction (sympathy) is dual to repulsion (antipathy), stretch is dual to squeeze, push is dual to pull -- forces are dual. All forces are dual in physics hence energy must be dual:- Energy = force * distance -- simple physics. Energy is duality, duality is energy. Potential energy is dual to kinetic energy -- gravitational energy is dual. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual. Curvature or gravitation is dual. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). The point duality theorem is dual to the line duality theorem. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. Classical reality is dual to quantum reality synthesizes true reality -- Roger Penrose using the Hegelian dialectic.

[@wargreymon2024]

I see Roger I press like

[@hyperduality2838]

Twistor space is an equivalent or dual description of space/time. Space is dual to time -- Einstein. Space/time symmetries are dual to mobius maps -- stereographic projection (conformal invariance). Points are dual to lines -- the principle of duality in geometry. Null points (twistor space) are dual to null lines (light rays in space/time). Clockwise is dual to anti-clockwise -- helicity. Spinors (matter, Fermions) are dual to vectors (waves, Bosons) -- wave/particle duality. Duality means that there is a 4th law of thermodynamics:- Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! All observers make predictions to track targets, goals and objectives -- a syntropic process, teleological. Teleological physics (syntropy) is dual to non teleological physics (entropy). Thesis (action) is dual to anti-thesis (reaction) creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic. Action is dual to reaction -- Sir Isaac Newton or the duality of force. Attraction (sympathy) is dual to repulsion (antipathy), stretch is dual to squeeze, push is dual to pull -- forces are dual. All forces are dual in physics hence energy must be dual:- Energy = force * distance -- simple physics. Energy is duality, duality is energy. Potential energy is dual to kinetic energy -- gravitational energy is dual. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual. Curvature or gravitation is dual. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). The point duality theorem is dual to the line duality theorem. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. Classical reality is dual to quantum reality synthesizes true reality -- Roger Penrose using the Hegelian dialectic.

[@antondevey]

Starts at 1:45

[@wintc]

This reminds me a lot of string theory, at least at a very high conceptual level. Both are derived by postulating that the fundamental unit is not a particular point, but is instead a complex, conformal surface created by worldlines.

[@JAYMOAP]

Yes he doesn't realise but he essentially working in the framework of string theory. Ironically, he criticised string theory for higher dimensions but he does the same haha

[@aldendaniels7259]

@@JAYMOAPyea, he claims he strictly works in 4 dimensions…. but they’re complex dimensions 😅

[@skhotzim_bacon]

​@@JAYMOAPString Theory deserves criticism, not because of its ideas or beautiful mathematics, but because of the theory itself. What's problematic is that whenever the theory fails to match observed phenomenæ, instead of abandoning it and pursuing other avenues, proponents resort to elegant mathematical solutions that merely cover up its gaping fundamental flaws. The theory's reliance on an absurd plethora of dimensions renders it untestable and unfalsifiable, while also monopolizing and wasting the attention of brilliant minds in physics for decades. The only redeeming quality of String Theory lies in its beautiful ideas and mathematical constructs, which have contributed to advancements in other fields. It's perfectly reasonable for Penrose to critique the theory while acknowledging its positive aspects and leveraging its insights for alternative purposes. That's why he is brilliant.

[@JAYMOAP]

@@skhotzim_bacon penrose Bitwistors are 6 dimensional. Why you think he forced to get to this direction? Because the single case doesn't work. I don't you understand why higher dimensions pops up in math. It's relevant in other areas as well, for example condensed matter physics.

[@skhotzim_bacon]

@@JAYMOAPI know they are. I never implied they weren't. Are you having trouble with basic literacy and reading comprehension? Let me reiterate what I said: "The only thing good about it are its beautiful ideas and mathematics that contribute to other fields. I see nothing wrong with Penrose criticizing the theory but also recognizing its beautiful ideas and utilizing them for a better purpose." The logical fallacies you used were: *Strawman Fallacy:* You misrepresented my argument by suggesting that I'm unaware of why higher dimensions are used in mathematics and other fields. *Red Herring:* You introduced the topic of the use of multiple dimensions in Penrose's work on twistor theory as a distraction from the legitimate criticism of String Theory.

[@SpotterVideo]

What do the Twistors of Roger Penrose and the Hopf Fibrations of Eric Weinstein and the "Belt Trick" of Paul Dirac have in common? It takes two complete turns to get down the "rabbit hole" (Alpha Funnel 3D--->4D) to produce one twist cycle (Quantum unit). Can both Matter and Energy be described as "Quanta" of Spatial Curvature? (A string is revealed to be a twisted cord when viewed up close.) Mass= 1/Length

[@LibrawLou]

It seems the Clifford-Hopf geometry would be best viewed (as are Spinors & Maxwell's single equation) by Hestenes' Geometric Algebra -- a unifying mathematical formalism of physics.

[@LibrawLou]

The problem of various pseudo & non-physical formalisms arising might be avoided by casting Twistor theory from the start in the purely physical formalism of Geometric Algebra, a specialized Clifford algebra.

[@SpotterVideo]

@@LibrawLou Many of the drawings of Twistors, like those seen here, reveal cycles of "twist". Is each twist cycle the source of the "Quantum", being proportional to Planck's Constant?

[@LibrawLou]

@@SpotterVideo Interesting -- that does seem a connection to tiny, gravity-"free" spacetime of particles? And how does the twist relate to the double spin required to complete a rotation of a lepton? And could recasting Twistors in Hestenes' Geometric Algebra allow connecting to C. Furey's view of Octonions as the fourth Division algebra necessary to be a major explainer of physics, and not just an interesting math subject?

[@SpotterVideo]

@@LibrawLou If we add one extra spatial dimension, can we get an Alpha funnel soliton which requires two rotations at the 3D boundary to produce one complete twist cycle at the smaller end of the funnel?

[@chevasit]

Very good 👍👍👍

[@patrickmendoza3210]

Roger Penrose: "But anyway, he was very useful to me." LMFAOOOOOO

[@brendawilliams8062]

To me also, I need to revisit his lectures more often. I’ve been lately in 1001600474/ 1001603612 and the relationship of them to 35/ 34999.. and the place 112 if divided by the associated 1014 no.s will move the motion of the no.s from left towards the right. Things that interest me.

[@brendawilliams8062]

I mean that 3168 has a 1014 no. In a 10125 space. The 1014 will go upwards 18 spaces to 35 no.

[@brendawilliams8062]

I don’t see the sphere packing unless you moving left to 101010.. no.s To the left

[@EsdrasOlivaresPcmasterrace]

Penrose's mind is a godam powerhouse same as Eric Weinstein

[@Killer_Kovacs]

I came to nearly the same conclusions. Can hyperspheres be monopolar?

[@KipIngram]

You know, I SO wish I could find an introduction to this stuff that actually taught it from first principles. Everything I've been able to find is already just a little out of my reach, and I haven't been able to find the "ladder rungs" I need to cover the gap yet. 😞

[@RandomNooby]

Calculus, linear algebra, Scalars, vectors, manifolds, geometry, group theory ,etc at postgrad level would be a good start point I think...

[@KipIngram]

@@RandomNooby Well, I'm fairly familiar with a lot of those things. It's just when I start to get to the twistor stuff itself that it doesn't "gel" for me. I may be getting closer - I did find a video series on spinors recently that I got a lot out of, so hopefully that has moved me a little closer. Out of your list group theory is probably the thing I need the most shoring up on.

[@antronixful]

​@@KipIngram grad school is a great place tu study it. Differential Geometry, abstract algebra and algebraic structures are prerequisites to understand spinors. Also, to get the "why" of this topic, you should get to study modern formalisms of both general relativity and quantum mechanics, involving groups and weird algebras and geometric interpretations. Then I think you can jump towards this topic, because it involves a deep understanding in both fields.

[@hyperduality2838]

Twistor space is an equivalent or dual description of space/time. Space is dual to time -- Einstein. Space/time symmetries are dual to mobius maps -- stereographic projection (conformal invariance). Points are dual to lines -- the principle of duality in geometry. Null points (twistor space) are dual to null lines (light rays in space/time). Clockwise is dual to anti-clockwise -- helicity. Spinors (matter, Fermions) are dual to vectors (waves, Bosons) -- wave/particle duality. Duality means that there is a 4th law of thermodynamics:- Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! All observers make predictions to track targets, goals and objectives -- a syntropic process, teleological. Teleological physics (syntropy) is dual to non teleological physics (entropy). Thesis (action) is dual to anti-thesis (reaction) creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic. Action is dual to reaction -- Sir Isaac Newton or the duality of force. Attraction (sympathy) is dual to repulsion (antipathy), stretch is dual to squeeze, push is dual to pull -- forces are dual. All forces are dual in physics hence energy must be dual:- Energy = force * distance -- simple physics. Energy is duality, duality is energy. Potential energy is dual to kinetic energy -- gravitational energy is dual. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual. Curvature or gravitation is dual. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). The point duality theorem is dual to the line duality theorem. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. Classical reality is dual to quantum reality synthesizes true reality -- Roger Penrose using the Hegelian dialectic.

[@LibrawLou]

@Hyperduality : Dialectic rules under ever-onward Synthesis!

[@AA-gl1dr]

Wonderful

[@hyperduality2838]

Twistor space is an equivalent or dual description of space/time. Space is dual to time -- Einstein. Space/time symmetries are dual to mobius maps -- stereographic projection (conformal invariance). Points are dual to lines -- the principle of duality in geometry. Null points (twistor space) are dual to null lines (light rays in space/time). Clockwise is dual to anti-clockwise -- helicity. Spinors (matter, Fermions) are dual to vectors (waves, Bosons) -- wave/particle duality. Duality means that there is a 4th law of thermodynamics:- Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! All observers make predictions to track targets, goals and objectives -- a syntropic process, teleological. Teleological physics (syntropy) is dual to non teleological physics (entropy). Thesis (action) is dual to anti-thesis (reaction) creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic. Action is dual to reaction -- Sir Isaac Newton or the duality of force. Attraction (sympathy) is dual to repulsion (antipathy), stretch is dual to squeeze, push is dual to pull -- forces are dual. All forces are dual in physics hence energy must be dual:- Energy = force * distance -- simple physics. Energy is duality, duality is energy. Potential energy is dual to kinetic energy -- gravitational energy is dual. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual. Curvature or gravitation is dual. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). The point duality theorem is dual to the line duality theorem. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. Classical reality is dual to quantum reality synthesizes true reality -- Roger Penrose using the Hegelian dialectic.

[@gene4094]

Interesting, as this could describe “Worm Holes”. I’m not a physicist, but I am curious if the positive and negative hemispheres, could be obtained by the use of, synthetic materials, negative refractive indexed meta materials (left-handed electromagnetic rotation); and natural material, (right-handed electromagnetic rotation). Just a thought…

[@UnforsakenXII]

@@hyperduality2838 : /

[@18890426]

​@@hyperduality2838 Always two there are! Nice quote!!

[@hyperduality2838]

Positive is dual to negative -- charge, numbers or curvature. Twistor space is an equivalent or dual description of space/time. Space is dual to time -- Einstein. Space/time symmetries are dual to mobius maps -- stereographic projection (conformal invariance). Points are dual to lines -- the principle of duality in geometry. Null points (twistor space) are dual to null lines (light rays in space/time). Clockwise is dual to anti-clockwise -- helicity. Spinors (matter, Fermions) are dual to vectors (waves, Bosons) -- wave/particle duality. Duality means that there is a 4th law of thermodynamics:- Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! All observers make predictions to track targets, goals and objectives -- a syntropic process, teleological. Teleological physics (syntropy) is dual to non teleological physics (entropy). Thesis (action) is dual to anti-thesis (reaction) creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic. Action is dual to reaction -- Sir Isaac Newton or the duality of force. Attraction (sympathy) is dual to repulsion (antipathy), stretch is dual to squeeze, push is dual to pull -- forces are dual. All forces are dual in physics hence energy must be dual:- Energy = force * distance -- simple physics. Energy is duality, duality is energy. Potential energy is dual to kinetic energy -- gravitational energy is dual. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual. Curvature or gravitation is dual. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). The point duality theorem is dual to the line duality theorem. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. Classical reality is dual to quantum reality synthesizes true reality -- Roger Penrose using the Hegelian dialectic.

[@JAYMOAP]

Ironic but Penrose complained against string theory dealing with higher dimensions, but here you are he construct 6 dimensional model haha. On the other hand, his gluing is essentially a gauging procedure so he derive gauge theory just like string theory does. He doesn't realise this but he essentially indeed in the framework of string theory.

[@hyperduality2838]

Twistor space is an equivalent or dual description of space/time. Space is dual to time -- Einstein. Space/time symmetries are dual to mobius maps -- stereographic projection (conformal invariance). Points are dual to lines -- the principle of duality in geometry. Null points (twistor space) are dual to null lines (light rays in space/time). Clockwise is dual to anti-clockwise -- helicity. Spinors (matter, Fermions) are dual to vectors (waves, Bosons) -- wave/particle duality. Duality means that there is a 4th law of thermodynamics:- Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! All observers make predictions to track targets, goals and objectives -- a syntropic process, teleological. Teleological physics (syntropy) is dual to non teleological physics (entropy). Thesis (action) is dual to anti-thesis (reaction) creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic. Action is dual to reaction -- Sir Isaac Newton or the duality of force. Attraction (sympathy) is dual to repulsion (antipathy), stretch is dual to squeeze, push is dual to pull -- forces are dual. All forces are dual in physics hence energy must be dual:- Energy = force * distance -- simple physics. Energy is duality, duality is energy. Potential energy is dual to kinetic energy -- gravitational energy is dual. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual. Curvature or gravitation is dual. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). The point duality theorem is dual to the line duality theorem. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. Classical reality is dual to quantum reality synthesizes true reality -- Roger Penrose using the Hegelian dialectic.

[@hywelgriffiths5747]

There are still only four spacetime dimensions. The rest are the spinor degrees of freedom. String theory has extra spacetime dimensions

[@timohelenius1482]

you must think sometimes with brains, its most difficult situation in your life

[@lizhang9807]

abraxes

[@timohelenius1482]

in finland the more the more cloud the singer will gave loud, it is the best in finland, but i like more mozart

[@KaliFissure]

Surface(cos(u/2)cos(v/2),cos(u/2)sin(v/2),sin(u)/2),u,0,2pi,v,0,4pi I suggest applying twistor theory to the above manifold.

[@hyperduality2838]

Twistor space is an equivalent or dual description of space/time. Space is dual to time -- Einstein. Space/time symmetries are dual to mobius maps -- stereographic projection (conformal invariance). Points are dual to lines -- the principle of duality in geometry. Null points (twistor space) are dual to null lines (light rays in space/time). Clockwise is dual to anti-clockwise -- helicity. Spinors (matter, Fermions) are dual to vectors (waves, Bosons) -- wave/particle duality. Duality means that there is a 4th law of thermodynamics:- Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! All observers make predictions to track targets, goals and objectives -- a syntropic process, teleological. Teleological physics (syntropy) is dual to non teleological physics (entropy). Thesis (action) is dual to anti-thesis (reaction) creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic. Action is dual to reaction -- Sir Isaac Newton or the duality of force. Attraction (sympathy) is dual to repulsion (antipathy), stretch is dual to squeeze, push is dual to pull -- forces are dual. All forces are dual in physics hence energy must be dual:- Energy = force * distance -- simple physics. Energy is duality, duality is energy. Potential energy is dual to kinetic energy -- gravitational energy is dual. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual. Curvature or gravitation is dual. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). The point duality theorem is dual to the line duality theorem. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. Classical reality is dual to quantum reality synthesizes true reality -- Roger Penrose using the Hegelian dialectic.

[@timohelenius1482]

if you google robots sight from japan then follow ...

[@thomasolson7447]

The tangent plane on the sphere is where the algebra happens. Instead of Pythagoren Theorm, it's the quadratic formula.

[@hyperduality2838]

Lie groups are dual to Lie algebras. Twistor space is an equivalent or dual description of space/time. Space is dual to time -- Einstein. Space/time symmetries are dual to mobius maps -- stereographic projection (conformal invariance). Points are dual to lines -- the principle of duality in geometry. Null points (twistor space) are dual to null lines (light rays in space/time). Clockwise is dual to anti-clockwise -- helicity. Spinors (matter, Fermions) are dual to vectors (waves, Bosons) -- wave/particle duality. Duality means that there is a 4th law of thermodynamics:- Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! All observers make predictions to track targets, goals and objectives -- a syntropic process, teleological. Teleological physics (syntropy) is dual to non teleological physics (entropy). Thesis (action) is dual to anti-thesis (reaction) creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic. Action is dual to reaction -- Sir Isaac Newton or the duality of force. Attraction (sympathy) is dual to repulsion (antipathy), stretch is dual to squeeze, push is dual to pull -- forces are dual. All forces are dual in physics hence energy must be dual:- Energy = force * distance -- simple physics. Energy is duality, duality is energy. Potential energy is dual to kinetic energy -- gravitational energy is dual. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual. Curvature or gravitation is dual. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). The point duality theorem is dual to the line duality theorem. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. Classical reality is dual to quantum reality synthesizes true reality -- Roger Penrose using the Hegelian dialectic.

[@thomasolson7447]

@@hyperduality2838 that's what I said!