# Stuff and Things > HISTORY, veterans & science >  projective space

## nonsqtr

Fun with eyeballs and cameras!

Let's talk about a super-important geometry called "projective space".

It's different from Euclidean space. In normal Euclidean space, parallel lines never intersect. But I'm projective space, parallel lines intersect at exactly one point, called the "point at infinity". Kind of like this:



There is another real-world example that everyone knows about, it's the pinhole camera. Here's how that looks:



If you're clever you've already noticed that the point at infinity has become the origin! In other words, in the first pic, there are things "beyond infinity".

You can intuitively see in the pinhole camera pic, that a "projection" of a point P onto the image plane (in this case the film) is basically a vector passing through the origin (which in this case is the focal point).

The salient observation is that any point along the vector will project to the same point in the image plane. Thus the 3-dimensional real world maps to a 2-dimensional image plane.

It turns out, projective space is what they call a "quotient space" in topology. In this example it is the quotient of 3-d space with an equivalence class defined by scalar multiplication. The points lying along a vector are P = k X, where X is any other point on the same line.

https://en.m.wikipedia.org/wiki/Proj...20through%20it.

Turns out, this is a great way to learn algebraic topology. Projective manifolds are generally "non-orientable". (An example of a non-orientable manifold is a Moebius strip). The topology drops out of the algebra, which starts with Rings. If you're interested, the goal would be to understand this vocabulary:

Section 27.13 (01ND): Projective spaceThe Stacks project

Why is this important? The answer is, be sure there are "spaces that are locally projective", just like there are spaces that are locally Euclidean. Physicists love the ones that are locally Euclidean because they're easy to do math on. (The math can get a little weird in projective space).

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Neo (10-03-2021),QuaseMarco (10-01-2021),UKSmartypants (10-02-2021)

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## QuaseMarco

Pictures I can deal with but formulas................... ah........ no.    :Smiley20:

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crcook84 (10-01-2021),Northern Rivers (10-01-2021)

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## crcook84

> Pictures I can deal with but formulas................... ah........ no.


Right. Projective Space is an interesting idea. But, despite being into computers, I'm rather lousy with math. Mind you, I can make complex equations in Excel so that, as I alter values, the equations will automatically recalculate the new values so that I don't have to do it all over again.

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## Physics Hunter

Not deeply interesting, the "Projective Space" is not actually a space, it is actually a perception or perspective and might be cast as a transform, since you can walk the RR Tracks and climb the tree in real space.  

I always wonder where you are going with this "topology" stuff.  The only way that I have heard of topology in the field of geo-positioning or electro-magnetic fields, which we used in the Modeling and Simulation field.

The pinhole camera has a singular sample perspective and converts 3D to 2D.

Two human eyes convert 3D space to two 2D perspectives that are correlated and can be reconverted into a 3D model.

However, make no mistake that the image of the tree, or mental 3D model of the rail tracks are models of the real space, they have no independent reality.

If viewed as a Transform, such as transforming a Spherical spatial model into a Cylindrical one, the math is not done, but the groundwork is all long finished.

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Neo (10-03-2021)

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## Big Dummy

2CF03BFE-27D7-422E-9EE7-E39EE2396DF3.jpeg

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## UKSmartypants

> Not deeply interesting, the "Projective Space" is not actually a space, it is actually a perception or perspective and might be cast as a transform, since you can walk the RR Tracks and climb the tree in real space.  
> 
> I always wonder where you are going with this "topology" stuff.  The only way that I have heard of topology in the field of geo-positioning or electro-magnetic fields, which we used in the Modeling and Simulation field.



ahhh dont knock nonsqtr, Topology is the key to understanding space.  Manifolds are sexy in cosmology.  Projective Geometry appears in Cosmology in a thing Ive never mentioned here yet, (mainly because it makes my brain fry) called F-Theory, which is the maths behind projective duality and modular forms.  F-theory is formally a 12-dimensional theory, but the only way to obtain an acceptable cosmological background is to compactify this theory on a two-torus. By doing so, one obtains type IIB superstring theory in 10 dimensions. This allows you to describe  elliptically fibred Calabi–Yau four-folds, which are the things lurking at every point in space below the Planck Length, which i have pontificated at length about  :Big Grin: 


https://en.wikipedia.org/wiki/Modular_form  -  you figure it out, does my head in.  Modular forms appear EVERYWHERE in maths. Its what helped solve Fermets Last Theorem

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## Wildrose

> Fun with eyeballs and cameras!
> 
> Let's talk about a super-important geometry called "projective space".
> 
> It's different from Euclidean space. In normal Euclidean space, parallel lines never intersect. But I'm projective space, parallel lines intersect at exactly one point, called the "point at infinity". Kind of like this:
> 
> 
> 
> There is another real-world example that everyone knows about, it's the pinhole camera. Here's how that looks:
> ...


The first one is just an optical illusion as they never really touch. It's just the further out you go it appears they get closer and closer.  Your camera example is problematic because the lines are never parallel.

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## Physics Hunter

> ahhh dont knock nonsqtr, Topology is the key to understanding space.  Manifolds are sexy in cosmology.  Projective Geometry appears in Cosmology in a thing Ive never mentioned here yet, (mainly because it makes my brain fry) called F-Theory, which is the maths behind projective duality and modular forms.  F-theory is formally a 12-dimensional theory, but the only way to obtain an acceptable cosmological background is to compactify this theory on a two-torus. By doing so, one obtains type IIB superstring theory in 10 dimensions. This allows you to describe  elliptically fibred Calabi–Yau four-folds, which are the things lurking at every point in space below the Planck Length, which i have pontificated at length about 
> 
> 
> https://en.wikipedia.org/wiki/Modular_form  -  you figure it out, does my head in.  Modular forms appear EVERYWHERE in maths. Its what helped solve Fermets Last Theorem


I didn't knock anyone.  Real scientists don't take honest criticism of their work/thoughts personally.

Do you actually assert that my post was incorrect?

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## UKSmartypants

> I didn't knock anyone.  Real scientists don't take honest criticism of their work/thoughts personally.
> 
> Do you actually assert that my post was incorrect?



no not at all  :Big Grin:   but you underestimate the importance of projective geometry on the Cosmos   :Big Grin:   M Theory is screwed without it  :Big Grin:

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## Captain Kirk!

> no not at all   but you underestimate the importance of projective geometry on the Cosmos    M Theory is screwed without it


You don't know if he underestimated jack.  You just put people down because you are that kind of self centered person.

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## UKSmartypants

> You don't know if he underestimated jack.  You just put people down because you are that kind of self centered person.



post reported, pointless abuse and ad hominem attack, Go away, your wrecking Nonsqrt thread.

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## Captain Kirk!

> post reported, pointless abuse and ad hominem attack, Go away, your wrecking Nonsqrt thread.


Perhaps if you stop being rude?

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## UKSmartypants

> Perhaps if you stop being rude?



blocked, you need to apologise to nonsqrt for wrecking his thread. Note any further pms will be copied to Mods. not intersted.

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## Authentic

> Perhaps if you stop being rude?


Or if he could learn how to spell nonsqtr.

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## Captain Kirk!

Poor little crying butthurt jerk.

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## Call_me_Ishmael

> Not deeply interesting, the "Projective Space" is not actually a space, it is actually a perception or perspective and might be cast as a transform, since you can walk the RR Tracks and climb the tree in real space.  
> 
> I always wonder where you are going with this "topology" stuff.  The only way that I have heard of topology in the field of geo-positioning or electro-magnetic fields, which we used in the Modeling and Simulation field.
> 
> .


What?   :Shocked: 


I thought topology was all about this guy 
MV5BNTI4NTQwNzYxOV5BMl5BanBnXkFtZTcwMDQzNzYzNA@@._V1_UY264_CR77,0,178,264_AL_.jpg

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Physics Hunter (10-02-2021)

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## Physics Hunter

> no not at all   but you underestimate the importance of projective geometry on the Cosmos    M Theory is screwed without it



Out of my specialization.  I only study Astro for fun, and I am not ardent about it.

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Neo (10-03-2021)

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## UKSmartypants

> Out of my specialization.  I only study Astro for fun, and I am not ardent about it.


Absolutely. I cant get my head round some of the maths Nonsqtr posts up.  We cant all know everything. I bet even Stephen Hawking didnt know anything about 14th Century Etruscan Pottery.

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## Neo

The overriding force in the universe is gravity, understanding gravitational forces and equating them into modelling and simulation fields must be problematic?

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## UKSmartypants

> The overriding force in the universe is gravity, understanding gravitational forces and equating them into modelling and simulation fields must be problematic?



well arguably only on a large scale. Gravity is irrelevant at a  quantum scale. You can argue at 10^-35 cm  the Casimir-Poulder force  is where its at.  This is part of the failure of Quantum theory, its doesnt account for gravity or time.

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Neo (10-03-2021)

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## nonsqtr

Okay people, it's all good, this is an engineering discussion, I have a thick skin, I expect that others have lofty scientific egos just like mine...  :Grin: 

PH has a good point, which is that geometries are "views", they're ways of looking at things. What's interesting though, is how the relationships change. (Math is all about relationships). As far as what is "real", how do you know? If you were a creature living in ProjectiveLand how could you tell? If all your rulers and angles were projective, they might "appear" to be Euclidean. It's kind of the same concept as gravity actually being "curvature of the universe". 

In chemistry (physics) there are "orbits" of electrons around nuclei which look very much like gravitational interactions, they occur at certain preferred distances (and on certain preferred paths) which are "equilibria" where angular momentum matches an attractive force.

Sometimes, if you work stuff out in projective space it becomes a whole lot simpler. The idea that Smarty was referring to is not what we in 3d are projecting ourselves onto a 2d piece of film - it's that WE are the piece of film. In other words, instead of 3 dimensions projecting onto 2, what we have instead is 4 dimensions projecting onto 3. What that fourth dimension is, is anyone's guess - but the math says "no matter what it is", it has to obey certain rules.

The concept of modular forms is especially interesting because it relates to the issue of circle packing ("space filling processes") we discussed in another thread. Consider for example, the concept of 'self similarity' in fractal geometry. There are MANY examples where the view as one zooms in or out can be described as a scalar multiple of the view at another. So the exact of zooming in and out becomes very much like a projection. One view is just like another, up to a scale factor.

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Physics Hunter (10-03-2021)

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## UKSmartypants

> Sometimes, if you work stuff out in projective space it becomes a whole lot simpler. The idea that Smarty was referring to is not what we in 3d are projecting ourselves onto a 2d piece of film - it's that WE are the piece of film. In other words, instead of 3 dimensions projecting onto 2, what we have instead is 4 dimensions projecting onto 3. What that fourth dimension is, is anyone's guess - but the math says "no matter what it is", it has to obey certain rules.


Im clear in my brain about this. we exist on a 10D manifold, this is what String theory demands.

In WW2, USAAF bomber forces flew in "three lots of three lots of three"  The bomber flew in formations of 3,  three formations of three flew in a larger copy of the same formation and three larger formations flew in the same pattern so a flight 3 x3 x3 planes, 27 planes flew together.

The way I see it working is that we exist physically in a flight of three - x y and z axis of our projective plane.  that's D1 D2 and D3   the next 7 D are folded up, compactified, except one, D4.

Our 3D projective plane world sheet travels along the single uncompactified D4 dimension. We cant observe it, because to do so requires us to move in D5, which we cant, since its still compactified. So we travel along D4 one single planck frame at a time. This we see as 'time', but it isnt, its an illusion, and in manifests in the phenomena of entropy. Time does not exist.

Take an analogy. A 3D sphere passing thru a 2D plane, to the local 2D observer, looks like a dot that gets bigger and bigger, turns into a circle, then shrinks back down to a dot and vanishes. The 2D observer would say "thats a circle travelling in time", but hes wrong, its a sphere travelling in a dimension higher. We are in exactly the same position. We cannot observe the spatial movement  in D4 so we call it time.

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## Call_me_Ishmael

> Im clear in my brain about this. we exist on a 10D manifold, this is what String theory demands.
> 
> In WW2, USAAF bomber forces flew in "three lots of three lots of three"  The bomber flew in formations of 3,  three formations of three flew in a larger copy of the same formation and three larger formations flew in the same pattern so a flight 3 x3 x3 planes, 27 planes flew together.
> 
> The way I see it working is that we exist physically in a flight of three - x y and z axis of our projective plane.  that's D1 D2 and D3   the next 7 D are folded up, compactified, except one, D4.
> 
> Our 3D projective plane world sheet travels along the single uncompactified D4 dimension. We cant observe it, because to do so requires us to move in D5, which we cant, since its still compactified. So we travel along D4 one single planck frame at a time. This we see as 'time', but it isnt, its an illusion, and in manifests in the phenomena of entropy. Time does not exist.
> 
> Take an analogy. A 3D sphere passing thru a 2D plane, to the local 2D observer, looks like a dot that gets bigger and bigger, turns into a circle, then shrinks back down to a dot and vanishes. The 2D observer would say "thats a circle travelling in time", but hes wrong, its a *sphere travelling in a dimension higher*. We are in exactly the same position. We cannot observe the spatial movement  in D4 so we call it time.


Your example. You use the word "traveling". That would imply a velocity along that dimension. Call that dimension ♧ . The velocity magnitude, speed in that dimension, along that axis would be d♧/dt. That is the definition of speed.  Time does exist. Or did you have some other definition for the word "traveling".

The bigger issue in the thread of higher dimensions projecting onto lower dimension manifolds is just not a big deal... albeit complex to compute for some - if not most - topologies of the lower dimensional manifolds.

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## Physics Hunter

> The overriding force in the universe is gravity, understanding gravitational forces and equating them into modelling and simulation fields must be problematic?


Actually Electro Magnetic, and Fluid flow modelling is more difficult than 3D Near Earth motion modelling.
I have done some of the former, and lots of the latter.  

I always got called into Sim and Modeling programs after some dipshit with a CS degree tried to model some kind of physics that mattered to the outcome of the study and failed miserably.

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## nonsqtr

> Your example. You use the word "traveling". That would imply a velocity along that dimension. Call that dimension ♧ . The velocity magnitude, speed in that dimension, along that axis would be d♧/dt. That is the definition of speed.  Time does exist. Or did you have some other definition for the word "traveling".
> 
> The bigger issue in the thread of higher dimensions projecting onto lower dimension manifolds is just not a big deal... albeit complex to compute for some - if not most - topologies of the lower dimensional manifolds.


I think it's a bigger deal than you think.

Here is the classical example of what Smarty was talking about, it's called the Riemann sphere.



https://en.m.wikipedia.org/wiki/Riemann_sphere

The idea is, you set the point at infinity (the focal point in the pinhole example), to be the north pole of the sphere. Now you the observer are standing on the north pole of the sphere, which equates with the focal point in the pinhole example - and you're looking out "through" the sphere, onto the piece of film. So, define a location for your film (any perpendicular plane will do, defined by "distance from the north pole"), and trace any vector from the north pole "through" the sphere onto the image plane.

You'll see as you do this, that the Riemann projection "covers" the image plane. Completely. It fills R(2).

In the classic example the third dimension that the sphere lives in is a Complex dimension. (That is to say, "imaginary", using complex numbers). This ends up being very similar to the situation in electrical engineering, where we only see the real "part" of the wave even though the imaginary part has an impact on it.

In this example of projective space, ask yourself, what happens if we deform the sphere? Let's say it's not a perfect sphere anymore, how does that affect the outcome?

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## Call_me_Ishmael

> I think it's a bigger deal than you think.


If you say so. I have not worked with these mappings for ages but as I refreshed my memory with your link, conformal mapping and analytic functions started to come to me. I suspect I lost alot of my handiness with the math but was certainly comfortable working in the complex plane. Wrapping it on a sphere and adding a place for infinity just seems like a mapping.... one that can go either way even though you express it in the direction from the sphere to the plane.  It seems as natural as apple pie going the other way for me... plane to sphere.  I'm sure I'm missing something though. 

Adding any concave deformation on the sphere certainly does put a fly in apple pie though. The mapping from plane to concave deformed sphere is not single valued in the deformed sphere coordinate system. Non-concave deformations don't seem to be a problem so long as you have the mathematical description of the deformed sphere... like an ellipsoid. 

I'm sure I'm trivializing something that is not trivial. I sometimes forget where I put my keys and the names of long time acquaintances.  I can't even remember who the author was of my Tensor analysis text although I often made reference to ... was it Synge and Shield? Or something like that. 


Anyways ... a Riemann_sphere was no doubt a topic in either a complex variables class or my tensor class.  But as an engineer working certain transmission line problems we needed practical tools. One tool was a mapping that we knew as "the Smith chart" .. it maps the right hand side of the complex plane (infinity included) onto the unit circle.  It is a stereographic projection as well .. although we never called it that. It was just a conformal mapping to us engineers. 


http://delta.cs.cinvestav.mx/~mcinto...lex/node7.html

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Physics Hunter (10-04-2021)

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## Physics Hunter

So, SOMEBODY PLEEEEEEESE,  put the power to the ground here.

Why should I give a shit?

My inner Engineer shieking out in pain...

(I ack that UKsmarty tried.)

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## nonsqtr

Let's take this another step.

Let's say for the sake of discussion, that Smarty is correct and what we see in 3d is the result of some oddball machinations in a higher dimension 

But we know that certain things are true in our 3-d reality, like for instance, there is stability. If an object is 'moving' like in Ish's example, we know that it retains its structure as it moves. Well, that is a "constraint" for our higher dimensional process: things have to come back in the same way they went in.

So, how many ways can that happen?

One way to answer, is to "lift" a 3d object into the higher dimension, using an inverse projective map. (This is kinda what our brains do). Once the object is in up-space though, there's only certain things you can do to it, to still retain its size and shape. An example of an applicable set of transforms might be the Unitary Group which consists of rotations and translations. Are there more? Yes there are. Figuring out what they are, is not easy.

In general, projective mappings (and inverse projective mappings) are useful any time we want to consider what might happen in higher dimensions. One of the big advantages of projection mappings is the metrics end up being Hermitian, which makes makes the mathematics of evolution pretty straightforward. (If you're interested, look up Kahler manifolds and the Fubini-Study metric).

Note that ALL of this depends on the definition and existence of 'points at infinity'. Infinity in this case is defined by the complement of sets, so in the railroad track example it would be "any point we can not see", which mathematically translates into the concept of 'sufficiently far away'.

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## nonsqtr

> So, SOMEBODY PLEEEEEEESE,  put the power to the ground here.
> 
> Why should I give a shit?
> 
> My inner Engineer shieking out in pain...
> 
> (I ack that UKsmarty tried.)


Here's why you should care:

The Kodaira embedding theorem states that every compact Kahler surface is a deformation of a projective Kahler surface.

https://en.m.wikipedia.org/wiki/Koda...edding_theorem

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## Physics Hunter

> Here's why you should care:
> 
> The Kodaira embedding theorem states that every compact Kahler surface is a deformation of a projective Kahler surface.
> 
> https://en.m.wikipedia.org/wiki/Koda...edding_theorem



I take it that you don't have to write proposals to win money for your projects?

Feature:Benefit...

What is the benefit?

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## Call_me_Ishmael

> Note that ALL of this depends on the definition and existence of 'points at infinity'. Infinity in this case is defined by the complement of sets, so in the railroad track example it would be "any point we can not see", which mathematically translates into the concept of 'sufficiently far away'.


We always defined infinity as a limit. And we defined it as either a right hand limit or a left hand limit, depending on whether one approached it from increasing positive numbers or from increasing negative numbers. Your terminology seems more geometrical though.

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## nonsqtr

> So, SOMEBODY PLEEEEEEESE,  put the power to the ground here.
> 
> Why should I give a shit?
> 
> My inner Engineer shieking out in pain...
> 
> (I ack that UKsmarty tried.)


A real world example would be:heirarchical motor control in self organizing systems.

If you know about robotics and motor control, you probably know about forward and inverse kinematics, and the idea of mapping those to an "orientation space" suitable for navigating maxes and such.

In such systems, the numbers of possible trajectories are so vast as to be infinite - and the REPRESENTATION of orientability occurs in an infinite dimensional space. 

If you consider the "capability space" in robotic motor systems, this is a perfect example of what projective representations are good for. The Fubini-Study metric applies to infinite-dimensional Hilbert space, and if you check into it you'll come across the Hodge operators and you'll discover how to build an evolution equation from them.

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## nonsqtr

> We always defined infinity as a limit. And we defined it as either a right hand limit or a left hand limit, depending on whether one approached it from increasing positive numbers or from increasing negative numbers. Your terminology seems more geometrical though.


Well, if you consider the Riemann sphere, the North Pole ends up not being in the set. (Of points projecting to the plane). The "point at infinity" in the higher dimensional system doesn't really "exist", it has to be created from that which does.

So, a plane in R(2) has no infinities, it only has real numbers. The "point at infinity" is something we can't see, but the system behaves "as if" it's there.

The geometric definition of infinity comes from compactification. I like it because it's an "operational" definition.

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## Call_me_Ishmael

> Here's why you should care:
> 
> The Kodaira embedding theorem states that every compact Kahler surface is a deformation of a projective Kahler surface.
> 
> https://en.m.wikipedia.org/wiki/Koda...edding_theorem


And as a working stiff (way back when) - if I understand this - I would only care if I had used some transformation to simplify a problem and the problem was now in what you call a Kahler surface and I wanted to make sure I didn't trip over some singularity. If I had analytic functions in my original space, I would want to make sure that it was analytic in this new space as well. 

Either that's true or I have no idea why this would matter, except to a mathematician engaging in mental masturbation.

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## nonsqtr

> And as a working stiff (way back when) - if I understand this - I would only care if I had used some transformation to simplify a problem and the problem was now in what you call a Kahler surface and I wanted to make sure I didn't trip over some singularity. If I had analytic functions in my original space, I would want to make sure that it was analytic in this new space as well. 
> 
> Either that's true or I have no idea why this would matter, except to a mathematician engaging in mental masturbation.


This is how memory traces get stored in your brain.

"Higher dimensional representations".

Consider: a baby learns to talk by babbling. This activity self-organizes the "capability space", so the organism can elocute on demand (or "at will").

The process of self organization makes use of the difference between anterior and posterior probabilities. Think about that, in a context where representations have to be time independent, in other words, sequences have to read out correctly regardless of speed.

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## Physics Hunter

> A real world example would be:heirarchical motor control in self organizing systems.
> 
> If you know about robotics and motor control, you probably know about forward and inverse kinematics, and the idea of mapping those to an "orientation space" suitable for navigating maxes and such.
> 
> In such systems, the numbers of possible trajectories are so vast as to be infinite - and the REPRESENTATION of orientability occurs in an infinite dimensional space. 
> 
> If you consider the "capability space" in robotic motor systems, this is a perfect example of what projective representations are good for. The Fubini-Study metric applies to infinite-dimensional Hilbert space, and if you check into it you'll come across the Hodge operators and you'll discover how to build an evolution equation from them.


Ok, that computes.

I always had an argument with the robotics guys, with stepper motors the movement space was not technically infinite.  I won a fair amount of drinks with that one.  They were labrats that did not often step out into the real world.

But in a DARPA mobility challenge sense, vehicles moving over rough terrain and dirt roads with limited sensors, the space is at least virtually infinite.
The winners won, with a Toureg, realizing that it was not the offroad prowess of the vehicle that made the difference (versus an Oshkosh...) but the ability to accelerate once a decision was made.

I have acknowledged that if one is considering a sensor limited system, and I will extend that to an effector limited system, that the concept of a perception space is important and useful.

As noted previously, we spent a ton of time in undergrad Physics doing spatial transforms to gain an advantage on some particular problem that lent itself to some particular coordinate representation.  This came in quite handy when someone asked me if I could simulate Synthetic Aperture Radar (SAR).

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## nonsqtr

Cool! You'd enjoy the presentation on one of my web sites, it's kind of a lo-tech version of what the gubmint does. 

Coupla observations - first, the Riemann construction happens to be useful because it lends itself to conformal mapping, but it doesn't have to be a sphere, right? The projection metric could be anything, the shape could be a triangle instead of a sphere.

The sphere is useful because the point at infinity is conveniently defined in terms of a radius from the origin, and the origin is conveniently colocated with the point being studied.

Then secondly, one could envision a collection of such spheres with different radii, all centered at the same point. What does "radius" really mean in this context? 

Thirdly - conic sections! Let's say you have an hourglass shape like a relativistic light cone, and you position your image plane so it slices through the cone (resulting in either symmetrical or asymmetrical conic sections).

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Physics Hunter (10-04-2021)

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## Call_me_Ishmael

> This is how memory traces get stored in your brain.
> 
> "Higher dimensional representations".
> 
> Consider: a baby learns to talk by babbling. This activity self-organizes the "capability space", so the organism can elocute on demand (or "at will").
> 
> The process of self organization makes use of the difference between anterior and posterior probabilities. Think about that, in a context where representations have to be time independent, in other words, sequences have to read out correctly regardless of speed.


...regardless of speed *and* the direction from which you approach the representation of interest. 

I think you said the practical equivalent of what I said but you parameterized some "capability space" and prefer to talk about that while I simply was talking about the number space onto which you parameterized your "capability space". 

Your lingo is unfamiliar to me so I should just stop .

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## UKSmartypants

> Your example. You use the word "traveling". That would imply a velocity along that dimension. Call that dimension ♧ . The velocity magnitude, speed in that dimension, along that axis would be d♧/dt. That is the definition of speed.  Time does exist. Or did you have some other definition for the word "traveling".
> 
> The bigger issue in the thread of higher dimensions projecting onto lower dimension manifolds is just not a big deal... albeit complex to compute for some - if not most - topologies of the lower dimensional manifolds.



You are getting fixated on specific words. Use any other word than 'travelling' if you like.


4D is a single world line. On compactified dimensions there is no such concept as distance  or time, its meaningless.


Time is an arrow that defines the direction entropy grows, and as Ive also said before its wrong because its entirely based on fallacious assumptions. The Second law of Thermodynamics cannot be applied to the universe as a whole because a) there is no proof the quantity of energy (and by that we mean mass/energy/information) is fixed, a basic requirement of entropy  - in fact we know it isnt fixed, what about all the information that vanishes into black holes, and b) entropy also requires a closed system, again we have no proof the universe is a closed system. Plus time is a nonsense concept at quantum levels. Movement along the D4 therfore does not necessarily  involve time ,entropy or the expenditure of energy, at a quantum level, ona higher dimension world line. A lot more scientists are coming round to this way of thinking, but for other reasons, mainly to do with how the universe creates reality and how wave function collapse occurs.  The fact is 'time' and 'entropy' are more likely to be secondary local phenomena - Einstein has already proved that by default, but theres nothing in physics that demands a time dimension, not anywhere.

I reject time as a concept, theres no proof its necessary for anything within the framework of a 10D Manifold, its just  an illusion in 3D.

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## Call_me_Ishmael

> Ok, that computes.
> 
> I always had an argument with the robotics guys, with stepper motors the movement space was not technically infinite.  I won a fair amount of drinks with that one.  They were labrats that did not often step out into the real world.
> 
> But in a DARPA mobility challenge sense, vehicles moving over rough terrain and dirt roads with limited sensors, the space is at least virtually infinite.
> The winners won, with a Toureg, realizing that it was not the offroad prowess of the vehicle that made the difference (versus an Oshkosh...) but the ability to accelerate once a decision was made.
> 
> I have acknowledged that if one is considering a sensor limited system, and I will extend that to an effector limited system, that the concept of a perception space is important and useful.
> 
> As noted previously, we spent a ton of time in undergrad Physics doing spatial transforms to gain an advantage on some particular problem that lent itself to some particular coordinate representation.  This came in quite handy when someone asked me if I could simulate Synthetic Aperture Radar (SAR).


Did you work DARPA's MSTAR program?  That was long ago but they did SAR simulation as well.

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## Call_me_Ishmael

> You are getting fixated on specific words. Use any other word than 'travelling' if you like.
> 
> 
> 4D is a single world line. On compactified dimensions there is no such concept as distance  or time, its meaningless.
> 
> 
> Time is an arrow that defines the direction entropy grows, and as Ive also said before its wrong because its entirely based on fallacious assumptions. The Second law of Thermodynamics cannot be applied to the universe as a whole because a) there is no proof the quantity of energy (and by that we mean mass/energy/information) is fixed, a basic requirement of entropy  - in fact we know it isnt fixed, what about all the information that vanishes into black holes, and b) entropy also requires a closed system, again we have no proof the universe is a closed system. Plus time is a nonsense concept at quantum levels. Movement along the D4 therfore does not necessarily  involve time ,entropy or the expenditure of energy, at a quantum level, ona higher dimension world line. A lot more scientists are coming round to this way of thinking, but for other reasons, mainly to do with how the universe creates reality and how wave function collapse occurs.  The fact is 'time' and 'entropy' are more likely to be secondary local phenomena - Einstein has already proved that by default, but theres nothing in physics that demands a time dimension, not anywhere.
> 
> I reject time as a concept, theres no proof its necessary for anything within the framework of a 10D Manifold, its just  an illusion in 3D.


The only reason you say there is no time is because your 10D space is the static container of all that has happened and all that will ever happen.  As I said before, you have integrated out time from negative infinity to positive infinity. 

But whatever. It's academic in the truest sense of the word.

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## Authentic

This thread is turning into a science project.

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## UKSmartypants

> The only reason you say there is no time is because your 10D space is the static container of all that has happened and all that will ever happen.  As I said before, you have integrated out time from negative infinity to positive infinity. 
> 
> But whatever. It's academic in the truest sense of the word.


its more complex than that.

Our 4D projective space is part of a 10D Calabi Yau manifold, and exists on a 2D brane. There may well be an infinite number of such Calabi Yau manifolds, each one hosting a complete universe. There is no concept of time or distance on this Brane, because it is not part of our universe. We are, if you like a hologram projected onto it. Thus the movement along the uncompactified D4 world line does not entail time or motion as concepts as we understand them. Such concepts are meaningless in this context, because there is no concept of entropy outside our universe, on the brane.

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## nonsqtr

> theres nothing in physics that demands a time dimension, not anywhere.


Yes, there is. You're just not looking for it in the right place.

You're looking for it "out there", whereas the answer is "in here".

'Out there" turns out to be rather mundane and uninteresting, compared to what happens "in here".

Physics in it's present form is very primitive. Biophysics is actually considerably more advanced in the fundamental areas of the structure of random processes. Biologists can describe space filling processes, whereas cosmologists can't (yet - although you're alluding to the beginnings of it).

The easy piece is this: the purpose of the brain is to optimize the behavior of the organism IN REAL TIME. So for example, I'd encourage you to check out the various forms of learning. There are time independent forms, and time dependent forms. You can get a neural network to self-organizes emitted sequences if you set the time constants just right, but you can't get time-independent sequences without REPRESENTING the information in time-independent bases.

In a real brain, there are dedicated processes and circuits that "extract causality from time series", it happens in real time with several million time series at once. The information is encoded in an unknown manner - we have basically "no idea" how it gets from the orientation buffer into the global store. We do know that it's encoded into "objects and events" before that happens.

Here's a hint: there is no first-derivative information in cosmology. It's all second-derivative stuff. The wave equations aren't damped.

However in the brain fully HALF of the circuitry specifically handles the first derivative.

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## nonsqtr

What is interesting too, is to study the language and grammar of causation.

The Grammar of Causation and Interpersonal Manipulation | Edited by Masayoshi Shibatani

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## nonsqtr

> Time is an arrow that defines the direction entropy grows


We disagree.

Entropy is a MEASURE, not a "thing".

Moreover, it is a NON-LOCAL measure. Yes, in ALL cases. Defining entropy locally is entirely nonsensical. (Because information can not be defined locally).

My guess is that time is also a process, not a thing. It's an optimization, in the same sense that entropy is an optimization. The simplest example is quantum evolution, which is an optimization over the available states.

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## UKSmartypants

> Yes, there is. You're just not looking for it in the right place.
> 
> You're looking for it "out there", whereas the answer is "in here".
> 
> 'Out there" turns out to be rather mundane and uninteresting, compared to what happens "in here".
> 
> Physics in it's present form is very primitive. Biophysics is actually considerably more advanced in the fundamental areas of the structure of random processes. Biologists can describe space filling processes, whereas cosmologists can't (yet - although you're alluding to the beginnings of it).
> 
> The easy piece is this: the purpose of the brain is to optimize the behavior of the organism IN REAL TIME. So for example, I'd encourage you to check out the various forms of learning. There are time independent forms, and time dependent forms. You can get a neural network to self-organizes emitted sequences if you set the time constants just right, but you can't get time-independent sequences without REPRESENTING the information in time-independent bases.
> ...



But its all a macro phenomenon. Nothing in M theory or Quantum theory demands time. Quantum processes are time symmetric.


Entropy is a local phenomena, it must be, since there's no proof the conditions required by the second law apply to the universe as a whole. There's no proof the universe is a closed system. Theres no proof the universe has a fixed amount of energy + mass + information.  Theres simply no proof entropy applies to the observable universe. And theres nothing to contradict the proposition 'time' is an illusion created by observing d4 one Planck frame at a time.


Consider
-The universe may be an open system within the multiverse.
-The small size of the early universe may have limited entropy to be low. This compact universe may have been an effective entropy reset. As it has expanded, by inflation, entropy has been much more "room" to increase. 
-The second law is a statistical law that predicts that entropy will usually increase. On a local macro scale, it basically always does increase. Given an infinite time, global entropy will decrease from time to time. This is basically Poincaré recurrence theorem.
-There may be an unknown physical process that causes entropy to decrease. For instance the arrow of time could reverse at maximum expansion. This is Liouville's theorem.

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## Physics Hunter

> Did you work DARPA's MSTAR program?  That was long ago but they did SAR simulation as well.


Nope, my work was before that, and lower fidelity for a vastly different purpose.

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## nonsqtr

> But its all a macro phenomenon. Nothing in M theory or Quantum theory demands time. Quantum processes are time symmetric.


Well gee, here's a novel thought - quantum processes operate within constraints, geometric and otherwise.




> Entropy is a local phenomena, it must be, since there's no proof the conditions required by the second law apply to the universe as a whole.


/Slap!

Entropy is a MEASURE, not a "phenomenon".




> There's no proof the universe is a closed system. Theres no proof the universe has a fixed amount of energy + mass + information.  Theres simply no proof entropy applies to the observable universe.


Sure there is. It is a repeatable and independently observable process. It obeys well known statistics. But if you're looking for an "it" you won't find one.




> And theres nothing to contradict the proposition 'time' is an illusion created by observing d4 one Planck frame at a time.


Irrelevant. A rose by any other name is still a rose.

I gave you the answer a long time ago. That which distinguishes time is it's not reversible, and the only physical process we know of that's not reversible is choice. It is impossible to describe a stochastic process without referencing time, because stochastic choice is not reversible. There is mounting evidence that the concepts "precedes" and "results from" are hardwired into the brain.

The act of CHOOSING FROM A SET is not reversible. It has nothing to do with spacetime, it's more fundamental than that. Every stochastic process is a choice from a set. The entirety of quantum mechanics depends on this concept, it wouldn't work at all without it.

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## UKSmartypants

yes, but as always the argument between you and me resolves down to you talking about brains and me talking about the Multiverse. I dont think the two relate. What goes on in your brain is an infinitesimal part of the universe as a whole. Its a local thing.

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