Reciprocal System Research Society

After recently reading about Projective geometry, a few thoughts occurred to me.

One thing I noticed, and was also briefly mentioned by the author, that Projective Geometry is actually a basis for Euclidean geometry. In the same manner that Logic is really a basis for mathematics.

Euclidean geometry deals with congruence and similarity in specific figures. While Projective geometry deals primarily with the formation of such figures.

I know Projective geometry is used mostly in applications beyond parallel geometry, but, Projective geometry CAN be used to form parallel forms.

On that ending note, I also wanted to share something I discovered.

In reference to Larson's RS theory, if in Projective geometry you have parallel lines it is assumed that there is a 'common' point that does not lie on a line (for they are parallel and do not meet). The author suggested thinking of it to infinity (never ending). However, I feel they are making an assumption that is completely wrong! If using Larson's idea of inverse relationships, then a much more accurate concept of this 'common' point of two projected parallel lines can be made.

If the parallel lines are a motion moving outside from their center in two opposite directions, and the 'common' point is given an inverse relationship, then it is a motion that moves inside the point's center in two attractive directions.

Beyond that I am actually having trouble finding words to describe the full concept. I just hope that helps to describe it.

It really started boggling my head when I applied the principle of duality (for projective geometries, where the point and the line interchange) to that concept above... Makes my head spin!

Then I got into some very exhaustive thoughts about the double helix. The 'common' point of 'life', the Phi-ratio, and etc...

What does every one think?

Tulan wrote:

I know Projective geometry is used mostly in applications beyond parallel geometry, but, Projective geometry CAN be used to form parallel forms.

Be warned... there are so many variations of projective geometry, you can get really lost. But the basic concept is that "projective geometry" deals with transforming a simple concept of "ratio" (or "motion", in Larson's terms) into other geometries thru the process of making assumptions.

As an analogy, look at the XML and XSL transformations I taught you, for web programming. You start with basic XML data elements, which have no format, design or layout. It is just "raw data", tagged to say what it is. Thru the use of the X Stylesheet Language (XSL), you provide a series of instructions (assumptions) that allow you to create formatted documents, that convert the data into something presentable, whether that be HTML, printed output, PDF files or whatever. The "stylesheet" are the assumptions you place on the raw data, to get the desired output.

In Projective Geometry, the "strata" are the stylesheets... those instructions (assumptions) that allow you to convert the raw data (cross-ratio, or Larson's "scalar motion") into something presentable, like affine geometry, metric geometry, or Euclidean geometry.

A precise analogy would be: XML (projective) transforms thru XSL to raw HTML (affine). That raw HTML then has a cascading stylesheet (CSS) applied to it (metric), and the browser grabs the images and text and displays it (Euclidean). Notice that each step adds a layer of assumption, affine being HTML, metric being CSS and Euclidean being "pixels" on a screen.

It is the process that builds the worlds of illusion, from the bland, raw data.

I can also draw a musical analogy... "Notes" are the projective layer, which combine to form chords (affine), which are used to form movements (metric), which creates the song (Euclidean). Each step, again, has a specific assumption. Chords require notes to be played in simultaneously. Movements are groups of chords. Songs are a mixture of movements.

The assumptions that we pile up to get Euclidean geometry, from the "raw data" of projective geometry, are:

1: Projective; the "raw data" is the cross-ratio, a ratio of ratios. Nothing else.

2: Affine; Introduction of the duality of zero/infinity, with a point at zero, and a plane at infinity (in 3D, points and planes are duals; in 2D, points and lines are duals). This creates parallel and orthagonal relationships.

3: Metric: the absolute conic (think of the field of vision), which creates the concept of relative measurement -- you can find the "percentage" size of things, but not the exact size. In other words, you can say "this is half as tall as that", but you cannot say, "this is 6 feet high."

4: Euclidean: scale FIXED at Unity, so you know how big 1 foot is, and CAN say, "this is 6 feet high." (This is a problem for Larson, because you cannot postulate Euclidean geometry and scalar motion at the same time, as Euclidean geometry is scale-invariant!)

Tulan wrote:

In reference to Larson's RS theory, if in Projective geometry you have parallel lines it is assumed that there is a 'common' point that does not lie on a line (for they are parallel and do not meet). The author suggested thinking of it to infinity (never ending). However, I feel they are making an assumption that is completely wrong! If using Larson's idea of inverse relationships, then a much more accurate concept of this 'common' point of two projected parallel lines can be made.

There is a difference between "infinity" and "unbounded". What is interesting is that parallel lines DO meet at infinity, no matter where you draw them. To better conceive this, takes its inverse -- rather than a point at zero and a plane at infinity, take the counterspace view, a plane at zero, and a point at infinity. Picture it as a sphere, the surface being "zero" and the center, "infinity." When you draw "parallel lines", the become radii... separate at the origin, but all merge into the point at infinity.

Tulan wrote:

It really started boggling my head when I applied the principle of duality (for projective geometries, where the point and the line interchange) to that concept above... Makes my head spin!

Since you are in to computer models, you will find it easier to think in 3D terms, than 2D ones... the duals are then the point and plane, and the line is common to both. A set of (x,y,z) coordinates simply refers to locations on the X, Y and Z axes. If we find where they meet in 3-space, we get a point. OR, we can draw a triangle connecting these three locations on the axis, and define a plane. That is the nature of this duality.

Hopefully, this will help you clarify your thoughts.

That did so very much, it cleared up alot of mis-interpretations I had. Thank you very much! Lots for me to chew on now!

The parallel lines is still bugging me though. If you have say train tracks, and they are in a straight line all the way across the earth's surface, they will end up meeting themselves as they loop back around - I understand that. But the 'common' point for where you were to 'project' the train-tracks - because the two seperate tracks do not have a point in euclidean space where they intersect to form a the 'common' point.

So the author as I was reading (and interpreted it) suggested thinking of this 'common' point at infinity (he is talking of unbounded for this infinity), as though you were to stand on train tracks and look down the center of them (disregarding the idea that they can loop around and meet themselves) and at some point see the tracks 'merge' at unboundedly.

I don't like it though, because technically there ither is NO point at all or the 'common' point is the center of a 'ring of tracks'. One CAN project a 'common' point into a center of a sphere or circle made of parallel lines.

Sometimes I forget to not connect to the archive when I am thinking of this stuff - because I end up getting flooded with concepts! lol.

I really appreciated those analogies, thank you for helping me.

Tulan wrote:

The parallel lines is still bugging me though. If you have say train tracks, and they are in a straight line all the way across the earth's surface, they will end up meeting themselves as they loop back around - I understand that. But the 'common' point for where you were to 'project' the train-tracks - because the two seperate tracks do not have a point in euclidean space where they intersect to form a the 'common' point.

He is talking about the "vanishing point", which is that point where parallel lines meet at infinity. It is what gives art that "perspective" so you can look at a 2D image and interpret it as 3-dimensional.

Tulan wrote:

I don't like it though, because technically there ither is NO point at all or the 'common' point is the center of a 'ring of tracks'. One CAN project a 'common' point into a center of a sphere or circle made of parallel lines.

Look at the attached image of railroad tracks, in exactly the configuration you were talking about. Are the two rails of the tracks parallel? If you say "yes", then you have made an ASSUMPTION about the 3-dimensional nature of the lines on the page, when they are, in fact, 2-dimensional.

Take out your ruler, and measure the distance between the two rails at the bottom of the image. Now, go about 1/4 way up, and measure again. I'll bet you lunch that the measurement is SMALLER than it was at the bottom. Now go up and measure where they are working on the rails in the distance... I'll bet the two rails TOUCH each other there, because that is the "vanishing point", the place where these parallel rails meet.

So yes, the rails DO MEET, in the REALITY of a 2-dimensional drawing, but DO NOT MEET, in the ILLUSION of a 3-dimensional interpretation.

That is the essence of projective geometry... you add an assumption, and create an illusion. Ever wonder why Ra and the Confederation gang call our "reality" an "illusion"? This is exactly why.

So, do parallel lines actually meet? Since the concept of "parallel" is the FIRST ASSUMPTION that takes projective geometry to affine geometry, they DO MEET in the reality of cross-ratio, but DO NOT MEET in the illusion of the Affine.

It's all a matter of keeping your "perspective" on things!

Awesome, thank you, it is giving me alot to munch on. Brings up alot of thoughts on personal perspective and how interchange between individuals can be difficult unless some sort of abstract form of communication is available (like mathematics).

A picture might be worth a thousand words but if the perspective of the person looking at it is completely different from the author's, then it isn't worth anything, unless there is some kind of unifying element so the viewer can say, "Oh now I know what he is seeing".

Concepts can also be hard to desribe with still life pictures. A good example I remember is trying to draw the concept of "The point of no return." to a toddler who knew nothing of the concept of "The point of no return." (without using any spoken or written words).

Then of course this brings me round-robin to the question you posed (how would you communicate with a being that evolved in a completely different logos), and the episode of stargate with the 'elemental' language.

Then recursion happens...

I appreciate your response, thank you.

Tulan wrote:

Brings up alot of thoughts on personal perspective and how interchange between individuals can be difficult unless some sort of abstract form of communication is available (like mathematics).

You have to go back to archetypal symbolism and motif. That is the root language. I've always considered math a poor choice, because of its preference for reductionism. 10/5 = 2, mathematically, but then I've lost the 10 and the 5, and what they originally meant. That is why my computer simulations of RS2 never reduced anything -- everything worked from "speeds", not displacements.

Tulan wrote:

A picture might be worth a thousand words but if the perspective of the person looking at it is completely different from the author's, then it isn't worth anything, unless there is some kind of unifying element so the viewer can say, "Oh now I know what he is seeing".

That's why they call it "consensus reality."

Tulan wrote:

Concepts can also be hard to desribe with still life pictures. A good example I remember is trying to draw the concept of "The point of no return." to a toddler who knew nothing of the concept of "The point of no return." (without using any spoken or written words).

As Wiley Coyote knows, "One step past the edge of a cliff." And usually followed by an anvil.

Cartoonists are experts at expressing concepts in "still life" by depicting that "moment of realization", done to humor.

Tulan wrote:

Then of course this brings me round-robin to the question you posed (how would you communicate with a being that evolved in a completely different logos), and the episode of stargate with the 'elemental' language.

Then from Larson we know that the elemental language is composed of 128 base concepts, their inverses (at least in 3rd density), since that is the total, possible combinations of sub-atomic and atomic motions.

Words would be formed by joining compatible concepts, like chemical formula.

The unconscious aspect of yourself knows the elemental symboilsm of the Universe, because it is part of it. All it needs is a way to connect that unconscious knowledge to conscious knowledge, like watching Babylon 5... the brain matches the patterns in the unconscious, and assigns them a corresponding symbol. From that point on, language can be used to communicate, because of the consensus reality created.

The more symbols you have to choose from, the better you can communicate.

Mmmkay.

The more symbols you have, the more analogies you can draw with the symbol sets of other people, using language or visual art (consensus reality) to express them.

What I am grappling with now, is mathematics. Obviously, mathematics is useful in designing a wooden desk without using trial and error of materials; therefore saving time, materials and money. But some mathematics seems SO abstract that it is completely seperated from anything remotely related to existence (what is the point?).

So truly, no specific subject is really priority. If math serves to abstract concepts of physical objects for manipulation and study, it is but just one aspect of an experience. Then there are symbols and motifs, created to help me understand myself through communication (obviously, example of B5).

So my question is, what is the point of all the higher mathematics? Beyond using elementary math - Geometry - Trig - Algebra - for their usefulness in manipulating abstract concepts of physical objects, is 'higher' mathematics really worth the study?

Quote:

So my question is, what is the point of all the higher mathematics? Beyond using elementary math - Geometry - Trig - Algebra - for their usefulness in manipulating abstract concepts of physical objects, is 'higher' mathematics really worth the study?

From the way I see it, this higher math stuff reminds me of the epicycles idea prevalent quite long back, about the movement of the planets around the earth. Till the people figured out that there are just too many complications for it to be a genuine idea, the epicycles concept prevailed….

The point being, the higher math is required for people to venture into areas unexplored, and deal with the situations, TILL an insight comes to simplify the mathematics due to a different way of looking at things. It may serve as a temporary mess pending further simplification.

If we do not like the abstract math, there are many other ways, I suppose…as the other 'aspects of experience'.

Cheers,

Gopi

Tulan wrote:

What I am grappling with now, is mathematics. Obviously, mathematics is useful in designing a wooden desk without using trial and error of materials; therefore saving time, materials and money. But some mathematics seems SO abstract that it is completely seperated from anything remotely related to existence (what is the point?).

I agree with what Gopi posted; it is just a device that is used to attempt to conceive the inconceivable. I don't use "higher math", because I've always believe the Universe is a simple place. I think it all goes back to what Larson said, "complexity is entertaining, simplicity is not." Higher maths are very entertaining.

Tulan wrote:

So my question is, what is the point of all the higher mathematics? Beyond using elementary math - Geometry - Trig - Algebra - for their usefulness in manipulating abstract concepts of physical objects, is 'higher' mathematics really worth the study?

Like any "device", it can sometimes help you see connections that you could not see before. The Clifford Algebra and Bott periodicity, for example, pointed out the "how to" for things I knew must be there, but could not derive from the bottom, up.

You can learn something from anything; just don't obsess on only a single viewpoint. Look at it all.

That is pretty much how I felt about it.