Reciprocal Geometry

bperet · Post by **bperet** » Sat Sep 04, 2004 6:50 pm

The primary difference between RS and RS2 is the inclusion of various strata of geometry, and their polar inverses.

This has a number of consequences that lead to a slightly different set of conclusions than Larson's original statement of Reciprocal System theory.

First, one must understand that geometry is based upon perception -- how we view things with our senses. With normal, binocular vision, we see a Euclidean world -- one in which we can tell that 1 meter in length is 1 meter in length, regardless of where it is placed in relation to us (providing it is within the tolerance of our vision to perceive). This gives us the absolute measures we need to function in this world. This is WHY Euclidean geometry is scale invariant.

Close one eye, and the Euclidean world vanishes, becoming a Metric world, where only relative measurements exist. If one were to place a yardstick in an otherwise featureless room, you would only know it was a yard long because you would recognize the yardstick. If someone placed a 6-inch long yardstick in the room, that looked exactly like a regular yardstick, you would still perceive it to be a yard long, except at a further distance away, since you lost the triangulation that binocular vision provides. Metric geometry is scale VARIANT--in other words, the scale of an object can change, resulting in a perceived change in location. Under metric perception, you could not tell the difference if I shrunk the yardstick to half size, or moved it twice as far away. It would look exactly the same in both cases.

With that understanding, one realizes that once we leave the Euclidean realm of "coordinate time-space" (what Larson calls "extension space"), we have lost absolute measurements, and only have measurements that make sense relative to each other. Therefore, we can conclude that:

1) All scalar motion is non-Euclidean, since it is scale variant.

2) All scalar motion exists outside extension (coordinate, vector) space.

3) Only extension space has absolute measurement.

4) Scalar motion is relative to other motions; their ratio being invariant (absolute).

Larson identified the natural datum of scalar motion to be Unity, 1 unit of space per 1 unit of time being the reference point for speed. This is known as a "unit boundary" ("unit" meaning "1").

When we move from the time-space of the material sector into the space-time of the cosmic sector, the aspects of space and time invert. A material speed of s/t becomes a cosmic speed of t/s, perceived in the material sector as "energy".

Likewise, the geometry of the "other side" becomes the polar inverse of the geometry we are measuring with. In our normal, Euclidean realm of extension space, the cosmic sector appears as polar-Euclidean -- what we would describe mathematically with "imaginary" numbers. Our translational motion in space appears as rotational motion in the cosmic sector. The same holds true for the time and space regions.

The time region is the region within 1 unit of space. The magnitude of motion in the spatial aspect of motion is fixed at unity. In other words, 1/t. Only "time" can increase, not space. The boundary of the time region is therefore 1/1 -- a unit boundary, so we also see the configuration space of the atom as polar-Euclidean to our perception, which is known as counterspace.

All measurements in counterspace are based in rotation. Translation does not exist in counterspace, except as a compound rotation.

In normal, extension space, we see rotation as a shear in two dimensions. Move something in one direction (translation); push it in another direction at the same time, and it follows a curve. In counterspace, the principle is the same... rotation something in one direction, push it again in another, and it follows a straight line. We see all motion in both the cosmic sector, and the time region as having a primary motion of ROTATION, whereas in the material sector, the primary motion is TRANSLATION.

This brings us back to the natural datum of unity. In the material sector, we see outward translational motion in 3 dimensions, which results in the progression of the natural reference system -- everything moving away from everything else.

BUT, when we view the time region that defines atoms, we see an outward ROTATIONAL motion in 3 dimensions, which results in Larson's "rotational base" with NOTHING rotating, because rotation occurs naturally in the region.

The natural consequence of this is that "linear vibration", upon which Larson builds the photon, DOES NOT OCCUR, because you cannot have "linear" motion in a polar region, unless there are TWO dimensions involved -- just as it requires TWO translations to rotate an object in "normal" space.

At this time, we only have unit-speed rotation. The next step is to move away from Unity, and see what happens.

Alluvion · Post by **Alluvion** » Wed Nov 03, 2004 10:54 pm

if there is one axis/vertex there is a unified structure for rotation. The "farthest" geometric condition of two axes would be perpendicularity - that 90 degree shift would produce a cubic intersection that would not be an "inverse" but shows up in the spiral patterns of DNA (graphed as a wave phase shift in trigonometric systems).

I am just going to share some thoughts and let them be:

spiral motion is the combination of both linear (intellectual) motion and rotational motion (intuitive). The intellect resolves the axis between two points as the most true path between any two points or conditions - this relates to the concept of time/space units in that the intellect operates with a translational/axial charachter. The spirit/heart/intuition resolves rotation about an axis or condition where space is a fixed radius of condition (perhaps awareness, beleif, etc) and only the time portion of the units flows.

Spiral motion mediates between these two, in that the intellect resolves the primary axial path, but is balanced with the rotational path of intuition (rotation equated with awareness, integration, etc). The more I read about these ideas and concepts the clearer it becomes to me that these conditions/structures become only more connected and less distinct. THe motion of space and the motion of time are resolved in the perfect proportions of the golden rectangle/spiral - where intellect and intution, dark and light, are in pure equanimity. Perhaps any spiral dervied of a similar geometric application would be (for ex, how might one compose a golden triangular pattern?) - what is the 3-dimensional form of spiral motion? the vortex?

bperet · Post by **bperet** » Fri Nov 05, 2004 9:02 am

polakie11 wrote:

if there is one axis/vertex there is a unified structure for rotation. The "farthest" geometric condition of two axes would be perpendicularity - that 90 degree shift would produce a cubic intersection that would not be an "inverse" but shows up in the spiral patterns of DNA (graphed as a wave phase shift in trigonometric systems).

In the rotational space of counterspace, the axes are "imaginary" (polar) and aren't straight lines, but arcs that require an infinite angle to complete. Applying "cubic" intersection in counterspace results in a spherical volume, not a cube, since each axis IS a rotation, not an "axis OF rotation".

polakie11 wrote:

spiral motion is the combination of both linear (intellectual) motion and rotational motion (intuitive).

You could also refer to "linear" as spatial and "rotational" as temporal.

This agrees fairly well with the observed valuing systems of thinking as both rational and discrete (object oriented, local) and feeling as rational and functional (relation oriented, non-local). (I believe you are referring to the rational function of "feeling", not "intuition" in the above, since intuition, like "sensation", is classified as "irrational").

polakie11 wrote:

The intellect resolves the axis between two points as the most true path between any two points or conditions - this relates to the concept of time/space units in that the intellect operates with a translational/axial charachter. The spirit/heart/intuition resolves rotation about an axis or condition where space is a fixed radius of condition (perhaps awareness, beleif, etc) and only the time portion of the units flows.

I would suggest checking out this link: http://www.anth.org.uk/NCT/ which is Nick Thomas' discourse on projective geometry and counterspace. You may find the concept of rotational (non-linear) spaces very interesting.

polakie11 wrote:

Spiral motion mediates between these two, in that the intellect resolves the primary axial path, but is balanced with the rotational path of intuition (rotation equated with awareness, integration, etc).

Do you have a system that can resolve the infinite angles of counterspace into the concept of "rotational depth" -- angles of more than 360 degrees -- as part of this spiral function?

polakie11 wrote:

THe motion of space and the motion of time are resolved in the perfect proportions of the golden rectangle/spiral

It makes sense that the projective intersection of space and time would produce a helical result, I cannot see where the "golden" aspect comes into play. How did you come to this conclusion? Why would it follow this proportional system instead of just a simple 1:1 correspondence?

polakie11 wrote:

what is the 3-dimensional form of spiral motion? the vortex?

If the spiral is the projection of a line and a circle (one dimension each, since the circle is a counterspatial turn), then the 3d form would be the projection of the cube and sphere.

But also consider that your point of view is in the Euclidean "linear" system when viewing the spiral. What does the spiral become when viewed from the counterspatial "polar" geometry system? Rather than a bounded circle being drawn out along an unbounded line, it would be a bounded line drawn out around an unbounded circle?

Alluvion · Post by **Alluvion** » Sat Nov 06, 2004 11:37 am

>>>>since each axis IS a rotation, not an "axis OF rotation".

oh ok, I hadn't had this idea before. I'll think about this.

>>>>You could also refer to "linear" as spatial and "rotational" as temporal.

This agrees fairly well with the observed valuing systems of thinking as both rational and discrete (object oriented, local) and feeling as rational and functional (relation oriented, non-local). (I believe you are referring to the rational function of "feeling", not "intuition" in the above, since intuition, like "sensation", is classified as "irrational").

I think the terms irrational/rational overlap in some ways. I don't think I know enough to understand this though.

>>>Spiral motion mediates between these two, in that the intellect resolves the primary axial path, but is balanced with the rotational path of intuition (rotation equated with awareness, integration, etc).

>>>Do you have a system that can resolve the infinite angles of counterspace into the concept of "rotational depth" -- angles of more than 360 degrees -- as part of this spiral function?

fractline, linear movment - the spiral progress in a linear fashion, forever inward you might say. An infinite angle is one of motion rotationally, and stillnes linearly.

So if a spiral mediates between the operations of the spatial/temporal,an of an infinite angle and depth, mediates the temporal/spatial - space/time vs time/space - what do you think? does this make sense?

>>>

It makes sense that the projective intersection of space and time would produce a helical result, I cannot see where the "golden" aspect comes into play. How did you come to this conclusion? Why would it follow this proportional system instead of just a simple 1:1 correspondence?

perhaps I just don't understand this clearly, but a 1:1 ratio would produce either a straight line , or a pure arc (understood as spatial: understood as temporal) the golden mean deviates from this 1:1 by allowing some "room"/time" that produces a measured spiral, and intelligent spiral - the mind and the heart balanced in both measures of their respective qualities. The golden-rectangle-spiral is self generating, self inducing. THe only change, which produces a change in space and time, is a change in scale - if the spiral is charted from the increasingly small and proportiante golden rectangle, the spiral moves towards unity, or inwardness, along with these changes in scale. I see it as the equanimity between mind/heart. I feel like I am rather unclearly communicating this.

>>>If the spiral is the projection of a line and a circle (one dimension each, since the circle is a counterspatial turn), then the 3d form would be the projection of the cube and sphere.

But also consider that your point of view is in the Euclidean "linear" system when viewing the spiral. What does the spiral become when viewed from the counterspatial "polar" geometry system? Rather than a bounded circle being drawn out along an unbounded line, it would be a bounded line drawn out around an unbounded circle?

Motion I think.

davelook · Post by **davelook** » Tue Jul 17, 2007 11:20 am

polakie11 wrote:

THe motion of space and the motion of time are resolved in the perfect proportions of the golden rectangle/spiral

Bruce wrote: "It makes sense that the projective intersection of space and time would produce a helical result, I cannot see where the "golden" aspect comes into play. How did you come to this conclusion? Why would it follow this proportional system instead of just a simple 1:1 correspondence?"

She might be onto something here....

After reading this page... http://milan.milanovic.org/math/english ... alpha.html

...I'm finding strange instances where simply inserting factors of ten gives you constants in other areas. I'll give a list soon, but here's an example...

classical electron radius = alpha^2 X bohr radius = 2.81794028E-15

charge^2 to mass ratio = e^2 / mass_e = 2.81794028E-8

This reminds me of the work of Bruce Cathie, where the decimal's position isn't as important as the agreement between the significant digits.

Dave

davelook · Post by **davelook** » Mon Aug 06, 2007 9:16 pm

davelook wrote:

classical electron radius = alpha^2 X bohr radius = 2.81794028E-15

charge^2 to mass ratio = e^2 / mass_e = 2.81794028E-8

OK, here's what I think is going on here...

Larson uncharged electron mass= .00057870

Larson Charged electron mass = .000548741

uncharged/Charged ratio = 1.054595884

h-bar value = 1.054571628 X 10^-34 J*s

1.054595884 / 1.054571628 = 1.000022997E+034

Or, as mentioned on a John Baez page http://math.ucr.edu/home/baez/lengths.h ... wavelength....

"The Compton wavelength depends only on hbar, c, and m. The classical electron radius depends only on e, c, and m. Nice set-up, huh? I suppose I should relent and tell you that this mysterious number 1/137, the fine structure constant, is just

e^2/hbar c.

It's a dimensionless constant depending only on hbar, e, and c. In this respect it's more fundamental than any of the length scales mentioned, because all the length scales mentioned involve the electron mass, and one could work them out for particles other than the electron, whereas

e^2/hbar c

is truly universal, once you remember that the "electron charge" is nothing specific to the electron but is a basic aspect of electromagnetism that applies to all charged particles. (Yes, quarks apparently have charge 1/3, but that doesn't really affect my point.) In other words, the fine structure constant is a dimensionless measure of how strong the electromagnetic force is, and we have seen that it sets the ratio of 3 important length scales.

bperet · Post by **bperet** » Tue Aug 07, 2007 12:55 am

Dave, have you examined these numeric similarities when expressed in natural units of space and time only, rather than conventional units?

Horace · Post by **Horace** » Wed Aug 08, 2007 7:00 pm

Bruce,

I tried making an animation of a teapot shrinking smaller and smaller until it entered counterspace. All viewed by a stationary camera outside of "counterspace".

However, I failed misreably, mainly because:

Quote:

points are separated in counterspace by a quantity which is never infinite

Do you know how to get around that?

Horace

P.S.

I also tried to keep the teapot stationary in space while shrinking the camera until it was completely in counterspace, "looking out" at the teapot outside. Same problems.

bperet · Post by **bperet** » Wed Aug 08, 2007 10:48 pm

Horace wrote:

I tried making an animation of a teapot shrinking smaller and smaller until it entered counterspace. All viewed by a stationary camera outside of "counterspace".

However, I failed misreably, mainly because:

Quote:
points are separated in counterspace by a quantity which is never infinite
Do you know how to get around that?

When you cross the unit boundary into counterspace, you need to "dualize" the affine assumptions, and switch the point at zero to a plane at zero, and the plane at infinity to a point at infinity. (In essence, turning things inside-out). The closest thing I've found to accomplishing that with stock modeling software is to change the camera characteristics from a perspective (rectangular) projection to a "fisheye" (spherical) projection, then dualize the geometry of the object being modeled.

I haven't tried anything as fancy as a teapot, but when using a box, in "space" the box is defined by its corner points, creating an interior volume (solid model). When you cross into counterspace, the box needs to be defined by the union of 6 planes, defining an exterior volume (the inside of the box appears open, even though it is technically the outside because of the zero-infinity flip). In your teapot, each point defining its surface would have to be transposed to its dual, a plane where the volume is defined along the surface normal for that point, and a union of the resulting planes used to define the counterspace object.

In simple terms an "object" becomes a "mold" when entering counterspace.

(By "plane", I am using the solid modeling term, where the plane divides "empty" from "full", and a normal is used to determine which side is which).

You may find the visualization a bit confusing, because the "outside" is near the middle of the projective sphere, with the teapot sort of splattered around the interior of the sphere (plane at zero). Kind of like looking in a kaleidoscope.

davelook · Post by **davelook** » Sun Aug 19, 2007 12:41 am

bperet wrote:

Dave, have you examined these numeric similarities when expressed in natural units of space and time only, rather than conventional units?

Well, since Larson says space and time are reciprocal, I tried sqrt(10) as space and 1/sqrt(10) as time, since 3.162... and .3162 are perfect reciprocals of each other.

No real dice that I could see.

But the way I set up my huge Excel spreadsheet of all constants allows me to change just the unit space conversion factor, and have that change cascade through all the constants.

So I set SPACE to the Golden Ratio, 1.618..., the conversion factor is 4.5563E-08 X 3.5117E+07 = 1.618...(Phi=SPACE), and it's reciprocal, .618... (phi=TIME). (Phi & phi are another set of "perfect" reciprocals)

I realize that when Larson says space and time are reciprocally related, he doesn't mean in a strict mathematical sense. But I'm just having fun, so I can try pet theories and see what I come up with. And I'm fascinated by Phi.

Anyway, recall this relationship I discovered...

classical electron radius = alpha^2 X bohr radius = 2.81794028E-15

charge^2 to mass ratio = e^2 / mass_e = 2.81794028E-8

I didn't think much of it at the time, thinking it was just alpha rearing it's head in a roundabout way.

But if you simply re-normalize the classical electron radius to the unit of space as simply Phi, the radius becomes, (.....drum roll please....) 1.00069 E-07!

So maybe Phi/4.55633 provides the dimension shift between charge, magnetism, and mass! This is all very preliminary, of course.

Reciprocal System Research Society