sarada wrote:
Projective Geometry is a system that allows one to project scalar values thru a series of assumptions that results in a Euclidean perspective. Each of these "assumptions" forms a "stratum" of geometry. Normally, there are 4 strata:1. What is meant by Geometric Strata?
1. Projective stratum
The only invariant (assumption) is the cross-ratio (Larson's "scalar speed"; see forum article on "Scalar Motion" for a detailed explanation of this stratum.)
Here, the only thing that "exists" geometrically is the relationship between points on a single line. Relationships between lines are undefined, so there is no dimensional structure.
2. Affine stratum
Incorporates the assumption of a "plane at infinity", a mathematical device to allow parallel lines to intersect, and the "inverse" of the plane at infinity: the point of origin. (They are called "duals" of each other, since they are just two reciprocal aspects of the same thing).
The consequence of an origin and plane at infinity is a geometric relationship between "cross-ratios" -- parallelism and orthagonal structure. In other words, you can have "shape", but cannot determine the lengths or angles of any component, since the only points of measure are zero and infinity.
If you have some familiarity with Art, Pablo Picasso's "Cubist" painting style is a form of affine projection -- the shapes are there, but appear to have a bizarre correlation since length and angle are ignored.
3. Metric stratum
Introduces the concept of an "absolute conic" which has the consequence of RELATIVE lengths and angles (groups of similarities). The projective cone give fixed relationships between length and angle, but is only applicable to a structural grouping -- consistent within an object, but no connection between objects (scale variance).
Basically, it is the view you get if you look at the world with one eye closed -- you can see shapes and can determine relative dimensions of the shapes (like the base is twice the height), but cannot determine distance to an object, its actual size, or the relation between objects (no "perspective", since the scale can be different for each object).
4. Euclidean stratum
Introduces the assumption of Scale=1.0, with the consequence of absolute measurement. Our system of binocular vision uses this assumption to determine distance. Internally, our brain adjusts the perceived angle and distance so that the object has a scale of 1.0 (in all dimensions), to match everything else. This give us our "perspective" view of the world.
In Summary:
Projective: cross-ratio invariant (only).
Affine: adds invariants of relative distances along a direction, parallelism, plane at infinity.
Metric: adds invariants of relative distances between directions, angles, absolute conic.
Euclidean: adds scale invariance and absolute distance.
sarada wrote:
Yes, when viewed from the "outside".2. Is not the inside of the quantum of space a region of inverse space, i.e., polar-Euclidean?
Remember that projections, Affine, Metric and Euclidean, are like the shadows cast on a wall -- not the original object, but a perception of scalar motion built upon assumptions.
The region you take measurements from always appears to be Euclidean, since that is how our consciousness perceives the universe about us. But every time you cross a unit boundary (space, time or speed), it is like looking thru a distorting lens, and the other side is polar.
sarada wrote:
There are polar-Metric and polar-Affine geometries. The Projective stratum is the inverse of itself, since there is no system of measure, either absolute or relative. It is like taking the reciprocal of "1" -- you get "1" again. The same rules apply to Projective or polar-Projective; there is no difference in treatment.3. Like polar-Euclidean, are there inverse-Metric, inverse-Affine and inverse-Projective Geometries?
sarada wrote:
Assuming the Euclidean point , the polar is the plane passing thru , which can also be expressed in counterspace as three polar rotations (a quarternion) rotating the plane of origin into that position. So can represent either/both a translation along an axis in 3 dimensions (x,y,z=length), or a rotation about an axis in 3 dimensions (x,y,z=angle).4. In Euclidean stratum, co-ordinates are x, y and z. In the corresponding polar-Euclidean what are the relevant co-ordinates?
sarada wrote:
The degrees of freedom are components/variables that are not fixed to a specific value by assumption.5, In connection with the geometric strata, what does one mean by "degrees of freedom"?
The projective stratum, in 3 dimensions, is represented by a 4x4 matrix, where ONE value is fixed. This leaves 15 variables free, which they call degrees of freedom.
In the affine stratum, the plane at infinity is defined by [ 0 0 0 1 ], with the unity value providing the projective stratum requirement of one value being fixed. This leaves 12 degrees of freedom (16-4=12).
The metric stratum adds the absolute conic, which imposes constraints on the remaining 12 variables for rotation, translation and scale. In the normal projective system, this leaves you with 3 variables to define rotation about axes, 3 to define translation and 1 for scale. Total, 7 degrees of freedom. (Though you can use 9 degrees of freedom by applying scale independently to each dimension, rather than the same scale for each dimension).
The Euclidean stratum fixes scale at unity, so it is reduced to 3 rotation and 3 translation, total 6 degrees of freedom.
sarada wrote:
Think of real numbers as a "linear" measurement, and imaginary numbers as a "polar" measurement -- same concept, different (inverse) geometries. Euclidean measurements are taken by "real" numbers, and polar-Euclidean measurements are taken by "imaginary" numbers. Nehru wrote a good paper on the imaginary operators; it is on the forum if you don't have a copy. I found it very helpful.6. You associated Euclidean Geometry with real numbers and polar-Euclidean wiith imaginary numbers. Rather, I would think polar-Eclidean to be complex numbers, since a mere imaginary number, as a concept, does not exist. The 'mere' imaginary number you talk of can only be a complex number with its real component zero. Am I correct?
"Real" and "Imaginary" are just systems of measurement; one is as valid as the other. The reason they are called "imaginary numbers" is because our consciousness cannot see, measure and comprehend polar geometries directly thru observation -- we can only do it in our "imagination."
I see no harm in treating a purely imaginary measurement as a complex number with the real component at zero, nor treating a purely real measurement as a complex number with the imaginary component at zero.
But the problem with using complex numbers to represent the system is that it can only represent a 1-dimensional function--one "length" in space, one "angle" in counterspace, and the system is three-dimensional.
The secondary problem is that "orientation" is not preserved going across the unit boundary -- only the net magnitude can be transmitted to the outside region. Thus a 3-dimensional "imaginary" motion within the quantum of space can only be measured by its net effect across the unit boundary in "real" space. I suppose it would be possible to incorporate all three imaginary dimensions in a complex number, retaining the single "real" value to account for this, as: R + f(iX,jY,kZ), but that starts looking like the quaternion [R iX jY kZ].
sarada wrote:
I do not know the history of the development of projective and synthetic geometry in detail. I believe it was discovered by Poncelet and developed by Gergonne and Jakob Steiner, based on two theorems that were discovered independently by Pascal and Brianchron on the 2-dimensional relationships between lines and circles.7. You wrote : Inverses of geometric objects: How were they arrived at ?
Note that the duals I listed are for a 3-dimensional system. In a 2-dimensional one, such as a drawing on a piece of paper, points and lines are duals of each other. This is what Poncelet discovered -- Pascal and Brianchron's theorems where the same thing -- just geometric inverses of each other. This lead to point-line duality, and when extrapolated into three dimensions, the point-plane duality.
sarada wrote:
Volumes are a geometric property, not a geometric object (like "area" is a geometric property of a plane, not the geometric object of the plane, itself). They cannot be projectively transformed -- only the planes (surfaces) enclosing them can be. This restriction comes from the Affine assumption of the plane at infinity -- not a "volume at infinity."We [Nehru & Sarada] wonder whether the inverse of a volume is also a volume? Are not the
following more logical inverses?
point volume
line plane
plane line
volume point ?!
Length, Area, and Volume are geometric properties (absolute measurements) that cannot be transformed, since they are the result of a transformation. Points, Lines and Planes are geometric objects (zero or infinite structures, no measurements) that can be transformed.
I suppose this is also the reason why we don't see any "3-dimensional rotation" in Larson's atomic system -- only 1-dimensional (electric) and 2-dimensional (magnetic).
BTW, those duals you listed are duals in a 4-dimensional system. Nehru precluded this arrangement with his discovery of the dimensional relationship "n(n-1)/2=n", which limits the universe to 3 independent dimensions. (For this arrangement to work, you would have to alter the "plane at infinity" to a "volume at infinity" to allow for the geometric construct of a volume, and thus would need to be represented in a 5x5 matrix, requiring 4 independent dimensions).
sarada wrote:
I use the term "area" because it is a 2-dimensional surface (planes with normals) that separate what is enclosed from what is exposed. This seperating surface can be transformed to points thru duality, but not what is enclosed or exposed, because that is all a matter of perspective.8. When you mentioned "Enclosed area Exposed area", could this area be composed of planes ---- like the inside of a cubical? But then the inverses have to be points.
So the walls of your cubical can be transformed to points, but at the same time, what you considered to be "inside" your cubicle is now on the outside -- even though nothing never moved.
Lately, I've been completely ignoring the concept of "volume" in research, because you can get the same properties just from the area enclosing a volume. If I have a blob of jelly that has a surface area of 6 square centimeters, I can squish it into any shape I want -- and it will always retain the same area and volume. It seems all the physical properties are attached to lengths and areas -- not volumes.
