Discussion in 'Ancient and Original Native and Tribal Prophecies' started by CULCULCAN, Sep 26, 2014.
Other hexagrams can be constructed as a continuous path.
unicursal hexagramTwo uniform star-polyhedra have hexagramvertex figuresOne star polyhedronhas hexagram facesD2 symmetryD3 symmetry
Great ditrigonal icosidodecahedron
Great triambic icosahedron
Pascal line GHK of hexagon ABCDEF inscribed in ellipse.
Opposite sides of hexagon have the same color
Hexagrammum Mysticum Theorem
In projective geometry, Pascal's theorem
(also known as the Hexagrammum Mysticum Theorem)
states that if six arbitrary points are chosen on a conic
(i.e.,ellipse, parabola or hyperbola)
and joined by line segments in any order to form a hexagon,
then the three pairs of opposite sides of the hexagon
(extended if necessary) meet in three points which lie on a straight line,
called the Pascal line of the hexagon.
The theorem is valid in the Euclidean plane,
but the statement needs to be adjusted to deal with the special cases
when opposite sides are parallel.
The most natural setting for Pascal's theorem is in a projective plane since all lines meet
and no exceptions need be made for parallel lines.
However, with the correct interpretation of what happens when some opposite sides of the hexagon
are parallel, the theorem remains valid in the Euclidean plane.
If exactly one pair of opposite sides of the hexagon are parallel, then the conclusion of the theorem
is that the "Pascal line" determined by the two points of intersection is parallel to the parallel sides
of the hexagon. If two pairs of opposite sides are parallel, then all three pairs of opposite sides form pairs of parallel lines and there is no Pascal line in the Euclidean plane (in this case, the line at infinity of the extended Euclidean plane is the Pascal line of the hexagon).
This theorem is a generalization of Pappus's (hexagon) theorem
– Pappus's theorem is the special case of a degenerate conic of two lines.
Pascal's theorem is the polar reciprocal and projective dual of Brianchon's theorem.
It was formulated by Blaise Pascalin a note written in 1639 when he was 16 years old
and published the following year as a broadside titled "Essay povr les coniqves. Par B. P.".
A degenerate case of Pascal's Theorem (four points) is interesting;
given points ABCD on a conic Γ, the intersection of alternate sides, AB ∩ CD, BC ∩ DA,
together with the intersection of tangents at opposite vertices (A, C) and (B, D) are collinear
in four points; the tangents being degenerate 'sides', taken at two possible positions on the 'hexagon'
and the corresponding Pascal Line sharing either degenerate intersection.
This can be proven independently using a property of pole-polar.
If the conic is a circle, then another degenerate case tells us that for a triangle,
the three points that appear as the intersection of a side line with the corresponding side line
of the Gergonne triangle, are collinear.
Six is the minimum number of points on a conic about which special statements can be made,
as five points determine a conic.
The converse is the Braikenridge–Maclaurin theorem, named for 18th century British mathematicians William Braikenridgeand Colin Maclaurin (Mills 1984), which states that if the three intersection points
of the three pairs of lines through opposite sides of a hexagon lie on a line,
then the six vertices of the hexagon lie on a conic; the conic may be degenerate,
as in Pappus's theorem.
The Braikenridge–Maclaurin theorem may be applied in the Braikenridge–Maclaurin construction,
which is a synthetic construction of the conic defined by five points, by varying the sixth point.
The theorem was generalized by Möbius in 1847, as follows:
suppose a polygon with 4n + 2 sides is inscribed in a conic section,
and opposite pairs of sides are extended until they meet in 2n + 1 points.
Then if 2n of those points lie on a common line, the last point will be on that line, too.
If six unordered points are given on a conic section, they can be connected into a hexagon
in 60 different ways, resulting in 60 different instances of Pascal's theorem
and 60 different Pascal lines. This configuration of 60 lines is called theHexagrammum Mysticum.
As Thomas Kirkman proved in 1849, these 60 lines can be associated with 60 points in such a way
that each point is on three lines and each line contains three points.
The 60 points formed in this way are now known as the Kirkman points.
The pascal lines also pass, three at a time, through 20 Steiner points.
There are 20 Cayley lines which consist of a Steiner point and three Kirkman points.
The Steiner points also lie, four at a time, on 15 Plücker lines.
Furthermore, the 20 Cayley lines pass four at a time through 15 points
known as the Salmon points.
Pascal's original note has no proof, but there are various modern proofs of the theorem.
It is sufficient to prove the theorem when the conic is a circle, because any (non-degenerate) conic
can be reduced to a circle by a projective transformation.
This was realised by Pascal, whose first lemma states the theorem for a circle.
His second lemma states that what is true in one plane remains true upon projection to another plane.
Degenerate conics follow by continuity (the theorem is true for non-degenerate conics,
and thus holds in the limit of degenerate conic).
A short elementary proof of Pascal's theorem in the case of a circle was found by van Yzeren (1993), based on the proof in (Guggenheimer 1967).
This proof proves the theorem for circle and then generalizes it to conics.
A short elementary computational proof in the case of the real projective plane
was found by Stefanovic (2010)
We can infer the proof from existence of isogonal conjugate too.
If we are to show that X = AB ∩ DE, Y = BC ∩ EF,Z = CD ∩ FA are collinear for conconical ABCDEF,
then notice that ADY and CYF are similar, and that X and Z will correspond to the isogonal conjugate
if we overlap the similar triangles. This means that angle DYX = angle CYZ, hence making XYZ collinear.
A short proof can be constructed using cross-ratio preservation. Projecting tetrad ABCE from D onto line AB, we obtain tetradABPX, and projecting tetrad ABCE from F onto line BC, we obtain tetrad QBCY.
This therefore means thatR(AB; PX) = R(QB; CY), where one of the points in the two tetrads overlap, hence meaning that other lines connecting the other three pairs must coincide to preserve cross ratio. Therefore XYZ are collinear.
Another proof for Pascal's theorem for a circle uses Menelaus' theorem repeatedly.
Dandelin, the geometer who discovered the celebrated Dandelin spheres, came up with a beautiful proof using "3D lifting" technique that is analogous to the 3D proof of Desargues' theorem.
The proof makes use of the property that for every conic section we can find a one-sheet hyperboloid which passes through the conic.
There also exists a simple proof for Pascal's theorem for a circle using Law of Sines and similarity.
Proof using cubic curves
Pascal's theorem has a short proof using the Cayley–Bacharach theorem that given any 8 points in general position, there is a unique ninth point such that all cubics through the first 8 also pass through the ninth point. In particular, if 2 general cubics intersect in 8 points then any other cubic through the same 8 points meets the ninth point of intersection of the first two cubics. Pascal's theorem follows by taking the 8 points as the 6 points on the hexagon and two of the points (say, M and N in the figure) on the would-be Pascal line, and the ninth point as the third point (P in the figure). The first two cubics are two sets of 3 lines through the 6 points on the hexagon (for instance, the set AB, CD, EF, and the set BC, DE, FA), and the third cubic is the union of the conic and the line MN. Here the "ninth intersection" P cannot lie on the conic by genericity, and hence it lies on MN.
The Cayley–Bacharach theorem is also used to prove that the group operation on cubic elliptic curves is associative. The same group operation can be applied on a cone if we choose a point E on the cone and a line MP in the plane. The sum of Aand B is obtained by first finding the intersection point of line AB with MP, which is M. Next A and B add up to the second intersection point of the cone with line EM, which is D. Thus if Q is the second intersection point of the cone with line EN, then
Thus the group operation is associative. On the other hand, Pascal's theorem follows from the above associativity formula, and thus from the associativity of the group operation of elliptic curves by way of continuity.
Proof using Bézout's theorem
Suppose f is the cubic polynomial vanishing on the three lines through AB, CD, EF and g
is the cubic vanishing on the other three lines BC, DE, FA.
Pick a generic point P on the conic and choose λ so that the cubic h = f + λg vanishes on P.
Thenh = 0 is a cubic that has 7 points A, B, C, D, E, F, P in common with the conic.
But by Bézout's theorem a cubic and a conic have at most 3 × 2 = 6 points in common,
unless they have a common component. So the cubic h = 0 has a component in common
with the conic which must be the conic itself, so h = 0 is the union of the conic and a line.
It is now easy to check that this line is the Pascal line.
A Property of Pascal's Hexagon
Given hexagon ABCDEF, let AC meet BD at G, BE meet CF at H, AE meets DF at I:
Then, as well known, the six vertices of the hexagon lie on a conic if the points G, H, I are collinear.
In addition, the two conditions are equivalent:
Degenerations of Pascals's theorem
Pascal's theorem: degenerations
There exist 5-point,4-point and 3-point degenerate cases of Pascal's theorem.
In a degenerate case, two previously connected points of the figure will formally coincide
and the connecting line becomes the tangent at the coalesced point.
See the degenerate cases given in the added scheme and the external link on circle geometries.
If one chooses suitable lines of the Pascal-figures as lines at infinity one gets many interesting figures
on parabolas and hyperbolas (see the German sites Parabel and Hyperbel).
Coxeter & Greitzer 1967, p. 76
Young 1930, p. 67 with a reference to Veblen and Young, Projective Geometry, vol. I, p. 138, Ex. 19.
Wells 1991, p. 172
Mosaic swastika in excavated Byzantine(?) church in Shavei Tzion (Israel)
"The swastika has 13 bars,
six (6) seen, and, six (6) unseen + The Core aka 1 aka ONE for 6+1+6 = 13,
which is part of The Creaton of The Thirteen (13 Keys...
which is also symbolically represented as: thirteen (13) poles in the tipi aka teepee.
The four (4) diamond shapes aka The Whirling Fireplace is what connects
the twelve (12) energetic bodies that form The Original Spark...
which flows through:
zero-point to the triad, trine and trinity through each of the four (4) levels of three (3) points,
which is a discovery of Susan Lynne Schwenger.
PHYSICAL BODY (1)
EMOTIONAL BODY (2)
INTELLECTUAL BODY (3)
(which connects through)
TRIAD (to the)
ETHERIC BODY (4)
LOW CAUSAL BODY (5)
HIGH CAUSAL BODY (6)
(which connects through)
TRINE (to the)
HIGHER SELF BODY (HSB) aka (HS) (7)
ESSENCE BODY aka THE CORE BODY (8)
The Past, The Present, and, The NOW,
through THE PIVOT of THE NOW,
(which connects through)
TRINITY (to the)
LOW CAUSAL BODY (10)
MID CAUSAL BODY (11)
AND, Finally to the:
HIGH CAUSAL BODY (12)
which, when pulled full -circle or 360 degrees
attaches, all of it, to:
ZERO-POiNT aka 0-POiNT,
which completes THE ORiGiNAL SPARK
known as The Thirteen (13) Keys to YOU."
~ Susan Lynne Schwenger
The Final Synthesis - eXKavier -
The Thirteen (13) Keys to THE ORiGiNAL SPARK
All of these terms, or word phrases were coined by Susan Lynne Schwenger
between 1975 - 1987
DISCOVERY by Susan Lynne Schwenger
-The Four (4) Flying Triangles
- The Original Spark in CUNEIFORM SCRIPT
DISCOVERY by Susan Lynne Schwenger-The Four (4) Flying Triangles - The Original Spark in CUNEIFORM SCRIPT
The symbols and meaning of Hammurabi's name are:
ha + am + mu + ra + bi
('fish' + 'wild bull' + 'year' + '?' + 'innkeeper')
SO, which symbol above is:
- i wrote to this site to get them to give me that info
after, i receive it, i'll show you all something very interesting
Mesopotamian Cuneiform (3100 BC)
Cuneiform was a combination writing system composed of pictograms andideograms (idea symbols) and phonograms (sound symbols). The cuneiform system began as pictures and continued, in part, to convey meaning the way pictures do.
The symbol for king is:
Originally a picture of a king, it had the same form but was said differently in many different languages: Lugal in Sumerian,Šarru in Assyrian and Babylonian,Haššu in Hittite.
Because of the need to write names unambiguously, the phonographic system, or writing by speech sounds, forced its way into cuneiform. As words were written, only their sounds were considered, not their meanings.
The symbols and meaning of Hammurabi's name are:
ha + am + mu + ra + bi
('fish' + 'wild bull' + 'year' + '?' + 'innkeeper')
A Sumerian school, the earliest on earth, called an edubba (tablethouse).
Here boys studied 12 years to become scribes in the cuneiform script.
Ti - Arrow of Life
Artist’s rendering of the amplituhedron, a newly discovered mathematical object
resembling a multifaceted jewel in higher dimensions.
Encoded in its volume are the most basic features of reality that can be calculated
— the probabilities of outcomes of particle interactions.
Illustration by Andy Gilmore
Susan Lynne Schwenger says:
"this is what it looks like:
when you apply:
The Thirteen (13) Keys to YOU
The Original Spark
The "four" 4 whirling fireplaces aka
The "four" 4 flying triangles"
- A Discovery by Susan Lynne Schwenger
New Discovery Simplifies Quantum Physics
That’s right ladies and gentlemen, quantum mechanics just got easier to understand. A team of physicists have released a paper showing their discovery of a jewel-like geometric structure that takes equations, which can be thousands of terms long, and simplifies them to a single term. This discovery is poised to dramatically simplify the equations particle physicists use when calculating particle interactions. It also proposes the uncomfortable idea that space and time are not fundamental aspects of our reality, and it brings us much closer to unifying gravity and quantum theory under one comprehensive model.
The discovery comes on the heels of decades of research in particle interactions. Particle interactions are some of the most basic and common events found in nature. Traditionally, these interactions have been very difficult or even impossible to calculate. Scientists required the use of the world’s most powerful computers to calculate even the simplest interactions. This new geometric structure, called the amplituhedron, is so simple that a particle physicist could calculate these interactions, by hand, on a single sheet of paper.
That, in case you were wondering, is insanely impressive. Harvard University theoretical physicist, Jacob Boujaily, and founder of this idea, said, “The degree of efficiency is mind-boggling. You can easily do, on paper, computations that were unfeasible even with a computer before.”
The Basic Idea
This theory is revolutionary on a number of fronts. At the moment, it’s being catapulted into the forefront of grand unified theory research. Some physicists suspect that a geometric object similar to the amplituhedron could eventually lead to a bridge that connects the physics of the very large with the physics of the very small. To date, all of the unified theories that are proposed are riddled with serious and deep-rooted problems, such as paradoxes and infinities. To unify macro and micro physics, the amplituhedron is paving the way to eliminate two of physics deeply rooted points and some of quantum theory’s central pillars: locality and unitarity.
Image Credit: Georg Johann
Simply put, unitarity is the idea that the sum of all probabilities describing every potential outcome of any quantum event is always equal to one (yes, that was the simple was of saying it). This places an inherent restriction on the amount of evolution that is allowed in any quantum system. Following the same “simple” trend, locality is basically the idea that particles can only interact with, and be influenced by, particles occupying space immediately surrounding them. It’s important to note that locality exists in quantum mechanics largely because special relativity insists upon it. Experimentally, we have shown through quantum entanglement that there seems to be a way to get around locality in the quantum world. In contrast, unitarity is a mathematical construction that helps to make nice round equations. In quantum field theory, both locality and unitarity are central concepts, but there is a catch. When attempting to add gravity to quantum theory, under certain situations, these two pillars (locality and unitarity) break down and stop working. This presents some amount of evidence that neither principle is a fundamental aspect of nature.
This is where the amplituhedron comes in. This geometric shape isn’t constructed by using the probabilities innate to spacetime, but instead suggests that the nature of spacetime is an attribute of the geometry of the amplituhedron. Our idea about the fabric of reality is just that– fabricated, an imaginary construct we have laying over the deeper and more fundamental construction of spacetime. According to David Skinner, a theoretical physicist who calls the Cambridge University home, “It’s a better formulation that makes you think about everything in a completely different way.”
The Complicated World of Particle Interactions
The amplituhedron is a very menacing, beautiful, complicated, multifaceted object that exists in higher dimensions. In principle, you can use the volume of this object to calculate all of the most basic features of reality, known in quantum mechanics as “scattering amplitudes.” This computation describes the probabilities of particles changing into other particles when colliding. These types of calculations are routinely made and tested at particle colliders such as the Large Hadron Collider (LHC). To understand the importance of the amplituhedron, we must first look at where it all began, 60-years ago with the development of Feynman diagrams.
Named after the Nobel winning Richard Feynman, these diagrams describe all of the ways a particle could scatter, and then the likelihood of any given outcome actually occurring. Feynman diagrams range from the trivially simple to the impossibly difficult. The simplest Feynman diagrams resemble trees, while the more complicated ones have one or more loops that explain particles turning into a virtual particle. A virtual particle is interesting because they aren’t observed in nature, but many physicists have regarded them as a mathematical necessity because they were required to achieve unitarity.
Though Feynman’s diagrams were a stroke of genius , they were simply the wrong tool to use to calculate nuclear particle interactions. In fact, the fact that we are able to compute anything at all is the prime discovery of the computer age; the number of diagrams required to describe something as simple as the 2-gluon to 4-gluon interaction gets so explosively large that scientists didn’t start those computations until the age of computers. You see, to describe the collision of two gluons that result in four gluons in a lower energy state, particle physicists at the LHC require the use of 220 Feynman diagrams. Together, these diagrams represent thousands of terms involved in the computation that are necessary to determine the scattering amplitude. In short, scientists have realized that Feynman diagrams, though beautiful, are effective ways to calculate a single mathematical object–they are laborious, require many different pieces, and are so numerous that it makes it difficult to do computations even with computers. Physicists are trying to move from that “incalculable” process to a single calculation that (thought difficult) is possible for humans to do (and certainly much easier for computers).
This started with theoretical work preparing for the completion of the Superconducting Super Collider (SSC) that was to be built in Texas (but eventually canceled). Physicists wanted to create a background framework describing scattering amplitudes with which to test the SSC and look for exotic or interesting signals. Physicists quickly determined that creating such a framework for even simple 2-gluon to 4-gluon interactions was so complicated that “they may not be evaluated in the foreseeable future.” Then, in the 1980s, this gluon interaction was simplified from an equation containing several billion terms to a single formula 9-pages long. This was an expression computers of the time could handle, and quantum field theory got a little more manageable. This type of simplifying laid the groundwork for the amplituhedron.
Enter: The Amplituhedron
Though the gluon simplification was achieved in the mid-1980s, it took a couple of decades for particle physicists to really start putting that revolution to use. This started in the mid-2000s when physicists started to find patterns in the scattering amplitudes – and you know how much physicists like patterns. This started the general trend of thought that an underlying mathematical structure might be supporting quantum field theory.
Credit: Arkani-Hamed et al.
Eventually, twistor variables and their corresponding diagrams were developed, which attempted
to simplify Feynman diagrams even further. These diagrams moved away from describing particle interactions
in familiar variables, such as time and position, and used twistor variables instead.
These diagrams worked, and gained rapid acceptance among particle physicists, but scientists didn’t understand how they worked, why they worked, or what made them so simple. Arkani-Hamed provides a colorful description by saying, “The terms in these relations were coming from a different world, and we wanted to understand what that world was.”
The amplituhedron didn’t start coming to light until December of 2012 with the discovery of the positive Grassmannian.
This geometric object is the result from studying the relationship between recursion relations and their corresponding twistor diagrams. According to the paper, these diagrams act as an instruction manual for calculating the volume of portions of the positive Grassmannian. This object consists of a region of N-dimensional space bounded intersecting planes (where N is the number of interacting particles).
This geometric structure was exciting, but incomplete. The positive Grassmannian’s construction was being restricted
by locality and unitarity. Instead of falling together as eloquent things tend to fall together, something was missing.
The prevailing idea was that determining the scattering amplitude had to be the answer to some other mathematical question. It turns out, that idea was right.
Credit: Nima Arkani-Hamed
The scattering amplitude was determined to be the volume of the amplituhedron. Natalie Wolchover from
the Simon Foundation best describes this mathematical structure,”
The details of a particular scattering process dictate the dimensionality and facets of the corresponding amplituhedron.
The pieces of the positive Grassmannian that were being calculated with twistor diagrams and then added together by hand were building blocks that fit together inside this jewel, just as triangles fit together to form a polygon.”
To reiterate the awesomeness of this achievement, the diagram pictured here is a sketch of an amplituhedron
depicting an 8-gluon particle interaction. If you were to attempt to use Feynman diagrams to represent this,
you’d be dealing with about 500 pages of algebra.
If the discovery of the amplituhedron wasn’t cool enough, physicists have also discovered a “master amplituhedron.”
This object has an infinite number of sides (similar to how a circle has an infinite number of sides in two dimensions) and it can, in theory, describe every possible physical process. All of the amplituhedra that exist in lower dimensions should exist on one of the master’s facets. Skinner describes this structure as having powerful calculational ability
and talks of it’s incredible suggestiveness since “they suggest that thinking in terms of spacetime
was not the right way of going about this.”
Quantum Gravity: The Future of Physics
This idea has very profound implications. Thus far, all of our theories attempting to unify gravity with quantum mechanics have failed.
Because of this, scientists have an impossible time describing the internal workings of black holes, the singularity that started the big bang, and other important objects and events. Ideas like string theory are at the forefront of this research, but they tend to be confusing or unproven/unprovable (or both). According to Arkani-Hamed,
” We can’t rely on the usual familiar quantum mechanical space-time pictures of describing physics
We have to learn new ways of talking about it. This work is a baby step in that direction.”
Image Credit: Charles Imes
It’s very important to note that the amplituhedron, even though it doesn’t include unitarity and locality,
also doesn’t include gravity. Physicists are in the middle of working on that very problem. It’s possible the amplituhedron contains the answer to quantum gravity, finally unifying the four fundamental forces of physics,
or it’s possible the final geometric shape we seek is a little different.
This work is fantastic, very exciting, and moving along very quickly.
As physicists seek to understand the meaning of the amplituhedron the rest of the world gets to wait
with bated breath to learn of their findings. It’s possible we could have another Einsteinian-type revolution
of our understanding of the nature of reality within our lifetimes. Wouldn’t that be exciting?
Susan Lynne Schwenger says:
"on each level there is both male, and, female aspects...
m, f, mf, fm, mffm, fmmf, and,
then mffmfmmf & fmmfmffm
(the first six (6) groups)
which blends to form:
The Seven (7) Levels of:
mf fm fm mf fm mf mf fm fm mf mf fm mf fm fm mf "
(note the 18 aspects - 18 are in you, along with 18 in your Essence Twin aka ET)
- Susan Lynne Schwenger
~ THE RETURN OF THE HiGH SPiRiT MAGiC ~ eXKavier
CiRCUM FERRE 360° ~ 360 Degrees ~ Full Circle
~ The "we CAN" freedom of 'total' being, known as: KAN-Zalyintána
~ Susan Lynne Schwenger
Separate names with a comma.