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 Posted:
Thu Nov 24, 2005 1:08 am Post subject:
Tim's golden Coordinates (polysigned news) |
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Admiring a danceress is partly beautifull because of her coordination
of movements of different parts of her body and with or against gravity
and the movement of other dancers. Our sphaerical coordinate system
stems from observing the harmonic and sometimes disturbed motion of
heavenly bodies, in the first place from the rotation of earth itself.
It has one axis, fixed at a point, the centre of earth, one orientation
of rotation, a starting point for this is given by the direction to the
sun, when in spring this direction is in the plane given by the equator
(So only one reference point not "intrinsic" to earth). And one has the
latitude, the difference of direction from the centre of earth to the
earth axis, with again an orientation by differing north from
south-pole. An other visualization is an umbrella, which one can open
more or less, and rotate too - a cone with centre in the middle of
earth.
A cartesian coordinate-system gives more equality to directions, just
three of them, starting at a common point. In contrast to the
female-round spherical system it's more male-cubic with three right
angles and linear addition. It's our personal reference-system, body
axis (most often aligned to gravity), our symmetrie gives us left and
right (most often aligned to the line of our two eyes) and front and
back. In taking a deeper look into it, there's another orientation
hidden in it: In assigning a positive meaning to three of the six arms
of this rectangular system one chooses a clockwise or anticlockwise
orientation. Let a ray start at the origin with equal distance from all
three postive arms and look from the centre.
Nature gave us and all (?) the animals another a cartesian like
orientation organ in the inner ear, three loops of tube filled with
liquid in three planes, the normals pointing cartesian, perpendicular
at right angles.There's a more advanced version of this given in
regarding the x, y, z -axes as axes of a true rotation, as i, j,
k-axes. By this we get an extra multiplication and as these quaternions
relate to the rotation of fluids in the loops of our inner ear, we must
have quaternion-software build in our brain. In space-ships they have
these "inertial platforms" with three gyros at right angles performing
the same job. As both cartesian systems exists in two versions, a left
one and a right one, it might not be a surprise, that nature provided
us with both types of "inertial platforms" (as we have two ears) and
this has some advantages like redundancy, like extra information from
differencing both inner-ear-informations. (Actually, as we have two
stations in Europe sending a time signal, in Rugby and in Frankfurt, we
could make use of it as a GPS-independent system with two points of
reference).
A line is determined by one point and one direction or by two points
with differing first from second point - a plane by one point and two
directions and left or right orientation or by three points (marked on
paper either with a left turn or a right one), so - okay, we need four
points for space, just the origin in cartesian coordinates with a point
at equal distance from it on each positive axis. Connecting these gives
us a tetrahedron, but off centre and with unequal distances between the
points. Tim Golden took a tetrahedron with four points of equal
distance as basis for a new kind of coordinate-system, making use of
the normals of the four sides. The elements of this 4 mathematical
dimension vectorspace (in the common three space-dimensions ) he called
polysigned numbers:
http://bandtechnology.com/PolySigned/PolySigned.html
Don't forget to click on
FourSigned.html
The four normals define by their opposites to the inside a fifth point,
a centre, a starting point for four directions. So just like the
spherical coordinates, they only come in positive numbers. Compared
with cartesian, it has some redundancy, so for example no "gimbal
lock"( i hope). The four axes, denoted with " + ", " - ", " * " and " #
" come with a already worked out addition, multiplication and more.
There are redundancy remowing rules.
(Introducing a logarithmic or power operation, might differ negative
inner ( log c, when 0 < c < 1, from positive extension through the
surface of value 1= e^0).
A tetrahedron is a nice X-mas cadeaux for a mathematician, connected
with the proposition, to cut it into two equal halves. And having these
two parts lying on the desk is nice, when seeing people trying to make
a pyramid from it. Cutting a model of the six edges into two equals of
three edges, shows that's these can be equal ( triangles with one side
bent ouside the plane - both path's connecting all four corners), but
only with differing orientation, which gives the possibility to differ
left from right. In giving a consecutive order 1,2,3 4 to the four
corners one can see, that two types of tetrahedron are possible, but
the two pathes can provide enough info, so we only need one tetra.
As for the X in X-mas, cut it into two, in a V and a /\ and turn one
against the other by an right angle and You 'll get from the endpoints
a tetrahedron. (these four arms can also be taken as a
coordinate-system, as they match to the first one, when prolonged to
the opposite side).So there are two planes perpendicular hidden in it.
The multiplication has conical properties too, see the website for it.
Putting a tetrahedron into a plastic sphere with the continents of
earth painted on ( i forgot the exact position of the four points, but
one was at the Falkland islands or so) one can see the four corners of
earth. Back to polysigned.
It's just the right mathematics for Laser Gyros, which in the advanced
form come in tetrahedron form (two rays of light "rotating" in each
boundary of the four surface triangles and with some vibrations to
prevent wave-lock of the opposing rays. By the way, do they make use of
two rays with polarisation at a right angle, or of the change of
polarisation caused by the movement of the platform ?).
By fittig a tetrahedron into another one with the same center You can
build a nice eight-pointed star, a golden one of course.
Ho,Ho,Ho
Hero |
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Timothy Golden http://www
Guest
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| Posted:
Thu Nov 24, 2005 7:27 am Post subject:
Re: Tim's golden Coordinates (polysigned news) |
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Hero.van.Jindelt@gmx.de wrote:
| Quote: | Admiring a danceress is partly beautifull because of her coordination
of movements of different parts of her body and with or against gravity
and the movement of other dancers.
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Thanks Hero for the polysigned prose.
I'd like to state more about the polysigned numbers.
The same identity principle that allows four-signed numbers to generate
the tetrahedral coordinate system apply in other dimensions. So for
example in four dimensions a five-hedron exists that accurately
represents the five-signed number system.
Before you dismiss this sign game, don't forget that three-signed
numbers, which are two-dimensional are homomorphic to complex numbers.
Two-signed numbers are the real numbers.
So a natural construction extends the real numbers up to the complex
numbers without any use of i or square roots of minus one or any
complication except generalizing sign.
I can't say things so artfully as Hero.
They are plain and simple but you have to be willing to question sign.
Should you do so almost certainly you will arrive at the same
definition.
-Tim |
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Guest
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| Posted:
Mon Nov 28, 2005 1:08 am Post subject:
Re: Tim's golden Coordinates (polysigned news) |
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Timothy Golden http://www.BandTechnology.com wrote:
| Quote: | ... polysigned numbers....
They are plain and simple but you have to be willing to question sign.
Should you do so almost certainly you will arrive at the same
definition.
Signs are derived from directions in space and time, giving a direction |
to counting. And they have operational properties, minus can change (in
the older systems) left to right, inner to outer (mirroring through the
surface of a sphere, like (1/3 ) ^ (-1) inverts around the 1 to 3), a
vector can operate with * (times) from left on a matrix,.. so we are
flexibel. So why not let a sign denote a position in a quadruple, like
sqrt( -1) denotes the second position in (a, b) the vectorspace ( RxR,
+ , *) over (R, + , * ), so i* b = ( o , b ) and at the same time
operates as sqrt ( - 1 ) = ( 0 , 1 ) = i on a pair ( 3, 4 ) by
addition, changing it to ( 3, 4 ) + i = ( 3 , 5 ) or by multiplication
rotating by 90 degrees left ( 3, 4 ) * i = ( - 4 , 3 ).
I did "discover" the cone in the cartesian coordinate system too. I
wrote:
| Quote: | okay, we need four points for space, just the origin in cartesian
coordinates with a point at equal distance from it on each positive axis.
Connecting these gives us a tetrahedron, but off centre and with
unequal distances between the points.
But this tetrahedron 0 > x=1 > y=1 > z =1 ist a quarter of a bigger |
tetrahedron, which is nice on center with 0 and with equal sides and
angles. Let a ray start at the origin with equal distance from all
three negative arms with length 1 and connect to x=1 > y=1 > z =1. A
tetra with side length sqrt ( 8 / 3 ).
Now, if You rotate round the axis of equal distance to all three axes,
You get a cone with tip at zero and an angle of arcus tan sqrt 8,
between the rotation axis and the surface.
One can repeat - or learn or teach - quite a lot of formulas about
triangles with a tetrahedron. Go from one corner along a side, over a
plane, and up the negative ray extension to the center, You will met
the relations 1 : 1, 1: 2 and 1 : 3. And one learns quite a lot about
3D, of course You did succeed already in cutting the tetra into two
equal solids with a cut, a square plane, and where this is inside the
tetra. May be You found the line, connecting the middles of two edges,
which have no common point. Or expresssing it with the twisted X: the
middle of the line, which conects the upper tips of V, with the middle
of the line, which connects, the lower tips of /\.
One can start with numerologie or pyramid-powers ( to get away from it
with some realistic stuff from its core ). A diamond comes in a shape
like the Cheops-pyramid doubled, connected with its square bases. Or
connect on the six directions in cartesian coordinates, three axes with
plus and minus, six points with equal distance from the zero. On a
dollar-note it is depicted as an eye inside a square. In some inner
circles of vip-societies ( i for immoral) they reveal the meaning:
Mer-Ka-Ba, or simply a tetrahedron, somehow made out of light, shall be
in the core of the pyramid. In the most inner "circle", they are more
down to earth and talk about keeping up the profits by monopolizing the
diamond-trade. And this matches to the egocentrism of the "lady"
(nowadays), behaving like a four or eight tentacled octopus with a huge
eye, organizing a tetra-structured business from-out the center.
A stone-cutter or a chemist can tell us better: diamond is made up of
pure carbon C, which has four electrons in the outer sphere. Now make
up a crystal-lattice one can really see tetrahedrons of four C-atoms,
with a fifth in the middle. But a better look, not thinkig of corners
only, gives as element not a four-sided solid, but an atom with four
arms extending, joining arms with other C-atoms.
Is C more a four-arm beeing, so is water half a tetra, half four armed.
An oxygen-atom in the middle of a tetra, two corners with
hydrogen-atoms, and the other two corners are two electron-pairs, thus
two arms. Only one arm has NH3, ammoniak, three H's as a base, N in the
middle, and the forth corner an electron-pair. Only Methan is a true
tertra, C in the middle, four corners of H makes CH4.
| Quote: | From this one can proceed to the four letters in the alphabet of RNA,
which created it's information storage medium, the DNA - or one can |
proceed to the beauty of snow-flakes (H2O), yesterday arriving at my
place. And this long blue hours of dawn gives opportunity to speculate
about the prism properties of a tetra, about Fourier's and Fermat's
principles.
Or one can find a nice solution to avoid the wave-lock in modern
tetra-laser gyros (1996 in sci.optics in the thread "ring laser gyro
ques" ) by thinking about pathes along the six edges of a tetra for
light of changing colours and polarisation creating a pulsating gem for
our orientation.
Beautifull math's, isn't it ?
Hero |
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Hero
Guest
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| Posted:
Mon Nov 28, 2005 5:08 pm Post subject:
Re: Tim's golden Coordinates (polysigned news) |
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Hero.van.Jindelt@gmx.de wrote:
| Quote: | A stone-cutter or a chemist can tell us better: diamond is made up of
pure carbon C, which has four electrons in the outer sphere. Now make
up a crystal-lattice one can really see tetrahedrons of four C-atoms,
with a fifth in the middle. But a better look, not thinkig of corners
only, gives as element not a four-sided solid, but an atom with four
arms extending, joining arms with other C-atoms.
Actually a perfect crystal with a surface of no interaction with other |
elements can not grow, so how does it got it's size ? Now, if a carbon
atom is a four arm beeing, the four arms are elctrons, so in the centre
of the four planes the positive core can attract electrons of other
atoms. It forms with four other C-atoms a tetrahdron But each C atom
at a corner is pointing with it's electron arms into one region, making
it a point of high interest for electrons outside. And when an atom
comes near the corner, the corner C disconnects the binding to the
inner C by bringing one elctron to the outside. This happening at all
four corners will free the inner C, not from it's place but from it's
electron connection with the corners. Now we have a tetra with a free
center and this can become a corner for four other tetras. And at the
same time some corners can become a center for another tetra - which
results in a double Cheops-pyramide, an eight-hedron. One can count
eight tetras in it, of course with common corners and edges. Eight C's
inside and six outside. How can this grow ? If You take a second look,
the eight insiders are forming now a cube, a six-hedron, and by
disconnecting and establishing new mutual relationsships this cube has
eight arms free, extending through the surface of the eight-hedron.
Here my tale about the tetras ends. It is part of the story of crystals
with lots of math and orientation in space in it. NB, the most
developed form of crystals, the RNA is the basis of organic life.
Back to signs.
A kind of isomorphie exists between the polysigned numbers and
bookkeeping. In both one has only numbers of positive value. In
bookkeeping the position where a number appears gives it the meaning of
debt or money spent and so forth. In polysigned the sign is giving a
number it's place.
The keyword in bookkeeping is "Dopik" and there is a text in the
websites of ams on the web about this.
Have fun
Hero |
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Hero
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| Posted:
Mon Nov 28, 2005 5:08 pm Post subject:
Re: Tim's golden Coordinates (polysigned news) |
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Hero van Jindelt at gmx.de wrote:
If You take a second look,
| Quote: | the eight insiders are forming now a cube, a six-hedron, ..
I forget to tell You, that ,when the carbons form the cube, there is no |
loger a central carbon.
And seeing a cube with it's six plane square surfaces - drawing on each
a diagonal in the right order will give You a tetra of four of the
eight corners.
And one more from physics, the bigger elementary particles like protons
are modelled with three different quarks and their three opposites or
anti-quarks. Place these on the axes of a cartesian coordinate system.
The quarks are differed by four properties: spin, isospin, strangeness
and charge (the last one is +1 for protons, zero for neutrons and -1
for electrons, but the electrons are not considered here). These form a
kind of four-signed numbers. So assign to every of the four rays or
axes from the center of a tetrahedron a sign (property). No let the
six rays or three axes of the cartesian go through the middle of the
six edges of the tetra.
A vector of four properties will represent a vector of up to three
quarks.
There is a dependency between the four properties. That for a start,
then proceed to addition, multiplication and log. I don't have much
knowledge of physics, but someone might like to work this out. (May be
there's a relation to Feynman diagramms too, but this is far beyond my
horizon. )
This idea is just based on my pattern recognition.
Have some pleasure with solving this particle-sudoku
Hero |
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| Posted:
Fri Dec 02, 2005 11:41 pm Post subject:
Re: Tim's golden Coordinates (polysigned news) |
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Hero wrote:
....
Inside a cube is a tetra: connect four of the eight corners by
diagonals on the sides.
This connects to the thread " determinants of nxnxn matrices"
Hero |
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