Re NEW COORDINATE SYSTEM IN PHYSICS
Porat
My own four directional coordinate system need not use negative numbers
but is perfectly capable of using them. It is a simpler, and, I believe,
in some important respects a more veridical coordinate system than the
standard rectangular coordinate system. The new system uses axes that
are four rays emanating from an origin all at the approximate angle of
109.471220634491 degrees from one another. The exact angle is given by
measure of the obtuse interior angle of an isosceles triangle with a
base measuring twice the square root of two units and sides measuring
the square root of three units. If Mr. Porat is reading this he would
benefit from noting that this angle is very important to his description
of the alpha. Porat has often stated that the alpha is a three
dimensional cross, and that the particles come together at the angle of
120 degrees. A brief analysis will show that four objects cannot come
together at 120 degrees to one another. The maximum number of objects
that can come together at 120 degrees to one another is three. In three
dimensional space the maximum number of objects that can come together
all at the same angle to one another is four. There is only one angle
at which four objects can all intersect each other equally, that angle
is approximately 109.471220634491 degrees. There seem to be few people
grasping this concept, I would like to correspond with those who do.
This concept is relevant to fusion, but it is necessary to grasp the
concept before one can grasp how it relates to fusion. Is there anyone
out there who understands what a four directional coordinate system is?
Aloha,
D. Lloyd Jarmusch
-----------------------------------------------------------------------
D LL Jarmusch i take off my hat for you !.
you introduced something and it seems that you are not aware
of how important it is.
first the personal aspect: i had always the impression that
i am 'shouting in the desert' and no one gives a damn to my threads.
now i realize that there is one single person who not only is
listening to me but even remembers it in exact details- even a long
time later. so you give me some optimism.
btw did you remembered my claims all by yourself or was it someone
who helped you with this. i'll tell you why:
because in the exemplar that i sent to Jim Carr i added especially for
him
in my hand writing a sketch that describes the Alfa as a tetraheder.
he deserves it because it was his remark about his colleague Don Robson
that started a model of alfa clusters for simple nuclei.
(it seams that he got stuck after a few light nuclei while
i went on (independently) (without getting stacked) and solved
*all the periodic table with the alfa clustering.
btw it is not only alfas but also others anyhow from many points of
views
i went on much further than Don Robson.
anyway in the above sketch i indicated that angle between the four
arms of the tetraheder exactly the same angle 109.4712206 etc etc
anyone who knows some stereometry can calculate it.
you can ask Jim he will confirm all the above *story*.
actually i remember now that there was another Guy in this newsgroup
who corrected my "120' degrees to those '109' sory i cant
remember now his name i wish he will show up to share his credit for it.
now back to our subject. it never occurred to me that only four
*lines* (let's speak for simplicity about lines - moreover
*lines* are not very far away in simple description of protons and
neutrons). -- only four lines can convert to a point with
(and this is important to emphasis) with a constant angle
between all of them!).
if that statement of you is correct then you have no idea
how important it is.
it is important because it gives an additional fortification to
the structure of the alfa which i discovered through *other *
considerations so it is another cross reasoning of it.
the reasoning that you added is a geometric one i would say
geometric-phisical one. --
because one might ask : why on earth should those bloody particles
combine as a tetraheder why not another possibility?
now you come and say : actually *there is no other posibility*-
from geometric reasoning!. which for me at least is good as a
mathematical
proof.
the protons and neutron in the alfa are similar in structure and
mass therefore they should be distributed in space as equal partners
sharing the space *equally*. with a constant angle (and constant
distances)
or if you want in other words; maximum geometric balance gives also
maximum
stability as the alfa is known for. maximum geometric balance
gives also balance in the interior forces between them.
btw if you will substitute the *beame* or *stick* model
with a * ball or a sphere then you cannot go far with it describing
in a verified way the structure of *all* nuclei. and you get an infinite
number of possibilities which is not backed up in experimental data. -
experimental data indicates only a limited numbers of possibilities.
note that according to my model all the four particles(proton proton
neutron neutron combine all of them in a *point* (a very little volume
and are spread outwards like beams to four directions -exactly
as you describe your four axis coordinate system so
one would ask why only four particles and not more of them?
the answer is in that 'point' in one point you cannot combine a great
number
let's make it cleare Lloyd : did you say that no more that four
lines can be combined with equall angles between them ?.
actually even four in a point might be questionable but there is
some answer in addition to LLoyds answer
(not in this thread which becomes long)
so that structure is not because choosing one or another coordinate
system but because of more *objective* reasons-phisical reasons
which do not depend on human invented coordinate system-
nature does not know anything about coordinate systems -coordinate
systems are only human inventions to facilitate the description
of space and transferring information between people -it is a sort of
a human language. therfore any valid coordinate system is good.
perhaps you might claim that your language * is more practical
easier etc. the same way as the polar system of coordinates
is more convenient for some cases, while it is less convenient foe
other cases.
your fore coordinate system is good for describing the alfa but
might not be convenient to describe the nuclei of lead for instance.
you ask if the structure of alfa is relevant for fusion.
is there any doubt about it?!.
Aloha lloyd and thanks again.(i like your head and spirit)people like
you give some hope.
and remember what old 'Catto' said
the Alfa is a cross like shape (tetraheder)
the tritium a Y like (also three dimensional)
the Deuteron -- -- as before
and most nuclei are 'rectangular pipes'
and that's enough for one thread.
Y.Porat
D. Lloyd Jarmusch
for me on the crackpot index with this post. Thanks -D.L.J.)
Porat wrote:
> D.LLoyd Jarmusch wrote:
>
> ....four directional coordinate system.... uses axes that
> are four rays emanating from an origin all at the approximate angle of
> 109.471220634491 degrees from one another.... If Mr. Porat is reading
> this he would
> benefit from noting that this angle is very important to his description
> of the alpha.....
> D LL Jarmusch i take off my hat for you !.
>
> you introduced something and it seems that you are not aware
> of how important it is.
That is true, I probably do not know how how important it is, but I do
believe, and have believed for the past seventeen years, that the fact that
space is four directional is of exceptional importance in understanding and
solving problems involving fundamental physical processes, especially
fusion.
> first the personal aspect: i had always the impression that
> i am 'shouting in the desert' and no one gives a damn to my threads.
> now i realize that there is one single person who not only is
> listening to me but even remembers it in exact details- even a long
> time later. so you give me some optimism.
> btw did you remembered my claims all by yourself or was it someone
> who helped you with this.
Yes, I remember your claims quite well, all by myself.
> ....anyway in the above sketch i indicated that angle between the four
> arms of the tetraheder exactly the same angle 109.4712206 etc
> etc...actually i remember now that there was another Guy in this newsgroup
>
> who corrected my "120' degrees to those '109' sory i cant
> remember now his name i wish he will show up to share his credit for it.
I vaguely remember some post like that but I noticed the angle discrepancy
right away, I would have make the correction sooner but I was busy with
other things.
> now back to our subject. it never occurred to me that only four
> *lines* (let's speak for simplicity about lines - moreover
> *lines* are not very far away in simple description of protons and
> neutrons). -- only four lines can convert to a point with
> (and this is important to emphasis) with a constant angle
> between all of them!).
> if that statement of you is correct then you have no idea
> how important it is.
My statements about the angle are quite correct, and I do have an inkling of
how important it is.
> it is important because it gives an additional fortification to
> the structure of the alfa which i discovered through *other *
> considerations so it is another cross reasoning of it.
> the reasoning that you added is a geometric one i would say
> geometric-phisical one. --
> because one might ask : why on earth should those bloody particles
> combine as a tetraheder why not another possibility?
> now you come and say : actually *there is no other posibility*-
> from geometric reasoning!.which for me at least is good as a mathematical
> proof.
> the protons and neutron in the alfa are similar in structure and
> mass therefore they should be distributed in space as equal partners
> sharing the space *equally*. with a constant angle (and constant
> distances)
> or if you want in other words; maximum geometric balance gives also
> maximum
> stability as the alfa is known for. maximum geometric balance
> gives also balance in the interior forces between them.
>
> btw if you will substitute the *beame* or *stick* model
> with a * ball or a sphere then you cannot go far with it describing
> in a verified way the structure of *all* nuclei. and you get an infinite
> number of possibilities which is not backed up in experimental data. -
> experimental data indicates only a limited numbers of possibilities.
>
> note that according to my model all the four particles(proton proton
> neutron neutron combine all of them in a *point* (a very little volume
> and are spread outwards like beams to four directions -exactly
> as you describe your four axis coordinate system so
> one would ask why only four particles and not more of them?
> the answer is in that 'point' in one point you cannot combine a great
> number
> let's make it cleare Lloyd : did you say that no more that four
> lines can be combined with equall angles between them ?.
>
That is not exactly how I would put it, but yes you have the idea, that is
what I am saying, and it is true. I would put it more like this: Four is
the maximum number of rays that may emanate from a common origin such that
each ray ("line") forms the same angle with all other rays. From the
standpoint of sphere-packing, four is the minimum number of spheres that may
be packed in a stable configuration. Notice that the maximum number possible
for all being at the same angle is also the minimum possible for stability,
that is to say four. These are somewhat metaphysical points, so we discuss
them here at great risk of being branded crackpots. But this stuff is for
real, it is important, and where else are we going to discuss it? Anybody
want to come to my house in Hawaii and discuss it and other aspects of
fusion and metaphysics? We can go down to the beach and kick back in the
shade ....
> actually even four in a point might be questionable but there is
> some answer in addition to LLoyds answer
> (not in this thread which becomes long)
> so that structure is not because choosing one or another coordinate
> system but because of more *objective* reasons-phisical reasons
> which do not depend on human invented coordinate system-
> nature does not know anything about coordinate systems
Yes, this is true, yet the tetrahedral coordinate system does help
illustrate the fact that our normal three-dimensional space may be
profitably described as four directional, and that description indicates a
real and fundamental quality of nature, just as the concept
"three-dimensional" describes a real fundamental quality of nature.
> -coordinate
> systems are only human inventions to facilitate the description
> of space and transferring information between people -it is a sort of
> a human language.
Exactly, but some people think that their language is the only language fit
for proper human discourse, and that their language is not only the 'best'
language but the 'best' of all possible languages. Let us continue to use
the most practical language for our purposes but let us not deny that other
languages, even the most exotic, may contribute useful ideas to totality of
human discussion.
> therfore any valid coordinate system is good.
> perhaps you might claim that your language * is more practical
> easier etc. the same way as the polar system of coordinates
> is more convenient for some cases, while it is less convenient foe
> other cases.
> your fore coordinate system is good for describing the alfa but
> might not be convenient to describe the nuclei of lead for instance.
>
All I want to claim right now is that my coordinate system is useful for
revealing a fundamental aspect of physical space, or at least is is useful
for revealing a fundamental aspect of normal non temporal Euclidean
mathematical space. Space is four directional, and recognizing that fact
inspired me to invent a new approach to inducing fusion.
> Aloha lloyd and thanks again.(i like your head and spirit)people like
> you give some hope.
I enjoy your contributions Mr. Porat, and I like your spirit too. Keep up
the good work, I am beginning to get a better idea of your conception of the
alpha and why you think it is important. Thank you.
Aloha,
D. Lloyd Jarmusch
Peter Hanely
your proposed coordinate system (face normals on a tetrahedren)
has only 3 independent axis's, the 4th being a function of the
first 3. This system may be convienent for sphere packing
problems, but I am not shure how well it would work for
nuclear physics.
--
Peter Hanely =/\=
pha...@jps.jammed.net
(email has been "jammed")
D. Lloyd Jarmusch
post it to the group. I find the most interesting part to be the last bit by
Kirby Urner.
Aloha - DLJ
crystal wrote:
" Hi, Everybody.
I like to point out that coordinate systems are basically VISUAL
SYSTEMS. That is, a situtation can be expressed in graphical
presentations - if only for the aid of comprehension. It would be
useful.
Yes. Calculation wise, it doesn't make a difference. BUT using a
mathematical modelling software - the same problem can look very
different.
There are possible applications: control engineering, technical analysis
(specific to the coordinate system), beautiful art pictures (Lots of
profits here for artists, just kidding.)...
Incidentally, there was a dicussion about >3-dimensions coordinate
systems on another list.
Excerpt from the info file
in Majo...@teleport.com,
by sending "info synergetics-L" in the body:
>
> SYNERGETICS-L is a list for the discussion of R. Buckminster
> Fuller's Synergetic Geometry.
>
> For Fuller, Synergetics was embedded in the context of a design
> science revolution which, within a decade of its inception, could
> launch humanity on a sustainable evolutionary path with high
> living standards for all aboard Spaceship Earth. So jumping back
> and forth between Synergetics and its applications and related
> individual initiatives is not out of bounds.
>
> For example, threads around design science projects, such as
> STRUCK for creating and exploring elastic interval geometry,
> related to tensegrity, and DOME for outputting VRML, CAD and
> ray-tracable geodesic domes, also intertwine.
* SYNERGETICS-L MAILING LIST
Single
Email: majo...@lists.teleport.com
Body : subscribe synergetics-l
Otherwise, the 4-coordinate positive integer system is also refered by
some as a 60 degress coordinate system, in the discussions on the Syn-l
list. (Thread: Re: CubeWorld: addinng dimensions) Yes, the spelling is
right. Hmm...I think they might spliter into other threads on this topic
soon.
As for the alpha particles, the 4-coordinate system can have
mathematical applications esp. in quantum physics.
Best Regards,
Crystal.
Subject:
Re: Syn-l: Magnetic Fields, Control Engineering & Quick Pattern
Recognition
Date:
Tue, 7 Jul 1998 10:50:32 -0500
From:
wiz...@prysm.net
To:
cry...@postme.net
>> O'scopes can display the relationship of several signals
>> compared to other signals.
>
>But wouldn't it be a 2-d axis display?
>
Yes, but all displays are 2-d, we actually add more stuff (vector
transformations) to the scan to make then look 3-d.
>Analogue input electric signals vs digital electronic signals? I was
>referring to sensor outputs from measuring devices such as valves,
>meters, pumps, gauges, solenoid valves, tachometers, etc.
>
>These are devices that have variable electrical/eletronic signal outputs
>depending on their readings.
>
>Can these be interfaced to the vector scope?
Yes, you have given me more information. Slow moving signals are better on
computers. However, digital signals can be interfaced to an oscilloscope
with D/A converters to produce analog signals to be put into the O'scope. I
have done this to create a display for computer memory debugging. It is a
very effective tool for this purpose. By interpreting the shape of the
display, one may determine the status of memory. This is very effective for
hardware development or repair.
Command and control signals have many different kinds of electronic states.
Interfacing them to computers requires condition circuitry (filters, level
converters, current-to-voltage, A/D etc.) PLCs merely provide ladder logic
to command and control circuitry. Computers can provide the ladder logic
also. This is probably, I guess, to what you are referring. Proviging 3-d
display of command and control systems requires custom programming. There
is expensive (and expensive) software available for this purpose.
Subscribe to Measurement & Control News (ISSN 0194-1461).
Subscription ac...@mac-med.com
>
>How about A/D boards and PLCs? Aren't they designed just specifically
>for this purpose?
The main purpose of using an oscilloscope is to be able to interpret faults
and irregular conditions in complex circuitry in real time. Slow moving
signals can be interpreted by a computer. Your application can be enhanced
with an oscilloscope. Any electronic facility should have an oscilloscope.
Without an oscilloscope (and an experienced technician) you run the risk of
not being able to see how signals are behaving (to enhance or repair
circuitry). A computer is not an oscilloscope. Computers were designed and
built with oscilloscopes. Computers are nice for command and control, but
they cannot see signals in real time like an oscilloscope (unless they have
an oscilloscope peripheral attached). I have worked for a government agency
that operated a large facility with industrial control systems. They did
not have the experience to know that electronic systems require an
oscilloscope to service the electronics. They did not understand that in
order to debug problems with the control systems the technician (me) needed
an oscilloscope.
Good luck with your endeavor.
Jay Salsburg, Design Scientist
Creating that which all others have yet to imagine.
Subject:
Syn-l: An Overview of the Quadray Coordinate System
Date:
Thu, 09 Jul 1998 16:50:42 GMT
From:
pd...@teleport.com (Kirby Urner)
Reply-To:
synerg...@teleport.com
Organization:
4D Solutions
To:
synerg...@teleport.com
Newsgroups:
sci.philosophy.meta, bit.listserv.geodesic, sci.math,
alt.education, k12.chat.teacher, k12.chat.senior
AN OVERVIEW OF THE QUADRAY COORDINATE SYSTEM
by Kirby Urner
4D Solutions
July 9, 1998
NeoCartesian Coordinates
Quadrays comprise a NeoCartesian coordinate system with four
positive basis rays (1,0,0,0) (0,1,0,0) (0,0,1,0) (0,0,0,1)
from an origin (0,0,0,0). Whereas the conventional XYZ
apparatus uses 3-tuples and permuted signage to map volume,
quadrays assign a unique address to each point using only
positive scalars or zero. This is made possible by the
fact that positive vectors point symmetrically in four
directions from the origin, meaning we can approach the
origin (0,0,0,0) from any direction by means of a positive
basis vector sum. Put another way: a vector pointing 180
degrees from a positive vector is likewise a positive vector
e.g.
-(1,0,0,0) = (-1,0,0,0) = (0,1,1,1)
Because (1,1,1,1) is a sum of four vectors pulling equally
and symmetrically away from they origin (e.g. is (1,0,0,0)
+ (0,1,0,0) + (0,0,1,0) + (0,0,0,1)) it is equivalent to
(0,0,0,0), i.e. these vectors placed tip-to-tail take us
right back to the origin. Indeed, any (n,n,n,n) = (0,0,0,0).
This identity property allows us to reduce any quadray 4-tuple
(a,b,c,d) to lowest terms by subtracting the lowest valued
scalar (n,n,n,n) from the vector sum (a,b,c,d), where n is
nonzero (if the lowest coordinate is already 0 -- i.e. none
are negative -- then we're done).
By these means, all quadrays are reduced to 4-tuples consisting
of one or more zeros, with the rest of the coordinates all
positive scalars. For example:
(-10, 5, 0, 8) - (-10, -10, -10, -10) = (0, 15, 10, 18)
( 5, 9, 1, 6) - ( 1, 1, 1, 1) = (4, 8, 0, 6)
-10 and 1 are the "lowest scalars" in their respective
4-tuples, hence the "identity element" reduction to lowest
terms (akin to reducing fractions to lowest terms -- for
pedagogical purposes).
Polyhedra in the IVM
Given that quadrays point to the 4 vertices of the regular
tetrahedron of edge length 2, with an angle between them
of acos(-1/3), polyhedra embedded in the face-centered-cubic
lattice (fcc) or isotropic vector matrix (ivm) tend to have
whole number coordinates. Indeed, all fcc vertices are
themselves linear combinations of {2,1,1,0} where { } means
"all permutations of".
These are the coordinates of a regular Octahedron, dual to
the duo-tet cube, in quadray 4-tuples (a,b,c,d):
POINTID ACOORD BCOORD CCOORD DCOORD
O1A 1.0000000 0.0000000 0.0000000 1.0000000
O1B 1.0000000 0.0000000 1.0000000 0.0000000
O1C 1.0000000 1.0000000 0.0000000 0.0000000
O1D 0.0000000 1.0000000 0.0000000 1.0000000
O1E 0.0000000 1.0000000 1.0000000 0.0000000
O1F 0.0000000 0.0000000 1.0000000 1.0000000
The duo-tet cube (C1) consists of a tetrahedron (T1) and
its dual (T2) with the following quadray coordinates:
POINTID ACOORD BCOORD CCOORD DCOORD
T1A 1.0000000 0.0000000 0.0000000 0.0000000
T1B 0.0000000 1.0000000 0.0000000 0.0000000
T1C 0.0000000 0.0000000 1.0000000 0.0000000
T1D 0.0000000 0.0000000 0.0000000 1.0000000
T2A 0.0000000 1.0000000 1.0000000 1.0000000
T2B 1.0000000 0.0000000 1.0000000 1.0000000
T2C 1.0000000 1.0000000 0.0000000 1.0000000
T2D 1.0000000 1.0000000 1.0000000 0.0000000
The space-filling rhombic dodecahedron, being the set of vertices
of the duo-tet cube + its dual, the octa O1, has vertices of
these two sets combined.
The above listings are from ALLPOINTS (catalog of points). To
generate polyhedra, sets of edges connecting these cataloged
points are required.
Here are the edges of the rhombic dodecahedron, taking from
SHAPES, a related table. Columns ID1 and ID2 are populated
with vertices already cataloged (above) in ALLPOINTS.
SHAPEID ID1 ID2
R1 O1A T1A
R1 O1A T2B
R1 O1B T1A
R1 O1B T2B
R1 O1A T2C
R1 O1A T1D
R1 O1F T1D
R1 O1F T2B
R1 O1B T2D
R1 O1B T1C
R1 O1F T1C
R1 O1F T2A
R1 O1D T1D
R1 O1D T2A
R1 O1D T2C
R1 O1D T1B
R1 O1C T1A
R1 O1C T2D
R1 O1C T1B
R1 O1C T2C
R1 O1E T1C
R1 O1E T2D
R1 O1E T1B
R1 O1E T2A
R1 O1E T1C
Volumes of IVM Polyhedra
The quadrays volume method is defined around Euler's for the
volume of a tetrahedron given its 6 edges, as modified by de Jong
to return a value of 1 for the 2x2x2 tetrahedron, where '2' is in
fcc sphere radii. If we call the distance from 1 fcc sphere center
to its neighboring center (12 options) an interval of measure 1,
then our tetrahedron is 1x1x1. It has a volume of 1, as per
Fuller's modeling of 3rd powering in his synergetic geometry
(a philosophical language more than a math).
Given the above volume method, the volumes of the so-far introduced
polyhedra is as follows:
Tetrahedron (T1 and dual T2): 1
Duotet Cube C1 (T1+T2): 3
Octahedron O1 (dual of duo-tet): 4
Rhombic dodecahedron R1 (C1+O1): 6
Quadrays were originally introduced on Synergetics-L by David Chako,
and evolved in a neoCartesian direction by Kirby Urner.
Kirby
Major references:
http://www.teleport.com/~pdx4d/quadrays.html
http://www.teleport.com/~pdx4d/quadshapes.html
http://www.teleport.com/~pdx4d/quadvols.html
http://www.teleport.com/~pdx4d/quadphil.html
---------------------------------------------------------
Kirby T. Urner http://www.teleport.com/~pdx4d/kirby.html
4D Solutions http://www.teleport.com/~pdx4d/ [PGP OK]
---------------------------------------------------------
"
D. Lloyd Jarmusch
Peter Hanely wrote:
> a Mathimatical caviot:
> your proposed coordinate system (face normals on a tetrahedren)
> has only 3 independent axis's, the 4th being a function of the
> first 3.
A philosophical/mathematical caviot: It seems to me that the normal
Cartesian system has only two independent axes since the z axis is a
function of x and y where x = 0 and y = 0. But I'm not sure what any of
this has to do with the fact that space is four directional which is my
major point in regards to fusion.
Apparently others have come up with the exactly the same tetrahedral
system I find so fascinating. Truth cannot be suppressed indefinitely,
if you stomp on it it comes oozing out from under your shoe. I would
like to post the following from Kirby Urner, thanks to Crystal:
Rob Burgis
use of negative numbers. Nonetheless, I agree that this coordinate system
seems superfluous for most problems.
Being an engineer (with a coincidental math degree), however, the only real
problem which I can see this being useful for is propulsion systems in
space, where you only have "pushes" (i.e. positive impulses), and no
"pulls". I'd have to take it at face value that there may be some physics
problems where such a system may be useful.
Cheers,
Rob Burgis
Peter Hanely <pha...@jps.jammed.net> wrote in article
<35AA31...@jps.jammed.net>...
> a Mathimatical caviot:
> your proposed coordinate system (face normals on a tetrahedren)
> has only 3 independent axis's, the 4th being a function of the
Peter Hanely
> Cartesian system has only two independent axes since the z axis is a
> function of x and y where x = 0 and y = 0. But I'm not sure what any of
> this has to do with the fact that space is four directional which is my
> major point in regards to fusion.
let me try that again. any spacial coordinate (real space) can be
given by a position along 3 independent axes. The position along
any 4th axis is a function of the position along the other 3.
space is 3 dimensional, and any 4 axis system must have a redundent
axis.
space-time is 4 dimensional, but I have seen no indication that
is what you are trying to describe.
Robin van Spaandonk
[snip]
>> let's make it cleare Lloyd : did you say that no more that four
>> lines can be combined with equall angles between them ?.
>>
>
>That is not exactly how I would put it, but yes you have the idea, that is
>what I am saying, and it is true. I would put it more like this: Four is
>the maximum number of rays that may emanate from a common origin such that
>each ray ("line") forms the same angle with all other rays. From the
>standpoint of sphere-packing, four is the minimum number of spheres that may
>be packed in a stable configuration. Notice that the maximum number possible
>for all being at the same angle is also the minimum possible for stability,
>that is to say four. These are somewhat metaphysical points, so we discuss
>Yes, this is true, yet the tetrahedral coordinate system does help
>illustrate the fact that our normal three-dimensional space may be
>profitably described as four directional, and that description indicates a
>real and fundamental quality of nature, just as the concept
>"three-dimensional" describes a real fundamental quality of nature.
I think it's the three dimensional nature of flat space that is
responsible for the fact that 4 is the maximum number of rays (as
above). In fact to carry it a step further, I think the maximum number
of such rays in any dimensional system, is 1 more than the number of
dimensions.
Regards,
Robin van Spaandonk <rvan...@vic.bigpond.net.au>
D. Lloyd Jarmusch
that space is four directional is for the most part off-topic for s.p.f but
I believe it is important background information for understanding my
approach to inducing fusion. Whether or not it is relevant to fusion, it is
at least entertaining and enlightening, entertaining for me anyway, and
enlightening for those who do not yet understand that space is
four-directional. For Ian and others who do not like this topic they do not
have to read those threads devoted to it.
Peter Hanely wrote:
> any spacial coordinate (real space) can be
> given by a position along 3 independent axes.
I agree, but that does not have any bearing on the fact that space is four
directional.
> The position along
> any 4th axis is a function of the position along the other 3.
The truth of that statement depends upon what you use for your axes, lines or
rays. In a system that uses rays as axes it is still true that the location
of any point (in "real space") can be given by its coordinates along 3
independent axes, yet you need a fourth independent axis to map ALL points in
"real space" with spatial coordinates. That fourth axis is not a function of
the other three axes since those axes are rays.
Your statement would be correct, even for systems using rays, if you allow
the use of negative numbers and assume the existence of three additional rays
which point in the opposite directions of the first three, which is in fact
the case in the normal rectangular Cartesian system. It is important to note
that the normal rectangular Cartesian system is a SIX-directional system. The
normal rectangular system has two additional directions beyond the necessary
four. I am not saying there is anything wrong with that. I am just saying
that the normal system, with its six directions, obfuscates the fact that
only four directions are needed to adequately model static three-dimensional
space.
> space is 3 dimensional, and any 4 axis system must have a redundent
> axis.
> space-time is 4 dimensional, but I have seen no indication that
> is what you are trying to describe.
Correct, I was not trying to describe four-dimensional space-time, but rather
non-temporal space. Nonetheless, a four-dimensional space-time in which one
can only move forward in time and not backward is also a four-directional
space, since moving forward in time is a function of movement along the four
directions. A four-dimensional space-time in which one could move forward and
backward in time is a five-directional space.
To get a more intuitive grasp of what I mean picture the one-dimensional
space constituted by a line.
Figure 1.
<____________________>
-2 -1 0 1 2
Now picture a zero-dimensional creature (a point) living on that line.
Figure 2.
<_____________.____>
-2 -1 0 1 2
The creature's position on that line is given by a single coordinate
magnitude that is either positive or negative and indicates the creature's
distance from the origin in one direction or the other. In Fig. 2 the
creature is at about positive 1.5. Now that we have this picture we may
represent the creature's movements in time by graphing those movements in two
dimensions. Picture the two-dimensional graph plotted on a coordinate system
that consists of three rays emanating from a common origin all at 120 degrees
from one another.
Figure 3.
^ A ^ B
\ /
\ /
\ /
\ /
\ /
\ /
l
l
l
l
l
l C
v
Let us name the rays A, B, and C. Let A represent the negative direction of
Fig. 1 and let B represent the positive direction, C will represent the
direction of motion backward in time.
Figure 4.
^ A ^ B
\ /
\ /
\ /
\ /
\ /
<__________ \ /______________>
-3 -2 -1 0 1 2 3
l
l
l
l
l C
v
If we imagine our point-creature at time t1 starting at the origin we have
the following:
Figure 5a.
^ A ^ B
\ /
\ /
\ /
\ /
\ /
\ . /
l
l
l
l
l C
v
On the one-dimensional graph the creature's position looks like this:
Figure 5b.
<_________.________>
-2 -1 0 1 2
If we imagine our point-creature moving one unit in the positive direction we
have the following at time t2:
Figure 6a.
^ A ^ B
\ /
\ /
\ /
\ /
\ ./
\ /
l
l
l
l
l
l C
v
On the one-dimensional graph the creature's position then looks like this:
Figure 6b.
<____________._____>
-2 -1 0 1 2
If we imagine our point-creature then moving one unit in the negative
direction we have the following at time t3:
Figure 7a.
^ A ^ B
\ /
\ /
\ /
\ . /
\ /
\ /
l
l
l
l
l
l C
v
On the one-dimensional graph the creature's position again looks like this:
Figure 7b.
<_________.________>
-2 -1 0 1 2
Notice that, since the creature's movements are limited to motions in
direction A or B, no matter how the creature moves it will remain in the
section of the two-dimensional graph between rays A and B and all its future
movements will be limited to a similar v-shaped section originating from its
present location. Limited to motion in two directions the creature cannot go
back in time.
Positions of the point-creature in one-dimensional space graphed in two
dimensions to show its movement through a two-dimensional space-time for
times t3 through t6:
Figure 8.
^ A ^ B
\ t6 . /
\ t5 . /
\ t4 . /
\ t3 . /
\ /
\ /
l
l
l
l
l
l C
v
If our zero-dimensional point-creature, living in its one dimensional space,
with the added dimension of time had the option of moving in direction C,
then it could go back in time, and it therefore would have the capacity of
unlimited access to all points of its two-dimensional space-time. Yet if our
zero-dimensional creature could not move back in time it would be limited to
only a partial section of its two-dimensional space-time plane. At any give
time its possible future locations would always be limited to the v-shaped
partial plane in front of it which lies within the larger two-dimensional
space-time.
Similarly, we three-dimensional creatures, since we cannot go back in time,
are limited to motion in four directions within our four-dimensional space
time, and are therefore limited to a certain section of the larger space-time
in which we live. If we could move in the fifth direction we could visit any
point in the whole of our four-dimensional space-time.
Robin van Spaandonk wrote:
" I think it's the three dimensional nature of … space that is
responsible for the fact that 4 is the maximum number of rays (as
above). In fact to carry it a step further, I think the maximum number
of such rays in any dimensional system, is 1 more than the number of
dimensions."
What Robin said echos what Dr. Carlson has said on this subject. Dr. Carlson
said
> : if a space can be spanned by N vectors with unrestricted coefficients,
> : then (N+1) vectors will be required if the coefficients are restricted
> : to non-negative values. ...
What they have said seems to be true for unbound spaces, but it is not
strictly true for bound spaces that if a space can be spanned by N vectors
with unrestricted coefficients, then (N+1) vectors will be required if the
coefficients are restricted to non-negative values. For instance any convex
half plane that is bound by two non-collinear rays may be spanned by two
vectors whether or not the coefficients are restricted to non-negative
values. Likewise the four-dimensional space-time in which we live appears to
be bound by a restriction on traveling back in time, and such a bound
space-time may be represented by a four-directional coordinate system.
Aloha,
D. Lloyd Jarmusch
D. Lloyd Jarmusch
on your desktop)
Peter Hanely wrote:
> let me try that again. any spacial coordinate (real space) can be
> given by a position along 3 independent axes. The position along
> any 4th axis is a function of the position along the other 3.
My system is so natural others have independently come up with exactly the
same system and dealt with the same questions. I like Kirby Urner's answer
to this particular question quoted from one of his web pages:
" So the quadray player doesn't see a fourth vector as a linear
combination of the other three, but on the contrary,
as the very
reason why "negative mirroring" is inessential. It's
the xyz system
of 6 spokes which looks overbuilt and dependent upon
redundant "mirror opposites" to map volume. And it's
the xyz
apparatus which therefore seems less "basic" or
"primitive"
overall, vis-a-vis the quadray system of only 4
spokes and no
negative mirroring. Plus quadrays easily pair to
give six vectors
in the xyz orientation, which gives xyz the
appearance of a
construction within quadray space. xyz as a whole is
derivative."
For clarity the full text of the relevant webpage from Kirby Urner follows:
Investigations into
the Linear Algebra Concepts
used in
the XYZ and Quadray Language Games
by Kirby Urner
First posted: November
20, 1997
Last updated: November 20,
1997
In xyz, the 3 positive basis vectors line the edges
of one of eight
sectors -- lets call it the alpha sector (not
'quadrant' because we
have eight of them -- octant OK). To have
tip-to-tail vector
addition get us outside the alpha sector and into
the rest of
space, we need to reverse the positive vectors,
which reversal
operation we accomplish by multiplying the positives
by -1. The
result is 6 rays (3 positive and 3 negative)
arranged like a "jack"
(as in the game). If you paint them all black, it's
not evident which
three are positives, nor which is the alpha sector.
You see 6
spokes stemming from a common hub i.e. from the
origin, or
(0,0,0).
In the game of quadrays, 4 positive basis vectors
carve space
into four quadrants. To get beyond a given sector,
simply scale
and add the vector already outside it. No
multiplication by -1 is
needed to map all of volume. If you paint the rays
black, it's not
evident which quadrant is called alpha but then it
doesn't really
matter, since all 4 rays are positive; the symmetry
is simpler than
in xyz, wherein you have this "pure" +++ --- versus
"mixed" ++-
+-+ -++ --+ -+- +-- signage in the various octants.
You see 4
spokes stemming from a common hub i.e. from the
origin, or
(0,0,0,0).
An xyz player might say 3 quadrays are basis vectors
(any 3) and the fourth is a linear combination
of those 3. This assumes we're following the rule
that vector reversal is just another scalar
operation which introduces no new basis rays (only 3
of the xyz spokes are considered 'basic' i.e.
the positive ones). But in quadrays, any vector
pointing oppositely to a basis vector is itself a linear
combination of the other three basis vectors e.g.:
-(1,0,0,0) = (0,1,0,0) + (0,0,1,0) +
(0,0,0,1) = (0,1,1,1)
This means "vector reversal" is accomplished by the
original basis rays themselves, not through
introducing multiplication by -1. The fourth quadray
is a linear combination of the other three only if
vector reversal is slipped in as primitive and
essential to the game. But in quadrays, we don't need
to give vector reversal such a primitive status. By
making all four rays equally basic, negative
numbers are what become inessential.
In other words, all points P(a,b,c,d) in the
surrounding volume may be readily addressed by
elongating/contracting the four basis rays and
adding them tip-to-tail, with no negatives entering into
the coordinate addressing scheme. A preliminary step
involving negative mirroring to get additional
spokes, as in xyz, is not required.
So the quadray player doesn't see a fourth vector as
a linear
combination of the other three, but on the contrary,
as the very
reason why "negative mirroring" is inessential. It's
the xyz system
of 6 spokes which looks overbuilt and dependent upon
redundant "mirror opposites" to map volume. And it's
the xyz
apparatus which therefore seems less "basic" or
"primitive"
overall, vis-a-vis the quadray system of only 4
spokes and no
negative mirroring. Plus quadrays easily pair to
give six vectors
in the xyz orientation, which gives xyz the
appearance of a
construction within quadray space. xyz as a whole is
derivative.
In sum, the xyz player needs two operations: vector
grow/shrink and vector reversal, in support of
tip-to-tail vector addition, in order to reach all
points, whereas the quadray player only needs the
grow/shrink operation, and not vector reversal, to
have vector addition perform adequately.
Of course, the xyz player lumps both vector reversal
and vector elongate/contract under the
heading of a single operation, called scalar
multiplication. But the argument can be made that an
operation which reverses a vector's direction is
simply not of the same kind as one which alters the
length but not the orientation of a vector. The
reorienting "mirroring" function should be
distinguished from the non-reorienting "scaling"
function.
To reiterate, the negation
operator (-) performs a distinctly
different role in xyz
space, creating mirror vectors without
which the positives alone
are helpless to reach allspace. We
should perhaps treat with
some suspicion this claim that a
mirrored basis ray is not
itself a basis ray, since once created
there's no way to tell
them apart, or at least there shouldn't be,
since the positive half of
the number line is as much a "mirror" of
the negative half as the
other way around.
Take a jack and spray
paint one vector red and the oppositely
pointing vector green.
It's not clear why one has priority or
should be seen as having
produced the other. So why should
only the red (or green)
vector be designated a "basis vector"
and the other as the
operationally derived "reflection" of the one
true oppositely pointing
twin?
Cartesianism seems laced with a cultural "positives
are better" bias, which leaves the negatives out
of accounts as "derivative" or "second rate", even
though these negatives are equally relied upon
and betray no detectable differences other than in
sign and orientation. Perhaps its the asymmetry
of (-)(-) = + and (+)(+) = + that makes positives
and negatives come across as having a higher and
lower rank. To nod twice is to be even more
affirmative whereas to say "not no" is to negate the
negation and return a positive. Positives don't self
negate while negatives do. ~(~p) = p, but so
does +(+p), if defined at all.
Turning to philosophy, we find "truth" playing the
role of the "positive" with the operation of
negation conventionally depending upon positive or
affirmative statements as targets to operate
upon. The proposition "All seagulls eat rats" (p) is
conventionally negated as "Some seagulls don't
eat rats" (~p). The negation lacks the power and
authority of the positive, comes across as a mere
"exception to the rule" -- enough to negate the
truth of the positive obviously, but not to come off
as symmetrically bold in its own right, something
more like "All rats eat seagulls" or "All seagulls
avoid rats like the plague".
Basically, negative numbers play second banana to
the positives in much of the curriculum,
following this paradigm of symbolic logic, are
regarded less as numbers in their own right than as
positives being operated upon by a symbol (-). When
moved into the realm of geometry, this
convention makes less sense, because both +x and -x
are equally important, as directions, and
there's no clear reason to regard one as
operationally more dependent upon the other than vice
versa. Yet in conventional linear algebra this is
made to be the case. The xyz positives are
predefined as "basis vectors" while the negatives
are seen as operationally derived by vector
reversal and are therefore not considered equally
basic.
Quadrays put all four directions on an even playing
field, not marking any as less important, less
basic, more derivative. Vector reversal is defined,
but as shorthand for a positive operation making
no use of "second banana" mirror vectors.
For further reading:
An Introduction to Quadrays
Quadrays and the Concentric Hierarchy
Memo re this paper to Dr. Benacerraf, Chair,
Princeton Philosophy (Nov 30, 1997)
Memo to Dr. Paul Ernest, University of Exeter
(April 10, 1998)
On Ludwig Wittgenstein's Contribution to a
Pragmatic Philosophy
Memo re teaching linear algebra and computing
Synergetics versus HyperCross Dogmatics
Synergetics on the Web
maintained by Kirby
Urner