Object Convolution
Timothy Golden
Two geometrical objects can be joined via an arithmetic product operation.
Each object can be regarded as a set of points:
O1 = p11, p12, p13, ..., p1n
and likewise for O2 a series of points p21 ... p2m.
The resultant object
O3 = p11 p21, p11 p22, p11 p23, ... p11 p2m,
p12 p21, p12 p22, p12 p23, ... p12 p2m,
p13 p21, p13 p22, p13 p23, ... p13 p2m,
... ,
p1n p21, p1n p22, p1n p23, ... p1n p2m
In the complex plane line segments will yield areas with interesting boundaries. This geometric product could have some relation to spatial curvature interpretations whose constituents are flat.
Under the polysign construction such geometric products can be carried out in any dimension. It would seem that if this math were already built it would be built on the complex plane. What area of mathematics is this? I don't see these geometrical objects as functions. Though they are nearby they are simpler. Their continuous form still lends themselves to this computation. For instance a point set which is connected (meaning that paths or solids may be approximated) can maintain a coherent image in its approximated form.
This is a simplistic construction that relies upon a geometric arithmetic product. Not much to it. Is this overlooked in existing mathematics? Point products in real and complex spaces are common enough. This is merely a generalization of that concept.
-tpg20070624
tommy1729
this is more or less what i had in mind when i mentioned mye bubbles.
the infinite dimensions are represented bye the infinite points and therefore giving a shape.
the product of my bubbles would then give another shape or bubble.
you work on complex plane , whereas i considered polysigned.
strangely we switched here , since i usually do complex and you polysigned.
but thats not the main point , wheither or not we use complex or poly or whatever.
the concept of my bubbles is equivalent to your socalled object convulation.
so , im sorry cant give you credit since i had this in mind way before you.
but i can say im intrested.
tommy1729
Timothy Golden BandTechnology.com
> Two geometrical objects can be joined via an arithmetic product operation.
> Each object can be regarded as a set of points:
> O1 = p11, p12, p13, ..., p1n
> and likewise for O2 a series of points p21 ... p2m.
>
> The resultant object
> O3 = p11 p21, p11 p22, p11 p23, ... p11 p2m,
> p12 p21, p12 p22, p12 p23, ... p12 p2m,
> p13 p21, p13 p22, p13 p23, ... p13 p2m,
> ... ,
> p1n p21, p1n p22, p1n p23, ... p1n p2m
>
> constructions.
> -tpg20070624
So far the closest I have found to this concept is
http://reports-archive.adm.cs.cmu.edu/anon/2003/CMU-CS-03-181.pdf
but it is starkly different. Their concept of 'geometrical
convolution' involves taking an inverse, which by this definition
would be catastrophic unless one were to work in a very limited space.
I'll have to come up with some graphics to help convince others of the
value. The trouble with this topic is that in general an object
defined as a set of points is a broad concept. However this is exactly
how we view existence in its objectified form, witness any noun you
speak. Yes, the points have much more quality than merely being black
on white but the fundamental relation is accurate. Abstracted objects
such as 'beer' may perplex the enduser, but upon reflecting the users
definition such an object is apparently quantifiable. We can question
the user on every point and ask:
"Is that beer?"
Now the functional relation is named. However this is a fairly
abstract function. We are currently relying upon one user whose
interpretation may vary from another user. If I grow some wheat and
ferment it in water and make the user drink it and ask
"Is that beer?"
he may or may not taste it as beer.
Now, the geometrical object that is the defining component of object
convolution will arguably be less relativistic. Yet we observe that
the product upon the spaces which support such arithmetic products
hardly support relativity. For instance in the real numbers
( - 1 )( + 2 ) = - 2 ,
( + 2 )( + 5 ) = + 10 ,
( - 2 )( + 1 ) = - 2 ,
( - 3 )( 0 ) = 0 .
Here we see relative offsets of 0, +3, -1, and -2 respectively. The
resultants do not correspond to the sources (operands). Already we see
on this simple space that a relativistic interpretation of product
would need work to yield a coherent result. That work will not be
performed here. I don't know if it can be done. Regardless, back in
the superpositional space of the resultant we may accept the results
as coherent. A small change in a source will yield a small change in
the resultant. To claim that the resultant is in the same space as the
source does appear to be a mistake, yet they do share the same
dimension. Will relativity allow for the introduction of n-1
dimensional objects into an n-dimensional space? This would then allow
the line segment into the complex plane, and so a relativistic space
awaits this measly 1D object. The relativity is in 2D for the 1D
interval. This notion is new. That one line segment might look upon
another and observe it in a two dimensional space is in some regards
the ultimate Cartesian thought. However, the Cartesian system
constrains us to pondering these segments strictly on the real line.
And there also we can introduce the object convolution. But what is it
upon? What shall we convolve in the 1D space? Is the segment a valid
convolution source? According to this n-1 dimensional thinking the
restrictive position is no, that only points are going to be allowed.
Yes, the convolution operation will allow for more though the coverage
of such sets does look shall we say oblique. The n-1 principle is a
dimensional relation that goes without saying. For that lack of saying
perhaps we have paid a dear price. The n-1 relationship of the
dimensions is a thing which Cartesian mechanics has obscured by making
it too easy. Yes, the integral works and is beautifully behaved. But
is it too easy? We are caught in the present at a point where integral
relations have ruled and ruled well, but have have not answered the
final question, which is somewhat their discoverer's motivation of
them. Later, the integral answers strongly and purely to finer points
of optics, and all of reality to boot. Yet the original purpose of
that integral which led us from real valued function to two
dimensional area was based in time wasn't it?
It is a strange place that I am in. To think so large and to think it
so trivially makes one a blasphemer. Yet in riding such a wave I may
be finding the depth of those who constructed the things which I
criticise. Descartes dimensional relation and Newton's calculus are
tightly woven together. The slightest upset in the underlying
foundation of these two monoliths will have great significance. The
n-1 dimensional relationship is a constraint upon the convolution of
objects which bears looking into.
-tpg20070627
tommy1729
im sorry but if your claiming this theory as completely yours i must protest and claim my part in it.
>
> Now, the geometrical object that is the defining
> component of object
> convolution will arguably be less relativistic. Yet
> we observe that
> the product upon the spaces which support such
> arithmetic products
> hardly support relativity. For instance in the real
> numbers
> ( - 1 )( + 2 ) = - 2 ,
> ( + 2 )( + 5 ) = + 10 ,
> ( - 2 )( + 1 ) = - 2 ,
> ( - 3 )( 0 ) = 0 .
> Here we see relative offsets of 0, +3, -1, and -2
> respectively. The
> resultants do not correspond to the sources
> (operands). Already we see
> on this simple space that a relativistic
> interpretation of product
> would need work to yield a coherent result. That work
> will not be
> performed here. I don't know if it can be done.
what about changing the operator from
a*b -> a*b +a +b +1
(which has -1 behaving like 0 and is still commutativ and is more dependant on sign )
tommy1729
Timothy Golden BandTechnology.com
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -
Hi Tommy.
I don't see that this concept has even hatched yet. Far more important
than trying to claim it would be making a contribution to it. It'll be
a few days before I can get to doing the graphics, and publishing them
is even more cumbersome. So there is a nice opportunity for you to
beat me to it. I believe that the line segment is the proper object
for study in the complex plane. So if you could post some links to
these graphics before me then you will have a claim. But Tommy
informationally where is your support for this being your idea? I see
no signal like object convolution in anything that you have posted
especially corresponding with the word bubble. Rather than fight about
such a silly thing why not just go forward and let it be? As we
discussed before, noone owns ideas. Upon sharing them they can be
adopted or even sprout variations (hopefully). If we were to start
fighiting over whose idea this was neither of us would seem credible
to the other. I think that your 2D version of the polysign system may
be of interest, but only if it has some consequence and
correspondence. Here we see the same problem. Yes object convolution
looks like a natural procedure (especially under polysign) but has it
any consequence? If 1D segments convolved in the complex plane yield
curved areas then there is enough to take this idea seriously. I'm not
even certain that it is true. So far my handwork in graphing them is
not very trustworthy. I found some graph paper the other day and
thought that would help me get an accurate graph but never got around
to it. Anyhow I'm pretty sure that they have curvature of some form.
Going point by point on one line the other is swept about in a
rotational fashion with a varying (scaled) length. That's just the
complex product at work. The resulting form is an area which is the
resultant of two 1D objects so this is arguably a new product form.
The object is an area so it's like a geometrical integral. When more
complicated objects are formed the surface areas taken segment by
segment could overlap so there is not necessarily a simple sum form
for the area of the resultant. Anyhow the segment composition is a
fundamental form. I'm really interested in getting together an
animation that characterizes it's behavior. Then we'll be able to see
if this procedure has legs.
-Tim
tommy1729
why didnt you reply to these last lines here ???
> >
> > (which has -1 behaving like 0 and is still
> commutativ and is more dependant on sign )
> >
> > tommy1729- Hide quoted text -
> >
> > - Show quoted text -- Hide quoted text -
> >
> > - Show quoted text -
>
> Hi Tommy.
>
> I don't see that this concept has even hatched yet.
i agree on that
> Far more important
> than trying to claim it would be making a
> contribution to it. It'll be
> a few days before I can get to doing the graphics,
> and publishing them
> is even more cumbersome. So there is a nice
> opportunity for you to
> beat me to it.
i dont have a website , but i do have a busy agenda.
I believe that the line segment is the
> proper object
> for study in the complex plane.
i might agree on that yes.
So if you could post
> some links to
> these graphics before me then you will have a claim.
> But Tommy
> informationally where is your support for this being
> your idea? I see
> no signal like object convolution in anything that
> you have posted
> especially corresponding with the word bubble. Rather
> than fight about
> such a silly thing why not just go forward and let it
> be? As we
> discussed before, noone owns ideas.
if you dont care , why you dont admit came from me or it was inspired bye me at least.
Upon sharing them
> they can be
> adopted or even sprout variations (hopefully). If we
> were to start
> fighiting over whose idea this was neither of us
> would seem credible
> to the other.
I think that your 2D version of the
> polysign system may
> be of interest, but only if it has some consequence
> and
> correspondence. Here we see the same problem. Yes
> object convolution
> looks like a natural procedure (especially under
> polysign) but has it
> any consequence?
the 2d version is what i mean with my bubble's.
the 2d version are an infinity of points on 2d.
behaving algebraicly kind a like your polysigned.
since the points are connected they describe a shape or form.
that is your "object".
and bubble operators like e.g. bubble multiplication would be your "object convolution".
you see ? it comes down to basicly the same.
when i mentioned it , you did see anything in it.
and now your considering it.
so points on 2d = bubble = shape = object.
succes with the theory and the graphs.
despite claiming my part in the idea , i do give my support to the concept.
keep me informed.
tommy1729
Timothy Golden BandTechnology.com
> > > what about changing the operator from
> > > a*b -> a*b +a +b +1
> why didnt you reply to these last lines here ???
Sorry Tommy. I must not understand it. If we add one to this operation
we'll literally be adding the identity point (@1) each time to the
graph. I do not see this as coherent. So I doubt that this is what you
meant when you wrote this. also we'll be adding the source objects
into the resultant graph. I am looking at them as two independent
spaces that happen to share the same dimension.
To place the sources into the resultant may not be wise.
I have been considering calculus for a long time on the polysign
domains. That thinking is the root of this idea. It has nothing to do
with constraining to 2D or anything that you have suggested as far as
I can tell. Still, I cannot lay down such a law. Here we communicate
in the open and under that paradigm all information that is shared is
open for scrutiny. In effect this means that we need not worry about
ownership. A traceable fingerprint is hopefully retreivable. In some
regards the object convolution idea is highly unoriginal since it is
merely a generalization of the arithmetic product that seems obvious
in hindsight. It also wouldn't surprise me if it has already been
studied and noone here has bothered to post their knowledge or this
topic simply goes unread. I have seen such poor performance before
from this media. For instance the tatrix format
a11
a21 a22
a31 a32 a33
...
matches the symmetric tensor, but whan I put out a query I never
received this answer back. What we try to do here is not guaranteed to
be good quality or receive good quality feedback. Still, it is better
than no interaction at all, and on occasion there are decent exchanges
so I like it enough to keep going this way. I'm not interested in
giving you ownership of this idea in part because ideas cannot be
owned. Some third party may be taking this thing to some extreme level
on some obscure branch of mathematics and running with it. Who cares?
It is their right to do so as far as I can see. The idea of not
getting credit- that would be disappointing, but this media does at
least give an uncensored marker in time. Just think of the crap that
could go on with a corrupt journal editor who abstracts one persons
rejected work to some other branch and passes it off to a friend.
There are no tracks to follow. How long and how much effort is it to
share work with others in journal format? An idea which takes fifteen
minutes to construct could take years to squeeze into someone elses
level of acceptance. Instead here we share for free and nearly
instantaneously. Uncensored and public. I like that. If I want chuck
in some rhetoric about what a chimp is in the oval office I can. I'm
in Kennebunkport now to protest him tomorrow. There will be a march
from downtown Kennebunkport to the Bush estate. Hopefully there will
be quite a turnout...
The crooks in the capitol
The cooks with the capital
The cocks with crap in their eyes
-Tim
Timothy Golden
>
> Two geometrical objects can be joined via an arithmetic product operation.
> Each object can be regarded as a set of points:
> O1 = p11, p12, p13, ..., p1n
> and likewise for O2 a series of points p21 ... p2m.
>
> The resultant object
> O3 = p11 p21, p11 p22, p11 p23, ... p11 p2m,
> p12 p21, p12 p22, p12 p23, ... p12 p2m,
> p13 p21, p13 p22, p13 p23, ... p13 p2m,
> ... ,
> p1n p21, p1n p22, p1n p23, ... p1n p2m
>
> In the complex plane line segments will yield areas with interesting boundaries. This geometric product could have some relation to spatial curvature interpretations whose constituents are flat.
I've verified curvature for some random segments. A part of the boundary of the area is sometimes curved. So far my segments are chosen by randomly choosing points on the unit circle so they are just a subset of the possibilities. For this subset a four vertex form is the resultant. It sometimes resembles a butterfly; that sort of X form seems to be general. The curvature is always concave and appears in the crotch of the X form of the resultant. Often the curvature is not apparent and the shape takes a simple X as two triangles of symmetrical area.
The coverage of the shape is clearly doubled especially for the strongly curved cases. Is this problematic for getting an area measure? There appear to be two area forms. Rather the area may have a density figure that would complete its characteristic. This sort of coverage would likely become extreme when convolving arbitrary paths as sources. A means of producing an area from geometry is present in this construction. Furthermore curvature is exhibited within that resultant. That is not surprising since the product of traditionally constructed segments would yield a parabolic curve. One difference here is that we are getting a more calculus oriented answer; an area rather than another path.
-tpg1183467861s
tommy1729
that seems intresting , but a bit vague...
got it on a website or pdf ??
tommy1729
Timothy Golden BandTechnology.com
I've published some graphics on my website:
http://bandtechnology.com/ObjectConvolution/index.html
These graphics are not a complete survey but I am fairly sure that
they expose enough for most to see what is going on. The four vertex
form has endpoints which are the endpoint products of the segments.
It's quite a diverse operator since it can hide within its own
coverage. I'd like to get this put in terms of relativity theory but
I'm afraid that it is just pure geometry until then.
-tpg1183808556s
Timothy Golden BandTechnology.com
when the product is taken as the complex product. See for instance
page 6 of
http://mae.ucdavis.edu/~farouki/covering.pdf
He does not detail the line segment convolution but does mention some
applications which sound impressive.
-tpg1183842159s
Timothy Golden BandTechnology.com
Here is another nice doc on the topic of Minkowski product:
http://www.math.hmc.edu/seniorthesis/archives/2003/msmukler/msmukler-2003-thesis.pdf
tommy1729
nice , but how about an integral ?
ive briefly looked at the pdf but did not find a closed form solution to an integral , nor a closed form for the surface described.
maybe i have overlooked ...
btw talking about calculus i have posted a tread for you ( about the calculus of polysigned )
as for my set theory , i know you are a computer man.
and it is striking that many people in that area work with data all the time.
yet in the actual mathematical version ( set theory and especially cantor set theory ) of sets of data , their is a different vision ... than what is thought at universities in the domain of computer science.( mereologie and ontology ea , although waved away as not 'set theory' or ' wrong ' set theory bye the pro-cantorians)
if you would have computer hardware based upon cantor set theory it would certainly crash.
for instance in cantor sets , the elements dont have a ' fixed ' place in a set. and even if they do , its not determined.
in my tommyian set theory (called like that bye others , ill just adopt it)
sets are like the memory in a computer.
or parts of the memory.
and (x)=x=((x)) is like a program referring to data in various ways ( to x , where x is not just a value but also a location ) therefore the 3 are equivalent.
but more important now is perhaps what i have written on the calculus of polysigned.
tommy1729
Timothy Golden BandTechnology.com
I've looked for your thread but don't see anything. What is the title?
Perhaps you could put a link to it here.
I don't believe that it is necessary to resolve the set issues that
are floating around. In terms of construction one is free to construct
what one wishes to and if it is consistent then it can stand. We can
squeeze and dissect such constructions but the reliance on one
dissection may not be as productive as investigating the nuances and
variations that are possible. Must every portion of mathematics be
derived from the natural numbers? No, certainly not. One might monkey
around and get to some such breakdown, but that breakdown does not
become a necessary part of the definition of such a construction, yet
that is the way that the teaching goes. Such breakdowns are poor bases
for communication. Real problems in axioms lead to real problems in
their consequences. When a concept is sharp it is so because it is
clearly defined. Whether that clarity extends to the next user is
problematic, but decomposing such a construction down to a natural
number basis will not necessarily yield a better understanding to that
user either. If anything the long way is going to be more problematic.
Effective communication is a large and universal problem.
-tpg1184193806s