Bo's new notation

Discussion of shapes with curves and holes in various dimensions.

Bo's new notation

Postby bo198214 » Mon Aug 07, 2006 12:13 pm

I would anyway alter the notation.

First of all the discussed # seems to me nothing more than the cartesian product of manifolds, i.e. we need no embedding space. The usual cartesian products of R<sup>i</sup> spaces is just the cartesian product of them as manifolds.

Then I would introduce the brackets [] for rectangular compositions and the parentheses () for spherical composition.
The spherical norm is s(x<sub>1</sub>,...,x<sub>n</sub>):=sqrt(x<sub>1</sub><sup>2</sup>,...,x<sub>n</sub><sup>2</sup>)
The rectangular norm is r(x<sub>1</sub>,...,x<sub>n</sub>):=max(|x<sub>1</sub>|,...,|x<sub>n</sub>|)
Though we could also introduce the lozenge norm l(x<sub>1</sub>,...,x<sub>n</sub>):=|x<sub>1</sub>|+...+|x<sub>n</sub>| which yields the same shape though rotated.
(Wendy pointed out these norms already in some thread).

The parametric functions associated are then
r(F<sub>1</sub>,....,F<sub>n</sub>)-R for [F<sub>1</sub>,....,F<sub>n</sub>]
s(F<sub>1</sub>,....,F<sub>n</sub>)-R for (F<sub>1</sub>,....,F<sub>n</sub>)
where F<sub>1</sub>,...,F<sub>n</sub> must have pairwise disjunct set of variables.

And the associated (hollow) shape is the zero set of the associated parametric function. And the associated solid shape is for F<=0.

Otherwise the notation is ambigous. If we omit the [], then we could (AB) interpret as ([AB]) which is not what was meant, though one can indeed also consider this shape ([AB])! Not only that but we can then also consider such shapes as ([AB][CD]). Which is impossible without []. On the other hand for topological considerations is [] equal to ().

For example (1)=[1], with F(x)=|x|-R. The associated (hollow) shape is two points. And the associated filled shape is a line segment.

Or more generally (F)=[F], so our above ([AB]) would be equal to [[AB]]. So (F) would be be hm, a pair of two equal shapes, more cant probably said because the shape of F does not determine F itself, the shape is only the zero set, so we dont know how the F=R and -F=R looks exactly.

So each RNS formula is indeed a family of shapes i.e. for each constant R the set F=R. Which then of course can again regarded as a shape in n+1 dimensions, which then looks interestingly like a cone. And we have the much discussed transformation from the shape (at t=R) to a point (t=0), with F=t, t>=0.

Split from "RNS and product notations" by Rob.
Last edited by bo198214 on Mon Aug 07, 2006 12:18 pm, edited 1 time in total.
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Postby Keiji » Mon Aug 07, 2006 12:17 pm

I don't understand any of that. :?
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Postby bo198214 » Mon Aug 07, 2006 12:22 pm

aehm, can you be a bit more specific?
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Postby Keiji » Mon Aug 07, 2006 12:29 pm

Maybe you could give some examples?
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Postby bo198214 » Mon Aug 07, 2006 4:36 pm

PWrong wrote:We used to use square brackets for RNS. They turned out to be redundant.

Its definitely not redundant. So how do you then write ([11][11]) in normal RNS notation? (This should be topologically equivalent to the duocylinder.)
Otherwise the notation is ambigous.

We can directly convert RNS to cartesian equations, so it's definately not ambiguous.

Its definitely ambigous. In this notation is [[11][11]] the same as [1111]. Da lachen ja die Hühner! (translated: There are laughing even the chickens).
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Postby Keiji » Mon Aug 07, 2006 4:42 pm

bo198214 wrote:
PWrong wrote:We used to use square brackets for RNS. They turned out to be redundant.

Its definitely not redundant. So how do you then write ([11][11]) in normal RNS notation? (This should be topologically equivalent to the duocylinder.)


(xy)(zw) :roll:
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Postby bo198214 » Mon Aug 07, 2006 4:44 pm

ja topologically equivalent does not mean the same.

Or is novadays (111) the same as 111? :roll:
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Postby Keiji » Mon Aug 07, 2006 5:33 pm

I wrote:Maybe you could give some examples?
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Postby bo198214 » Mon Aug 07, 2006 6:49 pm

Rob wrote:Maybe you could give some examples?


Hm, yes. I dont know what elementary functions povray has (especially whether it has the maximum function) but at least the similar lozenge norm should possible which needs only absolute value and additon.
So let us take the corresponding form of a torus, but rectangular:
[[11]1]
This implies the surface equation:
r(r(x,y)-r1,z)-r2 = 0
with lozenge norm:
||x|+|y|-r1|+|z|-r2=0
or with the rectangular norm
max(|max(|x|,|y|)-r1|,|z|)-r2=0

Have a look at the pictures (r1=20, r2=10) and you immediately get, what is meant and you can play a bit for your own with povray and such equations:
ImageImage
ImageImage

<s>PS: we should split this thread
1. bracket discussion
2. parametric and product description
3. tiger discussion

though we may omit 3. because there is already a tiger thread</s>
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Postby PWrong » Tue Aug 08, 2006 5:25 am

Those aren't rotopes. They're ugly square torii. What you're proposing isn't a replacement for RNS, it's a whole new set of shapes. Could you give a list of objects with names and your notation?
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Postby bo198214 » Tue Aug 08, 2006 10:21 am

Yes 111 isnt a rotope either.
But if we introduce rectangluar shapes then we should use the bracket.

1 is simply a variable its neither two points nor an interval.
(1) is two points or an interval by definition.

So if we write 111 I ask myself why I have to interpret it here as (1)x(1)x(1) where in (111) I have to interpret it completely different? This was already misleading to Rob, that the shape between the parentheses has to do with the shape including the parenthesis. So we see this notation is quite misleading.

If we speak about rotopes then they must have at least a pair parentheses around them.
If we speak about crossproduct then we should use it in the product notation.
If we speak about rectangular forms then they must at least one pair of [] have around.

No I dont give a list of names. We can call them simply rectangular or lozenge xyz. Especially the square, cube and tesseract is simply a rectangular circle, sphere and glome.
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Postby PWrong » Tue Aug 08, 2006 10:40 am

Yes 111 isnt a rotope either.
What do you mean? Of course 111 is a rotope, it's a cube. Your square torus has never been suggested as a rotope, and for good reason.

This was already misleading to Rob, that the shape between the parentheses has to do with the shape including the parenthesis. So we see this notation is quite misleading.

Lot's of notations are misleading. But this one works, it's simple, and it converts easily to cartesian/parametric equations.

We can call them simply rectangular or lozenge xyz.

What's a lozenge?
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Postby bo198214 » Tue Aug 08, 2006 11:25 am

Sorry PWrong but I have the impression that you generally have something against my suggestion here.

And if you would look independently and the RNS notation you would notice that exactly 111 or 1(11) does not convert to parametric equation.
This is another reason to use brackets, then we have an easy way to
convert [111] and [1(11)] to convert to parametric equations.

My suggestion was either
a) we allow rotopes only to be things with parentheses around
b) when we allow rectangular shapes (for example square, cube and tesseract - what have these todo with rotations?) for rotopes too, then we should use bracket notation.

And when we mean cartesian products then we should simply write the x. As is anyway currently already done in the product notation.

As for lozenge wikipedia is your best friend.
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Postby PWrong » Tue Aug 08, 2006 12:32 pm

Sorry PWrong but I have the impression that you generally have something against my suggestion here.

Well I don't fully understand it yet, but I happen to like our current notation (admittedly because Marek and I invented it).

a) we allow rotopes only to be things with parentheses around
b) when we allow rectangular shapes (for example square, cube and tesseract - what have these todo with rotations?) for rotopes too, then we should use bracket notation.

Rotopes is short for rotatopes, and it's the general name for anything formed by x and # products of k-spheres. I don't know why Marek chose that name.

And if you would look independently and the RNS notation you would notice that exactly 111 or 1(11) does not convert to parametric equation.


111:
x(t,u,v) = t
y(t,u,v) = u
z(t,u,v) = v

21:
x(r, theta, t) = r cos(theta)
y(r, theta, t) = r sin(theta)
z(r, theta, t) = t

This is another reason to use brackets, then we have an easy way to convert [111] and [1(11)] to convert to parametric equations.

How does that contain any more information than 111 and (11)1? You'll have to use better examples to show that there's a difference.
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Postby Keiji » Tue Aug 08, 2006 12:37 pm

PWrong wrote:Rotopes is short for rotatopes, and it's the general name for anything formed by x and # products of k-spheres. I don't know why Marek chose that name.


Rotopes include rotatopes, toratopes and tapertopes. It's not really a shortened version of rotatopes, because rotatopes are just a subset of rotopes...
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Postby bo198214 » Tue Aug 08, 2006 12:53 pm

With parametric equation I meant the usual one derived by the parenthesis expression. I.e. (A<sub>1</sub>...A<sub>n</sub>) is the parametric equation sqrt(A<sub>1</sub><sup>2</sup>+...+A<sub>n</sub><sup>2</sup>) - R = 0.

So if we would restrict the rotatopes to something that has a parenthesis around it, then we could derive the parametric equation for each RNS expression by one rule. (If we dont, we need at least an additional rule for 111 which is then though nothing more than the rule for [111].)

If we introduce the brackets we can also derive the parametric equation by this rule but only substituting the spheric norm, by the rectangular norm. So we had a quite consistent rule to derive the parametric equation for each RNS expression (including rectangular shapes).

You'll have to use better examples to show that there's a difference.


This were exactly the examples, that showed the difference.

But to be honest I am bit tired of doing that stuff. I think there are some frictions in your system that could be easily fixed, but if you find your system completly sound, then keep it.
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Postby PWrong » Tue Aug 08, 2006 1:11 pm

With parametric equation I meant the usual one derived by the parenthesis expression. I.e. (A1...An) is the parametric equation sqrt(A12+...+An2) - R = 0.

That's a cartesian equation or an implicit equation. Parametric equations have parameters.

111:
|x|< a
|y|< b
|z|< c

21:
x^2 + y^2 < r^2
|z| < a

I posted the rules for converting RNS to cartesian somewhere else.

(If we dont, we need at least an additional rule for 111 which is then though nothing more than the rule for [111].)

We already have that rule. The rule for [111] is nothing more than the rule for 111.
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Postby bo198214 » Tue Aug 08, 2006 1:35 pm

PWrong wrote:That's a cartesian equation or an implicit equation. Parametric equations have parameters.

ok, then replace everywhere I said parametric equation by implicit equation.
We already have that rule. The rule for [111] is nothing more than the rule for 111.


So do you also the rectangular norm there? I mean if this rule anyway already exists I dont see, why not applicate it also to inner brackets.

But as I said, if you dont want, best arguments wouldnt help either.
At least I have introduced the brackets. So anybody would understand when I use them.
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Postby Marek14 » Tue Aug 08, 2006 2:23 pm

PWrong wrote:
Sorry PWrong but I have the impression that you generally have something against my suggestion here.

Well I don't fully understand it yet, but I happen to like our current notation (admittedly because Marek and I invented it).

a) we allow rotopes only to be things with parentheses around
b) when we allow rectangular shapes (for example square, cube and tesseract - what have these todo with rotations?) for rotopes too, then we should use bracket notation.

Rotopes is short for rotatopes, and it's the general name for anything formed by x and # products of k-spheres. I don't know why Marek chose that name.


Did I? I'm not sure, but I might be the one to come with "rotopes" - although my version didn't include tapertopes which only started to appear later.
I'm only absolutely sure that it was me who invented the term "tiger", which I am glad to see universally accepted here :)

Too bad that my other idea to expand on rotatopes - the "graphotopes" disappeared in history :D
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Postby PWrong » Tue Aug 08, 2006 2:56 pm

Too bad that my other idea to expand on rotatopes - the "graphotopes" disappeared in history

Yes, although the crind is still on the wiki. And that first extension of the notation may have inspired everything else we've achieved.

I'm only absolutely sure that it was me who invented the term "tiger", which I am glad to see universally accepted here

I'm glad of that too. If you hadn't come up with that name, we'd have to call it "toraduocylinder", which is quite possibly the ugliest name in tetraspace.
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Postby Marek14 » Tue Aug 08, 2006 3:02 pm

PWrong wrote:
Too bad that my other idea to expand on rotatopes - the "graphotopes" disappeared in history

Yes, although the crind is still on the wiki. And that first extension of the notation may have inspired everything else we've achieved.

Funny that. I never heard of "crind" before people here told me - I came up with that shape on my own.

Actually, that was about a year ago now - I first found this forum around that time.
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Postby bo198214 » Tue Aug 08, 2006 3:06 pm

PWrong wrote:Rotopes is short for rotatopes, and it's the general name for anything formed by x and # products of k-spheres. I don't know why Marek chose that name.


If we take only take x and # products of k-spheres then we never get square, cube and tesseract. The only edgy thing we get is the 0-sphere which are 2 points and its cross products. So even this definition shows that they dont fit into rotatopes.
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Postby Marek14 » Tue Aug 08, 2006 3:58 pm

bo198214 wrote:
PWrong wrote:Rotopes is short for rotatopes, and it's the general name for anything formed by x and # products of k-spheres. I don't know why Marek chose that name.


If we take only take x and # products of k-spheres then we never get square, cube and tesseract. The only edgy thing we get is the 0-sphere which are 2 points and its cross products. So even this definition shows that they dont fit into rotatopes.


I think that the main problem is that we are inconsistent in whether we use "filled" or "hollow" spheres.

Allow me to elaborate. In spheration, the first term is always "hollow" - torus is spheration based on a circle, but NOT a disc. But the second term could be either filled or hollow - the resulting shape will look the same, it's just the distinction of torus surface vs. filled torus. The default mode for spheration is "hollow".

The thing is that in x product, we usually take "filled" as the default mode. (I just came up with a new word: "Defaulty" - meaning standard AND wrong :D) Cylinder, in hollow mode, it would be just pair of circles. While that's valid representation, we don't THINK of cylinder like that. We see cylinder as a solid, and that (and even thinking of it as a surface) requires using "filled" spheres (and filled 1D sphere is a line).

So, we see both 2x2 (duocylinder) and (2x2)#2 (tiger) as 4D solids - despite the fact that the 2x2 means something different in each of these cases to make it so - four-dimensional solid for duocylinder, and diframe for tiger.

So I think that this is the basic discrepancy - if you only use "hollow" spheres, it's of course correct to exclude cube, but then you have to exclude cylinder, too, or replace it with two circles. I propose to use "filled" spheres as a default mode, with spheration meaning spherating the "skin" or parent shape.
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Postby Keiji » Tue Aug 08, 2006 4:31 pm

By using CSG Notation we can avoid this :P
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Postby bo198214 » Tue Aug 08, 2006 5:15 pm

Let me explain my view:
We have the implicit equations gained from the RNS notation. If we set F=0 we get the hollow object and if we set F<=0 we get the filled object.

The transformation (A<sub>1</sub>...A<sub>n</sub>1...1) -> (A<sub>1</sub>x...xA<sub>n</sub>) # S<sub>n+d-1</sub> works only for the hollow interpretation. I mean there is no discussion about the cartesian product whether hollow or solid, its simply the cartesian product.

So I think that this is the basic discrepancy - if you only use "hollow" spheres, it's of course correct to exclude cube, but then you have to exclude cylinder, too

Yes, I would do so, I would only admit parenthesized RNS expression for rotatopes. For this class there is *one* rule to get the implicit equation, it has no arbitrarity at all. And the implicit equations must be regarded as the hollow versions, i.e. F=0, if we want the conversion to the product notation being valid (I strongly would disadvise to alter the well-defined cartesian product into hollow and non-hollow versions. )

I propose to use "filled" spheres as a default mode, with spheration meaning spherating the "skin" or parent shape.

Look, as long as we use RNS notation there is no problem, we interpret each implicit equation either =0 or <=0, so its only a matter of interpretation. But for the product notation there is no interpretation of the cartesian product, and no interpretation of the #.

So after repeating 3 times my point, of course the question remains how to obtain the cylinder for example. As far as I can see even the brackets dont help with this.

The implicit description of the cylinder is quite mixed what regards F=0 and F<=0:
(sqrt(x^2+y^2)=r and |z|<=r) or (sqrt(x^2+y^2)<=r and |z|=r)
I wonder whether we can make a new (another one!) product of it.
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Postby Marek14 » Tue Aug 08, 2006 6:18 pm

bo198214 wrote:The implicit description of the cylinder is quite mixed what regards F=0 and F<=0:
(sqrt(x^2+y^2)=r and |z|<=r) or (sqrt(x^2+y^2)<=r and |z|=r)
I wonder whether we can make a new (another one!) product of it.


Well, actually, it can be stated in a much simpler way:

max(x^2+y^2,z^2)=r^2

This is a form i played with during my graphotope era :)
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Postby bo198214 » Tue Aug 08, 2006 6:24 pm

perhaps then you should take second try to roll out your graphotopes - never heard of them.
It looks as if it is then representable with () and [] but some radius 0.

Yes, it is [(xy)z] , which is originally a torus (with one circle rectangular) with the implicit equation
max(|sqrt(x^2+y^2)-r1|,|z|)-r2 = 0
let now r1=0
max(sqrt(x^2+y^2),|z|)=r2
max(x^2+y^2,z^2)=r2^2

I.e. we contract the torus to the cylinder, the following picture shows the torus [(xy)z]:
Image

That means we could add a 0 notificator, applicable to parentheses and brackets, which means that the corresponding radius of the implicit equation is set to 0. For example
(1)<sub>0</sub> means a single point, where (1) means two points.
And I would then write the cylinder as [(xy)<sub>0</sub>,z].
Another example: ((xy)z)<sub>0</sub> would be degenerated to a circle. It looks as if this 0 indicator makes only real sense (i.e. does not degenerate to an already known rotatope) if applied inside.
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Postby Marek14 » Tue Aug 08, 2006 7:15 pm

bo198214 wrote:perhaps then you should take second try to roll out your graphotopes - never heard of them.


Try this old thread, then:

http://tetraspace.alkaline.org/forum/vi ... .php?t=350
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Postby moonlord » Tue Aug 08, 2006 7:28 pm

bo198214 wrote:(sqrt(x^2+y^2)=r and |z|<=r) or (sqrt(x^2+y^2)<=r and |z|=r)


That's why I coined SSET.

And by the way, there are names for the xD version of a body: nullframe, monoframe, diframe, triframe........
"God does not play dice." -- Albert Einstein, early 1900's.
"Not only does God play dice, but... he sometimes throws them where we cannot see them." -- Stephen Hawking, late 1900's.
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Postby bo198214 » Tue Aug 08, 2006 7:38 pm

After having a glance through it, I would say cool thing!
Are there still questions open?

I mean we have now really good expressivity.
We have the RNS products (), [], <>. With the 0 notificator we have also all the original Wendy's products (sorry but dont know how to call them). And with them all your graphotopes can be realized (am I right with this assumption?)
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