## Naming and notations

Discussion of shapes with curves and holes in various dimensions.
Admittetly the result of both A x B and B x A maybe geometrical the same object (btw A and B are by definition in a different subspace). At least its then not the triframe duocylinder, which is the union of two such geometrical identical objects.

Ok, I see what you mean. AxB is a rotated version of BxA.

So we have:
disk x circle = disk(x,y) x circle(z,w)
circle x disk = circle(x,y) x disk(z,w)

PWrong
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*nod* though I am currently not clear (and especially to lazy to verify) whether sometimes also mirroring must be involved to map AxB to BxA ...
bo198214
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well, I still don't understand the cartesian product thing. Maybe I'll look it up later on mathworld/wikipedia.

Do they contain a hole? Maybe we can answer this question by looking at the slices.

I understand that but I was talking about the tetraframe duocylinder. Is it any different?

From the slices on the rotatope pages on the mainsite, it doesn't look like it has a hole. But from the projections on that page I linked to before, it does look like it has a hole. That's just confusing.

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But from the projections on that page I linked to before, it does look like it has a hole.

I think the projections on that page are of the diframe duocylinder

PWrong
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Marek14 wrote:Do they contain a hole?

We can put a plane through the middle of duocylinder which will show in our slicing as an unmoving line through the centers of cylinder's circles. So this means there's a plane which doesn't pass through one specific tricylinder, but cannot be "pulled out" without intersecting it, which means that there is a hole.

What do you mean by hole?
The triframe duocylinder is the surface of the tetraframe duocylinder. The tetraframe duocylinder is convex as all cartesian products of convex sets are. Thatswhy the (triframe)surface is hollow but has no holes.

The axis line you talk of (the intersection of your plane with 3d space) does intersect the ends of the cylinder which belong to the projection of the triframe duocylinder to 3d.
bo198214
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bo198214 wrote:
Marek14 wrote:Do they contain a hole?

We can put a plane through the middle of duocylinder which will show in our slicing as an unmoving line through the centers of cylinder's circles. So this means there's a plane which doesn't pass through one specific tricylinder, but cannot be "pulled out" without intersecting it, which means that there is a hole.

What do you mean by hole?
The triframe duocylinder is the surface of the tetraframe duocylinder. The tetraframe duocylinder is convex as all cartesian products of convex sets are. Thatswhy the (triframe)surface is hollow but has no holes.

The axis line you talk of (the intersection of your plane with 3d space) does intersect the ends of the cylinder which belong to the projection of the triframe duocylinder to 3d.

There was a confusion of the terms. I thought that "triframe duocylinder" is one of the two "surfaces" that the duocylinder has. If the name covers the whole surface, then there is no hole.

Tetraframe duocylinder doesn't dontain a hole for the simple reason that it's convex.

Of course, in this case, triframe duocylinder is neither disk x circle nor circle x disk, but a union of both.
Last edited by Marek14 on Wed Jun 07, 2006 10:39 am, edited 1 time in total.
Marek14
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yes, then I do agree.
bo198214
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So by the CSG extended notation, the triframe duoclinder would be (LLxLE)+(LExLL) ("x" cartesian product).

We hit the directions again. The first disk and the second circle must be in the same plane, and the other two (the second disk and the first circle) must be in the same plane, independent from the first plane. How do we express that within CSG?
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moonlord
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(xyLL*xyLE)+(zwLE*zwLL) (* = cartesian product)

That would work. It does say on the page on the wiki that you can use lowercase letters to denote dimensions, and the default order is xyzw.

(a, b) x (c, d) gives all possibilities for (a, b, c, d)

surely it would have to be
xyLL*zwLE?

eh, I dunno..

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Rob wrote:(xyLL*xyLE)+(zwLE*zwLL) (* = cartesian product)

That would work. It does say on the page on the wiki that you can use lowercase letters to denote dimensions, and the default order is xyzw.

(a, b) x (c, d) gives all possibilities for (a, b, c, d)

surely it would have to be
xyLL*zwLE?

eh, I dunno..

He meant it like xyLL*zwLE. He said that the SECOND disk and the FIRST circle lie in the same dimensions, meaning disk from second cartesian product and disk from first.
Marek14
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Ah, I see. So it becomes:

(xyLL*zwLE)+(xyLE*zwLL)

Edit: I decided to add the *, # and C operations to CSG notation. See the wiki for what C does (note that anything that can be made using C, can be made without using C, but using C makes it remarkably shorter in some cases).

Now, would someone like to explain how matrices would work for the translation operations?

Keiji

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Now, would someone like to explain how matrices would work for the translation operations?

Well, I first learnt about matrices in year twelve. It's useful and not too difficult, but it's not the most interesting subject in maths.

But if you really want to know, here's a good introduction to matrices on Wikipedia.
http://en.wikipedia.org/wiki/Matrix_(mathematics)
And here's a page on transformation matrices.
http://en.wikipedia.org/wiki/Transformation_matrix
I might add this. You can't translate an object by multiplying by a matrix. You just have to add a vector to it.

EDIT: Just realised my url tag isn't working. I think it doesn't like brackets :?.

PWrong
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PWrong wrote:I might add this. You can't translate an object by multiplying by a matrix. You just have to add a vector to it.

Typically, one uses an (n+1)-dimensional transformation matrix to transform n-dimensional vectors. You add an (n+1)-th component to the vector. You make that new component equal to one. Then, you can use the (n+1)-st column of the transformation matrix to translate the vector.

You can use the (n+1)-st row of the transformation matrix to represent scaling.

Take, for example, the usual operation of rotating a vector, translating it, then projecting it to screen coordinates (x",y"). In equation form, this might look like:

x' = a1 * x + b1 * y + c1 * z + d1
y' = a2 * x + b2 * y + c2 * z + d2
z' = a3 * z + b3 * y + c3 * z + d3

x" = x' / z'
y" = y' / z'

In matrix form, you'd make a 4x4 matrix with rows:

a1 b1 c1 d1
a2 b2 c2 d2
a3 b3 c3 d3
a3 b3 c3 d3

Then, you take a vector < x, y, z, 1 ><sup>T</sup>, multiply
it by that matrix. To get < x', y', z', w' ><sup>T</sup>.

x" = x' / w'
y" = y' / w'

Now, this may not look like much of an advantage, but if you
have to compose rotations and translations and such, this is
the way to go (for dimensions beyond three... otherwise there
are more stable ways to do this).
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So, what should the conclusion be? Use the rules that Rob used in his last post? Or just look something else?
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moonlord
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I don't know exactly what you meant by that, but the CSG definition of a duocylinder is indeed (xyLL*zwLE)+(xyLE*zwLL).

Keiji

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What I meant is whether we should apply this notation (to all the bodies) or look for a better one. It seems you agree with it, and I believe you won't be the only. So there you go (ordered by frame dimensionality).

0D
point: "null"

1D
segment: xL, xE

2D
square: xyLD, xyLE+yxLE, xyEE
circle/disk: n/a, xyLL, xyEL

3D
cube: xyzLDD, xyzEDD+yzxEDD+zxyEDD, xyzEED+yzxEED+zxyEED, xyzEEE
cylinder: n/a, xyzLLD, xyzELD+xyzLLE, xyzELE
sphere: n/a, n/a, xyzLLL, xyzELL
torus: n/a, n/a, xyLL#LL, xyLL#EL

New operators included: D - makes two copies of previous body (same as H, but does not connect the resulting two shapes). How about this? Is it alright? The reason I don't use the H is because I find it counter-intuitive. Same for the M operator.
"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.
moonlord
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No offense, but what you posted is a load of bull.

For one, you do not have to include the xyzw if they are in that order, because it is the default.

And secondly, why complicate things? Just use H already. It then becomes easy:

cube: HHH, EHH, EEH, EEE.

As for the M operator, it's being considered to change all the transformations into one matrix operation.

Keiji

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Sorry for the late reply, but this thread appeared as already visited from last post to me.

Well, first of all I used them (xyzw) to avoid any possible misunderstanding. Secondly, as I already stated, I find H counterintuitive. That is, I understand what EHH gives, but can't imagine crap out of (triframe duocylinder)H.

I need to take it this way. The triframe duocylinder, abreviated 3-22 can be constructed this way: (xyLL*zwEL)+(xyEL*zwLL). H should extrude, so I need to get a 5D bounding space. H keeps the same net space, so all this means I'll get the triframe duocylinder-prism. Now this is scary...

Anyway, it seems the sole reason we're arguing is that I want to keep as few operators as possible, preferably only ones that I can simply apply mentally to a body, and you want to keep the notations simple... I won't be able to use the H until I understand it well, and this is not the case yet. On the other hand, you know what I mean. I think You should try to use, temporarily at least, the simpler system...
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"Not only does God play dice, but... he sometimes throws them where we cannot see them." -- Stephen Hawking, late 1900's.
moonlord
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moonlord wrote:I need to take it this way. The triframe duocylinder, abreviated 3-22 can be constructed this way: (xyLL*zwEL)+(xyEL*zwLL). H should extrude, so I need to get a 5D bounding space. H keeps the same net space, so all this means I'll get the triframe duocylinder-prism. Now this is scary...

How's that scary? Triframe duocylinder-prism is a pretty ordinary object.

Anyway, it seems the sole reason we're arguing is that I want to keep as few operators as possible, preferably only ones that I can simply apply mentally to a body, and you want to keep the notations simple... I won't be able to use the H until I understand it well, and this is not the case yet. On the other hand, you know what I mean. I think You should try to use, temporarily at least, the simpler system...

What's simpler, xyzEDD+yzxEDD+zxyEDD, xyzEED+yzxEED+zxyEED or EEH?

I think EEH.

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I would have to agree with Rob.

It is not difficult to understand Rob's method... study this chart that appears in the beginning of the thread, and you'll get it soon enough:

Rob wrote:EL = Solid circle (disk)
HL = Hollow circle

ELL = Solid sphere (ball)
HLL = Hollow sphere

EEL = Solid cylinder
EHL = Hollow cylinder
HHL = Wireframe cylinder (two circles)

EE = Solid square
EH = Hollow square
HH = Marked square

EEE = Solid cube
EEH = Hollow cube
EHH = Wireframe cube
HHH = Marked cube

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See you got it no more than I did? H and L give the same thing when applied to a point.

Now really, tell me the properties of the triframe duocylinder-prism. I mean, I cannot comprehend the results of the H operator when in more than 4D. I'll keep struggling, though...
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moonlord
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What do you mean, properties?

Keiji

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moonlord wrote:See you got it no more than I did? H and L give the same thing when applied to a point.

Now really, tell me the properties of the triframe duocylinder-prism. I mean, I cannot comprehend the results of the H operator when in more than 4D. I'll keep struggling, though...

I understand it alot more than you do... I understand the concept of it. Just because it isn't being used practically (at least in this case, in my opinion) doesn't mean I don't get it.
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Nick
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Bah, it must be the exams... I think I sounded idiot earlier.

The idea is I cannot visualise everything that H gives, but took me three posts to say this rightly. I sometimes complicate uselessly, and it seems this is the case. Anyway, sorry for any inconvenience created.

By 'properties', I was reffering to things like hypervolumes, whether it can roll, what kind of holes does it have (if any) and so on. In fact, I ask whether you can visualise it, but again that took more than normal to express. Sorry again.
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"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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### Re: Naming and notations

Going back to something noted earlier (I forget who by, apologies), I think that wire-frame ought to be the basis of the nomenclature, as a 3d wire-frame is something that can easily be understood by an amateur.

In this instance a 'frame' could be defined a figure constructed from elements two dimensions below it - i.e. wire [3-2d] frame [3d]

So this could indeed give plane-frame, realm-frame and flune-frame for [2,4], [3,5] and [4,6] respectively.

IIRC, Wendy talks about 'lace' polyhedra on another thread, I wonder is this pertains to these sort of realm-frames and flune-frames?
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### Re: Naming and notations

Some terminology exists for frames, etc, but most of these are specific to particular problems.

One should note that the standard terminology is so far from common usage (by way of meaning shift), that it by itself inhibits understanding of the abstractions. Take for example "ridge". In polytopes, the faces are bounded by structures that look like ridges. But this is the nature of the particular classes of polytope studied, not the general nature of face-boundaries: the term remains valid because the area of study has never left the intersection of the meanings of mountions and of margin. However, in tilings, the connection of faces is no longer a mountion-range, but now is flat with the thing. In something like a bathroom tiling, the "ridge" is actually a valley :S.

It is also the fate of general terms to become the specific for unnamed. So while we have polytope = line, polygon, polyhedron, ...., and people are only looking at 4d, it is natural to slip polytope there. Coxeter uses =cell to refer to the facet-count of any polytope, not just the 4d ones. But with just 4d in mind, we have cell migrating from its usual meaning (think cellular automata, cells in games etc), via face of N-dimensions, to the unnamed 4d thing to the name of a 3d surtope. Norman Johnson even suggested "cellule" to bear its original meaning! (One sees the same with "face" / "facet"). Hyperspace for 4d, becomes the same sort of dimension that things turn around on in six dimensions!

Once you look at the present terminology from six dimensions, it does look totally stupid, haphazard etc. What I have done in the Polygloss is to restore the meaning of stems to something that still means what they do outside polytopes.

The Polygloss 'fabric' and 'patch' idiom bases on the "fabric of space", ie a manifold. Since fabric can be of any dimension (button, thread, cloth, clay), the fabric is formed as "hedrix" for 2d manifold. A patch is part of fabric, eg "hedron". You do things with patches of space, to make a "polyhedron", or an "apeirohedron". Thus, while "polyhedron" is "many 2d patches", /poly/ also implies a closure or completeness (cf multi), so it is taken with a content. One can hardly use 'surface' for hedrix in 4d, since a surface divides, and a hedrix (2d manifold), does not. This is why, for example, you can render a klein bottle or whatever, without crossing. You can knot 2d surfaces in 4d and 5d, but only weave them in 4d.

A frame is a unclosed polytope (multitope) with all of the surtopes up to the named dimension. So a "latroframe", is a frame with vertices + edges marked. A "hedroframe" has also the 2d elements marked. &c. The process of adding higher surtopes to close the polytope is 'inno-analysis".

George Olshevsky uses the terms based on an army, which is similar to the frame-concept, except that it requires a leader to make the frame. For example, an Icosahedron can be a "general" of an "army", consisting of the various polyhedra that are inscribed in the 12 vertices of the icosahedron (teeloframe), or the "colonel" of a regiment that have the 12 vertices + 30 edges of the icosahedron (ie latroframe). This is the divide and conquerer method used by Jonathon Bowers to derive the uniform starry polychora. However, ye see that it supposes the existance of rank-holders, whereas latroframe etc do not.

The notion of 'spheration', first appeared here, in order to describe what happens when the vertices and edges are replaced by spheres and pipes. One might want to spherate the vertices to make them larger than the points on the line, for example, is still covered under spheration, since this is done separately to classes of surtope (eg vertices, edges, ...).

An other method is the form used by da Vinci, where the latroframe is shown as a wood frame, ready for the application of faces. It is as if the faces were cut out of a solid box. At the moment, a name has not been allocated here.

In regard to spheres, etc.

The notation I use for spheres is to treat them as a polytope in O, eg circle, {O), sphere (O,O), glome (O,O,O), and so forth. Placing semicolons in the midst makes way for the assorted ellipsoids (by way of increasing diagonal), so (;O,O;0} corresponds to rss(x,x,y,y), where x<y. One notes that this is not the duocylinder (which is (;O&;O)).

The distinction between surface (fabric) and content (disk), is made by noting in the former, the surface is the thing (glomohedrix), while in the latter, the surface is a patch that bounds something (eg glomohedron).

Extracting a subset of surtopes is a separate function, or can be done by the products of surfaces and solids.

eg bi-glomolatral prism = disk(2) * disk(2); glomolatral glomolatric prism disk(2) * ring(2) ; bi-glomolatric prism = ring(2) * ring(2).
GLX *# GLX (circle=surface * surface sphere Prism).

I have, for example, described the space of great arrows as a bi-glomohedric prism = ring(3)*ring(3).

Note that as yet i have no distinction between glomohedrix (E2) as a space, vs glomohedrix (sphere-surface), as say, a ring in 4d.
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