Cartesian Products of simple stuff

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Cartesian Products of simple stuff

Postby kingmaz » Fri Jan 09, 2009 10:35 am

As a start, I'll underline that I don't understand the mechanism of the Cartesian Product.

However, if the Cartesian Product of two circles is the duocylinder, does that mean the Cartesian Product of two triangles is a duopyramid?
Is this the same as Rotope 55?

Moving on from that, what is Cartesian Product of two pyramids called (Rotope 188?)?

I suppose the question, I'm leading towards is:
Do the 'Cartesian Squares' (in effect) follow a regular pattern or do each have to be considered on an individual basis?

This would lead onto a question regarding the use of indices/powers in generating higher dimensional figures, but I'll leave that for the moment... :\
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Re: Cartesian Products of simple stuff

Postby wendy » Fri Jan 09, 2009 11:16 am

The cartesian product is a feature of euclidean geometry. You can apply it by drawing shapes in different sets of axies, and then taking the common set.

For a hexagonal prism, you could cut a length from a hexagonal bar. This gives a prism as an off-cut (prisma means offcut!). Alternately, you could take a slab of constant thickness, and cut a pentagon from it. In either way, all space -> layer -> pentagonal prism.

In four dimensions, there are four axies, and so prisms can be cut from 2+2 axies, or 3+1. The former gives for example, polygon-polygon "duoprisms", such as the triangle-triangle (duo)prism. Circles behave as polygons, and a special name is used for the bi-circular prism (duocylinder), rather like a special name is used for the circular prism in 3d (cylinder).

"Duopyramid" is confusing in four dimensions, since you can erect a pyramid on a pyramidal base, or (as in 3d), erect a pyramid on each side of a 3d base. In any case, the term is depreciated. The figure here has no special name over bi-triangular (duo)prism.

Pyramids do not occur in the rotopes.

You can indeed use powers to generate higher polytopes. The products are all discovered in just this way.

PRISM => line ^n = line, square, cube, tesseract, ...
TEGUM => line ^n = line, rhombus, octahedron, 16ch, ...
CRIND => line ^n = line, circle, sphere, glome, ...
PYRAMID => point ^(n+1) = point, line, triangle, tetrahedron, pentachoron, ...
COMB = horogon ^n-1 = horogon, square lattice, cubic lattice, tesseract lattice, ...
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Re: Cartesian Products of simple stuff

Postby Keiji » Fri Jan 09, 2009 1:07 pm

kingmaz wrote:As a start, I'll underline that I don't understand the mechanism of the Cartesian Product.

However, if the Cartesian Product of two circles is the duocylinder, does that mean the Cartesian Product of two triangles is a duopyramid?


The cartesian product of two triangles is a 3,3-duoprism. There is no such thing as a duopyramid.

Is this the same as Rotope 55?


Rotope 55 is 5D (a 3,3-duoprism is 4D). R55 is a square pyramid prism pyramid which has nothing to do with a duoprism.

Moving on from that, what is Cartesian Product of two pyramids called (Rotope 188?)?


See my response to the next quote. Rotope 188 is not the cartesian product of two pyramids, it's a square pyramid diprism pyramid.

In general superscripts in rotopic group notation DO NOT imply Cartesian products; you seem to be assuming they do.

I suggest staying away from rotopic group notation as as I've said many times it's a huge mess. Use SSC2 instead.

Do the 'Cartesian Squares' (in effect) follow a regular pattern or do each have to be considered on an individual basis?


Yes, cartesian squares are indeed powertopes. The cartesian product of X and X can be called the Xar tetragoltriate. So your cartesian product of two pyramids (I'll assume square pyramids) would be called a square pyramidal tetragoltriate. Note that a duocylinder could be called a circular tetragoltriate and a tesseract could be called a square tetragoltriate.

This would lead onto a question regarding the use of indices/powers in generating higher dimensional figures, but I'll leave that for the moment... :\


This has already been looked into. See Powertopes.
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Re: Cartesian Products of simple stuff

Postby kingmaz » Sun Jan 11, 2009 2:26 pm

Thanks for the replies both. I do appreciate you taking the time to correct me. I'm very glad that the things that crop up in my mind have been dealt with in advance and you can explain where I'm going wrong in certain thoughts and assumptions. Still it's early days for me in this subject.

I'll take the point that rotopic notation is outdated though initially quite easy looking, it's just that SSC2 seems a bit like german - difficult to get used to the rules, but it's very regular and straightforward once you're there.
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Re: Cartesian Products of simple stuff

Postby kingmaz » Fri Jan 16, 2009 1:52 pm

It's bad form to reply to oneself, but...

The confusion over Rotope 55 arose due to me reading 11'1' as 1'1', apologies.

"Rotope 188 is not the cartesian product of two pyramids, it's a square pyramid diprism pyramid."
I was trying to use duopyramid for the fact that 11' followed by 11' appeared to be doing the same thing twice, rather than confusing ' with indices.
For a very easy looking operation 11'11' (yes I should use SSC2, I know) this seems a very long name, I do prefer pyramidal tetragoltriate.
Has an alternative naming system been proposed for successive tapered and linear extrusions?

(I bet Wendy's thought about it :) )
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Re: Cartesian Products of simple stuff

Postby Keiji » Fri Jan 16, 2009 4:07 pm

kingmaz wrote:For a very easy looking operation 11'11' (yes I should use SSC2, I know) this seems a very long name, I do prefer pyramidal tetragoltriate.


11'11' is NOT a pyramidal tetragoltriate - it's NOT the cartesian product of two pyramids!

11' is &G4 in SSC2, and the pyramidal tetragoltriate i.e. cartesian product of two pyramids is [&G4]x[&G4].

11'11' in SSC2 is &[[&G4]xG4], or &++&G4.

Concatenation in rotopic group notation does NOT necessarily mean cartesian product, where tapering is involved.

I would like to come up with a systematic naming scheme for these alternating taper/extrude sequences as it is very long winded to say p-pyramid q-prism r-pyramid s-prism and so on.
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Re: Cartesian Products of simple stuff

Postby kingmaz » Mon Jan 19, 2009 9:58 am

Hayate wrote:11'11' is NOT a pyramidal tetragoltriate - it's NOT the cartesian product of two pyramids!

Concatenation in rotopic group notation does NOT necessarily mean cartesian product, where tapering is involved.


Yes, you did say that once - sorry. :oops:
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