SSC3 discussion

Higher-dimensional geometry (previously "Polyshapes").

SSC3 discussion

Postby Keiji » Fri Mar 26, 2010 11:57 pm

It's been a while since I last worked on polytopes. Here are my latest thoughts on how to properly define a new version of SSC - which would be SSC3.

I shall define in three parts:
1. Bricks. The set of all bricks shall be denoted VB.
2. Standard convex polytopes (SCPs). The set of all SCPs shall be denoted VC.
3. Standard boundaries (SBs). The set of all SBs shall be denoted VS.
The set of all shapes within a certain other set V and of a certain dimension n shall be denoted Vn. Thus V = ∪{Vn|n ∈ ℕ ∪ {0}}.

I also reference the sets of tapertopes and toratopes (VTa and VTo) in my definition of standard boundaries, though we all know they are by now ;)

For clarity, ℕ refers to the set of natural numbers excluding zero.

We can now follow these rules:

Bricks
1. The following special cases are bricks: point, digon, dodecahedron, hecatonicosachoron (it is currently unknown whether the grand antiprism is also a brick)
-- {Pt, Dg, Ki1, Ks1} ⊆ VB
2. Any regular polygon with an even number of sides is a brick.
-- Gk ∈ VBk/2 ∈ ℕ ∧ k ≥ 3
3. The various truncations of a brick are bricks.
-- A Dx x ∈ VBA ∈ VBnx ∈ ℕ ∧ x < 2n
4. The brick product of a sequence (of correct order) of bricks is a brick.
-- ◊B<A1, A2, A3, ..., An> ∈ VBB ∈ VBn ∧ ∀iAi ∈ VB

SCPs
1. Any brick is an SCP.
-- A ∈ VBA ∈ VC --equivalently-- VB ⊆ VC
2. Any regular polygon is an SCP.
-- Gk ∈ VCk ∈ ℕ ∧ k ≥ 3
3. The following special cases are SCPs: snub dodecahedron, snub icositetrachoron, grand antiprism
-- {Ki0, Kk0, GAP} ⊆ VC
4. The various truncations of an SCP are SCPs. (like brick rule #3)
-- A Dx x ∈ VCA ∈ VCnx ∈ ℕ ∧ x < 2n
5. The pyramid of an SCP is an SCP.
-- &A ∈ VCA ∈ VC
6. The prismatoid product of a sequence of SCPs (of common dimensionality) is an SCP.
-- &<A1, A2, A3, ..., Ak> ∈ VC ↔ ∀iAi ∈ VCn
7. The brick product of a sequence (of correct order) of SCPs is an SCP. (like brick rule #4)
-- ◊B<A1, A2, A3, ..., An> ∈ VCB ∈ VBn ∧ ∀iAi ∈ VC

SBs
1. Any SCP is an SB.
-- A ∈ VCA ∈ VS --equivalently-- VC ⊆ VS
2. Any tapertope is an SB.
-- A ∈ VTaA ∈ VS --equivalently-- VTa ⊆ VS
3. Any toratope is an SB.
-- A ∈ VToA ∈ VS --equivalently-- VTo ⊆ VS
4. The brick product of a sequence (of correct order) of SBs is an SB. (like brick rule #4)
-- ◊B<A1, A2, A3, ..., An> ∈ VSB ∈ VBn ∧ ∀iAi ∈ VS

The set of standard boundaries contains every shape definable in SSC3. SSC3, like SSC2, does not care about deformations, angles, and so forth, so a rectangle is equivalent to a square, etc. You will however notice that since you can no longer take the pyramid of a toratope, ambiguous rotopes do not crop up.

Discuss.
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Re: SSC3 discussion

Postby anderscolingustafson » Sun Mar 28, 2010 3:14 am

Is a polytope the same thing as a 4d polygon, and what exactly does SSC stand for.
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Re: SSC3 discussion

Postby Keiji » Sun Mar 28, 2010 12:59 pm

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Re: SSC3 discussion

Postby PWrong » Fri Apr 02, 2010 8:01 am

Sorry, I don't remember much of this. I'll read it carefully and try to learn everything at some point, but I should be reading stuff for my PhD. It looks interesting though, and I can see it took a lot of work. I have a couple of questions and comments.

1. I'm not sure what's included in this scheme and what's not. It seems like trying to classify too many shapes at once, particularly shapes that have nothing to do with each other. I mean I can understand classifying shapes like polytopes, or toratopes, or cone-like objects separately. But it seems unnecessary to have a unifying scheme that includes the hecatonicosachoron, toraspherinder and a hexagonal pyramid, but excludes, say, a parabola. There's already a kind of "classification scheme" that covers every possible shape in any dimension, it's called equations.

2. How come a pentagon isn't a brick? Is there a simpler definition of a brick that these rules derive from? The "brick symmetry" definition would cover some very weird shapes. For example, draw any squiggly line in the first quadrant from (1,0) to (0,1) and reflect into the other quadrants appropriately, and that's a brick.

3. The notation is a bit difficult to follow, and probably even more so for forum members without a maths degree.
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Re: SSC3 discussion

Postby Keiji » Fri Apr 02, 2010 10:25 am

PWrong wrote:Sorry, I don't remember much of this. I'll read it carefully and try to learn everything at some point, but I should be reading stuff for my PhD.


Ah, my apologies :oops:

1. I'm not sure what's included in this scheme and what's not.


Well, anything that isn't explicitly allowed by the rules above isn't included. There are two reasons I didn't specify that in the first post:
- Writing it mathematically would be rather difficult and would clutter the existing rules.
- I might have left out something important which I would want to add at a later date.

It seems like trying to classify too many shapes at once, particularly shapes that have nothing to do with each other.


The main reason for this is I've never liked the idea that there is no "nice" set that contains all the uniform polytopes but also contains the tapertopes and arbitrary brick products etc. Ever since the CSGN (perhaps we should call that SSC version 0? ;) ) days, I've wanted to come up with a way to represent all the "interesting" shapes, which would naturally include pairs of completely unrelated shapes. And I've wanted to do it in such a way that it requires as few fundamental operations as possible.

But it seems unnecessary to have a unifying scheme that includes the hecatonicosachoron, toraspherinder and a hexagonal pyramid, but excludes, say, a parabola.


Well the simple reason for that is because the parabola goes off to infinity.

There's already a kind of "classification scheme" that covers every possible shape in any dimension, it's called equations.


Equations work great for surfaces without cusps. And they work okay for bracketopes. But what if I challenged you to find an equation for, say, the pyramid of the hexagonal brick product of a circle and a dodecahedron? Even if you could, it'd undoubtedly look a complete mess, a mess which says nothing about the actual structure of that 6D shape.

How come a pentagon isn't a brick?


Bricks have to have brick symmetry. You can't draw a non-degenerate pentagon such that point (x,y) implies point (+/-x, +/-y). Same argument for all the other polygons with an odd number of sides.

Is there a simpler definition of a brick that these rules derive from? The "brick symmetry" definition would cover some very weird shapes. For example, draw any squiggly line in the first quadrant from (1,0) to (0,1) and reflect into the other quadrants appropriately, and that's a brick.


Bricks - under this definition - cannot be curved. Yes, I did just realize I made a horrible omission which means that the crind isn't a standard boundary when it should be, and I will be fixing this shortly, so thank you for pointing that out. However, there will still be the set of non-curved bricks, which will obviously be used to construct the SCPs (as SCPs cannot be curved either).

In any case, assuming that you instead want to create a brick from an arbitrary path of straight lines (reflecting that appropriately), that would still not be a brick. SSC3 doesn't care about deformation, and because of that such a brick formed from reflecting an arbitrary path of k straight lines is completely equivalent to a regular 4k-gon.

3. The notation is a bit difficult to follow, and probably even more so for forum members without a maths degree.


That's why I included both plain English and strict mathematical forms of every rule. I wanted to use notation that is generally accepted and does not have any ambiguity. Even if someone might not understand it, they can still follow the "plain English" versions of the rules, and I can look at the notation to remind myself exactly what is and what isn't included.

Edit: I also just realized this is in the wrong forum, so I moved it.
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