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
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
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.