Keiji wrote:On the brick product:
When P is a convex polytope, you can simply split the vertices of P into those of a sequence of hypercuboids (each centred at the origin), make Cartesian products of the operands scaled by the edge lengths of those hypercuboids, and then take the convex hull to get the result of the brick product.
This is not new information, however it is important to realize that the brick product is still well defined even if P isn't convex or if it has curved elements, but using the "convex hull" technique described above doesn't work in the general case.
quickfur wrote:OK, so if P is a square, then the brick product is the same as the Cartesian product?
quickfur wrote:Well, the first thing is to understand what happens when P is, say, a diamond instead of a square. Hopefully that will help me understand what exactly the brick product does with the points in P, which will hopefully lead to a workable definition of the brick product for general P.
Keiji wrote:[...]
For a diamond you would simply orient one copy of each operand in perpendicular spaces and take the convex hull; that's a special case of the convex hull method since the diamond's vertices can be split into two perpendicular digons centred at the origin.
quickfur wrote:Hmm. I'm starting to get the impression that the brick product doesn't really do anything with P as a set of points, but rather with the way P was constructed.
This is the impression I'm getting from what you wrote above, and also what you said before about how a torus can be factored in 2 different ways, each of which produces a different brick product.
On another note, the standard constructions for the icosahedron/dodecahedron, if I'm not mistaken, have brick symmetry (the coordinates are taken as all changes of sign over some permutation of coordinates). What then does the brick product correspond with if P is, say, an icosahedron?
Keiji wrote:Great work! It took me a while to understand it all, but everything in your post so far is consistent with my understanding of the brick product.
[...] Regardless, I do very much like your second definition, it is a big step up from the current understanding which could deal with only convex shapes through the convex hull method.
Keiji wrote:Actually, now that I think about it some more, that one's solved too: if you think of the self-intersecting shape as being its surface, one dimension lower, and use your second definition, you'd get the correct result. Mainly my mistake here for treating bounding space as net space, which it is not.
Keiji wrote:Just so you know, I reverted your "improved" notation because it's rather bad form (not to mention confusing) to say that a vertices coordinates are half of its representative vector's values, and similarly, if we are treating a shape A as a set, the notation xA where x is a scalar ought to refer to some operation that doubles the cardinality of A, which scaling does not.
Keiji wrote:Hmm, I think it looks fine since it's rendered smaller than the bar used in set construction. I can't think of any better notation to use for it.
cprod(∅) = {()}
cprod(A ⋃ {a}) = cprod(A) × a
Keiji wrote:Before I read the rest of that post, I'd just like to say something about the point vs empty set topic. I know full well the Cartesian product of no operands is the point, however, you must then take the surface of that Cartesian product.
The neighborhood of the sole point in a point polytope is only going to contain the point itself. Therefore, it doesn't contain a neighbor not in the polytope, therefore, there are no points in the surface of the point, therefore, S(Point) = empty set (or null polytope).
Similarly, S(S(X)) would be the empty set for any X, since S(X) would always be some manifold with no boundaries. Which means S is not a function that decreases the net space by one: it only does that the first time it's applied and if applied twice, yields the null polytope. Whether we want to change the definition of S to suit what I wrote about it, or whether we just use the current S and correct that note in the article, is something to think about.
Edit: Alright, I corrected two of the cases you listed, other than that it's all correct but stuff I already knew since the results were almost all 3D or lower. The higher dimensional ones will be much more interesting I'm sure
Keiji wrote:As I understand it, P'{square, digon} should look like this:
I think P'{square, square} should have four squares oriented in zw separated in xy, all tapered to the origin, forming four square pyramids joined at their apexes.
This is just going by my intuition, though; any chance the definition can be amended to produce these figures?
I don't like the idea of a bag union, though... it's an unnecessary overcomplication.
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