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 V∟n. Thus V = ∪{V∟n|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 ∈ VB ↔ k/2 ∈ ℕ ∧ k ≥ 3
3. The various truncations of a brick are bricks.
-- A Dx x ∈ VB ↔ A ∈ VB∟n ∧ x ∈ ℕ ∧ x < 2n
4. The brick product of a sequence (of correct order) of bricks is a brick.
-- ◊B<A1, A2, A3, ..., An> ∈ VB ↔ B ∈ VB∟n ∧ ∀iAi ∈ VB
SCPs
1. Any brick is an SCP.
-- A ∈ VB → A ∈ VC --equivalently-- VB ⊆ VC
2. Any regular polygon is an SCP.
-- Gk ∈ VC ↔ k ∈ 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 ∈ VC ↔ A ∈ VC∟n ∧ x ∈ ℕ ∧ x < 2n
5. The pyramid of an SCP is an SCP.
-- &A ∈ VC ↔ A ∈ VC
6. The prismatoid product of a sequence of SCPs (of common dimensionality) is an SCP.
-- &<A1, A2, A3, ..., Ak> ∈ VC ↔ ∀iAi ∈ VC∟n
7. The brick product of a sequence (of correct order) of SCPs is an SCP. (like brick rule #4)
-- ◊B<A1, A2, A3, ..., An> ∈ VC ↔ B ∈ VB∟n ∧ ∀iAi ∈ VC
SBs
1. Any SCP is an SB.
-- A ∈ VC → A ∈ VS --equivalently-- VC ⊆ VS
2. Any tapertope is an SB.
-- A ∈ VTa → A ∈ VS --equivalently-- VTa ⊆ VS
3. Any toratope is an SB.
-- A ∈ VTo → A ∈ 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> ∈ VS ↔ B ∈ VB∟n ∧ ∀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.