I shall define in three parts:

1. Bricks. The set of all bricks shall be denoted V

_{B}.

2. Standard convex polytopes (SCPs). The set of all SCPs shall be denoted V

_{C}.

3. Standard boundaries (SBs). The set of all SBs shall be denoted V

_{S}.

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 (V

_{Ta}and V

_{To}) 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} ⊆ V

_{B}

2. Any regular polygon with an even number of sides is a brick.

-- Gk ∈ V

_{B}↔ k/2 ∈ ℕ ∧ k ≥ 3

3. The various truncations of a brick are bricks.

-- A Dx x ∈ V

_{B}↔ A ∈ V

_{B}∟n ∧ x ∈ ℕ ∧ x < 2

^{n}

4. The brick product of a sequence (of correct order) of bricks is a brick.

-- ◊B<A

_{1}, A

_{2}, A

_{3}, ..., A

_{n}> ∈ V

_{B}↔ B ∈ V

_{B}∟n ∧ ∀iA

_{i}∈ V

_{B}

SCPs

1. Any brick is an SCP.

-- A ∈ V

_{B}→ A ∈ V

_{C}--equivalently-- V

_{B}⊆ V

_{C}

2. Any regular polygon is an SCP.

-- Gk ∈ V

_{C}↔ k ∈ ℕ ∧ k ≥ 3

3. The following special cases are SCPs: snub dodecahedron, snub icositetrachoron, grand antiprism

-- {Ki0, Kk0, G

_{AP}} ⊆ V

_{C}

4. The various truncations of an SCP are SCPs. (like brick rule #3)

-- A Dx x ∈ V

_{C}↔ A ∈ V

_{C}∟n ∧ x ∈ ℕ ∧ x < 2

^{n}

5. The pyramid of an SCP is an SCP.

-- &A ∈ V

_{C}↔ A ∈ V

_{C}

6. The prismatoid product of a sequence of SCPs (of common dimensionality) is an SCP.

-- &<A

_{1}, A

_{2}, A

_{3}, ..., A

_{k}> ∈ V

_{C}↔ ∀iA

_{i}∈ V

_{C}∟n

7. The brick product of a sequence (of correct order) of SCPs is an SCP. (like brick rule #4)

-- ◊B<A

_{1}, A

_{2}, A

_{3}, ..., A

_{n}> ∈ V

_{C}↔ B ∈ V

_{B}∟n ∧ ∀iA

_{i}∈ V

_{C}

SBs

1. Any SCP is an SB.

-- A ∈ V

_{C}→ A ∈ V

_{S}--equivalently-- V

_{C}⊆ V

_{S}

2. Any tapertope is an SB.

-- A ∈ V

_{Ta}→ A ∈ V

_{S}--equivalently-- V

_{Ta}⊆ V

_{S}

3. Any toratope is an SB.

-- A ∈ V

_{To}→ A ∈ V

_{S}--equivalently-- V

_{To}⊆ V

_{S}

4. The brick product of a sequence (of correct order) of SBs is an SB. (like brick rule #4)

-- ◊B<A

_{1}, A

_{2}, A

_{3}, ..., A

_{n}> ∈ V

_{S}↔ B ∈ V

_{B}∟n ∧ ∀iA

_{i}∈ V

_{S}

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.