Group Theory Analysis

Symmetry structure of the mechanical category space

The Symmetry Group G = ⟨α, β, γ⟩

The mechanical category space M = {facing, rear, sitting, standing, side, inverted, oral, special} admits natural symmetry operations that form a permutation group G ≤ S₈.

Three generators act on M by swapping category pairs:
α (orientation reversal): facing ↔ rear
β (vertical inversion): sitting ↔ standing
γ (role exchange): facing ↔ inverted

The full group G = ⟨α, β, γ⟩ has order 12, generated by closing these transpositions under composition.

Generators

Generator	Cycle Notation	Order	Fixed Points
α (orientation)	`(facing rear)`	2	6
β (vertical)	`(sitting standing)`	2	6
γ (role exchange)	`(facing inverted)`	2	6

Full Group Elements (|G| = 12)

Element	Cycle Notation	Order	Fixed
g₀	`(identity)`	1	8
g₁	`(sitting standing)`	2	6
g₂	`(rear inverted)`	2	6
g₃	`(rear inverted) (sitting standing)`	2	4
g₄	`(facing rear)`	2	6
g₅	`(facing rear) (sitting standing)`	2	4
g₆	`(facing rear inverted)`	3	5
g₇	`(facing rear inverted) (sitting standing)`	6	3
g₈	`(facing inverted rear)`	3	5
g₉	`(facing inverted rear) (sitting standing)`	6	3
g₁₀	`(facing inverted)`	2	6
g₁₁	`(facing inverted) (sitting standing)`	2	4

Orbits of G on M

The orbit of x ∈ M is Orb(x) = {g(x) : g ∈ G}. Categories in the same orbit are "equivalent up to symmetry." Burnside’s lemma gives the orbit count: |orbits| = (1/|G|) Σ_g∈G |Fix(g)| = 5

Orbit	Categories	\|Orb\|
O₀	facing, rear, inverted	3
O₁	sitting, standing	2
O₂	side	1
O₃	oral	1
O₄	special	1

Orbit–Stabilizer Theorem Verification

For every x ∈ M: |G| = |Orb(x)| × |Stab(x)|
where Stab(x) = {g ∈ G : g(x) = x}.

Category	\|Orb\|	\|Stab\|	Product	\|G\|	✓
facing	3	4	12	12	✓
rear	3	4	12	12	✓
sitting	2	6	12	12	✓
standing	2	6	12	12	✓
side	1	12	12	12	✓
inverted	3	4	12	12	✓
oral	1	12	12	12	✓
special	1	12	12	12	✓

Stabilizer Subgroups

Category	\|Stab\|	Elements
facing	4	`(identity), (sitting standing), (rear inverted), (rear inverted) (sitting standing)`
rear	4	`(identity), (sitting standing), (facing inverted), (facing inverted) (sitting standing)`
sitting	6	`(identity), (rear inverted), (facing rear), (facing rear inverted), (facing inverted rear) ... (+1)`
standing	6	`(identity), (rear inverted), (facing rear), (facing rear inverted), (facing inverted rear) ... (+1)`
side	12	`(identity), (sitting standing), (rear inverted), (rear inverted) (sitting standing), (facing rear) ... (+7)`
inverted	4	`(identity), (sitting standing), (facing rear), (facing rear) (sitting standing)`
oral	12	`(identity), (sitting standing), (rear inverted), (rear inverted) (sitting standing), (facing rear) ... (+7)`
special	12	`(identity), (sitting standing), (rear inverted), (rear inverted) (sitting standing), (facing rear) ... (+7)`

Cayley Table

The Cayley multiplication table encodes the full group structure. Entry (i, j) = index of g_i · g_j in the group.

Cayley table for G (|G| = 12)

Cayley Graph

The Cayley graph Cay(G, S) has vertices = group elements, colored edges g → gs for each generator s ∈ S = {α, β, γ}. It encodes the group’s connectivity and structure visually.

Cayley graph Cay(G, {α, β, γ})

Quotient Structure

The group action partitions positions by orbit membership. The quotient set P/G groups positions whose categories are related by symmetry.

Orbit Class	Positions	Traditions
Orbit-0: facing+rear+inverted	107	perfumed_garden, fangzhongshu, greco_roman, ananga_ranga, shijuhatte, ratirahasya, kamasutra
Orbit-1: sitting+standing	39	perfumed_garden, fangzhongshu, greco_roman, ananga_ranga, shijuhatte, ratirahasya, kamasutra
Orbit-2: side	11	perfumed_garden, fangzhongshu, greco_roman, ananga_ranga, shijuhatte
Orbit-4: special	9	greco_roman, fangzhongshu, shijuhatte, kamasutra
Orbit-3: oral	15	fangzhongshu, greco_roman, shijuhatte, ratirahasya, kamasutra