From incidence to moves [frap-0002]
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To describe a puzzle move without committing to coordinates, we need an incidence record that says which pieces and labels belong together. Faces and incidence define an abstract polytope without choosing lengths or an embedding [schulte2017regular, Section 2]. Regularity supplies shape symmetries; the Rubik construction applies facet rotations locally to incidence positions [marchetti2025rubik, Definitions 5--6]. The sections below first make the shape visible in triangle and tetrahedron examples, then build this action and calculate its tetrahedron and cube consequences.

1. Faces, flags, and the symmetry of a shape [frap-0008]

A local turn needs a shape-level answer to two questions: which faces are incident, and which symmetries preserve those incidences? An abstract polytope retains the incidence of vertices, edges and faces without prescribing distances or angles. Schulte's flag formulation [schulte2017regular, Section 2] leads from these incidences to symmetries and to the extension of a facet symmetry used in local puzzle moves.

Definition 1.1. Abstract polytopes and flags [schulte2017regular, Section 2] [frap-000D]

An abstract $n$-polytope, for $n\geq 0$, is a partially ordered set $P$ of faces with a least face $F_{-1}$ and greatest face $F_n$. Every chain extends to a maximal chain, called a flag, with exactly $n+2$ faces. Faces in each flag have ranks $-1,0,\ldots ,n$, consistently across flags. Faces other than the least and greatest are proper; vertices, edges and facets have ranks $0,1,n-1$.

The diamond condition requires exactly two faces of rank $i$ between faces $F<G$ of ranks $i-1$ and $i+1$, for $0\leq i<n$. Two flags are adjacent if they differ in exactly one proper face. Strong flag connectivity requires any two flags to be joined by adjacent flags, each containing their intersection.

Thus, for each $0\leq i<n$, each flag has a unique adjacent flag differing in rank $i$. Strong flag connectivity restricts paths to these adjacent flags, each containing the endpoint flags' intersection; arbitrary paths through the least face would connect every pair without imposing an axiom. These conditions prescribe incidence without choosing lengths or angles.

A rank-zero polytope is a two-element chain. A rank-one polytope has two vertices between its least and greatest faces. These low-rank cases occur as sections of higher-rank polytopes.

Definition 1.2. Sections and vertex figures [schulte2017regular, Section 2] [frap-000E]

Let $P$ be a abstract polytope. For faces $F\leq G$ of ranks $i,j$, its section $G/F$ is the interval $\{H:F\leq H\leq G\}$, with ranks shifted so that $F$ has rank $-1$. For $F<G$, it is an abstract polytope of rank $j-i-1\geq 0$. For $F=G$, the singleton section is called the rank-$-1$ polytope. These low ranks allow all intervals, including vertex figures of polygons and intervals between consecutive faces.

The section $F_n/v$ at a vertex $v$ is its vertex figure. Its vertices are the edges through $v$; its faces record the incidences above $v$. A facet $H$ is viewed as the section $H/F_{-1}$.

Example 1.3. Flags of a triangle and a tetrahedron [frap-000F]

The face-poset construction of [schulte2017regular, Section 2] gives the following elementary examples of flags. A triangle has the empty face, three vertices, three edges and the whole triangle, of ranks $-1,0,1,2$. The flag $\varnothing <1<12<123$ has rank-zero neighbor $\varnothing <2<12<123$ and rank-one neighbor $\varnothing <1<13<123$. There are $3\cdot 2=6$ flags.

$\text{[math]}$

Each vertex in this flag graph records the vertex and edge of one flag; the common empty and greatest faces are suppressed. Edge labels give the rank changed by adjacency.

A tetrahedron has all subsets of $\{1,2,3,4\}$ as faces, ordered by inclusion; a nonempty face with $k$ vertices has rank $k-1$. Its flag $\varnothing <1<12<123<1234$ exhibits all five ranks. The vertex figure at $1$ is a triangle: its vertices correspond to the three incident edges. A cube also has a triangular vertex figure, although its facets are squares. Local and global shapes carry different information.

The triangle and tetrahedron calculation now supplies a concrete check: the flag graph records which flags are adjacent; a vertex figure records the local neighborhood seen from one vertex. With those objects in hand, the next steps ask which incidence-preserving maps act on all flags and how a symmetry of one facet can be extended.

Definition 1.4. Automorphisms and regularity [schulte2017regular, Section 2] [frap-000G]

An automorphism of a abstract polytope $P$ is a bijection $a:P\to P$ satisfying $F\leq G\iff a(F)\leq a(G)$. Automorphisms form a group $\Gamma (P)$ under composition; in $ab$, the map $b$ acts first. They preserve ranks and act on flags face by face.

The polytope is regular if this action is transitive on flags. The tetrahedron is regular: a permutation of its four vertices sends any flag to any other.

The underlying order automorphism is represented by OrderIso with equal source and target. This declaration does not impose the polytope axioms or assert regularity.

Lemma 1.5. A flag determines an automorphism [schulte2017regular, Section 2, p. 4] [frap-000H]

An automorphism of an abstract polytope fixing every face of one flag is the identity. Consequently the automorphism group of a regular polytope acts freely and transitively on its flags.

The cited paragraph preceding Lemma 2.1 states simple transitivity for regular polytopes. The elementary argument below also proves freeness without assuming regularity.

Proof.

For a flag adjacent to a pointwise fixed flag, only one rank has changed. The unchanged neighboring ranks bound a diamond condition with two possible faces. One is fixed, so the other is fixed too. The adjacent flag is therefore fixed pointwise. Propagating along the connected flag graph fixes every flag. Every face belongs to a flag, so every face is fixed. Transitivity is the definition of being regular.

Remark 1.6. Distinguished generators and the intersection property [frap-000I]

Choose a base flag $\Phi $ in a regular abstract polytope $P$ of rank $n$. By flag freeness, there is a unique automorphism $\rho _i$ taking $\Phi $ to its rank-$i$ neighbor, for $0\leq i<n$. The diamond condition and uniqueness give $\rho _i^2=1$. For $|i-j|>1$, the two rank changes are independent; $\rho _i\rho _j$ and $\rho _j\rho _i$ agree on $\Phi $, hence are equal. Connectivity shows that the $\rho _i$ reach every flag, so they generate $\Gamma (P)$. These are the distinguished generators of [schulte2017regular, Section 5, pp. 13--14].

For $I\subseteq \{0,\ldots ,n-1\}$, put $G_I=\langle \rho _i:i\in I\rangle $. They also satisfy

\[G_I\cap G_J=G_{I\cap J}.\]

This intersection property is part of the string C-group description [schulte2017regular, Section 5, equation (18)]. Involution and nonadjacent commutation relations alone do not reconstruct a polytope; the group-theoretic characterization includes this additional condition [schulte2017regular, Section 5, p. 14].

Lemma 1.7. Extending a symmetry of one facet [frap-000J]

Let $H$ be a facet of a regular abstract polytope $P$ of rank $n\geq 2$. Restriction gives an isomorphism

\[\operatorname {Stab}_{\Gamma (P)}(H)\cong \Gamma (H/F_{-1}).\]

In particular the facet is regular, and each of its automorphisms has a unique extension fixing $H$. This is a consequence of the flag action in [schulte2017regular, Section 2 and Lemma 2.1]; the argument follows.

Proof.

Extend a flag of the facet section by the greatest face of $P$. A stabilizer element restricting to the identity fixes this whole flag and is the identity by flag freeness. Restriction is therefore injective.

Given a facet automorphism $\alpha $, the regular action supplies an automorphism of $P$ taking the extended flag to the extension of its image under $\alpha $. It fixes $H$. Its restriction agrees with $\alpha $ on the chosen facet flag, so flag freeness in the facet makes these restrictions equal. This proves surjectivity. The same use of same regular action with any two facet flags shows that the facet is regular.

2. Turning a facet into a puzzle move [frap-0009]

Once the shape symmetries are known, the next question is local: how can a turn change only the positions incident to one chosen facet while carrying their labels along? A facet rotation extends to a symmetry of the whole shape, but a puzzle move applies it only at positions incident to the chosen facet. Marchetti's construction retains both entries of each incidence pair [marchetti2025rubik, Definitions 5--6]. We use finite regular abstract polytopes of rank at least three; the extension lemma itself does not require finiteness.

Definition 2.1. The rotation subgroup [marchetti2025rubik, Definition 4] [frap-000K]

Let $Q$ be a regular abstract polytope of rank $m\geq 2$. Choose a base flag and its distinguished generators (involutions) $\rho _0,\ldots ,\rho _{m-1}$. Its rotation subgroup is

\[\Gamma ^+(Q)=\langle \rho _{j-1}\rho _j:1\leq j<m\rangle .\]

The index starts at $1$: for a polygon, the single adjacent pair $\rho _0\rho _1$ generates its rotations. For triangles and squares these are the usual cyclic rotation groups.

Equivalently, $\Gamma ^+(Q)$ consists of the elements represented by even-length words in the distinguished involutions. Indeed every pair $\rho _i\rho _j$ is a product of adjacent pairs or their inverses, by cancellation along the index interval. Conjugating an even word by a distinguished involution again gives an even word, so this subgroup is normal. A change of base flag conjugates the distinguished generators and therefore leaves the subgroup unchanged. We do not require it to have index two for every abstract polytope.

Definition 2.2. Incidence positions and facet moves [marchetti2025rubik, Definitions 5--6] [frap-000L]

Let $P$ be a finite regular abstract polytope of rank $n\geq 3$, with least and greatest faces $F_{-1},F_n$. Write $P^\circ =P\setminus \{F_{-1},F_n\}$. The incidence positions are

\[\mathcal R(P)=\{(F,G)\in P^\circ \times P^\circ :F\leq G\}.\]

Choose a facet $H$ and an element $\alpha $ of the rotation subgroup $\Gamma ^+(H/F_{-1})$. The facet extension lemma gives its unique extension $\widehat \alpha \in \Gamma (P)$ fixing $H$. The corresponding facet move is

\[\mu _{H,\alpha }(F,G)= \begin {cases} (\widehat \alpha (F),\widehat \alpha (G)),&F\leq H,\\ (F,G),&F\not \leq H. \end {cases}\]

The first entry decides where the shape automorphism is applied; the second entry is transported with it. The facet is part of the move's name: one shape automorphism may stabilize several facets and give different local moves. Each move is a permutation of $\mathcal R(P)$.

Lemma 2.3. The inverse of a facet move [frap-000M]

Let $P$ be a finite regular abstract polytope of rank $n\geq 3$, $H$ a facet and $\alpha \in \Gamma ^+(H/F_{-1})$ a facet rotation. The facet move $\mu _{H,\alpha }$ is a permutation of $\mathcal R(P)$, with

\[\mu _{H,\alpha }^{-1}=\mu _{H,\alpha ^{-1}}.\]

This follows directly from the construction in [marchetti2025rubik, Definitions 5--6].

Proof.

An automorphism preserves proper faces and the order relation, so the moving branch remains in $\mathcal R(P)$. Since $\widehat \alpha (H)=H$, the condition $F\leq H$ holds exactly when $\widehat \alpha (F)\leq H$. Thus the moving and fixed subsets are preserved separately. On the moving subset the extension of $\alpha ^{-1}$ is $\widehat \alpha ^{-1}$ by uniqueness, and cancels the first map. On the fixed subset both maps are the identity.

The general construction combining permutations of complementary subsets is Equiv.Perm.subtypeCongr; its inverse formula is Equiv.Perm.subtypeCongr.symm. Here the predicate is $F\leq H$, the moving permutation is the restricted shape automorphism and the other permutation is the identity. These declarations supply that general combination; the polytope-specific preservation argument is the proof above.

The inverse lemma makes the local action reversible. The tetrahedron calculation that follows is a small witness for why the facet is part of a move's identity: two locally valid turns can act differently on the same incidence record and need not commute.

Example 2.4. Two turns of a tetrahedron [frap-000O]

Represent the faces of a tetrahedron by subsets of $\{1,2,3,4\}$. We calculate facet moves from [marchetti2025rubik, Definitions 5--6]. For the facet $H=123$, take an element of the rotation subgroup with global extension $\widehat \alpha =(123)$, fixing $4$. The vertex-edge records follow the cycle

\[(1,12)\longmapsto (2,23)\longmapsto (3,13)\longmapsto (1,12).\] $\text{[math]}$

Every arrow is the local move $\mu _{H,\alpha }$. The record $(4,14)$ stays fixed because $4\not \leq H$, although the global shape symmetry sends it to $(4,24)$. The diagram records this permutation of incidence positions, rather than the tetrahedron's face graph.

For records whose second entry is a facet, the same move gives

\[(1,124)\longmapsto (2,234),\qquad (12,124)\longmapsto (23,234),\qquad (4,124)\longmapsto (4,124).\]

The last record is fixed even though the global symmetry changes its second entry. Now let $K=124$ and take an element of the rotation subgroup with global extension $\widehat \beta =(124)$. Applying the $H$-move and then the $K$-move sends

\[(1,124)\xrightarrow {\mu _{H,\alpha }}(2,234) \xrightarrow {\mu _{K,\beta }}(4,134).\]

Reversing their order gives $(1,124)\mapsto (2,124)\mapsto (3,234)$, so the two moves do not commute. Each has order three: its cube is the identity on both branches, and its displayed action is nontrivial.

The tetrahedron witness motivates the generated move group: reachable labelled configurations are products of these reversible local turns. The ordinary cube calculation then shows how the same definition compares with visible stickers and which data it leaves out.

Definition 2.5. The Rubik group and labelled configurations [frap-000N]

For a finite regular abstract polytope $P$ of rank $n\geq 3$, the Rubik group is the subgroup generated by all facet moves:

\[\operatorname {Rub}(P)= \langle \mu _{H,\alpha }:H\text { a facet},\quad \alpha \in \Gamma ^+(H/F_{-1})\rangle \leq \operatorname {Sym}(\mathcal R(P)).\]

This is Marchetti's generated group of permutations [marchetti2025rubik, Section 3, p. 6, following Definition 6]. The inverse lemma ensures that the generators are permutations.

Give each incidence position its own initial label. A labelled configuration can be represented by the permutation carrying each initial label to its current position. The identity is the solved configuration; the reachable labelled configurations are the elements of $\operatorname {Rub}(P)$. A new move acts on the left: applying $\mu $ to configuration $g$ gives $\mu g$, with $g$ acting first. A visible coloring may identify several labelled configurations, so a solved visible coloring need not determine the labelled permutation.

Example 2.6. One ordinary cube face turn [frap-000P]

Realize the cube as $[-1,1]^3$ and let $H$ be the facet $z=1$. The facet move construction of [marchetti2025rubik, Definitions 5--6] gives the following explicit outer-face action. The shape rotation

\[q(x,y,z)=(y,-x,z)\]

is a clockwise quarter-turn when viewed from positive $z$, with $x$ rightward and $y$ upward. Its restriction to $H$ supplies our move. The corner $(1,1,1)$ goes to $(1,-1,1)$, and its color on $y=1$ goes to the color position on $x=1$. The corresponding corner at $z=-1$ is fixed by the puzzle move, although the global map $q$ moves it.

$\text{[math]}$

The signs give the $x,y$ coordinates of the four corners on $z=1$. The arrows show their clockwise motion; the central incidence record $(H,H)$ stays fixed. No orientation of a marked center is recorded.

Restricting the second entry to facets gives $8\cdot 3+12\cdot 2+6=54$ incidence positions: three per corner, two per edge and one per face center. The turn moves twelve corner stickers and eight edge stickers, fixing the other thirty-four positions, including the unmarked center of the turned face. This is the usual outer-face action on visible stickers of a $3\times 3\times 3$ cube in a fixed frame.

The full incidence set has $8\cdot 7+12\cdot 3+6=98$ elements: above each vertex are that vertex, three edges and three facets; above each edge are itself and two facets. The facet-colored subset is invariant under every move. Its restricted action supplies the comparison with ordinary stickers; a faithful restriction for all abstract polytopes is not needed. Marked center orientations and inner-layer turns are additional data, absent from this comparison.

The local action determines which positions participate and how their incident labels travel. Frames and orientation separate this transport into a destination and a local orientation.