Definition. Cubies, sticker states, and fixed local frames [frap-001D]
Definition. Cubies, sticker states, and fixed local frames [frap-001D]
A description with a fixed global viewing frame needs three pieces of vocabulary. A cubie is one physical unit of the layered cube, with a home position and the visible stickers incident to that position. For a three-layer cube these are the eight corners, twelve edges and six centres described in how the cube models connect. Bonzio, Loi and Peruzzi use this cubie decomposition in their general model [bonzio2018nxnxn, Sections 2-3]; the visible-sticker formulation in this fixed global viewing frame is our adaptation to the present cube convention.
For \(N=3\), let \(\mathcal S\) be the 54 labelled visible stickers and let \(\mathcal P=\{(p,n):p\in \{-1,0,1\}^3,\ \lVert p\rVert _\infty =1,\ n \text { is an incident outward normal at }p\}\). Thus \(\lvert \mathcal P\rvert =54\). A sticker state is a bijection \(s:\mathcal S\xrightarrow {\sim }\mathcal P\): every valid surface slot has exactly one occupant. The solved assignment \(s_0\) is the reference state, and \(s_0^{-1}\mathbin {\circ }s:\mathcal S\to \mathcal S\) records the state as a permutation of sticker labels, while \(s\mathbin {\circ }s_0^{-1}:\mathcal P\to \mathcal P\) records the same state as a permutation of the 54 slots. A move \(m:\mathcal P\to \mathcal P\) acts on the state by \(m\mathbin {\circ }s:\mathcal S\to \mathcal P\), so it left-composes the slot permutation by \(m\) and left-composes the sticker-label permutation by the transported move \(s_0^{-1}\mathbin {\circ }m\mathbin {\circ }s_0\). Marchetti's position-and-label transport gives the structural pattern [marchetti2025rubik, Section 3.1, Definition 7 and equation (5)]; the present 54-slot bijection is our adaptation of that pattern to the three-layer cube. The correspondence between these slots, cubie fibres, and the coordinates \((\sigma ,x;\tau ,z)\) is made explicit in how the cube models connect.
A fixed local frame specializes the local frame: it is a chosen reference ordering of the incident sticker directions at each cubie. A move records both the cubie's destination and how its labels are transported relative to that ordering. The six centre directions keep the global viewing frame fixed; a whole-cube rotation would change that frame and is a separate symmetry. These definitions distinguish the sticker state from a colour observation, which forgets labels that look alike.