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Use the coefficients in the authoritative PDF, including their factor $i$. Write $a=2\sqrt{3}i$ andThe numerator and denominator are coprime: a common zero would, by subtraction, require $z=0$, where both equal one. Therefore the degree of a rational map of the Riemann sphere is four. Direct algebra givesThe domain transformations $z\mapsto iz$ and $z\mapsto (iz+1)/(1-iz)$ are sphere rotations as special-unitary Möbius transformations: respectively a quarter-turn about the third axis and a one-third turn cycling the three coordinate axes. In particular the latter sends $0\mapsto 1\mapsto i\mapsto 0$, the north, first-axis and second-axis directions. They generate the order-24 rotational symmetry group of a cube, isomorphic to the symmetric group $S_4$. Their target transformations are rotations: $w\mapsto 1/w$ is a half-turn about the first target axis and $w\mapsto \omega w$ is a one-third turn about the third target axis. This proves whole-map combined equivariance, not just symmetry of selected roots.

The target rotation image is the order-six dihedral group $D_3$. The spatial half-turns about all three coordinate axes act trivially on the map: $R(-z)=R(z)$ and $R(1/z)=R(z)$ generate a Klein four-group kernel. Thus the full Skyrmion symmetry combines the spatial cubic rotations with compensating isospin rotations, while its angular energy and Skyrme baryon density have pure spatial cubic symmetry.

The critical directions make the geometry explicit. Its Wronskian of a rational map isThe five finite ramification points are $0,\pm 1,\pm i$; infinity supplies the sixth, since in the local coordinate $w=1/z$ the map starts with $R=1+2a{w}^2+O(w^4)$. These are the six coordinate-axis directions, or face centres of a cube. The angular Jacobian of a rational map vanishes there, consistent with a cube-shaped shell whose density is concentrated away from its face centres. Any further rotational equivariance would have to preserve this set, so the spatial proper rotation group is exactly the cubic group already generated above.

There is also a reflection relation $R(\overline{z})=1/\overline{R(z)}$. Together with the proper rotations, it makes the angular density invariant under the full order-48 symmetry group of a cube, usually denoted $O_h$. The target operation in this reflection relation is orientation reversing; it should not be mistaken for a proper isospin rotation. In the full Skyrme model, reflections are expressed using the field parity operation $U(\mathbf{x})\mapsto U^{†}(-\mathbf{x})$ together with a compensating isospin rotation.

This is the cubic charge-four rational-map ansatz:The cubic rational-map ansatz for four Skyrmions is a useful approximation and starting point for the cubic four-Skyrmion, whose lowest spin-zero, isospin-zero quantized state models an alpha particle. A radial minimization and, for precision, unrestricted field relaxation are still required. The TeX's missing $i$ changes this map; the six critical directions alone would not detect the error, because the same Wronskian zero set persists when $a$ is real. The actual rotational equivariance identities are the stronger check.

A space diagonal of the unit cube has displacement vector $(1,1,1)$. Its Euclidean norm, or two successive applications of the Pythagorean theorem, givesThe requested diagonal joins opposite vertices; $\sqrt{2}$ would be the diagonal of a face.

Three suitable examples are cubes bounded mainly by $\{100\}$ facets, octahedra bounded by $\{111\}$ facets, and truncated octahedra bounded by both $\{111\}$ and $\{100\}$ facets. These are crystallographically distinct shapes rather than three arbitrary rounded particle outlines.

At fixed volume the thermodynamic equilibrium shape minimizes $G_{\mathrm{surf}}={\sum }_i{\gamma }_i{A}_i$, with an edge/corner correction when particles are very small. In the Wulff construction, facet support-plane distances scale as their interfacial free energies. Silver's clean low-index energies, selective adsorption of ligands or ions, solvent chemical potential and possible support contact alter those energies. Crystal twinning and elastic strain can add further costs or stabilize multiply twinned forms. A face with slow attachment kinetics can also survive during growth even when that morphology is not the thermodynamic equilibrium Wulff shape. Homogeneous nucleation describes where nuclei form; it does not guarantee an thermodynamic equilibrium final shape.

For the inertia tensor of a uniform cube, reflection symmetry makes every off-diagonal element of the central inertia tensor vanish. Permutation of the three axes makes its diagonal elements equal. Since $M=8\rho a^3$ and the mean of each squared coordinate in the cube is $a^2/3$,Every central axis is therefore a principal axis, including the body diagonal. The angular momentum is parallel to the angular velocity, and its magnitude is

A polycube is a face-connected union of finitely many unit cubes in the three-dimensional cubic grid. Its exposed square faces determine its surface area.

A hard-sphere model relating the radius ratio of opposite-charge ions to a possible coordination number. For a cube of eight mutually touching anions around a touching cation, $r_c/r_a=\sqrt{3}-1$; real bonding and coordination-dependent radii limit its predictive power.

A rectangular parallelepiped is a parallelepiped whose three edge directions at each vertex are mutually perpendicular. Its faces are rectangles; if its edge lengths are $a,b,c$, its volume is $abc$. A cube is the special case $a=b=c$.

A cage ${\mathrm{R}}_8\mathrm{S}{\mathrm{i}}_8{\mathrm{O}}_{12}$ represented by a cube of eight silicon vertices with one bridging oxygen on each of twelve edges.