isohedral figure

Face transitivity definition

The working polyhedral interpretation requires the symmetry group to act transitively on faces; the reference definition of an isohedron additionally specifies convexity. Source: [Wolfram MathWorld: Isohedron](https://mathworld.wolfram.com/Isohedron.html).

1. Which reference supplies the definition used for this instance? provenance
2. Does that definition require convexity, or explicitly admit nonconvex figures? boundary

Faces versus tiles

In isohedral tilings, the symmetry action concerns tiles of the whole tiling. Any use of that interpretation here must be explicit. Source: [Wolfram MathWorld: Isohedral Tiling](https://mathworld.wolfram.com/IsohedralTiling.html).

1. Is the assertion about a polyhedron's faces or a tiling's tiles? definition
2. For a higher-dimensional figure, does 'face' mean a two-dimensional face or a facet? boundary

Designated faces

Record the adopted face decomposition, including whether coplanar subdivisions count as separate faces.

1. What boundary and incidence data identify each designated face? definition
2. Are coplanar subdivisions, holes or intersecting face boundaries admitted by the convention? boundary

Exactness and congruence

Record coordinates or defining constraints and compare face shapes as an initial obstruction test; congruence alone leaves the transitivity question open.

1. Which exact coordinates or geometric constraints define this realisation? provenance
2. How many face congruence classes are established, and with what numerical uncertainty? measurement

Symmetry group convention

Record whether all isometries are allowed and whether colours, labels or orientations are part of the preserved figure.

1. Must a symmetry preserve only geometry, or also face colours, labels and orientations? definition
2. Are orientation-reversing isometries permitted in the asserted classification? boundary

Orbit computation

For each verified symmetry generator, record its face permutation and compute the resulting orbits. Distinguish a known subgroup from the full permitted symmetry group.

1. Which whole-figure isometries generate the tested action, and how does each permute faces? measurement
2. How many face orbits result, and is the tested group known to be complete? measurement

Transitive subgroup witness

A verified subgroup acting transitively suffices for a positive certificate; identifying every symmetry is unnecessary for that conclusion.

1. What verified transformations carry a representative face to every designated face? measurement
2. How can an agent check that these transformations preserve the whole geometric realisation? action

Orbit separation certificate

Support non-isohedrality with a complete permitted-group computation or an invariant separating faces; otherwise retain an unresolved result.

1. Which invariant prevents two identified faces from being exchanged by any permitted symmetry? measurement
2. Could undiscovered symmetries merge the currently computed face orbits? boundary
3. What additional computation or exact geometric data would settle the unresolved claim? action

Independent transitivity claims

Record vertex, edge and flag transitivity separately, and do not treat abstract incidence automorphisms as geometric isometries without evidence.

1. Which vertex, edge or flag transitivity claims have been independently established? boundary
2. Does the available certificate concern geometric isometries or only incidence-preserving permutations? provenance

Post-transformation verification

After deformation, subdivision, decoration or dual construction, identify the resulting face set and verify the relevant symmetry action before transferring an isohedral classification.

1. Which previously verified face-mapping symmetries remain isometries preserving the modified figure? measurement
2. Does the operation change the designated faces or split their symmetry orbit? boundary
3. What new certificate is required before the modified figure can be classified as isohedral? action

Check these first

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these check these first hold for the sense of isohedral figure this model covers, and on what evidence? provenance

Kinds and varieties

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these kinds and varieties hold for the sense of isohedral figure this model covers, and on what evidence? provenance

Real-world use

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these real-world use hold for the sense of isohedral figure this model covers, and on what evidence? provenance

Typical measurements

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these typical measurements hold for the sense of isohedral figure this model covers, and on what evidence? provenance

Failure modes and hazards

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these failure modes and hazards hold for the sense of isohedral figure this model covers, and on what evidence? provenance

Neighbouring kinds and how to tell them apart

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these neighbouring kinds and how to tell them apart hold for the sense of isohedral figure this model covers, and on what evidence? provenance