software3d.com

I was reading

https://www.software3d.com/Millers5th.php

and I do disagree , because it does overlook at least one pair of enantiomorphous polyhedra.

I don't really know how to interpret the lines in the diagram. so maybe I am mistaken.

I see A, BL,BR CL and CR as independent parts. that all could be added to a polyhedron individually
they are only related that BL and BR are a pair of enantiomorphous cells, also CL and CR a pair of enantiomorphous cells.


My first interpretation is that Millers 5th rule doesn't really rule out including both polyhedra in a pair of enantiomorphous polyhedra.
I think the first line of rule 5 refers to icosahedra like Ag1 that combines the 2 unconnected icosahedra A and g1 both having full icosahedral symmetry and are unconnected.

the exception in line 2 refers to i think the icosahedra f1 that combines the icosahedra f1l and f1r (that are all only vertex connected)

I assume that in "the 59 icosahedra " the rule to include only one of a pair of enantiomorphous polyhedral is just implicit. (like it also makes other assumptions that e1,e2,f1,f2,g1,g2 are connected icosahedra)

so what now?
I think it is reasonable to include only one polyhedron of a pair of enantiomorphous polyhedra.

But in the given example it is interpreted to strong

Valid stellations, ignoring the exception and enantiomorphic repeats, are: CL, BL, A, CL-BL, BL-A, CL-BL-A, BL-A-BR, CL-BL-A-BR, CL-BL-A-BR-CR.
and you add then CL-CR, BL-BR, CL-BL-CR-BR.

this combines to the
reflexible stellations are A, BL-A-BR, CL-BL-A-BR-CR, CL-CR, BL-BR, CL-BL-CR-BR.

and the chiral stellations (preferring the L over the R cell) are CL, BL, CL-BL, BL-A, CL-BL-A, CL-BL-A-BR

and excludes the stellations CR, BR, CR-BR, BR-A, CR-BR-A, CR-BL-A-BR

I am not sure why he combination A-Cl is not included (I guess it comes that I see all cells as independently possible)

My worry is that no polyhedra of the pairs BR-CL , BL-CR and A-BR-CL , A-BL-CR included.

in my opinion at least one of each pair should

Miller was trying to reduce the set to something manageable, and I suppose reduce the number of impracticle polyhedra. Sure, you could allow all possible combinations of cells, and you'd get 2^N - 1 stellations, for N cells (presuming we skip the empty stellation with no cells selected), so it explodes pretty quickly. Even Miller's rules explode pretty quickly after the icosahedron.

There are also other rules to identify interesting sets of stellations, and there's no correct rules. Miller's rules is just one, and well-known thanks to Coxeter et al. Interpreting what Miller meant by his 5th rule is the question here.

The lines in the diagram mean they share a face. So A-CL would not be allowed because they don't share a face (would need BL to connect them). It would be two different cell types floating disconnected in space (but possibly sharing edges). It would be rules out by anyone's interpretation of Miller's rules. But you can still select those cells in Stella and see what it looks like if you want. Same for the other combinations you mentioned. They are still stellations in my book, but there's so many combinations that we have rules to reduce the number.