safety.nb

(* This program computes the genus of all configurations of a FCC 1-neighborhood. *)

rdpoly = PolyhedronData["RhombicDodecahedron", "Polyhedron"];

alloffsets = {{2Sqrt[2/3],0,0},{-2 Sqrt[2/3],0,0},{0,2Sqrt[2/3],0},{0,-2 Sqrt[2/3],0}, {Sqrt[2/3],Sqrt[2/3],2/Sqrt[3]},{-Sqrt[(2/3)],Sqrt[2/3],2/Sqrt[3]},{Sqrt[2/3],-Sqrt[(2/3)],2/Sqrt[3]},{-Sqrt[(2/3)],-Sqrt[(2/3)],2/Sqrt[3]}, {Sqrt[2/3],Sqrt[2/3],-(2/Sqrt[3])},{-Sqrt[(2/3)],Sqrt[2/3],-(2/Sqrt[3])},{Sqrt[2/3],-Sqrt[(2/3)],2/-Sqrt[3]},{-Sqrt[(2/3)],-Sqrt[(2/3)],-(2/Sqrt[3])}};

gennbrhd[offsets_]:=RegionUnion @@ Table[CanonicalizePolyhedron[Translate[rdpoly,offset]],{offset,offsets}];    

nbrhdgenus[offsets_]:=PolyhedronGenus[gennbrhd[offsets]];   

(* This shows that the genus for subsets of 1...5 always yield a genus 0 configuration always. *)
result_proof=Plus @@ nbrhdgenus /@ Subsets[alloffsets,{1, 5}] 

(* This shows that the genus for subsets of 6+ a genus 0 configuration is not certain. *)
disprove6 = Plus @@ nbrhdgenus /@ Subsets[alloffsets,{6}] 
disprove7 = Plus @@ nbrhdgenus /@ Subsets[alloffsets,{7}]  
disprove8 = Plus @@ nbrhdgenus /@ Subsets[alloffsets,{8}]  
disprove9 = Plus @@ nbrhdgenus /@ Subsets[alloffsets,{9}]  
disprove10 = Plus @@ nbrhdgenus /@ Subsets[alloffsets,{10}] 
disprove11 = Plus @@ nbrhdgenus /@ Subsets[alloffsets,{11}] 
disprove12 = Plus @@ nbrhdgenus /@ Subsets[alloffsets,{12}]  

(* This is exports figures for inspection or video rendering *)
counter = 0
Do [counter = counter + 1; Export[StringInsert["fig__lemma1.jpg", ToString @ counter, 5], Graphics3D[{Yellow, gennbrhd[i]}] ], {i, Subsets[alloffsets,{6, 11}]}]