“A non-trivial self-similar convex polyhedron (rep-8-tile, V=6 F=5 E=9) that I found using IFStile.” - Dmitry Mekhontsev
Here’s how to do it in WL:
In[]:=
wedge=ConvexHullMesh[{{0,0,0},{4,0,0},{4,4,0},{2,2,2},{4,2,2},{4,4,2}}];​​hull1=ConvexHullMesh[{{0,0,0},{2,0,0},{2,2,0},{1,1,1},{2,1,1},{2,2,1}}];​​hull2=ConvexHullMesh[{{1,1,1},{3,1,1},{3,3,1},{2,2,2},{3,2,2},{3,3,2}}];​​hull3=ConvexHullMesh[{{4,4,0},{4,2,0},{2,2,0},{3,3,1},{3,2,1},{2,2,1}}];​​hull4=ConvexHullMesh[{{4,0,0},{2,0,0},{2,2,0},{3,1,1},{2,1,1},{2,2,1}}];​​hull5=ConvexHullMesh[{{4,0,0},{4,2,0},{2,2,0},{3,1,1},{3,2,1},{2,2,1}}];​​hull6=ConvexHullMesh[{{4,0,0},{4,2,0},{4,2,2},{3,1,1},{3,2,1},{3,2,2}}];​​hull7=ConvexHullMesh[{{4,4,0},{4,2,0},{4,2,2},{3,3,1},{3,2,1},{3,2,2}}];​​hull8=ConvexHullMesh[{{4,4,0},{4,4,2},{4,2,2},{3,3,1},{3,3,2},{3,2,2}}];​​Graphics3D[{Opacity[0.9],Black,Thick,MeshPrimitives[wedge,1],​​White,Opacity[0.4],MeshPrimitives[hull1,2],​​MeshPrimitives[hull2,2],MeshPrimitives[hull3,2],MeshPrimitives[hull4,2],MeshPrimitives[hull5,2],MeshPrimitives[hull6,2],MeshPrimitives[hull8,2]},Boxed->False]​​
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Here’s a more compact form:
In[]:=
pts={{0,0,0},{2,0,0},{2,2,0},{1,1,1},{2,1,1},{2,2,1}};​​oct=FiniteGroupData["Octahedral","MatrixRepresentation"];​​xf={{1,{0,0,0}},{1,{1,1,1}},{8,{4,4,0}},{21,{4,0,0}},​​{16,{4,0,0}},{25,{4,0,0}},{44,{4,4,0}},{9,{4,4,0}}};​​hulls=pts.oct[[#[[1]]]]+Table[#[[2]],{6}]&/@xf;​​Graphics3D[{Opacity[0.9],MeshPrimitives[ConvexHullMesh[#],2]&/@hulls},Boxed->False]
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Or, even more compactly:
In[]:=
hulls=IntegerDigits[#,10,3]&/@#&/@{{000,200,220,111,211,221},{111,311,331,222,322,332},​​{440,420,220,331,321,221},{400,200,220,311,211,221},{400,420,220,311,321,221},{400,420,422,311,321,322},{440,420,422,331,321,322},{440,442,422,331,332,322}};​​Graphics3D[{Opacity[0.7],MeshPrimitives[ConvexHullMesh[#],2]&/@hulls},Boxed->False]
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Let’s repeat the mapping:
In[]:=
big={{0,0,0},{4,0,0},{4,4,0},{2,2,2}};bary=ResourceFunction["Areal"][big,#]&/@Flatten[hulls,1];​​wedges=Flatten[Table[Partition[ResourceFunction["Areal"][Take[hulls[[j]],4],#]&/@bary,6],{j,1,8}],1];​​Graphics3D[{Opacity[.4],MeshPrimitives[ConvexHullMesh[#],2]&/@wedges},Boxed->False,SphericalRegion->True]
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Let’s go another level with it:
In[]:=
big={{0,0,0},{4,0,0},{4,4,0},{2,2,2}};bary=ResourceFunction["Areal"][big,#]&/@Flatten[hulls,1];​​wedges2=Flatten[Table[Partition[ResourceFunction["Areal"][Take[wedges[[j]],4],#]&/@bary,6],{j,1,64}],1];​​Graphics3D[{Opacity[.3],MeshPrimitives[ConvexHullMesh[#],2]&/@wedges2},Boxed->False,SphericalRegion->True]
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CITE THIS NOTEBOOK

The Mekhontsev wedge: A 3D rep-tile​
by Ed Pegg​
Wolfram Community, STAFF PICKS, January 20, 2026
​https://community.wolfram.com/groups/-/m/t/3615445