Polyhedra of peculiar interest

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Why is something as mathematical as polyhera of any relevance?
Isn't all that polyhedra stuff usually in most cases just pointless mental gymnastic?
Well, yes and no. This page will focus just on a secific sub-class of polyhedra that
seem relevant in the context of gemgum tec for one reason or another (introduced below).
If other polyhera are added they'll go to the page: Random cool miscellaneous polyhedra.

This pages focus – Voxel like building blocks & Frame system nodes

This page focuses in particular on polyhedra that seem relevant as basis for …

(1) Basic primitive (voxel like) building blocks that …
– (1a) best match the various crystal structures of most in focus (diamondoid) base materials.
– (1b) best allowing transitions between different but compatible crystal stutructues.
… We don't want the crystal structures to be compatible but the building blocks thwarting that compatibility.

(2) Nodes of nano or microscale dedicated frame system.
Specific focus on the octet truss, as this is one of the rare cases where
natures math basically gives us just one single best self suggesting choice. All other options inferior.
Faces may be shifted inwards or outwards to asjust the sizes of the poyhedra faces.

Crystal system matching building blocks

Trapezo-rhombic dodecahedron

This polyhedron is rarely talked about due to
its lower symmetry compared to the normal rhombic dodecahedron
but it is highly relevant in the context of crystal systems as it is corresponding to
the lesser known hexagonal version of diamond that is called lonsdaleite.

The trapezo-rhombic dodecahedron is a natural building block (voxel) shape for lonsdaleite which also
gives a natural (the most natural) seamless transition over to normal cubic diamond (along the hexagonal main axis).
(wiki-TODO: Add visual illustration of that transition maybe in 4 of the 8 possible directions.)

One can think of this shape as the rhombic dodecahedron with one side rotated by 60°. So …
– there is a mapping between the hexagonal extended miller indices to the normal Miller indices.
(wiki-TODO: Uncover this mapping)
– only three of the twelve faces faces differ in surface normal to the faces of the rhombic dodecahedron
– (the six prism jacket faces differ in atomistic detail but not in surface normal.)

  • Wiegner-Seiz cell of hcp (hexagonal closed packed) crystal system
  • direction of faces (TODO)
  • spacefiller / 3D honeycomb
    shape compatible to rhombic dodecahedra along the main (111) direction

Rhombic dodecahedron

This shape is a natural building block (voxel) for normal cubic diamond and isostructural compounds.
It allows for a seamless transition to hexagonal diamond (lonsdaleite).
Along each of the the four (eight counting sign) (111) directions.

Cubic diamond is actually two fcc == ccp lattices interspersed.

This polyhedron is also related to the octet truss, as all the faces are normal to the struts of this truss.

  • Wiegner-Seiz cell of fcc == ccp (face centered cubic == cubic close packed)
  • All (110) faces and oone else.
  • spacefiller / 3D honeycomb
    shape compatible to trapezo-rhombic dodecahedra along the main (111) direction

Truncated octahedron

  • Wiegner-Seiz cell of bcc (body centered cubic) crystal system
  • All (100) faces and all (111) surfaces
  • spacefiller / 3D honeycomb

While normal cubic diamond is a variant on fcc = ccp it is not bcc.
Still it can perfectly well form (100) and (111) surfaces of these polyhedra.
So this would also be a sensible space filling building block for cubic diamond.
It does not provide a natural transition to hexagonal diamond though.
Heck it does not even provide a natural transition to the
rhombic dodecahedron building blocks of the same normal cubic diamond.

One can force a transition via the introduction of fused "twinned" building block voxels.
But one needs to take an arbitrary binary decision on a change in scale (larger or smaller).
The trick here is to fuse a truncated octahedon twinned and centered on the vertexes of a rhombic dodecahedron.
Or vice versa debending on the arbitrary binarc choice in scale-change taken.
(wiki-TODO: Add visual illustrations of the twinning based building block transition)

Just like bcc is dual to fcc (inverse space basis),
this is the dual polyhedron to the rhombic cuboctahedron.
Side-note: There are at least two ways to find that (both ways).
– Using the vertices of one polehdron as centers for Voroni cells to find the dual polyhedra.
– Forming the inverse basis and taking it's Wiegner Seiz cell (basically a Voroni cell too).

Side-notes: Trunctaing further in first
– yields a cuboctahedron and then
– yields a truncated cube.

Frame system relevant polyhedra

Octahedra and tetrahedra

  • Perhaps suprisingly not spacefillers on their own but forming a space filling set combinedly
  • The spacefill forms the empty voids in the octet-truss, or stated reversely: the edges form the octet truss

Both the tetrahedron and the octahedron is a delahedron.
This implies that the wireframe spacefill of them gives a spacetruss rather than a spaceframe.
meaning (when an external load is applied) the members only take tensile load or axial compression.
There are no bending (or torsion) loads on the nodes.

Chamfered cube and chamfered octahedron

Octet truss struts are all pointet in the six (or twelve counting sign) possible [110] directions.
On frame system nodes one might want surfaces that are normal to these directions to attach the struts to.
These are the (110) rhombic cuboctahedron faces.

The other faces can be added too to make them available if so desired.
Beveling is easier to specify than partial and direction dependent truncation thus this the nomenclature.
Adding cube faces leads to the (strongly) chamfered cube.
Adding octahedral faces leads to the (strongly) chamfered octagedra.
Adding both yields sort of a "fully truncated rohombic ddecagedron" and eventually an rhombicuboctahedron

Concenputally relevant polyhedra

Rhombicuboctahedron

This poylhedron features all the lowest Miller index surfaces with same emphasis.
Tht is (100) cube-, (110) rhombic dodecahedron-, and (111) octahedron- surfaces.
The truncated version (truncated rhombicuboctahedron)
adds next higher Miller index surfaces.

Related

External links (wikipedia)

Convrete polyhedra:

Concrete spacefills / 3D honeycombs:

space truss ~vs~ space frame:

  • One possible small set of bodies that fill space and can create flat surfaces by George W. Hart [1]
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