Percolation Theory
sujeet kumar
how site percolation threshold for diamond lattice is determined?
sujeet kumar
lattice?If it is there ,someone help me to find dual of diamond
lattice.
Walter Whiteley
It would be found using the points of your first lattice as
the centers of 'Voronoi cells' and seeing what the vertices
of this arrangement are.
The Voronoi Cell of a point p in a configuration is the set of
points closer to p that to any other point of the configuration.
That is what you do in the plane to change the triangular lattice to
the hexagonal lattice (and the reverse) and to change the square
lattice to the square lattice.
I have seen the list of 3-D lattices as duals, recently, but
did not put them to memory.
Something that may intrigue people
- it was a Chemist / Materials Design person who was talking
about this. They are busy making alternative forms of various
lattices out of mixed metal oxide and organic materials, for things
like hydrogen storage. The field is called 'Reticular Chemistry'.
For this, they want to know all the lattices or network like
patterns that the material might follow in self-assembly!
One of the questions they asked, at the workshop of mathematicians
and chemists, was
- How do we get more SYMMETRY into the curriclum!
Walter Whiteley
York University
John Conway
> Yes this is a 'dual' for a 3-D lattice.
[Walter then described the decomposition of space into
Voronoi cells of the lattice.]
This is indeed the geometrical dual of the standard decomposition into
Delaunay cells, and is probably the answer Sujeet wanted to his question:
> > Is there any concept of dual lattice in case of 3-dimensional
> > lattice?If it is there ,someone help me to find dual of diamond
> > lattice.
However, one should be aware that this is NOT the concept usually
known under the name "dual lattice". I'll expound a bit on both.
Geometers usually restrict the term "lattice" to a discrete set
(say L) of vectors that is closed under addition and subtraction. For
clarity, I'll call this an arithmetical lattice. They then use the
term "dual lattice of L" to refer to the set L* of all vectors that
have integral inner products with all vectors of L. This is also
called "the reciprocal lattice to L". It's really more of an algebraic
concept than a geometrical one - for clarity, I'll call it the
arithmetical dual. It does not usually have the same symmetries as L.
For example, the dual of the body-centered cubic (bcc) lattice is a
face-centered cubic (fcc) lattice, and these have distinct symmetry groups.
Other people use "lattice" more freely, as Sujeet does when he
refers to "the diamond lattice", by which I presume he intends the
arrangement formed by the individual carbon atoms in a diamond.
This is not a lattyice in the above arithmetical sense, but rather, the
union of two translates of an fcc lattice.
To understand the situation, it's best to take the fcc to consist
of all point (x,y,z) for which x,y,z are integers with even sum.
The dual bcc lattice consists of all (x,y,z) for which x,y,z are
EITHER all integers OR all halves of odd integers. This consists of
four cosets (translates) [0],[1],[2],[3] of the original fcc, according
as x+y+z is congruent to 0,1/2,1,3/2, respectively.
The diamond arrangement consists of two adjacent ones of these,
say [0] and [1]. Since it's not a lattice in the arithmetical sense,
its arithmetical dual is not defined. However, the points of space
that are furthest away from it are precisely those belonging to the
other two cosets [2] and [3], and so its dual in the sense Walter
gave is another copy of itself!
Regards, John Conway
sujeet kumar
sujeet kumar
In particular,what is the voronoi polyhedron for "Dihexagonal Prism"?
John Conway
> What is the voronoi polyhedron for the hexagonal(3D) lattice?
A hexagonal prism. John Conway
In general, there are only five possible topological shapes
the Voronoi cell of a 3-dimensional lattice can be, exemplified
by the truncated octahedron, the hexarhombic dodecahedron, the
rhombic dodecahedron, the hexagonal prism, and the cube. You
can find a proof in my book "The Sensual Form"
The Voronoi cell of the configuartion of carbon atoms in a diamond
(which is NOT a lattice) is a triakis truncated tetrahedron.
John Conway
John Conway
> Again I gave the incomplete question.
>
> In particular,what is the voronoi polyhedron for "Dihexagonal Prism"?
>
I'm afraid I don't understand either the terms "voronoi polyhedron of"
or "dihexagonal prism"! John Conway
Did you see my posting on the Voronoi and Delaunay tessellations
associated with the arrangement of carbon atoms in a diamond? I believe
it was prompted by one of your questions.
John Conway
sujee kumar
that are as near to the origin as to any other lattice point.
The Dihexagonal Prism is a 12 sided prism bounded by 12-faces ,each
parallel to the vertical axis.
sujeet kumar
knowledge I have gained by reading geometry book by H.S.M. Coxeter
that Voronoi Polyhedron is defined for any lattice so I wanted
to know Voronoi Polyhedron for Dihexagonal Prism.But I thought
Just writing hexagonl lattice can create confusion as many
lattice come under hexagonal system.So in my 2nd mail I made
the necessary change.
John Conway
> My knowledge about 3 dimensional are not very firm but whatever
> knowledge I have gained by reading geometry book by H.S.M. Coxeter
> that Voronoi Polyhedron is defined for any lattice so I wanted
> to know Voronoi Polyhedron for Dihexagonal Prism.
But "dihexagonal prism" sounds to me like the name of a polyhedron,
not a lattice (and I still don't know which polyhedron!).
> But I thought Just writing hexagonl lattice can create confusion as
> many lattice come under hexagonal system.So in my 2nd mail I made
> the necessary change.
Maybe that was a message of yours that I've already responded to?
The different possibilities for the topological shape of the vocell
(my abbreviation for "Voronoi cell") of a lattice are known up to
4 dimensions - in 1 2 3 4 dimensions the number of topological
shapes is exactly 1 2 5 52 respectively. Of these, the number of
"generic" ones is 1 1 1 3, and the number of generic ones is known
also in 5 dimensions, being 222.
I am very grateful to you for asking about "the dual of the diamond
lattice", since it taught me something I didn't know. As I remarked,
the carbon atoms in a diamond aren't arranged as a lattice. However,
the most natural tessellation corresponding to any discrete set of
points is the Delaunay one, whose dual is the Voronoi one. So I
think the best interpretation is that you were asking about the
Voronoi tessellation of the diamond, which in fact I did know - its
cells are "triakis truncated tetrahedra".
However, I realised that I didn't know the Delaunay tessellation,
which is into regular tetrahedra and squashed octahedra that have
two opposite equilateral faces and 6 obtuse ones whose obtuse angle is
109 degrees 28 minutes roughly. This squashing is just such as to make
the dihedral angles at the short edges be exactly 60 degrees. Each
squashed octahedron abuts regular tetrahedra at its equilateral faces
and other squashed octahedra at its obtuse ones. The short edges are
surrounded by 6 squashed octahedra, and the long ones by two squashed
octahedra and one regular tetrahedron.
The vertex figure, dual to the triakis truncated tetrahedron, is a
"tricated" (trigonally-truncated) triakis tetrahedron. I spent much of
yesterday trying to make a model of part of this tessellation. Thanks
again for introducing me to it! John Conway
John Berglund
>the most natural tessellation corresponding to any discrete set of
>think the best interpretation is that you were asking about the
>Voronoi tessellation of the diamond, which in fact I did know - its
>cells are "triakis truncated tetrahedra".
>However, I realised that I didn't know the Delaunay tessellation,
>which is into regular tetrahedra and squashed octahedra that have
>two opposite equilateral faces and 6 obtuse ones whose obtuse angle is
>109 degrees 28 minutes roughly.
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John Conway
So is apparently infinite. I'd call it a doubly-infinite
dodecagonal prism. Anyway, since it's not a lattice, the concept
of Voronoi cell doesn't apply to it.
John Conway
John Conway
You did astonishingly well!
The diamond atoms form two copies of the face-centered cubic lattice,
whose Delaunay cells are alternating tetrahedra and octahedra. Let me
call them the green and yello lattices (after the colors in my model).
Then we have the following "concentricities":
Green vertex, uptetrahedron, octahedron, downtetrahedron
Yellow uptetra, octa , downtetra, vertex.
JHC