[chimera-dev] Icosahedral lattice on a sphere

[Wei Zhang]

Hi Tom,

Thanks so much. I have found the Python file of Chimera on linux.  I
currently do not have ChimeraX yet.
I think if one version can be worked out, it is not too hard to make the
other version working.
I do not know Python.  I would need to ask my collaborator to look into
the code.

I am thinking for the cases k=0, or h=k, It is not too hard to implement a
spherical icosahedral cage.
This is the diagram for a planner icosahedron definition:
[image: Screen Shot 2020-12-09 at 9.02.59 PM.png]
We just need to think the H axis is an arc between two neighboring 5 fold
vertices, and K is another arc connecting the origin and the other nearby 5
fold vertice. So the positions of 1, 2, 3, 4 ... would be defined by angles
or length on the arc, rather than the length on the line connecting the two
5-fold points.

A quick test can be made by modifying the code of this routine (Linux
Chimera): Tools => Higher-order Structure => Icosahedron Surfaces. When the
sphere factor = 1. The subdivision can be considered to be a subdivision of
the arc that connects two 5 folds. I am not sure what is the best method to
draw the lattice lines when the sphere factor is >0 and less than 1, some
kind of interpolation function has to be developed.

I do not know how to handle the cases of skewed icosahedron either, ie. h
and k are different. It seems very difficult to define the references
points and determine how to draw those parallel lines on the spherical
surface. I am thinking we have to start with a big spherical triangle.
[image: Screen Shot 2020-12-09 at 9.56.03 PM.png]
For example, if we want to define h=3, k=2 lattice (red triangle), we need
to draw arcs on a sphere according to the purple triangle. The question is
if we think the red triangle is the outline of a spherical triangle of
three 5-fold vertices, where the vertices of the purple triangle are. The
current version of Chimera can draw any icosahedral lattice beautifully. I
think this must have been worked out for the planner cases. The python code
must have some clues. I have to think about it more.

Best regards,

Wei







On Wed, Dec 9, 2020 at 5:48 PM Tom Goddard <goddard at sonic.net> wrote:

> Hi Wei,
>
>   You are right the Chimera and ChimeraX hkcage command "sphere" option
> simply moves the cage vertices outward to lie on a sphere.  The sphere
> passes through the 12 vertices of a plain icosahedron so those 12 vertices
> don't get moved.  But the midpoint of one of the triangular faces of the
> icosahedron is at radius 0.79 (= sqrt( (1+2/sqrt(5))/3 )) if the vertices
> are at radius 1 (ie the ratio of inner icosahedron radius to outer radius
> is 0.79) so projecting a shape near the middle of a triangular icosahedron
> face will expand its size by 21% relative to if it is near an icosahedron
> vertex.
>
>   It is an interesting problem to try to reduce that distortion.  It looks
> impossible to remove it.  For that you would need an isometric map from an
> icosahedron triangle to a spherical triangle, but no such isometry exists
> since the triangle is flat and the sphere has curvature 1/R.  Let me know
> what you figure out.
>
>   The code for computing the cages in Chimera is all Python and is in your
> Chimera distribution, on Linux it is here
>
> chimera/share/IcosahedralCage/cages.py
>
> You would be better off working on the successor of Chimera, called
> ChimeraX.  If you improve hkcage we won't update Chimera as we no longer
> add new features to it since all development is on ChimeraX.  The ChimeraX
> hkcage command is in the ChimeraX distribution, on Linux it is here
>
> chimerax/lib/python3.8/site-packages/chimerax/hkcage/cage.py
>
>   Tom
>
>
>
>
> On Dec 8, 2020, at 6:58 PM, Wei Zhang <zhangwei at umn.edu> wrote:
>
> Hello,
>
> I use the command hkcage to generate an icosahedral lattice on a sphere:
>
> hkcage 5 7 sphere 1
>
> However, this only expands the strict icosahedral lattice (ie when using
> sphere=0) radially to match a spherical surface. As a result, the sizes of
> the hexagons and pentagons are not the same. The effect would be very
> dramatic if the parameter sphere is larger than 1.
>
> I am wondering if the code can be modified to generate a truly icosahedral
> lattice on a spherical surface, or if that piece of code is open to access?
>
> Thank you and best regards,
>
> Wei
>
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>
>
>
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