Re: Why no pentagon crystal lattices?

In sci.physics, Eric Gisse
<jowr.pi@xxxxxxxxx>
wrote
on 27 May 2007 13:58:55 -0700
<1180299535.063771.170210@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>:
On May 27, 8:21 am, hetware <massl...@xxxxxxxxxxxx> wrote:
This question appears in the problem supplement to the Feynman
Lectures: "Can you explain why there are no crystals which have the shape
of a regular pentagon? (Triangles, squares, and hexagons are common in
crystal forms.)"

I can flail my hands about and say something like, 'there are no
arrangements of pentagons which would result in a minimal energy
configuration', but that seems like begging the question. Is there a
geometrical explanation for this?

Ghost has given you the correct answer.

--http://www.vho.org/GB/c/DC/gcgvcole.htmlhttp://www.vho.org/GB/Books/dth/http://www.germarrudolf.com/http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm



There are, however, some interesting planar coverings if one is not
required to use a single shape. The pentagon can tesselate if it
uses a rhombus in concert, for example; a heptagon can tesselate with a
pentagon, though the latter is not quite regular. Of course one can
also cheat in a number of ways, by assembling shapes in a tesselable
geometry (box, triangle, hexagon) and repeating it; one can even get
creative and randomly rotate the shapes in some cases.

http://education.ti.com/educationportal/activityexchange/download_file.jsp?cid=US&fileurl=Math%2FGeometry%2F4600%2FGeoInvestigations_Voyage_Cabri_Act06.pdf

I'll admit I'm not sure why a pentagonal crystal cannot
fabricate itself from a seed, though (presumably that has
to do with atomic bonds and/or basic chemistry), but if
one assumes a grid of such seed crystals, there would have
to be gaps.

--
#191, ewill3@xxxxxxxxxxxxx
Windows. Because it's not a question of if.
It's a question of when.

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