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Questions tagged [crystallography]

This tag is for questions on mathematical crystallography.

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Why is $(A\rtimes_\alpha G)\cong (A\otimes K(L^2(G)))^{\alpha\otimes Ad\rho}$?

I found a very strange isomorphism that I did not know have to prove. Let $G$ be a finite discrete group. Let $Ad\rho_g(T)=\rho_g T\rho_g^*$ for all $T\in K(L^2(G))$ with $g\in G$ then: \begin{align*}...
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Difference between the wall-paper groups pg, pm and cm

I understand that the main difference between cm and pg and pm is that both given a line $l$ with a mirror symmetry $\rho$ in this line then either: \begin{align} t'=\rho t+t\text{ or }t'\neq\rho t+t \...
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1answer
58 views

How can we calculate the volume of a BCC Wigner-Seitz Cell? (Based on a imaginary cube)

Hello. As you can see in the picture, there’s this shape and shape’s surface consists of 6 square and 8 hexagon parts and I would like to know its volume but I don’t know where to start. The only ...
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30 views

Point groups where the tensor square of two-dim. irreps (over $\mathcal{O}(3)$) does not contain a two-dim. irrep in its decomposition

Which are the point groups where the tensor square of a two dimensional irreducible representation does not decompose into a sum that contains a two-dimensional irreducible representation? For ...
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1answer
105 views

Is there a “the” definition of a lattice?

I'm writing a paper about lattices in the complex plane, and while trying to explain the crystallographic restriction theorem, I realized that I never actually defined what a 'lattice' is. The ...
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1answer
65 views

How would I go about learning Quasi Crystal Mathematics?

I am trying to find out a path to learn quasicrystal mathematics. My math knowledge is only pretty basic however I am very much willing to work the way through up even undergrad parts and above if I ...
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2answers
178 views

What's wrong with this Penrose pattern?

I programmed the Penrose tiling by projecting a portion of 5D lattice to 2D space, by the "cut and project" method described in Quasicrystals: projections of 5-D lattice into 2 and 3 dimensions, H. ...
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1answer
31 views

Describe/name a cube in different poses

I have a relatively simple question: are there different names for a cube in different poses? Specifically, I need to distinguish a regular cube (i.e. a cube lying flat in the XY plane and parallel ...
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1answer
32 views

Setting up a mesh inside a parallelepiped

This is a spatial geometry/linear algebra question with direct applications in crystallography. However, knowledge of crystallography is NOT necessary to answer the question. I have the defining ...
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1answer
64 views

3D Cosine Function for an HCP Lattice

I would like to construct a 3D cosine function, $f$, which is scaled to the range $f\in[-1,1]$, and which has maxima which coincide with the HCP lattice. For example, here are the analogous 3D cosine ...
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Invariant Description of Displacements

Consider a collection of points, or particles (e.g., a crystal lattice). Is there a way to mathematically describe the Cartesian displacements of these points from their initial positions, such that ...
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38 views

How to create an irreducible representation of nonsymmorphic space groups, and go through an example

I want to create irreducible representations of nonsymmorphic space groups, specifically 2D space groups, pg, pmg, pgg, p4g. I've been reading some resources but the way explain them are too abstract ...
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2answers
37 views

Transformations commuting in 3D (crystallography)

I'm studying symmetry operations and trying to show that the symmetry operations of a crystal* are closed, so they form a group. A friend of mine said that these operations commute but I can't justify ...
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1answer
35 views

Line coordinates from plane intersection

Say we are given two planes denoted by their (h1, k1, l1) and (h2, k2, l2) Miller's indices. How to find the equation which will represent the line that is the intersection of these two planes?
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orientation-preserving subgroup of crystallographic space group

Let $G$ be one of the 230 crystallographic space groups and $G_0$ be the subgroup of orientation-preserving elements. Is it always true that $G_0$ is also one of the 230 crystallographic space groups? ...
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222 views

Riemann-Zeta Zeros and Quasicrystals

I came across quasicrystals in the Wikipedia page for the Riemann Hypothesis and then followed the references. On page 215 of Birds and Frogs Dyson makes the claim If the Riemann hypothesis is true,...
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2answers
150 views

Who coined the term “crystallographic root system”?

Who coined the term "crystallographic root system" and when? In particular is there a connection to applied 3D crystallography? It does not seem to be Killing or Cartan's terms (so presumably after ...
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2answers
2k views

What is a “convex” polyhedron?

I was doing a spot of light reading (crystallography), when the term "convex" polyhedron came up in a a section (very prominently) in conjunction with something else called the "Euler characteristic"....
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47 views

Definition of a lattice in an affine space

Studying crystals for solid state physics I figured that we must be able to define a crystal as an at most countable subset $C\subset M$ where $M$ is an affine space modeled after a vector space $V$ ...
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“Explaining” a coincidental connection between Pascal's triangle and symmetries of lattices

The $1$s on the edges of Pascal's triangle appear infinitely many times in the array. One number ‒ just one ‒ appears exactly once. Infinitely many have multiplicity $2.$ Infinitely many have ...