Questions tagged [crystallography]

This tag is for questions on mathematical crystallography.

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Where are the centered trigonal lattices among the 14 Bravais lattices?

I have identified the missing entries in the Bravais lattice table for seven crystal families as the already existing entries int the table. However I'm unable to do so just for the trigonal crystal ...
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20 views

Understanding Screw Axes in Crystallographic Space Groups

I'm having trouble understanding how a 'screw axis' actually acts as on $\mathbb{E}^3$ as an element of a space group. Everything I talk about here will be in 3 dimensions. nLab defines a space (or ...
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26 views

Why is the dimension of a crystallographic group unique?

The algebraic definition of a crystallographic group goes as follows: If a group $\Gamma$ fits into a short exact sequence $$0 \to \mathbb{Z}^n \overset{i}{\to} \Gamma \overset{p}{\to} G \to 1$$ such ...
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51 views

Information content of two bits

Let $X \in\{0,1\}$ and $Y \in\{0,1\} $ be two uniformly distributed bits. Let $B$ be an arbitrary random variable such that $I(X:B)=0$, $I(Y:B)=0$, and $I(X \oplus Y:B)=0$, then is it true that $I(X,Y:...
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28 views

Can I combine lattices (group) and Minkowski's Theorem with crystallography?

I am studying computer science and have an interest in material science. sadly I am only a layman in mathematics and physics. but I want to understand those better to be able to use them more freely ...
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19 views

Rotation of co-ordiante axes for a cubic crystal: Matrix rotation

When carrying out a rotation of the co-ordinate axes, x1 x2 x3 to say x1' x2' x3', When dealinf with the rotation of axes about a vector (using an orthogonal rotation matrix R), I was taught to use ...
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30 views

Are point groups subgroups of the Orthogonal group?

I have been reading D.R. Farkas’ “Crystallographic groups and their mathematics“, which seems like a reference introduction to the subject. In it, point groups are defined as the quotient of the ...
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2answers
29 views

How does one generate an image of a sine wave with (h,k) indices (2,2)?

I am trying to generate an image of a gradient that oscillates $h$ times along the $x$-axis and $k$ times along the $y$-axis. As an example, this image I generated below oscillates $5$ times along ...
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126 views

Crystallography Restriction theorem

I'm trying to understand the Crystallography Restriction theorem, but most proofs I have found include some assumptions that are non-obvious to me. For example, http://mathworld.wolfram.com/...
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124 views

Projecting a 3D lattice to 2D lattice considering a given plane

I have a lattice in 3D with $a$, $b$ and $c$ being its unit vectors. Say we have a simple cubic crystal then $a = (1, 0, 0)$, $b = (0, 1, 0)$ and $c = (0, 0, 1)$ are its building blocks. In principle, ...
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30 views

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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515 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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32 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
141 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
91 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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299 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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35 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
51 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
88 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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2answers
76 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
37 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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34 views

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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267 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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206 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
3k 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 ...
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69 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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94 views

“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 ...