Delone lattice studies in C3, the space of three complex variables

TL;DR

This paper investigates Delone scalars within the space of three complex variables, enhancing understanding of lattice reduction and classification in complex three-dimensional space.

Contribution

It introduces a study of Delone scalars specifically in ^3, linking them to lattice reduction and type determination in complex three-dimensional space.

Findings

Delone scalars are characterized in ^3

The structure of complex coordinate planes is elucidated

Implications for lattice classification are discussed

Abstract

The Delone (Selling) scalars, which are used in unit cell reduction and in lattice type determination, are studied in $\mathbb{C}^3$, the space of three complex variables. The three complex coordinate planes are composed of the six Delone scalars.

Tables

Table 1. Table 2: Phospholipase A2 unit cells

PDB id	Centering	a	b	c	$\alpha$	$\beta$	$\gamma$
1DPY	R	57.98	57.98	57.98	92.02	92.02	92.02
1FE5	R	57.98	57.98	57.98	92.02	92.02	92.02
1G0Z	H	80.36	80.36	99.44	90	90	120
1G2X	C	80.95	80.57	57.1	90	90.35	90
1U4J	H	80.36	80.36	99.44	90	90	120
2OSN	R	57.10	57.10	57.10	89.75	89.75	89.75

Full text

Note: In his later publications, Boris Delaunay used the Russian version of his surname, Delone.

{synopsis}

The space $\bf{C^{3}}$ is explained in more detail than in the original description. Boundary transformations of the fundamental unit are described in detail. A graphical presentation of the basic coordinates is described and illustrated.

1 Introduction

The scalars used by \citeasnounDelaunay1932 in his formulation of Selling reduction  [Selling1874] are (in the conventional order) $b\cdot c$, $a\cdot c$, $a\cdot b$, $a\cdot d$, $b\cdot d$, $c\cdot d$, where $d=-a-b-c$. (As a mnemonic device, observe that the first three terms use $\alpha$, $\beta$, and $\gamma$, in that order, and the following terms use $a$, $b$, $c$, in that order.)

\citeasnoun

andrews2019b chose to represent the Selling scalars in the space $\bf{S^{6}}$, {$s_{1}$, $s_{2}$, $s_{3}$, $s_{4}$, $s_{5}$, $s_{6}$} (defined in the order above), as a way to create a metric space for the measurement of the distance between lattices.  \citeasnounandrews2019b also considered the representation of this space as the space of three complex dimensions, $\bf{C^{3}}$ or $\{c_{1}$, $c_{2}$, $c_{3}\}$.

In $\bf{C^{3}}$, in terms of the Selling scalars, a vector is defined as {($s_{1}$,$s_{4}$), ($s_{2}$,$s_{5}$),($s_{3}$,$s_{6}$)}, where the real and imaginary parts of each are the “opposite” scalars according to the definition of \citeasnounDelaunay1932 (see \citeasnounandrews2019a). As a mnemonic device, note that the complex components involve ($\alpha,a$), ($\beta,b$), and ($\gamma,c$). Additionally, each complex term uses all four 3-space vectors; for example, $c_{1}$ is ($b\cdot c$, $a\cdot d$).

\citeasnoun

andrews2019b considered the matrix representations of the reflections in $\bf{C^{3}}$ (and in $\bf{S^{6}}$). This paper describes the boundary transformations at the edges of the fundamental unit of $\bf{C^{3}}$. In $\bf{S^{6}}$, the fundamental unit is the all negative orthant, which contains only and all of the reduced cells. In $\bf{S^{6}}$ and $\bf{C^{3}}$, the boundaries located where any $\bf{S^{6}}$ scalar (or correspondingly in$\bf{C^{3}}$, the real or imaginary part) equals to zero. The rationale for this work was that it might lend insights into the topology of the space of lattices.

2 Notation

Complex numbers will be represented in Cartesian format $(x,y)$, where $x$ is the real part and $y$ is the imaginary part.

We will represent a vector in $\bf{C^{3}}$ by {$(x_{1},y_{1})$, $(x_{2},y_{2})$, $(x_{3},y_{3})$} as an alternative to {($s_{1}$,$s_{4}$), ($s_{2}$,$s_{5}$),($s_{2}$,$s_{6}$)}.

Next we define the operators in $\bf{C^{3}}$ that will be used in the matrix descriptions of the transformations at the boundaries of the fundamental unit.

See Table  1.

3 Matrices of boundary transformations

For the boundary at $s_{1}$: (the real component of $c_{1}$).

\begin{bmatrix}\textfrak{M}{}_{r}&0&0\\ \textfrak{P}{}_{r}&\textrm{i}\Re{}&\Re{}\\ \textfrak{P}_{r}&\textrm{i}\Im{}&\Im{}\end{bmatrix} \begin{bmatrix}\textfrak{M}{}_{r}&0&0\\ \textfrak{P}{}_{r}&\textrm{i}\Im{}&\Im{}\\ \textfrak{P}{}_{r}&\textrm{i}\Re{}&\Re{}\\ \end{bmatrix} \begin{bmatrix}\textfrak{M}{}_{r}&0&0\\ \textfrak{P}{}_{r}&\Re{}&\textrm{i}\Re{}\\ \textfrak{P}{}_{r}&\Im{}&\textrm{i}\Im{}\end{bmatrix} \begin{bmatrix}\textfrak{M}{}_{r}&0&0\\ \textfrak{P}{}_{r}&\Im{}&\textrm{i}\Im{}\\ \textfrak{P}{}_{r}&\Re{}&\textrm{i}\Re{}\\ \end{bmatrix}

For the boundary at $s_{4}$: (the imaginary component of $c_{1}$).

\begin{bmatrix}\textfrak{M}{}_{i}&0&0\\ \textfrak{P}{}_{i}&\textrm{i}\Re{}&\Re{}\\ \textfrak{P}{}_{i}&\textrm{i}\Im{}&\Im{}\end{bmatrix} \begin{bmatrix}\textfrak{M}{}_{i}&0&0\\ \textfrak{P}{}_{i}&\textrm{i}\Im{}&\Im{}\\ \textfrak{P}{}_{i}&\textrm{i}\Re{}&\Re{}\\ \end{bmatrix} \begin{bmatrix}\textfrak{M}{}_{i}&0&0\\ \textfrak{P}{}_{i}&\Re{}&\textrm{i}\Re{}\\ \textfrak{P}{}_{i}&\Im{}&\textrm{i}\Im{}\end{bmatrix} \begin{bmatrix}\textfrak{M}{}_{i}&0&0\\ \textfrak{P}{}_{i}&\Im{}&\textrm{i}\Im{}\\ \textfrak{P}{}_{i}&\Re{}&\textrm{i}\Re{}\\ \end{bmatrix}

For the boundary at $s_{2}$ (the real component of $c_{2}$):

\begin{bmatrix}\textrm{i}\Re{}&\textfrak{P}{}_{r}&\Re{}\\ 0&\textfrak{M}{}_{r}&0\\ \textrm{i}\Im{}&\textfrak{P}{}_{r}&\Im{}\end{bmatrix} \begin{bmatrix}\textrm{i}\Im{}&\textfrak{P}{}_{r}&\Im{}\\ 0&\textfrak{M}{}_{r}&0\\ \textrm{i}\Re{}&\textfrak{P}{}_{r}&\Re{}\\ \end{bmatrix} \begin{bmatrix}\Re{}&\textfrak{P}{}_{r}&\textrm{i}\Re{}\\ 0&\textfrak{M}{}_{r}&0\\ \Im{}&\textfrak{P}{}_{r}&\textrm{i}\Im{}\end{bmatrix} \begin{bmatrix}\Im{}&\textfrak{P}{}_{r}&\textrm{i}\Im{}\\ 0&\textfrak{M}{}_{r}&0\\ \Re{}&\textfrak{P}{}_{r}&\textrm{i}\Re{}\\ \end{bmatrix}

For the boundary at $s_{5}$: (the imaginary component of $c_{2}$).

\begin{bmatrix}\textrm{i}\Re{}&\textfrak{P}{}_{i}&\Re{}\\ 0&\textfrak{M}{}_{i}&0\\ \textrm{i}\Im{}&\textfrak{P}{}_{i}&\Im{}\end{bmatrix} \begin{bmatrix}\textrm{i}\Im{}&\textfrak{P}{}_{i}&\Im{}\\ 0&\textfrak{M}{}_{i}&0\\ \textrm{i}\Re{}&\textfrak{P}{}_{r}i&\Re{}\\ \end{bmatrix} \begin{bmatrix}\Re{}&\textfrak{P}{}_{i}&\textrm{i}\Re{}\\ 0&\textfrak{M}{}_{i}&0\\ \Im{}&\textfrak{P}{}_{i}&\textrm{i}\Im{}\end{bmatrix} \begin{bmatrix}\Im{}&\textfrak{P}{}_{i}&\textrm{i}\Im{}\\ 0&\textfrak{M}{}_{i}&0\\ \Re{}&\textfrak{P}{}_{i}&\textrm{i}\Re{}\\ \end{bmatrix}

For the boundary at $s_{3}$ (the real component of $c_{3}$):

\begin{bmatrix}\textrm{i}\Re{}&\Re{}&\textfrak{P}{}_{r}\\ \textrm{i}\Im{}&\Im{}&\textfrak{P}{}_{r}\\ 0&0&\textfrak{M}{}_{r}\\ \end{bmatrix} \begin{bmatrix}\textrm{i}\Im{}&\Im{}&\textfrak{P}{}_{r}\\ \textrm{i}\Re{}&\Re{}&\textfrak{P}{}_{r}\\ 0&0&\textfrak{M}{}_{r}\\ \end{bmatrix} \begin{bmatrix}\Re{}&\textrm{i}\Re{}&\textfrak{P}{}_{r}\\ \Im{}&\textrm{i}\Im{}&\textfrak{P}{}_{r}\\ 0&0&\textfrak{M}{}_{r}\\ \end{bmatrix} \begin{bmatrix}\Im{}&\textrm{i}\Im{}&\textfrak{P}{}_{r}\\ \Re{}&\textrm{i}\Re{}&\textfrak{P}{}_{r}\\ 0&0&\textfrak{M}{}_{r}\\ \end{bmatrix}

For the boundary at $s_{6}$ (the imaginary component of $c_{3}$):

\begin{bmatrix}\textrm{i}\Re{}&\Re{}&\textfrak{P}{}_{i}\\ \textrm{i}\Im{}&\Im{}&\textfrak{P}{}_{i}\\ 0&0&\textfrak{M}{}_{i}\end{bmatrix} \begin{bmatrix}\textrm{i}\Im{}&\Im{}&\textfrak{P}{}_{i}\\ \textrm{i}\Re{}&\Re{}&\textfrak{P}{}_{i}\\ 0&0&\textfrak{M}{}_{i}\\ \end{bmatrix} \begin{bmatrix}\Re{}&\textrm{i}\Re{}&\textfrak{P}{}_{i}\\ \Im{}&\textrm{i}\Im{}&\textfrak{P}{}_{i}\\ 0&0&\textfrak{M}{}_{i}\\ \end{bmatrix} \begin{bmatrix}\Im{}&\textrm{i}\Im{}&\textfrak{P}{}_{i}\\ \Re{}&\textrm{i}\Re{}&\textfrak{P}{}_{i}\\ 0&0&\textfrak{M}{}_{i}\end{bmatrix}

4 Basics

The standard representation of the identity operation is

$\bf{c^{\prime}}$ = \begin{bmatrix}1&0&0$\\$0&1&0$\\$0&0&1\\ \end{bmatrix} $\bf{c}$.

The identity in $\bf{C^{3}}$ can also be written:

$\bf{c^{\prime}}$ = \begin{bmatrix}1&0&0$\\$0&\Re{}+\textrm{i}\Im{}&0$\\$0&0&\Re{}+\textrm{i}\Im{}\\ \end{bmatrix} $\bf{c}$.

\citeasnoun

Delaunay1932 does not consider the boundary transformations in detail. However, he uses them to define the process of Selling reduction. For example in $\bf{S^{6}}$, he lists the following as one of the possible results for a transformation on $s_{1}$: {-$s_{1}$, -$s_{1}$+$s_{2}$, $s_{1}$+$s_{3}$, $s_{1}$+$s_{5}$, $s_{1}$+$s_{4}$, $s_{1}$+$s_{6}$}. The third boundary transform for $s_{1}$ implements this operation and interchanges the real part of $c_{3}$ and the imaginary part of $c_{2}$:

\begin{bmatrix}\textfrak{M}{}_{r}&0&0\\ \textfrak{P}{}_{r}&\Re{}&\textrm{i}\Re{}\\ \textfrak{P}{}_{r}&\Im{}&\textrm{i}\Im{}\end{bmatrix}

Considered in $\bf{C^{3}}$, Delone’s alternate transformation for the $s_{1}$ boundary would exchange the real of $c_{2}$ with the imaginary part of $c_{3}$. That is the fourth matrix in the list for $s_{1}$. The other two transformations for $s_{1}$ can be generated from the two we have just displayed by the ”exchange operation” [andrews2019b] applied to the second and third $\bf{C^{3}}$ coordinates. Delone did not describe the latter two transformations, perhaps because even a single transformation was adequate to implement reduction. He had already listed two.

5 Graphical display of projections

The two-dimensional nature of the three coordinates of $\bf{C^{3}}$ suggests their use for graphical display.

As an example, we use Phospholipase A2 (retrieved from the Protein Data Bank [Bernstein1977]), which has had several similar or identical structures determined [LeTrong2007]. \citeasnounandrews2019b found additional cases (see Table 2)

Below, Figure 1 shows the unit cells as reported (the centering of lattices has not been removed). The following figures show various transformations and embellishments of the reported cells.

6 Summary

The transformation matrices shown above demonstrate the considerable regularity of Selling reduction, as used by Delaunay (1932), in comparison to Niggli reduction [Niggli1928]. While all the boundaries of the non-positive orthant of $\bf{S^{6}}$ are essentially the same (and related by reflections), the boundaries formed in Niggli reduction are of multiple types, and the fundamental unit of the space representing Niggli-reduced cells ($\bf{G^{6}}$, see \citeasnounAndrews2014) is non-convex.

The matrices also display one of the unique properties of $\bf{C^{3}}$, which makes it a useful conceptual representation. For the scalar being transformed, it and its opposite scalar create a unique row in each transformation matrix. These rows contain only the minus operator and zeros, highlighting the unique relationship of that pair of scalars.

Several aspects of $\bf{C^{3}}$ are evident from inspecting the matrices. First, each boundary has four possible transformations that can be applied. Since each of the transformations at boundaries are self-inverse, they are the same transformations that would be used in the process of cell (lattice) reduction. \citeasnounDelaunay1932 and \citeasnounDelone1975 give only two choices, presumably for simplicity, omitting transformations that use the ”exchange” operator (see \citeasnounandrews2019b).

7 Availability of code

The $C^{++}$  code for $\bf{C^{3}}$ is available in github.com, in https://github.com/duck10/LatticeRepLib.git.

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Acknowledgements

Careful copy-editing and corrections by Frances C. Bernstein are gratefully acknowledged. Our thanks to Jean Jakoncic and Alexei Soares for helpful conversations and access to data and facilities at Brookhaven National Laboratory.

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Funding information

Funding for this research was provided in part by: US Department of Energy Offices of Biological and Environmental Research and of Basic Energy Sciences (grant No. DE-AC02-98CH10886; grant No. E-SC0012704); U.S. National Institutes of Health (grant No. P41RR012408; grant No. P41GM103473; grant No. P41GM111244; grant No. R01GM117126, grant No. 1R21GM129570); Dectris, Ltd.