added an Idea-section with a reference to the Erlanger program

]]>added pointer to

- Peter Engel,
*Geometric Crystallography – An Axiomatic Introduction to Crystallography*, D. Reidel Publishing (1986) $[$doi:10.1007/978-94-009-4760-3$]$

where Bieberbach’s theorem is 3.1

]]>also added pointer to

- Leonard S. Charlap,
*Bieberbach Groups and Flat Manifolds*, Springer (1986) $[$doi:10.1007/978-1-4613-8687-2$]$

where it’s Theorem I 3.1.

]]>have added (here) one more sentence to the paragraph about “Bieberbach’s first theorem”, now mentioning the historically earlier, less explicit, definition of crystallographic groups.

(One could write this up more systematically than what we have here now, but I’ll leave it as is for the time being.)

]]>I have added publication data to

- Alejandro Tolcachier,
*Holonomy groups of compact flat solvmanifolds*, Geometriae Dedicata**209**(2020) 95–117 $[$arXiv:1907.02021, doi:10.1007/s10711-020-00524-8$]$

which used to be the entry’s main reference for “Bieberbach’s theorem” (here).

On this theorem I have now added also pointer to Farkas 81, Thm. 14,

and I have added Bieberbach’s original articles:

Ludwig Bieberbach,

*Über die Bewegungsgruppen des $n$ dimensionalen Euklidischen Raumes mit einem endlichen Fundamentalbereich*, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1910) 75-84 $[$dml:58754$]$Ludwig Bieberbach,

*Über die Bewegungsgruppen der Euklidischen Räume (Erste Abhandlung)*, Mathematische Annalen**70**(1911) 297–336 $[$doi:10.1007/BF01564500$]$Ludwig Bieberbach,

*Über die Bewegungsgruppen der Euklidischen Räume (ZweiteAbhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich*, Mathematische Annalen**72**(1912) 400-412 $[$doi:10.1007/BF01456724$]$

The first one of these makes explicit something at least close to what people attribute to Bieberbach, around its item III.

Haven’t dug out yet Frobenbius’s old article on the matter.

]]>typo: Sehoenflies -> Schoenflies

Mark John Hopkins

]]>If I follow, it should follow: compact + discrete = finite.

]]>Not sure if in addition we need to require the quotient group (the point group) to be finite, it feels like this should follow.(?)

]]>Thanks for catching. This should have been “discrete” of course, – have fixed it now.

]]>Perhaps a semidirect product of the lattice translation group (as normal subgroup) with a finite point group generated by reflections? Just guessing.

]]>It’s nice to see a mathematically precise treatment of space groups, aka crystallographic groups. But I don’t think this can be right:

Equivalently, a crystallographic group on a Euclidean space $E$ is a finite subgroup $S \subset Iso(E)$ of the isometry group of $E$ (its Euclidean group) that contains a lattice $N \subset E \subset Iso(E)$ of translations as a normal subgroup $N \subset S$, such that the corresponding quotient group, called the

point groupof the crystallographic group, is a subgroup $G \coloneqq S/N \;\subset\; O(E)$ of the orthogonal group.

How can $E$ be finite and contain a lattice of translations? Besides, crystallographic groups just aren’t finite.

I would just delete “finite” but I’m thinking there should be some other word here. So, I’ll just leave a warning.

]]>Okay, thanks for saying. True, the text didn’t say this well. I have now adjusted the text in the entry to bring this out more properly.

]]>Thanks a lot for clarifying that!

Sami Siraj-Dine ]]>

Thanks for your reply. Now I see more clearly what you are saying.

First regarding the “free”-ness condition in the normal subgroup: That’s just part of the very definition of what a discrete translation group should be. It just means that the translation group is isomorphic to $\mathbb{Z}^n$, and that’s what we want by definition when speaking about crystals.

Regarding your worries about the quotient: While I don’t know off the top of my head if the quotient of a crystallographic group by *any* normal translations subgroup is always guaranteed to be a subgroup of the orthogonal groups (you seem to suspect it is not), this is also not what is being claimed in the entry, is it. In the entry we demand explicitly the horizontal inclusions shown in the diagram reproduced now:

Or maybe that point needs to be stated better? If that’s what it is, I’d be happy to try to improve the text in this regard.

]]>Thank you for your answer!

I’m sorry, I am not sure I understand you properly. I tried reading the references you provide, and they support the definition you provide on the site, but I am afraid I am not able to wrap my head around the implications of requiring $N$ to be a free normal abelian subgroup (mostly whether “free” is a constraining hypothesis or not). But the question I am asking fortunately should not need this concept, if I got things right.

Let me rephrase my previous post to be sure we agree on the question, and that I am not making a mistake in my thinking.

Taking a non-symmorphic space group $S$ means that there is a symmetry operation $(R|a)$ that contains both an orthogonal $R$ and a translation part $t_a : x \mapsto x+a$, where the translation $t_a$ does not belong to the lattice $N$ (in a simple counter-example I saw, $a$ was half of a lattice vector, and that’s what I meant by fractional, it was not very clear, my bad).

Hence, the quotient $S/N$ would have an equivalence class corresponding to $(R|a)$, which is not an element of the orthogonal group. Is that correct?

That seems to be in contradiction with the definition of point groups I’ve read in crystallography physics books which requires them to be a subgroup of the orthogonal group, and to the inclusion that is written in the short exact sequences on the page.

It is probably a misunderstanding on my part, apologies if that is the case.

Best regards, Sami Siraj-Dine

]]>Thanks for your comment. If there is an error I’ll want to fix it.

Let me see…

Hm, checking again, it looks to me like the statement in the entry agrees with the terminology associated with Bieberbach’s first theorem; fully explicitly so in the way it is stated in the recent arXiv:1907.02021, Theorem 2.3 and paragraph below (on p. 4).

Also arXiv:1208.5055 seems to agree, less authoritatively but all the more seemingly regarding it as standard: around (0.2) on p. 3.

Finally, I wouldn’t think that what you mention regarding point groups possibly containing screw-axis transformaions etc. is in contradiction to this statement. I suppose the “fractional” in your “fractional lattice vectors” signifies this.

But if you think I am missing your point, please say so and let’s try to sort this out.

]]>First, great site. It’s been very useful to me in many situations.

The definition of the point group as the quotient S/N seems to me to be inequivalent to the standard definition (see Wiki for ex), because the point group usually is the the biggest subgroup of S to keep one point fixed, which, if I understand correctly, is only true if the associated short exact sequence splits.

In the case of a non-symmorphic space group, S might include screw-axis transformations and glide reflections, which can contain translations of fractional lattice vectors, hence should remain in the quotient. I could be missing something, I am fairly new to this topic.

Best regards,

Sami Siraj-Dine. ]]>

The term ’binary dihedral group’ shows up on the 3d page, but not given in Coxeter notation. Coxeter notation seems confusing.

]]>So I suppose I can easily check by direct inspection that the following are crystallographics groups $N \rtimes G$ in 4d, with $N = \mathbb{Z}^4$ the canonical translational subgroup of $\mathbb{R}^4$ and the point group $G$ acting on $\mathbb{R}^4\simeq_{\mathbb{R}} \mathbb{H}$ via their inclusion as subgroups of $Sp(1)$:

$N \rtimes G = ...$

$\mathbb{Z}^4 \rtimes \mathbb{Z}_2$

$\mathbb{Z}^4 \rtimes \mathbb{Z}_4$

$\mathbb{Z}^4 \rtimes Q_8$

So these must appear somewhere in these lists. But where? I must be missing something.

]]>Yes, it should be. But I have trouble parsing the entries.

Does $D_n$ denote the dihedral group? Does the binary dihedral group appear anywhere?

]]>Is it just a matter of going through a list? The list in ’Summary’ tells you which are non-crystallographic.

]]>added pointer to

- D. Weigel, T. Phan and R. Veysseyre,
*Crystallography, geometry and physics in higher dimensions. III. Geometrical symbols for the 227 crystallographic point groups in four-dimensional space*, Acta Cryst. (1987). A43, 294-304 (doi:10.1107/S0108767387099367)

am trying to find out which of the finite subgroups of SU(2) in the D- and E-series arise, via their canonical action on $\mathbb{C}^2 \simeq_{\mathbb{R}} \mathbb{R}^4$, as point groups of crystallographic groups in 4d.

]]>started to add various references.

Added a long quote from

- E. V. Chuprunov, T. S. Kuntsevich,
*$n$-Dimensional space groups and regular point systems*, Comput. Math. Applic. Vol. 16, No. 5-8, pp. 537-543, 1988 (doi:10.1016/0898-1221(88)90243-X)

on the history and current scope of the classification.

Not done yet, but need to interrupt now.

]]>expanded the Idea/Definition-section, in particular added this diagram:

$\array{ & 1 && 1 \\ & \downarrow && \downarrow \\ {\text{normal subgroup} \atop \text{lattice of translations}} & N &\subset& E & {\text{translation} \atop \text{group}} \\ & \big\downarrow && \big\downarrow \\ {\text{crystallographic} \atop \text{group}} & S &\subset& Iso(E) & {\text{Euclidean} \atop \text{isometry group}} \\ & \big\downarrow && \big\downarrow \\ {\text{point} \atop \text{group}} & G &\subset& O(E) & {\text{orthogonal} \atop \text{group}} \\ & \downarrow && \downarrow \\ & 1 && 1 }$Then I added the remark (here) that the normality of the subgroup $N$ is what makes the action of the point group $G$ descend to the torus $E/N$.

Is there a good name for these tori $E/N$ equpped with actions of Euclidean point groups $G$? They want to be called “representation tori” to go along with representation spheres, but I guess nobody says that.

]]>