All Questions
Tagged with euclidean-geometry gr.group-theory
16
questions
7
votes
1
answer
1k
views
An unpublished calculation of Gauss and the icosahedral group
According to p. 68 of Paul Stackel's essay "Gauss as geometer" (which deals with "complex quantities with more than two units") , Gauss calculated the coordinates of the vertices ...
- 1,969
12
votes
2
answers
1k
views
Group generated by two irrational plane rotations
What groups can arise as being generated by two rotations in $\mathbb R^2$ by angles $\not \in \mathbb Q\pi$?
If the centers of the rotations coincide, then the rotations commute and generate some ...
- 1,267
59
votes
4
answers
7k
views
Is orientability a miracle?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$This question is prompted by a recent highly-upvoted question, Conceptual reason why the sign of a permutation is well-defined? The responses made ...
- 80.3k
5
votes
1
answer
306
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Embedding an icosahedron
A transitive set in $\mathbf{R}^n$ is a finite set with a transitive group of symmetries. I want to understand how subsets of a transitive set constrain the group.
Let me start with the example of a ...
- 9,215
3
votes
0
answers
70
views
On isospectral planar domains (and a paper by Buser, Conway, Doyle and Semmler)
I have never seen a short, elegant way (from the viewpoint of a non-topologist) which constructs isospectral planar domains from Sunada group triples, although essentially those triples live at the ...
- 4,503
18
votes
2
answers
1k
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Emergence of the orthogonal group
Do we know what mathematician first considered, and perhaps named, what we call the group $\mathrm O(n)$, or $\mathrm{SO}(n)$, for some $n>3$?
I mean it specifically as group (not Lie algebra) ...
- 29.6k
6
votes
0
answers
366
views
Symmetry group and irreducible representation
Let $S$ be a bounded geometric shape in the Euclidean space $E=\mathbb{R^n}$. Assume that the origin of $E$ is a fixed point of every element of the symmetry group $G(S)$ of $S$, and assume that $G(S) ...
- 26.2k
2
votes
1
answer
243
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Even Isometries in neutral Geometry
Consider a Hilbert plane as in Hartshorne's 'Euclid and beyond' (axiomatic geometry), and its group of isometries f or 'rigid motion' generated by line reflections. Call f 'even' if it is the product ...
1
vote
0
answers
179
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Cocompact (finite covolume) lattices in euclidean groups
1) Is there a classification of cocompact ( or finite co-volume) lattices in Euclidean groups E(n)( motions of Euclidean space) ( especially in dimensions 2,3,4)?
2) Also what is (if any) the ...
- 11
7
votes
0
answers
313
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Status of an open question in Artin's "Geometric Algebra"
In Artin's book "Geometric Algebra", Chapter II, the author states some axioms for geometry (section 1) and then begins to prove some results about the symmetries of the geometry (section 2).
The ...
- 501
2
votes
0
answers
166
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Which Coxeter groups can be realized as affine reflection groups?
Every affine reflection group has a Coxeter presentation (https://en.wikipedia.org/wiki/Reflection_group#Relation_with_Coxeter_groups). How do you tell which Coxeter presentations arise from affine ...
- 2,100
3
votes
1
answer
161
views
Symmetry group for the frame bundle of a G-space
Let $Q$ be a smooth manifold, and let $G$ be a Lie group which acts smoothly on $Q$ on the left.
Question 1: does the group $G$ act naturally on the tangent bundle $TQ \to Q$?
My motivation here is ...
- 8,422
9
votes
4
answers
939
views
Applications of n-dimensional crystallographic groups
I would like to know what are the applications of the theory of $n$-dimensional crystallographic groups (aka space groups)
1) in mathematics
2) outside of mathematics,
besides the applications to $...
4
votes
0
answers
250
views
Finite subgroups of the unimodular group
This is related to this MO question (and others as well).
Hoping that this will not turn out to be too broad, I would like to know about the 'state of the art' of:
1) The problem of classifying ...
6
votes
4
answers
2k
views
Isomorphic but non-conjugate subgroups of $GL(n,\mathbb{Z})$ ?
I've been asked some questions by a friend interested in crystallography, and the following questions (I'm not an expert) came spontaneous to me:
1) Are there two finite subgroups $P,P'\subset\...
- 23.1k