All Questions
Tagged with rigidity mg.metric-geometry
9 questions
20
votes
0
answers
433
views
Is the dodecahedron flexible (as a polytope with fixed edge-lengths)?
Consider the (regular) dodecahedron $D\subset\Bbb R^3$. I want to continuously deform it so that throughout the deformation
it stays a convex polytope,
it stays a combinatorial dodecahedron (i.e. its ...
- 13.6k
13
votes
0
answers
378
views
Is a convex polyhedron determined by its edge lengths and angular defects?
Let's consider 3-dimensional convex polyhedra $P\subset\Bbb R^3$.
The angular defect at a vertex $v$ is $2\pi$ minus the sum of the interior angles of the incident faces at $v$.
Question:
Is a ...
- 13.6k
11
votes
1
answer
594
views
quantitative version of the rigidity of the 2-sphere
I am looking for a quantitaive version of the following theorem:
A compact surface with $K\equiv 1$ is isometric to the round sphere.
Of course I get the Berger, Brendle-Schoen Theorem which insures ...
- 914
11
votes
1
answer
654
views
How to correctly state Cauchy's rigidity theorem?
Cauchy's rigidity theorem is often stated briefly as
Any two (convex, 3-dimensional) polyhedra with pairwise congruent faces are themselves congruent.
As a more formal generalization to general ...
- 13.6k
6
votes
2
answers
168
views
Which criteria guarantee an orthogonal circuit in $\mathbb R^3$ to be rigid?
For $n\ge4$, define an orthogonal circuit or O-circuit as a closed circuit of $n$ unit segments in $\mathbb R^3$ such that any two neighboring segments form a right angle. (Physically this could be ...
- 13.4k
5
votes
3
answers
683
views
Alexandrov's generalization of Cauchy's rigidity theorem
Wikipedia states that A. D. Alexandrov generalized Cauchy's rigidity theorem for polyhedra to higher dimensions.
The relevant statement in the article is not linked to any source. The sources at the ...
- 13.6k
4
votes
2
answers
377
views
Isostatic graphs and the Henneberg conjecture
I have been reading "Combinatorial Rigidity" by Graver, Servatius and Servatius and I am interested in their chapter on rigidity in dimension $\geq$ 3. I have two questions.
What is the current ...
3
votes
5
answers
813
views
Is the following two-dimensional graph likely to be globally rigid?
Consider the two-dimensional non-planar graph $G$, with known topology and edge lengths $(r_1, r_2, ... r_N) \in R$, but unknown vertex coordinates. We further specify that:
All vertices within a ...
- 309
2
votes
0
answers
82
views
Infinitesimal rigidity vs local rigidity of isometrically immersed riemannian manifolds
I was reading the nice survey on rigidity, focusing on tensegrities by Connelly, and I'd like to know the status and feedback about a question he asks:
A theorem by Gluck and this work of Connelly, ...
- 2,041