All Questions
Tagged with rigidity discrete-geometry
7 questions
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
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
1
vote
0
answers
149
views
Ways of proving that a framework is locally rigid
Given a (bar-and-joint) framework/linkage, I would like to know what are possible ways of showing that the framework is locally rigid. Also, what is known about the computational complexity of ...
- 1,305
0
votes
0
answers
127
views
Frameworks in general position that are locally rigid but not infinitesimally rigid
The classical theorem of Asimow and Roth says that for a generic framework (i.e., coordinates of the nodes are algebraically independent), local rigidity and infinitesimal rigidity are equivalent. I ...
- 1,305
0
votes
0
answers
90
views
Which polytopes can be folded to an edge?
While playing with bar-and-joint linkages, I noticed that the skeleton of a regular 3-dimensional cube can be folded to a single edge (this can be achieved by first flexing the cube to bring it to a ...
- 1,305
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
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 ...
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