Paradoxical flexibility: frameworks and polyhedra
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21240%2F24%3A00380324" target="_blank" >RIV/68407700:21240/24:00380324 - isvavai.cz</a>
Result on the web
<a href="https://csgg2024.zcu.cz/komplet.pdf" target="_blank" >https://csgg2024.zcu.cz/komplet.pdf</a>
DOI - Digital Object Identifier
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Alternative languages
Result language
čeština
Original language name
Paradoxní pohyblivost: realizace grafů a mnohostěny
Original language description
One of the main questions of rigidity theory is whether a bar-joint framework, which is a graph with a realization of its vertices in the d-dimensional space, allows a continuous deformation preserving the distances between adjacent vertices. If yes, the framework is called flexible, otherwise rigid. For a fixed graph, either all generic frameworks are rigid, or all generic ones are flexible. However, non-generic realizations might behave differently yielding for instance paradoxical motions. A few years ago, we have characterized the existence of a (non-generic) flexible realization in the plane for a given graph in terms of special edge colorings, called NAC-colorings. Here we summarize this surprising interplay between combinatorics and geometry and its various extensions shall be presented. We focus also on polyhedra with triangular faces, which can be considered as bar-joint frameworks in the 3-space. In particular, we mention a new result on the smallest flexible polyhedron without self-intersections.
Czech name
Paradoxní pohyblivost: realizace grafů a mnohostěny
Czech description
One of the main questions of rigidity theory is whether a bar-joint framework, which is a graph with a realization of its vertices in the d-dimensional space, allows a continuous deformation preserving the distances between adjacent vertices. If yes, the framework is called flexible, otherwise rigid. For a fixed graph, either all generic frameworks are rigid, or all generic ones are flexible. However, non-generic realizations might behave differently yielding for instance paradoxical motions. A few years ago, we have characterized the existence of a (non-generic) flexible realization in the plane for a given graph in terms of special edge colorings, called NAC-colorings. Here we summarize this surprising interplay between combinatorics and geometry and its various extensions shall be presented. We focus also on polyhedra with triangular faces, which can be considered as bar-joint frameworks in the 3-space. In particular, we mention a new result on the smallest flexible polyhedron without self-intersections.
Classification
Type
D - Article in proceedings
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/GF22-04381L" target="_blank" >GF22-04381L: Paradoxical flexibility of frameworks</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Article name in the collection
10th Czech-Slovak conference on geometry and graphics 2024
ISBN
978-80-8208-145-2
ISSN
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e-ISSN
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Number of pages
10
Pages from-to
29-38
Publisher name
Vydavateľstvo Slovak Chemistry Library
Place of publication
Bratislava
Event location
Plzeň
Event date
Sep 9, 2024
Type of event by nationality
WRD - Celosvětová akce
UT code for WoS article
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