On the uniqueness of collections of pennies and marbles
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911182703558656 |
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| author | Dewar, Sean Grasegger, Georg Kubjas, Kaie Mohammadi, Fatemeh Nixon, Anthony |
| author_facet | Dewar, Sean Grasegger, Georg Kubjas, Kaie Mohammadi, Fatemeh Nixon, Anthony |
| contents | In this note we study the uniqueness problem for collections of pennies and marbles. More generally, consider a collection of unit $d$-spheres that may touch but not overlap. Given the existence of such a collection, one may analyse the contact graph of the collection. In particular we consider the uniqueness of the collection arising from the contact graph. Using the language of graph rigidity theory, we prove a precise characterisation of uniqueness (global rigidity) in dimensions 2 and 3 when the contact graph is additionally chordal. We then illustrate a wide range of examples in these cases. That is, we illustrate collections of marbles and pennies that can be perturbed continuously (flexible), are locally unique (rigid) and are unique (globally rigid). We also contrast these examples with the usual generic setting of graph rigidity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_03525 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the uniqueness of collections of pennies and marbles Dewar, Sean Grasegger, Georg Kubjas, Kaie Mohammadi, Fatemeh Nixon, Anthony Combinatorics Metric Geometry 05C10, 52C25, 52C17 In this note we study the uniqueness problem for collections of pennies and marbles. More generally, consider a collection of unit $d$-spheres that may touch but not overlap. Given the existence of such a collection, one may analyse the contact graph of the collection. In particular we consider the uniqueness of the collection arising from the contact graph. Using the language of graph rigidity theory, we prove a precise characterisation of uniqueness (global rigidity) in dimensions 2 and 3 when the contact graph is additionally chordal. We then illustrate a wide range of examples in these cases. That is, we illustrate collections of marbles and pennies that can be perturbed continuously (flexible), are locally unique (rigid) and are unique (globally rigid). We also contrast these examples with the usual generic setting of graph rigidity. |
| title | On the uniqueness of collections of pennies and marbles |
| topic | Combinatorics Metric Geometry 05C10, 52C25, 52C17 |
| url | https://arxiv.org/abs/2307.03525 |