RECORD
Goldberg, Fuller, Caspar, Klug and Coxeter and a general approach to local symmetry-preserving operations
Combinatorics · Submitted 8 May 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We will sketch the different approaches and generalize Goldberg's approach to a systematic one encompassing all local symmetry-preserving operations on polyhedra.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
Cubic polyhedra with icosahedral symmetry where all faces are pentagons or hexagons have been studied in chemistry and biology as well as mathematics. In chemistry one of these is buckminsterfullerene, a pure carbon cage with maximal symmetry, whereas in biology they describe the structure of spherical viruses. Parameterized operations to construct all such polyhedra were first described by Goldberg in 1937 in a mathematical context and later by Caspar and Klug -- not knowing about Goldberg's work -- in 1962 in a biological context. In the meantime Buckminster Fuller also used subdivided icosahedral structures for the construction of his geodesic domes. In 1971 Coxeter published a survey article that refers to these constructions. Subsequently, the literature often refers to the Goldberg-Coxeter construction. This construction is actually that of Caspar and Klug. Moreover, there are essential differences between this (Caspar/Klug/Coxeter) approach and the approaches of Fuller and of Goldberg. We will sketch the different approaches and generalize Goldberg's approach to a systematic one encompassing all local symmetry-preserving operations on polyhedra.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Cubic polyhedra with icosahedral symmetry where all faces are pentagons or hexagons have been studied in chemistry and biology as well as mathematics. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s1 | In chemistry one of these is buckminsterfullerene, a pure carbon cage with maximal symmetry, whereas in biology they describe the structure of spherical viruses. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s2 | Parameterized operations to construct all such polyhedra were first described by Goldberg in 1937 in a mathematical context and later by Caspar and Klug -- not knowing about Goldberg's work -- in 1962 in a biological context. | background 1.00 | 0.03 | 0.04 | asserted 1.00 |
| s3 | In the meantime Buckminster Fuller also used subdivided icosahedral structures for the construction of his geodesic domes. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s4 | In 1971 Coxeter published a survey article that refers to these constructions. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s5 | Subsequently, the literature often refers to the Goldberg-Coxeter construction. | background 0.98 | 0.05 | 0.04 | asserted 0.99 |
| s6 | This construction is actually that of Caspar and Klug. | result 0.55 | 0.28 | 0.23 | asserted 1.00 |
| s7 | Moreover, there are essential differences between this (Caspar/Klug/Coxeter) approach and the approaches of Fuller and of Goldberg. | result 0.46 | 0.42 | 0.19 | asserted 1.00 |
| s8 | We will sketch the different approaches and generalize Goldberg's approach to a systematic one encompassing all local symmetry-preserving operations on polyhedra. | contribution 1.00 | 0.81 | 0.47 | asserted 0.83 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
Cubic polyhedra with icosahedral symmetry where all faces are pentagons or hexagons have been studied in chemistry and biology as well as mathematics. In chemistry one of these is buckminsterfullerene, a pure carbon cage with maximal symmetry, whereas in biology they describe the structure of spherical viruses. Parameterized operations to construct all such polyhedra were first described by Goldberg in 1937 in a mathematical context and later by Caspar and Klug -- not knowing about Goldberg's work -- in 1962 in a biological context. In the meantime Buckminster Fuller also used subdivided icosahedral structures for the construction of his geodesic domes. In 1971 Coxeter published a survey article that refers to these constructions. Subsequently, the literature often refers to the Goldberg-Coxeter construction. This construction is actually that of Caspar and Klug. Moreover, there are essential differences between this (Caspar/Klug/Coxeter) approach and the approaches of Fuller and of Goldberg. We will sketch the different approaches and generalize Goldberg's approach to a systematic one encompassing all local symmetry-preserving operations on polyhedra.
Human review
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.