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On the number of facets of three-dimensional Dirichlet stereohedra II: Non-cubic Groups

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Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Daciana Bochis, Francisco Santos

Combinatorics · Submitted 18 Apr 2002

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that Dirichlet stereohedra for non-cubic crystallographic groups in dimension 3 cannot have more than 80 facets.

how it is stated · asserted 1.00 read the record →

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arXiv IDmath/0204231 (version 1)
Pith IDD3LONYUT, Pith's permanent record of this paper and its review
CitationDaciana Bochis, Francisco Santos. "On the number of facets of three-dimensional Dirichlet stereohedra II: Non-cubic Groups." arXiv:math/0204231 (2002). https://pith.science/paper/D3LONYUT
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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.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,976 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.63 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.13 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We prove that Dirichlet stereohedra for non-cubic crystallographic groups in dimension 3 cannot have more than 80 facets. The bound depends on the particular crystallographic group considered and is above 50 only on 9 of the 97 affine conjugacy classes of them. We also construct Dirichlet stereohedra with 32 and 29 facets for a hexagonal and a tetragonal group, respectively.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove that Dirichlet stereohedra for non-cubic crystallographic groups in dimension 3 cannot have more than 80 facets. result 0.98 0.96 0.94 asserted 1.00
s1 The bound depends on the particular crystallographic group considered and is above 50 only on 9 of the 97 affine conjugacy classes of them. result 0.97 0.84 0.09 asserted 1.00
s2 We also construct Dirichlet stereohedra with 32 and 29 facets for a hexagonal and a tetragonal group, respectively. result 0.77 0.96 0.04 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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  • arXiv listing abstract v1
not examined
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  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We prove that Dirichlet stereohedra for non-cubic crystallographic groups in dimension 3 cannot have more than 80 facets. The bound depends on the particular crystallographic group considered and is above 50 only on 9 of the 97 affine conjugacy classes of them. We also construct Dirichlet stereohedra with 32 and 29 facets for a hexagonal and a tetragonal group, respectively.

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