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Using symmetry to generate solutions to the Helmholtz equation inside an equilateral triangle

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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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Nathaniel Stambaugh, Mark Semon

Classical Physics · Submitted 29 Jun 2013

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

The authors' main claim We prove that every solution of the Helmholtz equation within an equilateral triangle, which obeys the Dirichlet conditions on the boundary, is a member of one of four symmetry classes.

how it is stated · asserted 1.00 read the record →

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arXiv ID1307.1023 (version 2)
Pith IDE5XBIAWX, Pith's permanent record of this paper and its review
CitationMark Semon, Nathaniel Stambaugh. "Using symmetry to generate solutions to the Helmholtz equation inside an equilateral triangle." arXiv:1307.1023 (2013). https://pith.science/paper/E5XBIAWX
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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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,608 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.16 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.72 · compares with prior results 0.14 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We prove that every solution of the Helmholtz equation within an equilateral triangle, which obeys the Dirichlet conditions on the boundary, is a member of one of four symmetry classes. We then show how solutions with different symmetries, or different energies, can be generated from any given solution using symmetry operators or a differential operator derived from symmetry considerations. Our method also provides a novel way of generating the ground state solution (that is, the solution with the lowest energy). Finally, we establish a correspondence between solutions in the equilateral and (30,60, 90) triangles.

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 every solution of the Helmholtz equation within an equilateral triangle, which obeys the Dirichlet conditions on the boundary, is a member of one of four symmetry classes. result 0.96 0.94 0.71 asserted 1.00
s1 We then show how solutions with different symmetries, or different energies, can be generated from any given solution using symmetry operators or a differential operator derived from symmetry considerations. method 0.67 0.94 0.15 asserted 1.00
s2 Our method also provides a novel way of generating the ground state solution (that is, the solution with the lowest energy). contribution 0.70 0.94 0.11 asserted 0.99
s3 Finally, we establish a correspondence between solutions in the equilateral and (30,60, 90) triangles. result 0.73 0.93 0.10 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.84 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • 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 every solution of the Helmholtz equation within an equilateral triangle, which obeys the Dirichlet conditions on the boundary, is a member of one of four symmetry classes. We then show how solutions with different symmetries, or different energies, can be generated from any given solution using symmetry operators or a differential operator derived from symmetry considerations. Our method also provides a novel way of generating the ground state solution (that is, the solution with the lowest energy). Finally, we establish a correspondence between solutions in the equilateral and (30,60, 90) triangles.

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