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Embedded flexible spherical cross-polytopes with non-constant volumes

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Alexander A. Gaifullin

Metric Geometry · Submitted 25 Jan 2015

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

The authors' main claim Hence these cross-polytopes give counterexamples to the Bellows Conjecture for spherical polyhedra.

how it is stated · asserted 1.00 read the record →

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arXiv ID1501.06198 (version 1)
Pith IDTVNWHXWM, Pith's permanent record of this paper and its review
CitationAlexander A. Gaifullin. "Embedded flexible spherical cross-polytopes with non-constant volumes." arXiv:1501.06198 (2015). https://pith.science/paper/TVNWHXWM
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 8,103 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.88 · states limitation 0.50 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.17 · compares with prior results 0.96 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We construct examples of embedded flexible cross-polytopes in the spheres of all dimensions. These examples are interesting from two points of view. First, in dimensions 4 and higher, they are the first examples of embedded flexible polyhedra. Notice that, unlike in the spheres, in the Euclidean spaces and the Lobachevsky spaces of dimensions 4 and higher, still no example of an embedded flexible polyhedron is known. Second, we show that the volumes of the constructed flexible cross-polytopes are non-constant during the flexion. Hence these cross-polytopes give counterexamples to the Bellows Conjecture for spherical polyhedra. Earlier a counterexample to this conjecture was built only in dimension 3 (Alexandrov, 1997), and was not embedded. For flexible polyhedra in spheres we suggest a weakening of the Bellows Conjecture, which we call the Modified Bellows Conjecture. We show that this conjecture holds for all flexible cross-polytopes of the simplest type among which there are our counterexamples to the usual Bellows Conjecture. By the way, we obtain several geometric results on flexible cross-polytopes of the simplest type. In particular, we write relations on the volumes of their faces of codimensions 1 and 2.

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 construct examples of embedded flexible cross-polytopes in the spheres of all dimensions. contribution 0.93 0.97 0.26 asserted 1.00
s1 These examples are interesting from two points of view. interpretation 0.92 0.27 0.03 asserted 0.97
s2 First, in dimensions 4 and higher, they are the first examples of embedded flexible polyhedra. interpretation 0.36 0.79 0.17 asserted 1.00
s3 Notice that, unlike in the spheres, in the Euclidean spaces and the Lobachevsky spaces of dimensions 4 and higher, still no example of an embedded flexible polyhedron is known. gap 0.80 0.08 0.05 asserted 1.00
s4 Second, we show that the volumes of the constructed flexible cross-polytopes are non-constant during the flexion. result 0.93 0.96 0.39 asserted 1.00
s5 Hence these cross-polytopes give counterexamples to the Bellows Conjecture for spherical polyhedra. interpretation 0.80 0.90 0.62 asserted 1.00
s6 Earlier a counterexample to this conjecture was built only in dimension 3 (Alexandrov, 1997), and was not embedded. background 0.68 0.04 0.09 asserted 1.00
s7 For flexible polyhedra in spheres we suggest a weakening of the Bellows Conjecture, which we call the Modified Bellows Conjecture. contribution 0.91 0.89 0.08 hedged possibility 0.65
s8 We show that this conjecture holds for all flexible cross-polytopes of the simplest type among which there are our counterexamples to the usual Bellows Conjecture. result 0.92 0.95 0.18 asserted 1.00
s9 By the way, we obtain several geometric results on flexible cross-polytopes of the simplest type. contribution 0.68 0.93 0.04 asserted 0.98
s10 In particular, we write relations on the volumes of their faces of codimensions 1 and 2. result 0.92 0.94 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.

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not examined
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  • relations to other papers
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  • T body sentences
  • C citations
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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 construct examples of embedded flexible cross-polytopes in the spheres of all dimensions. These examples are interesting from two points of view. First, in dimensions 4 and higher, they are the first examples of embedded flexible polyhedra. Notice that, unlike in the spheres, in the Euclidean spaces and the Lobachevsky spaces of dimensions 4 and higher, still no example of an embedded flexible polyhedron is known. Second, we show that the volumes of the constructed flexible cross-polytopes are non-constant during the flexion. Hence these cross-polytopes give counterexamples to the Bellows Conjecture for spherical polyhedra. Earlier a counterexample to this conjecture was built only in dimension 3 (Alexandrov, 1997), and was not embedded. For flexible polyhedra in spheres we suggest a weakening of the Bellows Conjecture, which we call the Modified Bellows Conjecture. We show that this conjecture holds for all flexible cross-polytopes of the simplest type among which there are our counterexamples to the usual Bellows Conjecture. By the way, we obtain several geometric results on flexible cross-polytopes of the simplest type. In particular, we write relations on the volumes of their faces of codimensions 1 and 2.

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