Standard

STEFFEN’S FLEXIBLE POLYHEDRON IS EMBEDDED. A PROOF VIA SYMBOLIC COMPUTATIONS. / Александров, Виктор Алексеевич; Волокитин, Евгений Павлович.

In: Journal for Geometry and Graphics, Vol. 29, No. 1, 04.08.2025.

Research output: Contribution to journal › Article › peer-review

Harvard

Александров, ВА & Волокитин, ЕП 2025, 'STEFFEN’S FLEXIBLE POLYHEDRON IS EMBEDDED. A PROOF VIA SYMBOLIC COMPUTATIONS', Journal for Geometry and Graphics, vol. 29, no. 1. https://doi.org/10.48550/arXiv.2508.02392

APA

Александров, В. А., & Волокитин, Е. П. (2025). STEFFEN’S FLEXIBLE POLYHEDRON IS EMBEDDED. A PROOF VIA SYMBOLIC COMPUTATIONS. Journal for Geometry and Graphics, 29(1). https://doi.org/10.48550/arXiv.2508.02392

Vancouver

Александров ВА, Волокитин ЕП. STEFFEN’S FLEXIBLE POLYHEDRON IS EMBEDDED. A PROOF VIA SYMBOLIC COMPUTATIONS. Journal for Geometry and Graphics. 2025 Aug 4;29(1). doi: 10.48550/arXiv.2508.02392

Author

Александров, Виктор Алексеевич ; Волокитин, Евгений Павлович. / STEFFEN’S FLEXIBLE POLYHEDRON IS EMBEDDED. A PROOF VIA SYMBOLIC COMPUTATIONS. In: Journal for Geometry and Graphics. 2025 ; Vol. 29, No. 1.

BibTeX

@article{15806c4e698f4612aac36af24b237639,
title = "STEFFEN{\textquoteright}S FLEXIBLE POLYHEDRON IS EMBEDDED. A PROOF VIA SYMBOLIC COMPUTATIONS",
abstract = "A polyhedron is flexible if it can be continuously deformed preserving the shape and dimensions of every its face. In the late 1970{\textquoteright}s Klaus Steffen constructed a sphere-homeomorphic embedded flexible polyhedron with triangular faces and with 9 vertices only, which is well-known in the theory of flexible polyhedra. At about the same time, a hypothesis was formulated that the Steffen polyhedron has the least possible number of vertices among all embedded flexible polyhedra without boundary. A counterexample to this hypothesis was constructed by Matteo Gallet, Georg Grasegger, Jan Legersk´y, and Josef Schicho in 2024 only. Surprisingly, until now, no proof has been published in the mathematical literature that the Steffen polyhedron is embedded. Probably, this fact was considered obvious to everyone who made a cardboard model of this polyhedron. In this article, we prove this fact using computer symbolic calculations.",
keywords = "Euclidean space, flexible polyhedron, embedded polyhedron, symbolic computations",
author = "Александров, {Виктор Алексеевич} and Волокитин, {Евгений Павлович}",
year = "2025",
month = aug,
day = "4",
doi = "10.48550/arXiv.2508.02392",
language = "English",
volume = "29",
journal = "Journal for Geometry and Graphics",
issn = "1433-8157",
publisher = "Heldermann Verlag",
number = "1",

}

RIS

TY - JOUR

T1 - STEFFEN’S FLEXIBLE POLYHEDRON IS EMBEDDED. A PROOF VIA SYMBOLIC COMPUTATIONS

AU - Александров, Виктор Алексеевич

AU - Волокитин, Евгений Павлович

PY - 2025/8/4

Y1 - 2025/8/4

N2 - A polyhedron is flexible if it can be continuously deformed preserving the shape and dimensions of every its face. In the late 1970’s Klaus Steffen constructed a sphere-homeomorphic embedded flexible polyhedron with triangular faces and with 9 vertices only, which is well-known in the theory of flexible polyhedra. At about the same time, a hypothesis was formulated that the Steffen polyhedron has the least possible number of vertices among all embedded flexible polyhedra without boundary. A counterexample to this hypothesis was constructed by Matteo Gallet, Georg Grasegger, Jan Legersk´y, and Josef Schicho in 2024 only. Surprisingly, until now, no proof has been published in the mathematical literature that the Steffen polyhedron is embedded. Probably, this fact was considered obvious to everyone who made a cardboard model of this polyhedron. In this article, we prove this fact using computer symbolic calculations.

AB - A polyhedron is flexible if it can be continuously deformed preserving the shape and dimensions of every its face. In the late 1970’s Klaus Steffen constructed a sphere-homeomorphic embedded flexible polyhedron with triangular faces and with 9 vertices only, which is well-known in the theory of flexible polyhedra. At about the same time, a hypothesis was formulated that the Steffen polyhedron has the least possible number of vertices among all embedded flexible polyhedra without boundary. A counterexample to this hypothesis was constructed by Matteo Gallet, Georg Grasegger, Jan Legersk´y, and Josef Schicho in 2024 only. Surprisingly, until now, no proof has been published in the mathematical literature that the Steffen polyhedron is embedded. Probably, this fact was considered obvious to everyone who made a cardboard model of this polyhedron. In this article, we prove this fact using computer symbolic calculations.

KW - Euclidean space

KW - flexible polyhedron

KW - embedded polyhedron

KW - symbolic computations

UR - https://arxiv.org/pdf/2508.02392

U2 - 10.48550/arXiv.2508.02392

DO - 10.48550/arXiv.2508.02392

M3 - Article

VL - 29

JO - Journal for Geometry and Graphics

JF - Journal for Geometry and Graphics

SN - 1433-8157

IS - 1

ER -

ID: 71522307