Geodesics on the regular tetrahedron and the cube

Diana Davis; Victor Dods; Cynthia Traub; Jed Yang

doi:10.1016/j.disc.2016.07.004

Geodesics on the regular tetrahedron and the cube

Diana Davis, Victor Dods, Cynthia Traub, Jed Yang

Mathematics

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

11 Scopus citations

Abstract

Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern–Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.

Original language	English (US)
Pages (from-to)	3183-3196
Number of pages	14
Journal	Discrete Mathematics
Volume	340
Issue number	1
DOIs	https://doi.org/10.1016/j.disc.2016.07.004
State	Published - Jan 6 2017

Funding

Our interest in this problem began at the American Mathematical Society’s Mathematics Research Communities workshop on Discrete and Computational Geometry in June 2012 at Snowbird. We thank Satyan Devadoss, Vida Dujmovic, Joseph O’Rourke, and Yusu Wang for organizing this workshop, and especially Joseph O’Rourke for introducing us to this problem. O’Rourke also made Fig. 5.6 . Travel funding for additional collaborations, during which we continued our work on this problem, was provided by the American Mathematical Society. We thank anonymous referees for suggesting additional references. Victor Dods partially supported by the 2011–2012 ARCS Fellowship. Jed Yang partially supported by NSF GRFP grant DGE-0707424 and NSF RTG grant DMS-1148634 .

Keywords

Cube
Geodesic
Regular tetrahedron
Stern–Brocot tree

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

Access to Document

10.1016/j.disc.2016.07.004

Cite this

@article{a035d8c723934ae6b59107ab3edd2993,

title = "Geodesics on the regular tetrahedron and the cube",

abstract = "Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern–Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.",

keywords = "Cube, Geodesic, Regular tetrahedron, Stern–Brocot tree",

author = "Diana Davis and Victor Dods and Cynthia Traub and Jed Yang",

note = "Funding Information: Our interest in this problem began at the American Mathematical Society{\textquoteright}s Mathematics Research Communities workshop on Discrete and Computational Geometry in June 2012 at Snowbird. We thank Satyan Devadoss, Vida Dujmovic, Joseph O{\textquoteright}Rourke, and Yusu Wang for organizing this workshop, and especially Joseph O{\textquoteright}Rourke for introducing us to this problem. O{\textquoteright}Rourke also made Fig. 5.6 . Travel funding for additional collaborations, during which we continued our work on this problem, was provided by the American Mathematical Society. We thank anonymous referees for suggesting additional references. Victor Dods partially supported by the 2011–2012 ARCS Fellowship. Jed Yang partially supported by NSF GRFP grant DGE-0707424 and NSF RTG grant DMS-1148634 . Publisher Copyright: {\textcopyright} 2016 Elsevier B.V.",

year = "2017",

month = jan,

day = "6",

doi = "10.1016/j.disc.2016.07.004",

language = "English (US)",

volume = "340",

pages = "3183--3196",

journal = "Discrete Mathematics",

issn = "0012-365X",

publisher = "Elsevier B.V.",

number = "1",

}

TY - JOUR

T1 - Geodesics on the regular tetrahedron and the cube

AU - Davis, Diana

AU - Dods, Victor

AU - Traub, Cynthia

AU - Yang, Jed

N1 - Funding Information: Our interest in this problem began at the American Mathematical Society’s Mathematics Research Communities workshop on Discrete and Computational Geometry in June 2012 at Snowbird. We thank Satyan Devadoss, Vida Dujmovic, Joseph O’Rourke, and Yusu Wang for organizing this workshop, and especially Joseph O’Rourke for introducing us to this problem. O’Rourke also made Fig. 5.6 . Travel funding for additional collaborations, during which we continued our work on this problem, was provided by the American Mathematical Society. We thank anonymous referees for suggesting additional references. Victor Dods partially supported by the 2011–2012 ARCS Fellowship. Jed Yang partially supported by NSF GRFP grant DGE-0707424 and NSF RTG grant DMS-1148634 . Publisher Copyright: © 2016 Elsevier B.V.

PY - 2017/1/6

Y1 - 2017/1/6

N2 - Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern–Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.

AB - Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern–Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.

KW - Cube

KW - Geodesic

KW - Regular tetrahedron

KW - Stern–Brocot tree

UR - http://www.scopus.com/inward/record.url?scp=84981288448&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84981288448&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2016.07.004

DO - 10.1016/j.disc.2016.07.004

M3 - Article

AN - SCOPUS:84981288448

SN - 0012-365X

VL - 340

SP - 3183

EP - 3196

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 1

ER -

Geodesics on the regular tetrahedron and the cube

Abstract

Funding

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this