Vol. 4, No. 4, 2011

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 4, 543–722
Issue 3, 363–541
Issue 2, 183–362
Issue 1, 1–182

Volume 16, 5 issues

Volume 15, 5 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 8 issues

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Editors' interests
 
Subscriptions
 
ISSN 1944-4184 (online)
ISSN 1944-4176 (print)
 
Author index
To appear
 
Other MSP journals
The Steiner problem on the regular tetrahedron

Kyra Moon, Gina Shero and Denise Halverson

Vol. 4 (2011), No. 4, 365–404
Abstract

The Steiner problem involves finding a shortest path network connecting a specified set of points. In this paper, we examine the Steiner problem for three points on the surface of a regular tetrahedron. We prove several important properties about Steiner minimal trees on a regular tetrahedron. There are infinitely many ways to connect three points on a tetrahedron, so we present a way to eliminate all but a finite number of possible solutions. We provide an algorithm for finding a shortest network connecting three given points on a regular tetrahedron. The solution can be found by direct measurement of the remaining possible Steiner trees.

Keywords
Steiner problem, length minimization, regular tetrahedron, piecewise-linear surface
Mathematical Subject Classification 2010
Primary: 05C05
Secondary: 51M15
Milestones
Received: 14 January 2011
Revised: 22 March 2011
Accepted: 24 March 2011
Published: 21 March 2012

Communicated by Frank Morgan
Authors
Kyra Moon
Mathematics Department
Brigham Young University
275 TMCB
Provo, UT 84602
United States
Gina Shero
Mathematics Department
Clarion University of Pennsylvania
189 STC
840 Wood Street
Clarion, PA 16214
United States
Denise Halverson
Mathematics Department
Brigham Young University
263 TMCB
Provo, UT 84602
United States