Vol. 4, No. 4, 2011

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The Steiner problem on the regular tetrahedron

Kyra Moon, Gina Shero and Denise Halverson

Vol. 4 (2011), No. 4, 365–404

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.

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

Communicated by Frank Morgan
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