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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
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Abstract |
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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
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Mathematical Subject Classification 2010
Primary: 05C05
Secondary: 51M15
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Milestones
Received: 14 January 2011
Revised: 22 March 2011
Accepted: 24 March 2011
Published: 21 March 2012
Communicated by Frank Morgan
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Authors |
| Kyra Moon | |
| Mathematics Department Brigham Young University 275 TMCB Provo, UT 84602 United States |
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| Gina Shero | |
| Mathematics Department Clarion University of Pennsylvania 189 STC 840 Wood Street Clarion, PA 16214 United States |
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| Denise Halverson | |
| Mathematics Department Brigham Young University 263 TMCB Provo, UT 84602 United States |
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