Another Proof of the Steiner Result for Three Equidistant Points in Euclidean Space and an Analogous Proof in Hyperbolic Space

Authors

Diana Dimond, Brigham Young UniversityFollow

Abstract

In the early 19th century, Jacob Steiner wanted to find the shortest path to connect three villages. He concluded that the shortest path depended on the angles of the triangle created. If all the angles were less than 120 then the shortest path involved a fourth interior point, a Steiner point, at which the segments from the vertices all meet at 120°. Our research group developed a new "slicing" method that can be used to prove which paths are minimal. To demonstrate this new method, we first give a new proof of a particular Steiner result. Then we use it to prove an analogous result in hyperbolic space; that is, the shortest path between three equidistant points in hyperbolic space is formed by hyperbolic geodesics that meet at 120°.

Author Bio

Diana is an undergraduate student in mathematics at Brigham Young University in Provo, Utah. I am originally from Madison, Wisconsin, so moving to the mountains was a welcomed change. I had the opportunity, because of a research mentoring grant in geometric analysis from BYU, to do research this past year. I worked with Dr. Michael Dorff, Dr. Denise Halverson, and Dr. Gary Lawlor on problems pertaining to minimal path networks. I found it to be a fascinating field, and one that I would like to continue doing research in for the next year and as a graduate student after I graduate from BYU in April of 2005.

Faculty Sponsor

Michael Dorff

Recommended Citation

Dimond, Diana (2004) "Another Proof of the Steiner Result for Three Equidistant Points in Euclidean Space and an Analogous Proof in Hyperbolic Space," Rose-Hulman Undergraduate Mathematics Journal: Vol. 5: Iss. 2, Article 1.
Available at: https://scholar.rose-hulman.edu/rhumj/vol5/iss2/1