Minimum Perimeter Developments of the Platonic Solids

A development of a convex polyhedron is a connected plane figure obtained by cutting the surface of the polyhedron and unfolding it. In this paper, we determine the length and configuration of a minimum perimeter development for each of the Platonic solids. We show that such developments are obtaine...

Full description

Saved in:

Bibliographic Details
Main Authors:	Akiyama, Jin, Chen, Xin, Nakamura, Gisaku, Ruiz, Mari-Jo P
Format:	text
Published:	Archīum Ateneo 2011
Subjects:	Geometry and Topology Mathematics
Online Access:	https://archium.ateneo.edu/mathematics-faculty-pubs/80 http://thaijmath.in.cmu.ac.th/index.php/thaijmath/article/view/70
Tags:	Add Tag No Tags, Be the first to tag this record!
Institution:	Ateneo De Manila University

id	ph-ateneo-arc.mathematics-faculty-pubs-1079
record_format	eprints
spelling	ph-ateneo-arc.mathematics-faculty-pubs-10792020-06-16T05:54:18Z Minimum Perimeter Developments of the Platonic Solids Akiyama, Jin Chen, Xin Nakamura, Gisaku Ruiz, Mari-Jo P A development of a convex polyhedron is a connected plane figure obtained by cutting the surface of the polyhedron and unfolding it. In this paper, we determine the length and configuration of a minimum perimeter development for each of the Platonic solids. We show that such developments are obtained by cutting the surface of the polyhedron along a Steiner minimal tree. We introduce the concept of Steiner isomorphism to develop a search algorithm for determining these Steiner minimal trees. Each of these trees is completely symmetric with respect to rotation around a fixed point. 2011-01-01T08:00:00Z text https://archium.ateneo.edu/mathematics-faculty-pubs/80 http://thaijmath.in.cmu.ac.th/index.php/thaijmath/article/view/70 Mathematics Faculty Publications Archīum Ateneo Geometry and Topology Mathematics
institution	Ateneo De Manila University
building	Ateneo De Manila University Library
country	Philippines
collection	archium.Ateneo Institutional Repository
topic	Geometry and Topology Mathematics
spellingShingle	Geometry and Topology Mathematics Akiyama, Jin Chen, Xin Nakamura, Gisaku Ruiz, Mari-Jo P Minimum Perimeter Developments of the Platonic Solids
description	A development of a convex polyhedron is a connected plane figure obtained by cutting the surface of the polyhedron and unfolding it. In this paper, we determine the length and configuration of a minimum perimeter development for each of the Platonic solids. We show that such developments are obtained by cutting the surface of the polyhedron along a Steiner minimal tree. We introduce the concept of Steiner isomorphism to develop a search algorithm for determining these Steiner minimal trees. Each of these trees is completely symmetric with respect to rotation around a fixed point.
format	text
author	Akiyama, Jin Chen, Xin Nakamura, Gisaku Ruiz, Mari-Jo P
author_facet	Akiyama, Jin Chen, Xin Nakamura, Gisaku Ruiz, Mari-Jo P
author_sort	Akiyama, Jin
title	Minimum Perimeter Developments of the Platonic Solids
title_short	Minimum Perimeter Developments of the Platonic Solids
title_full	Minimum Perimeter Developments of the Platonic Solids
title_fullStr	Minimum Perimeter Developments of the Platonic Solids
title_full_unstemmed	Minimum Perimeter Developments of the Platonic Solids
title_sort	minimum perimeter developments of the platonic solids
publisher	Archīum Ateneo
publishDate	2011
url	https://archium.ateneo.edu/mathematics-faculty-pubs/80 http://thaijmath.in.cmu.ac.th/index.php/thaijmath/article/view/70
_version_	1681506676856848384

Minimum Perimeter Developments of the Platonic Solids

Similar Items