Paper by Erik D. Demaine

Reference:

Erik Demaine, Martin Demaine, David Eppstein, Hiro Ito, Yuta Katayama, Wataru Maruyama, and Yushi Uno, “Geodesic paths passing through all faces on a polyhedron”, in Revised Selected Papers from the Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCGGG 2022), edited by Jin Akiyama, Hiro Ito, and Toshinori Sakai, Lecture Notes in Computer Science, volume 14364, Tokyo, Japan, September 9–11, 2022, pages 184–209.

Abstract:

The shortest path passing on the surface of a polyhedron is called a geodesic path. A geodesic path of a polyhedron has a property that it becomes a single line segment on a development. A geodesic path is the shortest path and it mostly passes a small number of faces. We, however, consider a problem “is there a case that a geodesic path passes all faces of a polyhedron?” For this problem the answer is “yes”: we found that a regular tetrahedron has such a geodesic path. The next question is “what polyhedra have such geodesic paths?” We define a face-guard geodesic path (FGG path, for short) as a geodesic path connecting two points on a polyhedron and passing through all its faces, call a polyhedron that has an FGG path an FGG polyhedron, and try to characterize FGG polyhedra. For this new problem, we prove that there exists an FGG n-hedron for any integer n ≥ 4, all tetrahedra and all triangular prisms with one exception are FGG polyhedra, and all cuboids and all regular polyhedra except regular tetrahedra are not FGG polyhedra.

Comments:

The paper is also available from SpringerLink.

Length:

The paper is 25 pages.

Availability:

The paper is available in PDF (751k).

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Related papers:

HamGeodesic_JCDCGGG2022 (Geodesic paths passing through all faces on a polyhedron)