Polyhedra: Solids or Surfaces?
Date: 06/06/2003 at 06:57:53 From: Zag Subject: Is a polyhedron always a solid figure? My question is: Is it necessary that a polyhedron should be a solid figure? Can we call a hollow figure a polyhedron? Cuboids and cubes are polyhedra, prisms and pyramids are polyhedra; if we construct a cube or cuboid or prism of hollow paper or cardboard, can we call that a polyhedron?
Date: 06/06/2003 at 12:55:27 From: Doctor Peterson Subject: Re: Is a polyhedron always a solid figure? Hi, Zag. You can find "polyhedron" defined both ways: as a solid whose surface is composed of polygons, and as a surface composed of polygons itself. Here are sample places I found that define it as a solid: Chapter 12 Lecture Notes - C.J. Dunham http://members.aol.com/frustum1/Chpt12.html Polyhedron - is a solid that is bounded by polygons (called faces), that enclose a single region of space. polyhedron - NIST http://www.nist.gov/dads/HTML/polyhedron.html Definition: The set of solutions to a finite system of linear inequalities on real-valued variables. Equivalently, the intersection of a finite number of linear half-spaces in R^n. and as a surface: Informal Geometry Glossary (AMS) http://www.ams.org/new-in-math/cover/geometry-glossary.html A polyhedron is a surface made up of a finite number of flat (plane) pieces. Some polyhedra may go off to infinity and others are bounded, that is, they will fit inside a large sphere. Glossary of Geometrical Terms - Graeme McRae http://mcraefamily.com/MathHelp/GeometryGlossary.htm polyhedron - a closed surface formed by polygonal plane faces, connected at the edges; a "solid polyhedron" is a solid (or the space) enclosed by a polyhedron. At least one site refers explicitly to the fact that both forms are used: George Hart: Glossary http://www.georgehart.com/virtual-polyhedra/glossary.html polyhedron - A three dimensional object bounded by polygons, with each edge shared by exactly two polygons. Various authors differ on the fine points of the definition, e.g., whether it is a solid or just the surface, whether it can be infinite, and whether it can have two different vertices that happen to be at the same location. So, how you define the term depends on your point of view; those who say that the polyhedron is the solid have to refer to "the surface of a polyhedron" when talking about its faces, and those who say it is the surface itself have to talk about "the solid bounded by a polyhedron" when they talk about its interior. I suspect that both groups have no trouble talking about a cardboard model as a polyhedron, regardless of the fact that its interior is air. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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