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/