what is a polyhedron?
The question seems trivial, but it's always better to know exactly what we speak about.
The polyhedra build a zoo difficult to conceive, so rich is it. The Herculean task to order it a little is naturally the responsibility of the mathematicians; they didn't lack of imagination to take up the challenge.
A polyhedron (from the Greek "poly" and "hedra": several basis) is a solid of the 3D space limited by a finite number of parts of planes, the faces, which are assembled along common segments, the edges ; the vertices are the points common to at least three faces (they are the end points of the edges). The faces are therefore plane polygons; the faces' sides are the polyhedron's edges and the faces' vertices are also the polyhedron's vertices.
Remark: "base" usually denotes the face on which the polyhedron rests on; thus each face may be a base (likewise each side of a triangle may be considered as "base"). If the polyhedron owns a face (or two parallel faces) which is easily to distinguish from the other faces it is also called "base" (examples: base of a pyramid, bases of a prism).
Thus a cylinder is not a polyhedron for several reasons: its lateral surface is not plane, its bases are not polygons, finally it has no vertex! No faces, no edges, no vertices, it couldn't be worse!
The simplest polyhedron is the triangular pyramid or tetrahedron (four triangular faces); thus the minimum is 4 faces, 4 vertices and 6 edges.
Remark : Some textbooks, not only don't give precise definitions for the vocabulary used to describe polyhedra, but also use this vocabulary for describing other solids (so the cone is supposed to have two "faces" and one "edge", the ball just one "face"...)
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here are three polyhedra
right regular prism - regular pyramid - "spherical" polyhedron | |
but these three solids are not polyhedra
circular right cylinder - circular right cone - ball |
• The polyhedra world is so vast and complex that it is often necessary to clarify the definitions to eliminate the "abnormal" cases:
• A polyhedron must be "bounded" (it can be completely enclosed in a sphere) and "closed" (its boundary must create two non connected regions, the inside and the outside).
• The interior must be connected (any two points can be connected with a path that is entirely in the interior); thus we eliminate for example the pairs of tetrahedra with only an edge or a vertex in common.
Each edge must be a side of TWO faces and the edges starting at a same vertex must define ONE sequence of faces where each edge is a side of two consecutive faces. Nevertheless two faces can intersect but the intersection is not an edge.
• If we want to avoid "holes" in the interior we need a connected boundary.
The toroidal polyhedra (doughnut shaped) don't lack of interest. A polyhedron without "tunnels" is said simply-connected (every closed loop in the interior can be shrunk in a continuous way to a point).
Even with all these precautions one is not certain to avoid all the "pathological" cases!
We see that it is not easy to give an "official" definition of a polyhedron; different definitions are adapted for specific needs. With a broad definition we can use the word "polyhedron" for all the numerous categories (like assemblings of polyhedra), included those which have not yet been discovered.
"The question "What is a polyhedron?" has never been fully answered. Definitions have ranged from solids to surfaces to skeletons to combinatorial point sets, with all sorts of features such as infinite extent, coincident elements and so on allowed by some investigators but not by others. Today, the debate rages perhaps stronger than ever." (Guy Inchbald)
At an elementary level it is thus better to avoid to be too precise by introducing useless restrictions and to be content with a simple definition easy to understand: a polyhedron is a solid bounded by a finite number of planar polygons.
A very simple definition by H. S. M. Coxeter: A polyhedron is a finite set of polygons such that every side of each belongs to just one other, with the restriction that no subset has the same property. The faces are not restricted to be convex, and may surround their centers more than once. Similarly, the faces at a vertex may surround the vertex more than once.
• "Geometrical solids can become the subject of a fascinating study. Not everyone, of course, will want to make the attempt to understand all the theorical mathematics involved in discovering and classifying those solids... But everyone surely can appreciate the beauty and symmetry of these solids, whose history is as ancient as Plato, Archimedes and Euclid..." Magnus Wenninger (The Mathematics Teacher - March 1965, page 244)
• "The polyhedra build an inexhaustible spring of inspiration for geometry teaching and to illustrate the research in mathematics. For education the polyhedra go along with multiple and all very rich activities (observations, manipulations, cardboard realizations, formulations of hypothesis, of counter-examples, etc…). For instance the use of nets may begin in the nursery school and go on in various ways to the university.
Furthermore polyhedra are a field deep-rooted in the history of mathematics. They are more or less present in all ages, included ours. This active field, far from being dried up, is thus a good choice for talks to popularize mathematics where the concrete and very familiar aspect of the studied object greatly simplifies the always delicate task of the popularizer who must again and again explain that, yes, mathematics are nice and alive." Thierry Lambre (September 2007, bulletin APMEP n° 471, page 533)
• "What kind of spatial understanding do first and second graders have? What do they see when they look at three-dimensional objects? What words do they use to describe what they see? How might their visualization skills be sharpened by building and describing three-dimensional structures? These questions emerged as we, a teacher educator and a classroom teacher, considered infusing spatial tasks into the primary school mathematics curriculum. We were concerned that our students were not given opportunities to develop their spatial abilities." (Karen Falkner)
• "Polyhedra have an enormous aesthetic appeal… The subject is fun and easy to learn on one's own… the best way to learn about polyhedra is to make your own paper models." (George W. Hart)
• "Hart brings the ideal forms of the polyhedra into the material world with wit and humor. The austere beauty of the polyhedra is thus made accessible and familiar." (Bob Brill)
• It is really better to construct your models yourself. As Pólya said, "Mathematics is not a spectator sport!"
| The notion of convexity is understandable by every person with some elementary knowledge in geometry: |
| 1. |
A polygon is convex if all its diagonals are inside.
Reminder: a diagonal of a polygon is a segment joining two vertices and which is not a side.
Remark: A triangle has no diagonals. |
| 2. |
A polyhedron is convex if all its diagonals are inside or on its surface.
Reminder: a diagonal of a polyhedron is a segment joining two vertices and which is not an edge.
Remark: A tetrahedron and the Császár's polyhedron have no diagonals. |
A general result interesting for polygons and polyhedra: the "Cavalieri's principle": |
| 1. |
If for two plane regions there exists an orientation and a line such that for any line parallel to it the slices have equal lengths in both regions, then these regions have equal areas. |
| 2. |
If for two solids there exists an orientation and a plane such that for any plane parallel to it the cross-sections have equal areas in both solids, then these solids have equal volumes. |
| examples: |
two parallelograms (or two triangles) with bases of the same length and heights of the same length have the same area (sections parallel to the bases)
two prisms (or two pyramids) with bases of same area and heights of same length have same volume (sections parallel to the bases) |
