Definitions as a Tool of Mathematics

Date: 03/04/2003 at 01:39:51
From: Tim
Subject: Defining polyhedra

Hello! I'm an editor at an educational publisher, working on maths 
books. Some of us have been talking about unusual polyhedra, and want 
to know how we can exclude 'doubled' shapes, e.g. two tetrahedra that 
join at a single vertex (sort of like an hourglass).

Under 

   Defining geometric figures
   http://mathforum.org/dr.math/faq/formulas/faq.figuredef.html 

Dr Math writes:

"If all the faces are plane regions, every edge is the edge of two 
faces, every vertex is the vertex of at least three faces, and no two 
faces cross each other, the figure is called a polyhedron."

I guess this boils down to asking what "no two faces cross each 
other" means, because a joined pair of tetrahedra seems to meet the 
other parts of the definition. I'm sure there's something really 
obvious we're missing.

The two-dimensional case is easy, because two triangles meeting at 
one vertex have one vertex as the endpoint of four sides instead of 
two: 

"If all the edges are segments, every vertex is the endpoint of two 
sides, and no two sides cross each other, the figure is called a 
polygon."

Similarly, in three dimensions, a pair of tetrahedra would violate 
Euler's Formula because one vertex is shared. But how does the 
definition of polyhedron exclude this figure?

Date: 03/04/2003 at 13:51:17
From: Doctor Tom
Subject: Re: Defining polyhedra

Hello Tim,

As you've noticed, it is an ugly situation!

You are correct to notice that the definition you found is deficient.  
Obviously, you'll have to tailor your definition somewhat to match the 
level of your proposed audience. If it's a bunch of third-graders, 
maybe even the Dr. Math definition is too sophisticated, and if your 
audience is professional topologists, maybe nothing would be good
enough.

You can get rid of the problem you mention by adding that the interior 
of a polyhedron is "connected." The interior is the set of points 
inside but not including the boundary. "Connected" means that you can 
connect any two points in the region with a path (curved or straight) 
that is entirely inside the region. If the two tetrahedra are touching 
at a point only or share an edge only, that edge or point is on the 
boundary and thus will not be part of the interior. Once you eliminate 
it, there is no path that will connect points in the two tetrahedra 
that does not pass outside the interior, at least at one point.

But this isn't good enough. Imagine a figure that looks like a ball 
(made of flat faces, of course) which is "pushed in" on both sides so 
that it touches itself on the inside, or if you form two "horns" that 
come out and touch at a point outside.

There is a great deal more ugliness in the definition if you go 
looking for trouble.

First of all, I would insist on a finite number of faces.

A polyhedron should be "closed" in the sense that the boundary divides 
the inside and outside into two regions that are not connected - any 
path from a point "outside" to a point "inside" must pass through
the boundary.

In fact, you'd better say that the inside is "bounded" - in other 
words, you can draw a box or sphere large enough to completely enclose 
it. Otherwise I can imagine an infinitely long tube made of faces.

Can a polyhedron have "holes" in it, in the sense of a doughnut-like 
think with flat faces?  The only way to insure that this doesn't 
happen is to say that every closed loop inside it can be shrunk in a 
continuous way, remaining inside, to a point. If the loop goes inside 
the doughnut and around the hole, it cannot be shrunk in this way.  
You have to decide whether you want these to be considered polyhedra 
or not. Figures without this sort of hole are called "simply-
connected."

If you toss out doughnut-like polyhedra, this eliminates the problem 
of the sort alluded to earlier where a sphere is pushed in to touch at 
the center, but not where horns touch on the outside.  To eliminate 
the "outside horn" problems, you need to insist that the boundary also 
be simply-connected.

Similarly, can it have a different sort of "hole" in the sense of 
having a box with a smaller box completely inside it where the 
interior of the polyhedron is considered to be inside the outer box 
and outside the inner? Is this a polyhedron or not? You have to 
decide.  If you don't want to allow this, you need to say that the 
boundary is also connected.

You had also better say that the faces are finite (or "bounded," if 

you like). Otherwise I can imagine a cube with faces running out to 
infinity in a bunch of directions so that they all meet another at any 
edge.

What's worrisome to me is that even with all the cautions I made 
above, I'm not certain I didn't leave out some pathological cases.

I'll leave this question open in case other math doctors would like to 
wade into this morass.

Good luck! You'll need it.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/04/2003 at 13:57:22
From: Doctor Peterson
Subject: Re: Defining polyhedra

Hi, Tim.

This is indeed tricky; it is hard to find a really good definition. 
Part of the problem is that polyhedra are studied from several 
different perspectives, each of which has slightly different needs; 
and also that at an elementary level one wants to avoid being too 
precise and making the definitions impossible to follow.

Your reference to Euler's Formula is interesting, because that 
actually falls under the study of topology, where many aspects of a 
polyhedron are irrelevant, but some things not normally considered 
essential to a polyhedron are essential. Here is a discussion of the 
conditions for that theorem and the related concept of the Euler 
characteristic:

   Euler's Formula Applied to a Torus
   http://mathforum.org/library/drmath/view/51815.html 

This shows how the Euler formula does not require flat faces, etc., 
but does require the right kind of connectedness (which, among other 
things, your two tetrahedra lack). But we can't include the latter 
issue in our definition of a polyhedron, at least in the same form, 
since you can have a toroidal polyhedron (at least in some fields).

As for a definition of the polyhedron that excludes two polyhedra 
that touch at a point, try this:

   Regular Polyhedra
   http://www.cut-the-knot.org/do_you_know/polyhedra.shtml 

  Definition 5
  Polyhedron is a solid (3D) body whose surface consists of a number
  of polygonal faces subject to two conditions to exclude some 
  "abnormal" cases. 

  * every side of every polygon belongs to just one other polygon
    (this precludes T-"intersections" and the like) 
  * the faces that share a vertex form a chain of polygons in which
    every pair of consecutive polygons share a side (this precludes
    the case of two pyramids touching at a vertex)

It's a little awkward, isn't it? That's why we more often just put 
the concept in the category of "I know one when I see one," when it 
comes to details at this level.

Finally, you asked about what it means for faces to "cross each 
other." That is illustrated by the Great Dodecahedron, whose faces 
are pentagons that pass through one another:

   Great Dodecahedron - Bruce Fast
   http://amath.colorado.edu/staff/fast/Polyhedra/gd.html 

   The Kepler-Poinsot Polyhedra - Tom Gettys
   http://home.comcast.net/~tpgettys/kepler.html 

Note that this IS allowed for a polyhedron! Here is a picture of many 
polyhedra, to show you the variety that is allowed under this broader 
definition:

   Uniform Polyhedra - V. Bulatov
   http://www.physics.orst.edu/~bulatov/polyhedra/uniform/ 

You may find the definitions here helpful, as well:

   Glossary - George W. Hart
   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.

   self-intersecting - A polygon with edges which cross other edges;
   a polyhedron with faces which cross other faces.

This doesn't say polyhedra can't have intersections; nor does it 
carefully exclude your twin tetrahedra.

Thinking about your question further, I have gained some insights 
into the nature of mathematical definitions that I want to share with 
you.

Comparing my answer to Dr. Tom's, I see that he approached the 
question as a true mathematician - not looking for an "official" 
definition, but making one up to meet your needs. We both recognize 
that there is not one standard definition of this term, but a variety 
of definitions all centering about the same concept; definitions are 
not carved in stone, but are adapted by the user to meet specific 
needs. Only the basic concept is common to all definitions of a 
polyhedron. In fact, I had a hard time finding careful definitions on 
the Web; usually the term is used almost informally, even by sites 
that focus on cataloging all polyhedra.

How can this be, when we know how important definitions are in 
mathematics?

I can compare a general concept such as polyhedra (defined simply as 
"solids bounded by planar polygons") with a place name such as "North 
America". We all know what it is; but if we were pressed as to the 
exact boundaries - does it include Central America, or not? - we 
would start to wonder if it is really defined at all. It is only when 
we start to prove theorems about polyhedra, or to write laws 
pertaining to North America, that we really need careful definitions 
that specify not only the general concept but also the precise 
boundaries. In law, I suspect that the term North America is left 
officially undefined in general, and is defined specifically for a 
particular law; for example, the North American Free Trade Agreement 
presumably specifies exactly which countries it pertains to, rather 
than just saying "North America." Similarly, theorems are not proved 
about polyhedra in general, but about specific types of polyhedra, 
such as convex polyhedra or regular polyhedra. The boundaries are 
much clearer for these "countries" than for the "continent" of all 
polyhedra.

In fact, those sites that list polyhedra do it by breaking them down 
into categories. And they probably don't want to define polyhedra 
restrictively, because they want to be able to add new categories of 
polyhedra freely. We need a general name that covers all of them, 
including perhaps kinds that haven't even been thought of. So a name 
like North America was useful even to explorers who didn't yet know 
how far the continent extends. If they had found that it was connected 
to Asia, they would eventually have had to clarify that boundary in 
order to talk about their explorations; but the connection to South 
America is small enough not to worry about - just as the pathological 
polyhedra at the limits of the definition, such as tetrahedra linked 
at a vertex, are few and seldom needed. We need to define the border 
only when there is a conflict there.

So I am perfectly satisfied to keep the broadest possible definition 
of polyhedra; in fact, I'd argue for weakening the definition in our 
FAQ, rather than strengthening it. In particular, I would not want to 
require simple connectedness, though requiring that the whole polygon 
be connected (which is usually assumed) and that its interior be 
connected (so that polyhedra touching at a vertex don't count as one) 
makes sense. Apart from that, "global" conditions (reflecting how the 
entire polyhedron works as a whole), such as simple connectedness, 
are inappropriate, since polyhedra that do not meet that requirement 
are very much worth discussing. The second restriction in the "Cut-
the-knot" definition makes sense; it is a "local" condition, saying 
only that at each point the polyhedron should behave like a piece of 
paper, on which you can walk completely around any vertex in a single 
circuit, rather than talking about what happens if you walk around 
the whole figure. But even that can be left out without losing much, 
especially when working with children, who aren't going to try to 
stretch the definition too far. (And if they are creative enough to 
do so, then they can benefit from a discussion such as this!) By 
keeping the definition simple and broad, you will avoid teaching 
unnecessary restrictions that might have to be dropped if a student 
gets into higher math. This keeps what you teach in line with actual 
usage.

This ties in with another issue that comes up frequently: does a 
cylinder or cone have faces, edges, and vertices? My usual answer is 
that the question should not really be asked at all, if the terms are 
defined as for polyhedra. Definitions that require a face to be a 
planar polygon, an edge the intersection of two planes, and a vertex 
the intersection of three or more planes are too restrictive when 
applied to more general surfaces that may be curved. (The result is 
the question, if a cylinder has no faces, then what do you call the 
circular and curved surfaces of which it is composed?) We need to 
recognize the proper context for our definitions, and not stretch 
them beyond their intended domain just because we want to bring 
mathematical precision into the classroom. Where definitions bring 
only confusion, and do not help in discussing the entity to which 
they are applied, we should replace them with definitions that fit, 
just as we ignore political boundaries when we study animal habitats. 
We might define Central America in one way on a political map, and in 
another way in talking about ecosystems. Definitions are a tool of 
mathematics, not its master.

There was one more comment I wanted to make. In George Hart's 
definition in my answer, he mentions the question "whether it can have 
two different vertices that happen to be at the same location." Only 
when I read Dr. Tom's discussion did I understand what case this 
covers: his "horns touching," where two vertices are stretched out so 
that they meet and share a point. Including such a restriction serves 
mostly to draw our attention to the fringes (if we understand it at 
all), and distracts from the main point, the central concept of the 
definition, which is what needs to be taught.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/06/2003 at 07:42:14
From: Tim
Subject: Thank you (Defining polyhedra)

Dear Doctor Tom and Doctor Peterson

Thanks so much for your responses to my question. You 
provide an excellent service! Our audience is mainly high 
school students, and we're probably going to go with a 
relatively basic definition, partly, as you suggest, to 
avoid being too restrictive. I can promise we'll keep 
coming back to your site for more useful tips.

Yours
Tim