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### 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
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

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

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

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
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
```
Associated Topics:
College Definitions
College Polyhedra
High School Definitions
High School Polyhedra

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