[Maths-Education] concentric and congruent circles

[Julian Gilbey]

On Thu, Jan 28, 2010 at 03:51:37PM -0500, Kate Mackrell wrote:
> I'm in the middle of quite an interesting experience regarding the
> definition of these mathematical terms, which I thought might be
> worth sharing.
> [...]
> One example is "adjacent angles".  To me, these are angles on a
> straight line, that  add up to 180.  To this company, these are
> angles with one common arm and a common vertex and no specific sum.

"Adjacent angles" seems a fairly vague term; I've never come across
it, and my personal preference is for their definition.  I think that,
were I ever to use it, I would define it each time I did so, as it is
not a standard term.

> Another example is of course the trapezoid/trapezium - does it have
> only one or at least one pair of parallel sides?

This one is simple: is a square a rectangle?  For mathematical
simplicity, the answer should be "yes": for every theorem which is
true for a rectangle is true for a square as well (except for those
which _assume_ that the side lengths are unequal).  This is an example
of a general principle in mathematical nomenclature, that a special
case of a general class is still a member of the more general class.
Geometry provides the most fertile soil for this, and examples include
things such as: a triangle is a polygon, a circle is an ellipse, and
so on.

> [...]
> To me (and Wolfram, and lots of other authoritative online
> dictionaries of mathematics), two circles are concentric if they
> have the same centre.  The radii of the circles are immaterial -
> hence two congruent circles may or may not be concentric.  To the
> people I'm working for, concentric circles necessarily have
> different radii.

[...]

> On the other hand, the company wants to still call two coinciding
> circles "congruent", which implies that there are still two circles
> in a relationship with each other - are they just being
> inconsistent?

Two coinciding circles are identical, and hence are both congruent AND
concentric.  The context will be the most significant in determining
which is the more useful description: are you more concerned about
them sharing a common centre or about them having the same radius?

> I also find the question interesting of when to consider it
> legitimate for a mathematical term to have a flexibly  determined
> meaning (such as trapezoid/trapezium or adjacent angles) and when to
> not consider it legitimate.  "Concurrent" has not got any of the
> mathematical inevitability of "-" x "-" = "+".  I shouldn't be
> concerned about its definition being changed.  But I find that I am
> very unwilling to change.

The meaning of a mathematical term is determined by convention.
Sometimes, terms are given different meanings by different people (the
most notorious is, I believe, "natural numbers": do these include zero
or not?).  Sometimes these are borderline cases (does a circle dragged
through a third dimension become a prism, or is a prism only created
when a polygon is so dragged?), other times (such as "adjacent
angles") it is when the term has not gained (near-)universal
acceptance.  But when a term has an accepted meaning across the whole
mathematical community, it would be very unwise to try changing it.
Examples of this are "concentric" and "congruent".

All my personal opinion, but hope it is of help.

   Julian