What is a vertex?

by Mark Dawes (January 2020)

For many people, maths is precise.  Mathematics is often exact.  Mathematical vocabulary however isn’t.

Years ago I was asked by a colleague “is a vertex where three or more faces meet?”.  I agreed that it was.  The next question was “what is the thing at the top of a cone called?”.  (My initial flippant response was “it depends which way up it is”, whereas the best answer is probably, “ice cream and a flake”).  It is, of course a vertex too.  We also use ‘vertex’ in lots of other mathematical situations, for example in graph theory, with intersecting lines, in 2D shapes and at the turning point of a quadratic graph.

This suggests that at least some mathematical words are used in different ways in different contexts, so it isn’t possible to have a single definition that will suit everyone.  There are also geographical differences (I gather that in the USA a ‘trapezium’ refers to a four-sided shape without any parallel sides, rather than one with a pair of parallel sides.)  Beyond this, finding a cast-iron definition for anything is difficult.

Is this shape a quadrilateral?  Or is it a pentagon?  Or a hexagon?

It was made using four straight lines, so that means it is a quadrilateral.  But does it have four vertices or five?  If it’s got a vertex at the crossing point then that surely means it’s a pentagon (5 vertices).  But then the two crossed lines have been split up, so there are actually six lines, so it’s a hexagon.  If we consider the four angles, they don’t add up to 360 degrees, but if we consider the two angles where the lines cross then we _do_ get a total of 360 degrees.  But in that case there _is_ a point where the lines cross, which means that while the angle now add up to 360 degrees it’s either a pentagon or a hexagon!  What is going on?  For what it’s worth, I think we need to redefine what ‘interior angles’ mean in this shape – so now a different mathematical concept and word needs to be explored further!

I raise this not because I particularly care whether this is or isn’t a quadrilateral, but to show how difficult it can be to define even simple mathematical concepts and ideas.

Help may well be at hand, though!

Cambridge Mathematics is crowd-sourcing definitions for its own glossary of mathematics.  A new word appears each week and we are asked to comment on the usefulness of each definition for different groups of learners of mathematics.  It’s easy to be part of.  Sign up here: cmdefineit.org or download the CM Define It app.

I have three take-aways from this:

1. It’s hard to define a word in a completely watertight way.

2. It might be necessary to have a different definition for, say, primary and secondary school pupils.

3. CM Define It is a good thing!

While CM Define It is not going to produce a single definition of each mathematical word (because of the need for different definitions for people at different stages and in different parts of the world), their new glossary will help to make defining and explaining mathematical vocabulary easier.

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