terminology.

the more i think about it, the more i suppose that my study seminar talk, last week, seemed like a 'comedy of errors.' i distinctly remember discussing the usual notion of a cubical subdivision of the plane, and then someone asked,

"so really, you're talking about squares, right?"

this comment dumbfounded me, for about a minute. squares? but then i finally realised his point:

"oh, right!" i answered, "yes, squares. i was talking about 2-dimensional cubes."

i also remember that it made the audience laugh, but that wasn't my intention. so let me explain.

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to the non-mathematician: if you spoke with a mathematician about shapes in two dimensions, you might soon be puzzled at his/her use of the word "circle." instead, s/he may often say the word "ball."

what's the difference, you may ask? a circle is a boundary; given a center point C and a radius r > 0, it consists of the points which are exactly distance r away from the point C. in set notation,

{ P ε R² : |P - C| = r }.

on the other hand, the boundary of a (2-dimensional) ball is a circle. it consists of the points which are distance r or less from the center point C. again, in set notation,

{ P ε R² : |P - C| ≤ r }.

why the difference, you may ask? a circle is a 1-dimensional object, whereas a 2-dimensional ball is .. er, 2-dimensional. also, a circle has a hole; a ball doesn't.

to the mathematician: yes, you understood the difference already, but you may ask: what does this have to do with cubes and squares?

i didn't realise it until now, but i reserve the keyword "square" for when i'm talking about boundaries of "cubes."

to me, a square in the plane is a boundary, consisting of four congruent line segments at right angles. the 2-dimensional object is a cube.

so in the language of the SATs [1] of yesteryear, this is how i use words:

circle : ball :: square : cube.

[1] to my non-american friends: SAT was once an abbreviation for 'scholastic aptitude test,' but now it means 'scholastic achievement test.' it's a standardised exam for high school students, so that colleges and universities have a criterion to reject the students whom they don't want.