You list some common conventions; the important point is that there is no *universal* convention, and only an *application* (such as your 4 apples) really provides a basis for identifying a multiplier definitively.

As for listing dimensions of a rectangle, that too varies greatly. My first thought is that in describing a matrix in math, we traditionally list rows x columns (that is, height first), even though in other areas we might list x by y. But looking at how dimensions of products are described, I get the impression that depends on what product you are talking about, and also whether it has a definite orientation. Boxes, for example, are commonly described as length (longest side) by width by height; there is no fixed orientation there. For printing, it appears that width (say, of a picture on a page, or of the page itself) is first, then height (as in 8 1/2 by 11 — except for 3×5 cards!). For windows, it seems to be the same; but I read that signs are traditionally listed as height by width.

Ultimately, this is not a math issue, and not a language issue, but a matter of convention among various communities. There is no naturally right way. And if the goal is to communicate, you just have to describe things clearly: 3″h x 4″w, for example.

]]>I think it was reasonably clear that this was what the questioner intended (and a further conversation would have revealed if it wasn’t), so it was not necessary to define it fully at that point. But you are right that some people make that mistake, thinking that if they can construct a trisection of, say, a 90 degree angle, then they have proved the mathematicians’ claim wrong. That’s why this point was made several times later in the article. We do have to clearly define exactly what we mean by trisection.

Your argument about arbitrary points similarly needs clarification. What Doctor Tom said was, “The straight-edge can only connect points already constructed, or use *arbitrary points*“. I would argue that when you draw a line with a straightedge, you are drawing *many* “arbitrary points”! But that’s not your issue.

My initial expectation on reading your comment was that you might be referring to something like the axiom of choice; but when I found the book you refer to, it was talking about something that I think is really irrelevant here.

To quote, Moise says, “Strictly speaking, random choices of points are not allowed in doing construction problems. The reason is curious: if they were allowed, then the so-called “impossible construction problems” to be discussed later in this chapter would be not quite impossible. For example, *an infinitely lucky person* might manage to pick, at random, a point on a trisector of any given angle” [my emphasis].

But that would not really make “trisection” as we are discussing it possible; if you *just happen* to pick the right point, then your result is not *provably correct*, which is essential in any construction. I think he is to some extent making the same mistake you felt Doctor Tom was making, in supposing that all we mean by a construction is to draw a line that trisects some particular angle, on one particular occasion.

To be honest, I’m not sure what Doctor Tom meant by “use arbitrary points”; he might mean drawing an arbitrary point along the straightedge, or he might mean drawing a line through an arbitrary point. I don’t know whether he had a particular kind of construction in mind. But he definitely didn’t mean that you could draw a line at random and hope it would be the trisector you are looking for.

]]>I read:

3

x 4

—-

as four groups of three, or four lots of three, or four 3s, so the second number is the multiplier

Due to the commutative property, it’s irrelevant which way around they are. I like the definition that says the concrete number is the true multiplicand; if you have 4 x apple, that’s fine – apples are concrete, so the multiplier is 4.

What I’m interested in is what the convention is when drawing 3 x 4 as a rectangle – is the vertical edge 3 and the horizontal edge 4, or are they in the same order as co-ordinates, so 3 is horizontal (x-axis) and 4 is vertical (y-axis)?

]]>However, in the second paragraph, he stated that arbitrary points may be used in a construction. Wrong, wrong, wrong! The ONLY points that can be used in a Euclidean construction are given points and those points derived from previous steps of the construction. If arbitraty points could be selected, any construction would be possible.

See, for example, the discussion following Problem 8 of Problem Set 19.1 in

“Elementary Geometry from an Advanced Standpoint” by Dr. Edwin E. Moise, published by Addison-Wesley, 2nd printing 1964.

Michael E Ochs BA Mathematics

Univ of Colo

I would be interested to know whether the dual usage confuses anyone in that context, or whether the word is just used in names of patterns and doesn’t cause any trouble because it is always pictured.

If an elementary teacher used quilts to demonstrate geometry, it could be a good opportunity to discuss with the children what all these “diamonds” have in common!

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