Is a Square a Rectangle?: Welsh edition


byrnes-euclid-rectangleAnswer:  No.

I’ve posted before about how even though in the US we define squares to be equilateral rectangles, there are many forces (often in the form of picture books on shapes) that treat squares and rectangles as distinct beings, and so it is really no surprise that many students reach college a little uncertain.  The mathematical definition doesn’t match the cultural one.

Anyway, this week in Geometry I was going over Euclid’s definitions, and I pointed out that he wrote explicitly that rectangles (oblongs) weren’t allowed to have four equal sides, which is different than the definitions we use today.

Then one of my students spoke up.  Katie is a senior math major getting certified to teach elementary & middle school, and this past fall she did part of her student teaching in Wales  (courtesy of a study abroad program here at the college).  She was in a 5th grade class, and according to their formal curriculum squares are not rectangles.  Indeed, Katie said that the definitions they used were pretty much the same as what appears in the translation of Euclid (rectangles have four right angles but can’t be equilateral; rhombi have  four equal sides but can’t have right angles; parallelograms have parallel sides, but can’t be equilateral or have right angles).

Similarly, when she taught about diamonds, she couldn’t call them squares even if they had four equal sides and four right angles.  She had to prove the same rules twice (say, that the diagonals of a square are equal, and that the diagonals of a equiangular diamond are equal) and when she drew them, she had to add a line to show whether the figure she was referring to was a square or a diamond.


That explains this Failblog post from last October:

more fail, owned and pwned pics and videos

(I’m not sure if this picture was from Wales, but she did see Shreddies in the store there.  She really liked the frosted kind.)

So now I’m wondering:  how are geometric figures (squares versus rectangles and the like) defined in other countries?

The image above is from Oliver Byrne’s way-cool color edition of Euclid’s Elements.

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4 Responses to “Is a Square a Rectangle?: Welsh edition”

  1. Dave Richeson Says:

    A similar question—a colleague asked me the other day if the positive x- or y-axes are include in “the first quadrant.” I don’t know (and haven’t investigated further).

  2. jd2718 Says:

    ask about trapezoids….

  3. Jason Dyer Says:

    You might like Jeff Miller’s section on trapezoids:

    TRAPEZOID. The usual definition of a trapezoid requires that it have exactly one pair of parallel sides. However, the UCSMP textbook Geometry (1997) has the definition “a quadrilateral with at least one pair of parallel sides.” (In this textbook, an isosceles trapezoid is defined as “a trapezoid with a pair of base angles equal in measure.”) According to Chris White, the Cresent Dictionary of Mathematics by William Karush (1962) defines trapezoid as “A quadrilateral which has exactly one pair of parallel sides (sometimes, the parallelogram, with both pairs of sides parallel, is included).”

    TRAPEZOID/TRAPEZIUM. In the United States, a trapezoid is generally defined as a quadrilateral with exactly one pair of parallel sides and a trapezium is a quadrilateral with no sides parallel. However, until the end of the eighteenth century, the definitions for these two terms were reversed, and outside the United States even today the definitions are often reversed.

  4. vlorbik Says:

    okay. i’m giving way to despair now. it’s hopeless.

    the axes aren’t in quadrants. you can quote me.

    on the other hand, i’m fanatical enough to believe
    that “noon” and “midnight” are neither AM nor PM
    (essentially because zero is neither positive nor negative).
    and even that moreover this should be obvious if one knows
    what the letters stand for.

    so maybe you’d better find another source.

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