# In what curricula are “rectangles” defined so as to exclude squares?

Most contemporary curricula define the word "rectangle" inclusively, so that all squares are automatically rectangles. Are there curricula in which this convention is not followed? That is, are there curricula in which "rectangle" is defined so as to explicitly exclude squares? International and cross-cultural perspectives, in particular, would be appreciated. Also, I would like to know if this is age-dependent, like when it comes to kindergartners.

Devil's advocate: Would you teach that the sun is white? (Analogy: Apparently, 'The Sun is yellow' is like squares are not rectangles.')

Some context: See previous revisions.

• There are occasionally advantages of defining rectangles to exclude squares. Example: All rectangles (other than squares) have two lines of symmetry; squares have four. – A. Goodier Mar 6 '18 at 18:01
• Just an anecdote: I was taught that squares were not rectangles, roughly up until I started taking programming courses, where Square was a subclass of Rectangle because it just made sense to teach things that way! – Cort Ammon Mar 9 '18 at 16:35
• @A.Goodier Squares also have two lines of symmetry. Plus two more. – Acccumulation Mar 9 '18 at 20:03
• @JW edited! thanks – BCLC Jan 24 at 23:55
• See the lower half of page 31 [= .pdf page 32] of this document. – Dave L Renfro Jan 25 at 9:33

I cannot answer the OP's question about cross-cultural/international perspectives, but here is a historical perspective that may be helpful. The issue here (whether the category "rectangles" includes or excludes the category "squares") is one aspect of a larger question having to do with whether the classification of quadrilaterals should be partitional or hierarchical. It turns out that until relatively recently (say, 120 years ago) partitional definitions were the norm.

For example, Euclid’s own definitions were unequivocally partitional: in the Elements, Book I, Def. 22 he wrote (translation is Fitzpatrick’s 2007 edition from Heiberg’s Greek text; emphasis added):

And of the quadrilateral figures: a square is that which is right-angled and equilateral, a rectangle that which is right-angled but not equilateral, a rhombus that which is equilateral but not right-angled

Thus, for Euclid a square was neither a rectangle nor a rhombus. Not surprisingly textbooks that followed Euclid closely tended to adopt this classification up through the middle of the 19th century. For example the 1849 edition of Playfair’s Elements of Geometry defined "square", "rhombus" and "oblong" (the word "rectangle" does not appear at all) as follows (emphasis added):

Of four sided figures, a square is that which has all its sides equal and all its angles right angles... An oblong is that which has all its angles right angles, but has not all its sides equal… A rhombus is that which has all its sides equal, but its angles are not right angles.

Similarly Brewster’s translation (1834 and 1851 editions) of Legendre’s Elements of Geometry defined the quadrilaterals as follows (emphasis added):

Among the quadrilaterals, we distinguish: The square, which has its sides equal, and its angles right angles. The rectangle, which has its angles right angles, without having its sides equal. The rhombus, or lozenge, which has its sides equal, without having its angles right angles. And lastly, the trapezoid, only two of whose sides are parallel.

Notice that in this last passage, in addition to defining "rectangle" and "rhombus" in an exclusionary way, the word "trapezoid" is also defined so as to exclude parallelograms.

However, in the second half of the 19th century this tradition of partitional definitions began to give way to the hierarchical classifications currently in use. The 1872 edition of Legendre’s Elements from the same publisher replaced the exclusive definition of rectangle quoted above with an inclusive one, defining a square as an "equilateral rectangle". Despite this, "rhombus" was still defined so as to exclude right-angled figures, and "trapezoid" was defined as "a quadrilateral which has only two of its sides parallel."

From ca. 1850 through 1910 texts were all over the place with respect to the definition and classification of quadrilaterals:

• Hill's (1884) A Geometry for Beginners used exclusive definitions in all cases: "The sides of a parallelogram may be either equal or unequal, and the angles may be either right or oblique ; so there are in all four kinds of parallelograms : the oblique unequal-sided parallelogram or RHOMBOID.. the oblique equal-sided parallelogram or RHOMBUS… the right unequal-sided parallelogram or RECTANGLE… and the right equal-sided parallelogram or Square."
• Hall & Stevens' (1891) A Text-Book of Euclid’s Elements for the Use of Schools defined "rhombus" so as to exclude squares, but "rectangle" so as to include squares. (In other words it followed the same conventions as the 1872 English-language edition of Legendre’s text.)

By 1900 or so most textbooks had settled on inclusive definitions for "rectangle" and "rhombus", so that a square was a species of both.

However even after this the convention persisted (and still persists!) to use an exclusive definition for "trapezoid". For example, Wells & Gart’s (1915) Plane and Solid Geometry explicitly stated that "a rectangle is a special parallelogram" and "The square is a special rectangle and also a special rhombus"; however, it continued to define "A Trapezoid is a quadrilateral which has one and only one pair of parallel sides". Similarly Auerbach's (1920) Plane Geometry had "A trapezoid is a quadrilateral with one, and only one, pair of parallel sides." The SMSG (1961) Geometry defines a trapezoid as "a quadrilateral in which two, and only two, opposite sides are parallel", and the two modern Glencoe-McGraw Hill Geometry textbooks that I have access to follow suit.

So, to make a long story short: the hierarchical definitions that are now commonplace first began to appear in the mid-19th century and became standardized in the late 19th and early 20th centuries, but the transition was a gradual and somewhat messy one, and is still incomplete today, at least with respect to trapezoids.

Having said all of this, it would not surprise me at all if there are places in the world where partitional definitions for rectangle are still in use – indeed in a setting where Euclid's Elements is still taught, I would expect that to be the case.

• The "hierarchical definition" issue you bring up reminds me of how the definition of the Borel set and Baire function hierarchies changed from the the first few decades of the 1900s to later (probably after WW 2). In earlier literature, a Baire class 2 function (say) is one that is a pointwise limit of Baire class 1 functions and not a Baire class 1 function. In later (and current) literature, Baire class 2 functions include Baire class 1 functions and Baire class 0 (= continuous) functions. The present version is much easier to use in stating theorems and such. – Dave L Renfro Mar 9 '18 at 18:17
• Devil's advocate: What colour is the sun? – BCLC Mar 20 '18 at 13:20
• What is most useful is to have separate exclusive terms for partitions based on some criteria, and inclusive terms which ignore those criteria, as would be the case with rational numbers, irrational numbers, and real numbers (the latter set including both of the former). Irrational numbers may then be partitioned into algebraic and transcendental, while rational numbers may be partitioned into integers and fractional numbers. Unfortunately, there's aren't separate terms to distinguish right-angle quadrilaterals with arbitrary sides from those with unequal sides. – supercat Jan 25 at 22:42
• @supercat amen to that – BCLC Apr 28 at 6:00

There is a model of how people progress towards abstract reasoning through the subject of geometry called the Van Hiele model. The model describes five levels: visualization, analysis, abstraction, deduction, rigor. It dates to the 1950's, and continues to influence curricula.

In the analysis level, children do not allow overlaps in categories, and will insist that a square is not a rectangle. So your question is at the heart of a developmental matter. Given that educators wish for their students to progress beyond the analysis level, I would be surprised to learn about a curriculum that rigidly defined squares to not be rectangles.

• What do you say in re mweiss' answer? – BCLC Mar 10 '18 at 16:48
• I see from mweiss answer that intellectually, society did not transition to the modern hierarchical definitions until the middle of the twentieth century, when the work of Piaget on cognitive development began a transformation in education. Teaching geometry using Euclid’s Elements is archaic, and we have moved on to using inclusive definitions. Note how the van Hiele model describes not only development in an individual’s life, but also intellectual development of civilization. You can see a societal transition from “visualization” to “analysis” in this discussion of quadrilaterals. – user52817 Mar 10 '18 at 21:26
• Another: what about Adam's? – BCLC Mar 19 '18 at 7:24
• Adam’s answer about programming is called the “circle-ellipse” problem. It points to a limitation of the object oriented programming paradigm for modeling. Marshall Cline, in his book C++FAQs, writes “Because a circle is a specialized ellipse (circles are ellipses with an extra symmetry constraint) does not imply Circle is substitutable for Ellipse.” – user52817 Mar 19 '18 at 15:22
• re Piaget, is Piaget in mweiss' answer? If so where please? If not, are you assuming it's not a coincidence that Piaget's work's transformation was the around same time as the beginning of the transition to modern hierarchical definitions? – BCLC Mar 19 '18 at 15:34

I just came across this discussion https://news.ycombinator.com/item?id=16605831 which reminded me of your question. The perspective isn't exactly math and isn't exactly not-math but rather, a question in object-oriented programming. Several of the comments (search for rectangle and square to get the relevant comments) there are worth reading on their own, but let me provide some context.

In object oriented programming, you define define types/classes which have properties and things you can do with them (methods). There is also a notion of "inheritance" where you take a class of type A and extend it to a class of type B. Since B is an extension of type A, it has all of the methods and properties of A. Hence any instance of type B can also play the role of an instance of type A.

One of the classic examples which is given in programming text books is that of squares and rectangles. Squares are a type of rectangle, so if we were to create classes of type Rectangle and Square, it makes a certain amount of sense that Square should extend Rectangle. Similarly, Rectangle should be an extension of ConvexPolygon which extends Polygon, etc.

However, squares are formed by constraining rectangles. They have a single side length. Rectangles have pairs of side lengths. The natural methods to implement for squares and rectangles should return different types: a single number vs. a pair. So perhaps having Square be an extension of Rectangle is not actually the best idea.

However, none of this would prevent a Rectangle (type) from being a square (math) incidentally.

• Wait why not just use oblong for pairs of side lengths? – BCLC Mar 19 '18 at 15:30
• What would that solve? – Adam Mar 20 '18 at 13:21
• Adam, I'm guessing there's some function 'Square' that takes one argument and another 'Rectangle' that takes two arguments that have to be distinct. Why not have a function 'Oblong' instead of 'Rectangle' ? – BCLC Mar 20 '18 at 13:32
• This sounds programmatically fraught. We want Square to be passable to any fn that takes any quadrilateral that Square can be. But this breaks: Rectangle.getTwoLongestSides() returns a pair of opposite sides. We could pick two arbitrary sides, but that assigns an orientation to Square that it should not have. And Kite.getTwoLongestSides() is expected to return two adjacent sides, so what do we do then? By allowing Square to be other things, we have to handle all these bizarre edge cases. Square has properties the others don't, and vice versa: it's just a bad fit for inheritance. – Dewi Morgan Jun 5 at 5:40
• (not a criticism of your answer, just a pondering on whether received wisdom about squares makes sense from a programming perspective, and how I could implement it) – Dewi Morgan Jun 5 at 6:19