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It seems to me that that set of all squares is essentially ${\Bbb R}$, having one degree of freedom. And the set of all rectangles exists in ${\Bbb R}$ x ${\Bbb R}$, having two degrees of freedom.

How can any element in ${\Bbb R}$ e.g. (3) be identical to an element in ${\Bbb R}$ x ${\Bbb R}$ e.g. (3, 3)? Dimensionality would seem to preclude this.

Therefore is it reasonable to say that we cannot precisely claim that a square is a special kind of rectangle?

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    $\begingroup$ Set of all pairs $(r,r)$ is a $1$ dimensional (vector-) subspace of $\mathbb{R}^2$ which is isomorphic to $\mathbb{R}$ via standard map $r\to(r,r)$ $\endgroup$ – user160738 Dec 24 '16 at 16:25
  • $\begingroup$ Do we then have different kinds of squares? Those that are represented by the subspace, and those that are represented by the reals? $\endgroup$ – Brad Thomas Dec 24 '16 at 16:32
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    $\begingroup$ How is the set of squares essentially $\mathbb R$? That sentence makes no sense to me. $\endgroup$ – user251257 Dec 24 '16 at 16:35
  • $\begingroup$ Do you feel that a point on the line $y=x$ is not really in the plane because the line only has one degree of freedom? But of course the pair $(x,x)$ is a perfectly good pair of real numbers for $x\in \mathbb R$. $\endgroup$ – lulu Dec 24 '16 at 16:39
  • $\begingroup$ In a different spirit...do you feel that rectangles with Length twice the Width aren't really rectangles either? After all, those only have one degree of freedom. What about $L=3W$? Or $L=\pi W$? Can you give an example of a real rectangle? $\endgroup$ – lulu Dec 24 '16 at 16:43
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It seems that you are confusing certain representations of squares and rectangles with the shapes themselves. In particular, we can define:

A rectangle is a quadrilateral with all right angles.

A square is a quadrilateral with all right angles and congruent sides.

Here, if we take any square, we see that it is a quadrilateral with all right angles, hence a rectangle.

What you are doing is coming up with a certain representation of squares and rectangles and claiming that, since these are not equal, squares are not rectangles. In particular, if we imagine looking at only squares and rectangles centered at the origin with sides parallel the axes, any pair of two positive real numbers can be associated to a rectangle and any single real number can be associated to a square - but these are just associations (i.e. functions from $\mathbb R$ and $\mathbb R\times \mathbb R$ to the set of squares and rectangles respectively), not equalities.

A similar argument could show that squares aren't squares: Clearly, a square centered on the origin with sides parallel the axes can be represented by a single real number giving its side length. But it can also be represented by a point on the line $x=y$ giving one of its corners! Since these representations are different, should we conclude that squares are not squares?

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"Dimensionality" is the wrong stick to measure this with. Nobody (hopefully) questions that a quadrilateral is "a special kind of" polygon, yet the two have quite different "degrees of freedom".

The question touches on the more general concept of embedding in mathematics. Often times, elements of a smaller structure can be "naturally" mapped to elements of an enclosing larger structure, in such a way that their defining properties are preserved. In cases where this canonical mapping is unique (up to some isomorphism of sorts), it is common to consider the corresponding elements in the two structures as identical, without restating each time that the "identity" is in fact based on the canonical embedding. For a couple of everyday examples...

  • Integers are embedded in the rationals by $\,\mathbb{Z} \to \mathbb{Q} :\, n \mapsto \frac{n}{1}\,$.

  • Reals are embedded in the complex numbers by $\,\mathbb{R} \to \mathbb{C} :\, x \mapsto x + 0 \cdot i\,$.

Those embeddings are so common that no one would blink an eye reading that $1 \in \mathbb{Q}$ or $2 \in \mathbb{C}$.

Much the same way, squares are a special kind of rectangles in the sense that the set of all rectangles "embeds" all squares as its subset of rectangles with equal sides.

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