Why closed moduli?
(From ``is a square a rectangle'' to ``compactification of moduli space'')
A hoary question is:
"Is a square a rectangle?"
(Do the set of rectangles include squares?)
Similarly "are circles ellipses? are equilateral triangle isoceles?"
The answer is:
"it's a choice of convention (and definition);
the most useful answer is \emph{yes}"
The underlying mathematical concept is of a \Def{closed} condition.
Defining rectangles in a way that includes squares is a closed condition,
and the interesting properties that rectangles satisfy
If you want to talk about ``rectangles excluding squares'',
a ``proper'' rectangle (meaning ``not a square'')
In equations,
a rectangle is a quadrilateral with all four angles right.
To define a proper rectangle requires saying:
a proper rectangle is a quadrilateral with all four angles right
and not all sides equal (i.e., a rectangle but not a square).
practically,
the statements we care about tend to be *closed* statements.
EG,
``In a rectangle, the diagonals are the same length''
or ``in a rhombus, the diagonals are perpendicular''
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BTW, give the straight answer:
formally, the answer to "is a square a rectangle"
is "depends how you define rectangle".
Further, "definitions including squares are more elegant and useful
than definitions excluding squares"
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In fact, this notion of "closed" important
For instance, $x^2-y^2=0$ is a ``degenerate'' conic
...and for instance
are contained in a 1-parameter family / are a limit of
the hyperbolae $x^2-y^2=c$
This leads to the notion of ``compactification of moduli space''.
Here's an elementary example
(This example may not seem interesting,
in that it doesn't tell you anything you don't already know.
The point is that this is an interesting concept in more complicated
settings, and this illustrates it in an elementary setting.
)
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Symmetry
A proper rectangle has $4$ orthogonal symmetries ($V=C_2 \times C_2$);
a square has $8$ ($D_4$).
More dramatically, a proper ellipse has exactly $4$ orthogonal symmetries (as the rectangle),
while a circle has infinitely many ($O(2)$).
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BTW, characterizing objects in terms of their symmetry groups is fraught:
the quadrilaterals with symmetry group $V$ are proper rectangles and proper rhombi.
There is a lesson in this:
the function "object -> symmetry group" is
semi-continuous
(in Zariski topology?)