The other day we, the various people who were around on Swhack, talked
about how area is squarish. This means that area is defined as how
many unit squares can fit in a space. I wondered whether other forms
of "area" were possible independently of squares. I wondered
especially about circular spaces, circular roots instead square roots,
and the like.

Today Noah and I discussed using equilateral triangles instead. We
realised that when you pack unit triangles into larger triangular
spaces, you get triangled(n) = squared(n). We worked out that this is
a consequence of the fact that tri(n) + tri(n - 1) = squared(n), where
tri is the conventionally defined triangular number:

http://en.wikipedia.org/wiki/Triangular_number

Our definition of triangled(...) here is of course different, but you
can see the relationship by taking a triangle of three unit length
sides, packing triangles inside, and then shading only those triangles
whose tips point upwards. Those are the tri(n) component, and the ones
whose tips point downwards are the tri(n - 1) component.

We thought about dimensional axes and what they mean. Why not have
three planar axes equally distributed from one another, so that they
are at 60 degree angles instead of 90 degree angles? We figured that
this was connected with our triangular tessellated areas, but then we
realised that four and five axes, and so on, would give some pretty
strange shapes with overlaps, and we didn't know what to make of them.

We also realised that there are only three kinds of shapes that work
in this way, giving the so-called square numbers: triangles, squares,
and diamonds. Perhaps there are others that we didn't think of. The
constraints are that super-shapes must be the same shape, and that
their sides must be equal and measurable with a ruler, i.e. straight.
Hexagons therefore don't work, and anyway they are composed of
triangles really, just like you can make bigger squares, and other
rectangles and such, with squares.

So this points to the fact that "square number" is a misnomer. One
could just as properly say triangular number, or diamond number. To
avoid confusing ourselves, we called them self-multiplied numbers. We
could have called them automultiplied numbers. These are numbers of
the form a * a, of course.

Perhaps you might scoff and say, okay, but a * a is still a ** 2,
which still points to squares as being fundamental. But consider how
one calculates a square root. Most modern calculators actually use
this identity to work out a square root:

sqrt(S) = e ** (1/2 ln(S))

Now consider what calculation there is for, as we defined it, a
triangular root. The calculation of triangular area is only linearly
related to square area anyway:

triangular area = square area * sqrt(3) / 4

And of course that's the same as:

triangular area = square area * triangular-root(3) / 4

So there's still no special primacy there.

In a way, this might help with my question about circular roots too. I
was wondering what a circular root might be, and derived it from A =
pi * (r ** 2), so that:

circular root = sqrt(area / pi)

This bugged me, since it felt wrong that a square root should appear
there. Why should squares still be fundamental? When I realised that
area was squarish, I figured that the sqrt must be introduced simply
because we're converting from a squarish area to a circular root. But
now I realise that you can write it like this:

circular root = triangular-root(squarish area / pi)

So then you could also ask why a triangular root appears. It would be
better to use the non-shaped definition as above, to introduce the
root of an self-multiplied or automultiplied number simply as a root
rather than a square root:

circular root = root(squarish area / pi)

And then this simply establishes that circles, at least in the way
defined above, don't belong to the class which probably enumerates out
to just triangle, square, and diamond.