Well, I am admittedly WAY over my head to even try and express one
perspective, but it's OT, so perhaps this "laymans" look might bring a
laugh:
Our very means of measuring is based on the concept of "linear"..
Numbers are taught to us first as places along a straight line, and
length as teh distance between those numbers. We later adding positive
and negative to broaden our perspective.
Further concepts like "Area" and "Volume" are direct products of "length".

So it's no great surprise that a circle blows our nice, neat tidy way
of expressing length, area and volume right out of the water.

What if we could turn it around and express area in terms described by
the diameter or radius of a circle? It's not at all "foreign" to
nature, even a caveman with a stick has the means to approximate a
circle with exceptional accuracy. Certainly easier than developing the
exact angles required for a square.
And to promote it to volume would be to express volume relative to some sph=
ere.

IF we could manage such a fundamental shift in our measuring means,
how often would the area of a square or the volume of a cube be
considered irrational?

So is the term irrational a feature of what we are measuring, or is it
a shortfall in our means of measurement? We are stuck in length, area,
and volume based on our concepts of "right angles".

More likely is just that I am undereducated and overly expressive at 3:AM.
Hope it was good for a laugh at least.
--=20
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