#generalised number

[isomorphismes]

Wiggly Numbers

http://en.wikipedia.org/wiki/Local_ring#Ring_of_germs:

we consider real-valued continuous functions defined on some open interval ... of ℝ. We are only interested in the local behaviour of these functions … ∴ identify two functions if they agree on some (possibly very small) open interval.... This identification defines an equivalence relation, and the equivalence classes are the "germs of real-valued continuous functions at 0". These germs can be added and multiplied and form a commutative ring.

(Of course 0 is just an arbitrary centre--but hey, why not 0?)

I feel like this is something I imagined, certainly not in this level of specificity, but at least wished for at some point in the past. Why should I be multiplying numbers like 13 and 27,714 when things in life are so less precise?

But we can still get the general idea of 13 and 27,714 without being so sticklerish about it. And no, the germ doesn't need to be defined on the frazzwangled continuum, it could be done on other topologies as well. (Wikipedia gets into it.)

Here's the only picture I could find of a germ online (the crayon splotches are my addition).

http://www.cs.bham.ac.uk/~sjv/GeoFuzzy.pdf :

Yes, it's no wonder this Grothendieck stuff is called an impressionistic mathematics.

[isomorphismes]

Pythagorean Theorem

This is how I first really understood the Pythagorean Theorem.

The outer circle looks just a little bit larger than the inner circle. But actually, its area is twice as large.

Kind of like the difference between medium and large soda cups, or how a tiny house still requires kind of a lot of timber, for how much air it encloses. If you buy a slightly wider pizza or cake it will serve proportionally more people; and if an inverse-square force (sound, radio power, light brightness) expands a little bit more it will lose a lot of its energy.

Ideas involved here:

scaling properties of squared quantities (gravitational force, skin, paint, loudness, brightness)

circumcircle & incircle

√2

This is also how I first really understood √2, now my favourite number.