[FRIAM] Photos of popped balloon

Nick Thompson nickthompson at earthlink.net
Mon Feb 4 15:48:42 EST 2019

Holy Moly, Lee.  With text books like that, I coulda beena mathematician
after all!


Nicholas S. Thompson
Emeritus Professor of Psychology and Biology
Clark University

-----Original Message-----
From: Friam [mailto:friam-bounces at redfish.com] On Behalf Of
lrudolph at meganet.net
Sent: Monday, February 04, 2019 9:33 AM
To: The Friday Morning Applied Complexity Coffee Group <friam at redfish.com>
Subject: Re: [FRIAM] Photos of popped balloon

> I think they were cylinders, not spheres, so there were two holes. 
> This is where we start talking about homology groups.

We don't absolutely *have* to.  The theories of Riemann surfaces and
algebraic functions got pretty far just having the (proto-homological, but
very ungroupy) notions of "simple connectivity" vs. "multiple connectivity".

[For those readers, possibly consisting of Nick alone, here's what that
means. Suppose you produce a thin sheet of copper by electroplating onto
some or all of the surface of a solid piece of wax that you then melt away.
For instance, you get a cylindrical surface if you start with a solid wax
cylinder and only electroplate onto its lateral surface, leaving the round
disks at its two ends unplated; and it will be possible to melt the wax away
without cutting a hole in the copper.  On the other hand, you get a
spherical surface if you start with a solid round ball of wax and
electroplate onto its entire surface (let's not worry about how you do
that...); in that case, you'll have to puncture the sphere (maybe cutting
out a little disk around the south pole) to let the melted wax escape.  
Just make one hole!  (And don't worry about possible difficulties draining
out all the wax, okay?)  For a third example, start with a piece of wax in
the shape of a donut (a so-called "solid torus" or, in a charmingly antique
idiom, an "anchor ring"); the resulting copper surface is a "torus" plain
and simple.  Again, a single hole will suffice to drain the
(idealized) wax; again, don't make any others.

Now take your pair of metal shears and start cutting somewhere on an edge of
the copper sheet.  In the cylinder example, you have two edges, each of them
a circle at one end of the cylinder.  In the sphere and torus examples, you
have a single (circular) edge, around the hole you drained the wax through.

It is a fact (which I hope you can imagine visually with no trouble, because
all this electroplating would be expensive and difficult) that no matter how
you the sphere-with-one-hole with your shears, starting and ending at edge
points, you will cut the copper into two pieces.  It is also a fact that on
both the cylinder and the once-punctured (i.e.,
drained) torus, there are *some* ways to cut from an edge point to another
edge point that do *not* cut the copper into two pieces.  (On the cylinder,
you have to start somewhere on one of the two circular edges and end
somewhere on the other: when you've done that you can unroll the cylinder
flat onto a table.  On the once-punctured torus, there are many very
different ways to make such a "non-separating" cut.)

Riemann and Co. described this qualitative distinction between the surface
of a sphere and the (lateral) surface of a cylinder (and torus, etc.,
etc.) by calling a sphere "simply connected" and the others "multiply
connected".  "Simple" here is like 0, and "multiple" like "a strictly
positive integer", which began the process of refining the qualitative
distinction into a quantitative distinction.  Very soon the quantitative
distinction was refined much more by making the various positive integers
distinct (so the "cut number" of the sphere is 0, the cut number of the
cylinder is 1, and--as it turns out--the cut number of the torus is 2).

Rather later, this quantitative distinction became more refined. 
Eventually it became *so* refined that "homology groups" appeared as the
best way to describe the refinements.

It is quite possible that the mathematical physicist John Baez, Joan's
younger cousin, wrote all this stuff up very clearly 15 or 20 years ago. 
If so, it would be findable with Google.]

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