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From: peno@enea.UUCP (Pekka Nousiainen)
Newsgroups: net.math
Subject: Euler's formula and arithmetic dimensions
Message-ID: <1095@enea.UUCP>
Date: Sun, 24-Nov-85 19:30:33 EST
Article-I.D.: enea.1095
Posted: Sun Nov 24 19:30:33 1985
Date-Received: Wed, 27-Nov-85 05:29:35 EST
Reply-To: peno@enea.UUCP (Pekka Nousiainen)
Organization: Enea Data, Sweden
Lines: 41

< line eater bait >

In the November issue of Science '85 Yu.I.Manin writes (p.91) that
Euler's formula

        6/PI^2 = (1-1/2^2)*(1-1/3^2)*(1-1/5^2)*...

(2,3,5,... = prime numbers, ^ = exponentiation) "gives the volume of a
simple solid in a space having one ordinary dimension and one arithmetic
dimension".  It sounds like he's talking about the space

	R x Spec(Z)   (R = reals, Z = integers)

where Spec(Z) is the algebro-geometric spectrum of integers.  Does
anybody know more about this?

Background for non-mathematicians:

Manin writes about the "odd" dimensions one can add to the ordinary
three dimensions of space.  Physicists are using such dimensions to
explain the behavior of elementary particles.

The most familiar (and not "odd") addition of dimension is the complex 
numbers: a + bi where i^2 = -1.  Similarly one can define numbers a + be 
where the new unit e (epsilon) satisfies e^2 = 0.  This space, usually 
denoted R[e], is already in common use in physics. If you think about 
how ordinary numbers behave - 0.1^2 = 0.01, 0.01^2 = 0.0001 - it makes 
sense to think of e as being an "infinitely small" number, so small that
its square actually is 0.  The resulting space R[e] is sort of real numbers
infinitesimally "thickened" into another dimension of space.

What Manin proposes above is still more abstract (and not yet used in 
physics).  The points in R x Spec(Z) would be pairs (a, p) where a is
an ordinary number and p is a prime number.  There'd also be special
points (a, 0) that sort of glue the whole thing together.

P.S.  I have NOTHING against Polar Bears but could we have a bit more
math in this group?  The recent posting on the independence of CH via
topos theory is one of the few articles that actually belongs here.
Anybody for net.math.expert?