Georg
Friedrich
Bernhard
Riemann was
a profoundly
original
thinker.
In the
span of
a very
brief life,
he left
us with
a host
of methods,
theorems
and concepts
named
after him,
including
the still
unproven
Riemann
hypothesis.
Riemann's
portrait includes
graphical
representations
of some
of Riemann's
most complex
concepts.
Riemann
surfaces,
an
instance
of one
overlaying
the lower
right of
Riemann's
portrait, are a Riemann
invention
arising
from multi-valued
functions
-- a not
uncommon
conundrum
in mathematics.
They resolve
problems
in which,
essentially,
two different
values
of 'y'
correspond
to the
same 'x',
as can
often arise
in
complex
operations,
and even
in relatively
simple
square
root operations
and logarithm.
Riemann
essentially
increased
the number
of 'x's",
so that
there were
enough
for each
possible
y.
The solution
is not
only ingenious,
but after-the-fact,
it is seems
somewhat
obvious
-- the
mark of
true insight.
Pictured
over
Riemann's
shoulder is another,
and quite
different
depiction
of this
very versatile
tool, the
Riemann
surface.
In
his 1859
paper
On
the Number
of Primes
Less Than
a Given
Magnitude,
Reimann dealt with a formula for the identification
and track
the occurrence
of prime
numbers
( a prime
number
being one
which has
no factor
except
itsef and
1). Prime
numbers not
only have
an almost
mystical
fascination,
but seem
to hold the
keys to many
perplexing
riddles.
Riemann
examined
the properties
of the Zeta
function, (inscribed
at the
lower
left
of Riemann's
portrait). And
he advanced
a riddle
of his
own,
the Riemann
hypothesis,
essentially
positing
that
the Zeta
function
has infinitely
many
non-trivial
roots,
with
the vast
majority
of
the roots lying between 0 and 1. This still unproven
hypothesis
remains
of the
most
important,
and beguiling,
unresolved
problems
in mathematics.
(And
if you
study the
portrait
for a moment,
you may
notice
in the
upper left
hand area
a "skewed"
rectangular
area which
hasn't
fully "shifted"
into place
-- an homage
to the
still unresolved
Riemann
hypothesis!).
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