July 04, 2018 Wednesday
Bedtime Story
'Application to Space' - Part 1
Riemann’s 1854 paper ‘On the Hypotheses
which Lie at the Bases of Geometry’ consists of following three parts:
(I) Notion of an n-ply extended magnitude
In the first part Riemann introduces and constructs
the notion of one-fold, two-fold and n-fold extended magnitude.
(II) Measure – relations of which a
manifoldness of n-dimensions is capable on the assumption that lines have a
length independent of position, and consequently that every line may be
measured by every other.
The second part is perhaps the heart of the
mathematics of manifold where he gives a mathematical expression for the line
element which is expressed as the square root of a quadratic differential.
He also defines flat-manifoldnesses (in
which the curvature everywhere is equal to zero) as a special case of
manifoldnesses with constant curvature.
(III) Application to space
I would prefer to skip the first two parts
but would like to quote the last part as it fascinated me the most because it
is the application of mathematics of topology to the physical space.
“ III. Application to space.
§1. By means of these enquiries into the
determination of of the measure-relations of an n-fold extent the conditions may
be declared which are necessary and sufficient to determine the metric
properties of space, if we assume the independence of line-length from position
and expressibility of the line-element as the square root of a quadric
differential, that is to say, flatness in the smallest parts.
First, they may be expressed thus: that the
curvature at each point is zero in three-surface directions; and thence the
metric properties of space are determined if the sum of the angles of a
triangle is always equal to two right angles.
Secondly, if we assume with Euclid not
merely an existence of lines independent of position, but of bodies also, it
follows that the curvature is everywhere constant; and then the sum of the
angles is determined in all triangles when it is known in one.
Thirdly, one might, instead of taking the
length of lines to be independent of position and direction, assume also an
independence of their length and direction from position.
According to this conception changes or
differences of position are complex magnitudes expressible in three independent
units.
§2. In the course of our previous
enquiries, we first distinguished between the relations of extension or
partition and the relations of measure, and found that with the same extensive
properties, different measure-relations were conceivable; we then investigated
the system of simple size-fixings by which the measure-relations of space are
completely determined, and of which all propositions about them are a necessary
consequence; it remains to discuss the question how, in what degree, and to
what extent these assumptions are borne out of experience.
In this respect there is a real distinction
between mere extensive relations, and measure-relations; in so far as in the
former, where the possible cases form a discrete manifoldness, the declarations
of experience are indeed not quite certain, but still not inaccurate; while in
the latter, where the possible cases form a continuous manifoldness, every
determination from experience remains always inaccurate: be the probability
ever so great that it is nearly exact.”
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night Mon Ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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