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Core Computational Project

https://sourceforge.net/p/corecomp

The aim of this project is to investigate the potential for using the pattern recognition
techniques available through vedic computation within the environment of modern
computation.

The software development is being facilitated through the Pure language, a functional
programming language with an excellent interface to the world of imperative programming.
Further information in this regard may be obtained by following the links at,
http://pure-lang.googlecode.com.

In the context of computation, the methods of vedic mathematics may seem a little obscure,
and outdated. However these methods are based on looking for patterns within computational
processes and exploiting these patterns to develop faster and more interesting techniques
than those available through the standardised approaches. In general vedic computation is
being used as a means to foster developments in mental computation. Further information
may be obtained by following the links at, http://www.vedicmaths.org/.

As well as looking at the initial areas of computational development, this project will
also be looking at the other end of the scale, languages, natural languages and
complexity. As part of this I wish to resurrect some research which I conducted into
Panni's grammar, apparently the world's first book of grammar and which, according to one
of the sources I consulted is laid out in a manner similar to a Backus Naur Form (BNF),
the modern method for prescribing a computational language. This brings up one interesting
point in relation to the development of mental techniques of computation, in that I
believe that the earliest knowledge in computation may similarly be prescribed, using the
language of Tirthaji's vedic mathematics. The essential difference goes back to the
earliest phase of computational development, wherein a student practices a mental
technique, a process of computation, rather than relying on memory or understanding. Both
of these develop, of course, but in a more natural context.

This project will involve going deeply into the area of computational complexity in order
to see if the insights provided through vedic, and other sources computation can shed some
light.

As an example there is an intriguing reference in Tirthaji's book on vedic mathematics,
that there exists a mental formula for Pi which allows for the expression of Pi to
whatever number of digits desired. This mental formula is expressed in sanskrit, hence the
interest in bringing natural language and computation together.

Finally, depending on how this work develops, there will be an opportunity to explore some
aspects of vedic geometry in a modern context. And, just to whet your appetite, the most
significant difference between modern geometry and vedic geometry, is in how the
relationship between the domain of a space and it's dimensions are considered.

In the standard approach to geometry, there is very little if no mention of the role of
the dimension of a space, as little more than a descriptor defining the number of degrees
of freedom afforded to objects, existing within or moving within a domain space. In a
1-dimensional domain, each object has a single degree of freedom, essentially
corresponding to extending or moving within a line. The only objects which can exist
within the 1-dimensional domain are 0-dimensional points, or 1-dimensional line segments.
Again, moving to 2-dimensions, there are two degrees of freedom, affording extension or
movement in two directions, essentially two degrees of freedom. The objects which occupy
this space are points, lines, both straight and curved, and plane bounded areas. This of
course gives rise to an extraordinarily rich area of computation, as does a three
dimensional geometrical perspective.

So far, so good, the difference between the vedic perspective and the modern perspective
based on Euclidean geometry are in apparent agreement. One of the reasons for this is that
from a vedic perspective Euclidean 1-space and Euclidean 2-space are a projection of
aspects of Euclidean 3-space, all affording the dimension of Euclidean 3-space, in that
the dimensional space itself is inherently of a single dimension.

Moving from 3-space to 4-space, however, the perspective changes. According to the
perspective based on the Euclidean approach, 4-space simply attaches another degree of
freedom of linear dimension to the mathematical model. Of corse we can speak of Riemann
space as well, involving curved spaces, but the essential aspect is that in each of the
higher spaces the associated dimensional space affords one degree of freedom, it is is
1-dimensional.

This contrasts with the hierarchal development of vedic spaces, which follows a very
simple prescription, which is to say that for any domain of dimension n, the dimensional
space admits its expression within a space of dimension n-2. Thus for 3-space the
dimensional space is 1-space, and there is agreement between the vedic and modern
perspectives. But once we move to 4-space the perspective changes, and the modern approach
considers a 4-dimensional domain prescribed by 4 linear degrees of freedom, in other words
4X1-dimensional spaces. In contrast, according to the vedic perspective, the 4-dimensional
domain space is prescribed using 4 dimensional spaces each admitting two degrees of
freedom, in other words 4x2-dimensional spaces. In other words the dimensional space of
4-space admits expression in terms of a planar space, and when we move up to 5-space, the
dimensional space admits expression in terms of a 3-dimensional space, the dimensions
themselves become solid.

So how does this difference manifest itself.

In terms of linear objects there is not necessarily much difference. Hyper solids in each
perspective may be considered to be the same, although detailed examinations may expose
subtle differences. For example the notion of a point in a 4-dimensional Euclidean space
will be replaced with the notion of a 4-dimensional space content infinitesimal element,
which although it may be infinitesimally small still carries with it the characteristics
of a 4-dimensional space. The space content of the domain will be different to the space
content of the boundary, which will be different to the space content of the dimensional
frame, effectively the boundary of the boundary.

One point to note here is that the perspective of a geometrical system where the space
content has extent, no matter how small, may enable a fresh approach to looking at the
singularities, which bedevil some developments in areas like string theory.

So that's it. In relation to geometry the first area which requires consideration, is how
to prescribe a hyper-sphere in a 4-dimensional space, where each dimension admits
expression as a 2-dimensional space. The difficulty here is how to define the metric of
the space, how to define the notion of distance between space elements. The normal
Euclidean norm doesn't work, as it is based on a linear measure, and the hyper-sphere is
prescribed as those points which are enclosed by a boundary all of whose points are
equidistant from some central point, i.e. the origin of the hyper-sphere.

Another possible approach is to define the hierarchy of spheres in terms of rotation.
Rotating a line segment out of 1-space to prescribe a disk, or a circle. Rotating a disk
out of 2-space to prescribe a sphere in 3-dimensions, and rotating a 3-dimensional sphere
out into a 4-dimensional space to prescribe a 4-dimensional hyper-sphere. At the moment it
is not obvious how this may be done so as to develop a consistent approach which may be
applied to a whole hierarchy of spaces.

There's a lot in this and we'll see how it goes.

Brian G. Mc Enery PhD

Rinneen Skibbereen Co. Cork Ireland

04 January 2013

briangmcenery@gmail.com