Attached is a power point presentation called "double vision" the reason I

was inspired to send this was because of how something may look very

different depending on far we are from that object.

In mathematics there is is the concept of Manifolds in which an object

appears to be one thing when the distance is small but something else when

the distance is large.

"A manifold is an abstract mathematical space in which every point has a

neighbourhood which resembles Euclidean space, but in which the global

structure may be more complicated. In discussing manifolds, the idea of

dimension is important. For example, lines are one-dimensional, and planes

two-dimensional."

For example in a manifold of 1 and object may appear to be a straight line,

but as the distance from the object increases it is seen to actually be a

curve. For example if we were to take a very large circle with a diameter of

10 miles then if we were to go to the circumference of that circle and take

a 6 inch segment of it to our vision it would appear to be a straight line.

In a manifold of 2 at a very close distance an object may appear to be a

flat plane, but as the distance increases we actually see that what we

thought was a flat plane is actually just a small portion of a large curved

object like a globe. For example a football field appears to be flat but in

reality it is part of the globe of the earth. It doesn't have to be a sphere

it can be any type of curved space like hyperbolic parabaliod (saddle

shape).

In a manifold of 3 an object can appear to be sphere for example but at a

greater distance it is something else--a lotus. In a lotus flower you will

see on close examination that there is a flat surface on which there are

many spheres supported by small columns. If you were to zoom in to one of

the small speres it would appear like a globe but as you zoom out you see it

is just one small sphere on the surface of the lotus.

This last example is of course useful regarding how the Bhagavatam describes

the universe as looking like a lotus flower. We can still have globe shapes

like the earth yet at same time retain the lotus shape. It all depends on

the perspective of teh viewer.

For more on Manifolds see: http://en.wikipedia.org/wiki/Manifold

The attached Power point presentation while having nothing to do with

Manifolds dramatically illustrates how an object may appear to be one thing

when viewed from a distance but something else when seen up close.

Food for thought when discussing what the universe looks like. The

Bhagavatam describes what the universe looks like to higher beings. From our

perspective it may appear to be different.