I found this info about Googolhedron here
Googolhedron - A 3 dimensional shape bounded by 1 x 10^100 similar polygons. This shape would look very much like a sphere. Having this many sides or facets would make it smoother than any man made object. Although, you could never have a googolhedron because there are not a googol particles in the universe.
Does it mean that a PERFECT sphere has an infinite number of polygons and hence does not exist?
How many faces does a sphere have ?
8 comments:
How many corners does a circle have?
Are you referring to limits by any chance? So a regular convex polygon tends towards a circle, when the number of points tends towards infinity. I think a similar arguement can be made for a sphere.
"Does it mean that a PERFECT sphere has an infinite number of polygons and hence does not exist". This is akin to asking the question - "Does it mean that a PERFECT line has an infinite number of points and hence does not exist".
Firstly, finite and infinite are our ways of guaging things. It does not affect reality whether there are finite (since "we" can count) or infinite (since "we" cannot count) number of entities as a part of that reality.
another dimension to it is that, the point of reference makes a difference in the perception of perfection.
a sphere from far would seem perfect until you came closer and realized it had some jagged edges.
the point about infinity can also apply to an imperfect sphere. it too can be seen as having infinite number of smaller faces. But it exists with its imperfection.
So, a perfect/imperfect large can always be seen as being constructed from an infinite number of smalls.
Okay I guess it is kind of convincing after those analogies you guys gave....
I wonder how they arrived on the volume and surface area of a sphere ?
Volume of a sphere = 2 * volume of a hemisphere ( Vh )
This can be got through integration.
Volume = Area * height.
Cylinder
Height varies from 0 to h
Area = pi * sqr(r)
Volume = pi * sqr(r) *h
Hemisphere
Height varies from 0 to r
Area = is also a function of height
Here by area, I mean the area of the circle. Volume is the just the sum of areas of circles stacked one on another.
Volume = Integrate b/w 0 to r,
pi * ( r*r - y*y ) * dy
{ The pi * (r*r - y*y) is the area of the circle at a height y }
= (pi/3) * cube(r) - pi * cube(r)
(Split the integral, since pi *r*r is a constant, can be moved outside ).
= (2/3)*pi*cube(r)
Volume of sphere = (4/3)*pi*cube(r)
We can solve similarly for surface area. I think we would consider circumferences instead of areas, and then integrate.
I bow to richie. Einstein, standing next to me, bows too.
Man.....That was one mathematical verbose entry! Amazing stuff....It took a while for me to understand this one !
:p,:p. In my defence, you asked for it :), and I kacchaxed.
Post a Comment