If the volume is equal to the numerical value of the Planck length then i am interested to know the dimensions of this geometrical solid.
An example: Suppose volume is a cube with side Planck length. Then the numerical value you get is about 10^-105. This is a lot smaller than the quantized value. Problems like this arise with any kind of geometrical object, including marshmellows, you can't even make a square with side Planck length without violating quantization.
Lets make the problem simpler so i can make my point without confusion. Pretend PL is .5 and you have a quantized system that does not allow smaller values. Then someone asks,'what happens if you square PL?'
You get .25 which is a smaller value than PL. 'No problem' you say, 'I'll just put .25 into my system as an exception.' What if you square the exception? then you get a smaller number still. You can repeat this process ad infinitem. You always get a number smaller than the previous and greater than zero. So you are rapidly creating an INFINITE number of exceptions approaching zero in your quantized system that was designed to have a FINITE number of points between zero and one. This is one of the reasons IMHO i believe quantization is dead.
--------------------------
Re: Agentredlum's example about size and dimensionality above:
I hope everyone will forgive me more of my words... I did not expect a geometric challenge to where those doggone irrational numbers are. At any rate:
1.616252*10-35 Planck-distance, in meters,
or “amount of line” covered by a Planck-distance, otherwise known as a Planck-length
(1.616252*10-35)*(1.616252*10-35) = 2.61227*10-70 = 1planck-area, in square meters,
or “amount of surface” covered by a Planck-area
(1.616252*10-35)*(1.616252*10-35)*(1.616252*10-35) = 4.22208*10-105 = 1planck-volume, in cubic meters, or “amount of space” contained in a Planck-volume – aka, a Planck-point as a 3-d quantized value based on the same 1-d quantized value as a Planck-length.
There isn't a reason to proceed with the calculation to higher dimensions because there is no evidence that there are any. However, if there were, the same logic would apply.
Math based on a number line with a finite number of points where those points are defined based on a Planck-length yields dimensional sizes which are comparably finite. Progressively larger, but not infinitely so. Only 70 orders of magnitude larger from a “line” to a volume… The difference with Agentredlum’s example is merely a difference in point of view, but an important difference.
So...the three standard issue multiplications above are different dimensional measurements using the same value, a Planck-length. Regardless of how it’s sliced, a Planck-volume is a Planck-length from each of its corners (planck-volume would be the smallest unit volume possible, mathematically analogous to a dimensionless point, but still one Planck-length per edge, visualized as a cube). There is no conflict in measurement from the “line” to the volume; it is but the same value measured in different dimensions. The dimensions being measured, however, are vastly different. Depending on the number of dimensions you wish to discuss, the value appears to get smaller according to the exponent, but in reality the dimensions are becoming larger. The same progression occurs whether the end volumes are cubes, spheres or marshmallows or universes – or, for that matter, however many dimensions you wish to consider. Also, with no infinite outcomes, which is unlike using math where infinite values exist in a line segment.
Intriguingly, 1.616252*10 to the +35th p-v’s laid one by one next to each other would make a line of p-v’s one meter long - a finite number of Planck-points. Using p-l’s and p-v’s suggests a means of distinguishing size between dimensions – where using traditional math to try and measure size differences between dimensions doesn’t work well, if at all. It is possible the CERN machine (LHC) may find real world evidence related to the question.
Speaking of dimensions, I’ve read that some consider time to be a 4th dimension. Usually it’s referred to as an extension of the third dimension, similar to the third as an extension of the second. I don’t think that’s the case… rather, time is an extra linear dimension similar to the single dimension but not as an extension of the three we experience. It has already been suggested that time is discrete in structure at a scale of about 1*10-43 seconds. Others have considered this question in relation the Zeno paradox and have come to the solution that time isn’t something we’re “in” that can be visualized statically like “in the present instant”. They assert time is something we and our world are passing through dynamically with no stop actions in the discrete instants. To me that is unsatisfying as it seems to mix discrete and continuous structure - but hard to refute.
cb
.
You were not talking about a single fraction -- you fairly explicitly listed a family of fractions.
