Regarding the measurement from the Planck length to the Observable Universe:
There are 202.34 Notations, by NASA scientist, Joe Koelecki (retired)
Back in December 2011, Bruce Camber and five high-school geometry classes in New Orleans involved themselves in an interesting little thought journey. When I was contacted by them, they were deeply into the process to discover that the number 2202.34 represents the ratio between the Hubble radius of the observable universe (according to the results in March 2012) and the Planck length (a number from modern quantum physics).
Here is how they did it:
1. The Hubble radius [astronomical measurement] is taken to be 1.31 x 1026 m and the Planck length [calculated] is 1.62 x 10-35 m. The Hubble radius comes from a recent estimate of the age of the universe published in Discover Magazine. The Planck length L may be calculated from: L = (hG/(2πc3))1/2where h is Planck’s constant, G is Newton’s gravitational constant, and c is the speed of light, all in appropriate units of measure.
2. The ratio between the two distances is then found to be:
1.31 x 1026 m / 1.62 x 10-35 m = 2202.34
This calculation arises from a related classroom activity, begun by Mr. Camber with those five geometry classes. The ratio is shown as a power of 2 (it could as well have been shown as a power of 10, or of any other number) in answer to the original class question, “How many times does one have to double the smallest known distance (the Planck length) to acquire the largest known distance (the present-day Hubble radius of the universe). I was consulted by Mr. Camber and assisted and advised him and his classes to produce the result shown above.
The significance of this result is that it displays the most extreme distance ratio imaginable in terms of a surprisingly finite number (202.34) of doublings. In a sense, it takes two quantities, neither of which can be adequately pictured in the mind, and shows them in ratio as a number that can be more easily pictured. I thought the exercise interesting and worth the effort and was happy to be called upon to contribute.
3. One additional note, the standard meter (1m) when compared to the Planck length corresponds to a ratio of 2115.57. We note that 2115 corresponds to 0.67m, and 2116 corresponds to 1.35m. In other words, the standard meter is not an even power-of-2 multiple of the Planck length. Mr. Camber and his classes have therefore suggested that a possible redefinition of the standard meter might be made by choosing one of these possibilities (i.e., 2115 or 2116times the Planck length) and used to replace the present-day standard. The present day standard is based on the wavelength of a particular atomic emission line. This new standard would be based on a purely theoretical concept.
Bravo to Mr. Camber and his classes for some very nice (and out-of-the-box) original thinking!!!