by Joe Kolecki, NASA scientist, (retired)

Mr. Bruce Camber and his five geometry classes recently involved themselves in an interesting little thought journey. They discovered that the number 2^{202.34} represents the ratio between the Hubble radius of the observable universe (according to the most recent results) and the Planck length (a number from modern quantum physics).

Here is how they did it:

The Hubble radius [astronomical measurement] is taken to be 1.31 x 10^{26} 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 (2011). The Planck length L may be calculated from: L = (hG/(2πc3))^{1/2} where h is Planck’s constant, G is Newton’s gravitational constant, and c is the speed of light, all in appropriate units of measure.

The ratio between the two distances is then found to be: 1.31 x 10^{26} m / 1.62 x 10^{-35} m = 2^{202.34}

This calculation arises from a related classroom activity, begun by Mr. Camber and his five geometry classes in a local high school in New Orleans. The ratio is shown as a power of 2 (it might 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 class 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.

One additional note, the standard meter (1m) when compared to the Planck length corresponds to a ratio of 2^{115.57}. We note that 2^{116} corresponds to 0.67m, and 2 ^{116}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., 2^{115}or 2^{116}times 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!!!

– Joe Kolecki, NASA scientist, retired

One of Joe’s pages developed at this time from which we learned and was grateful:

https://www.grc.nasa.gov/www/k-12/Numbers/Math/Mathematical_Thinking/calendar_calculations.htm