Concepts & Parameters & Questions

Sometimes the most simple concepts elude us

April 24, 2012, New Orleans: Would you agree that two of the most simple parameters of science and mathematics are: (1) smallest-to-largest and (2) multiplying-and-dividing by-2?

Then, why, as a well-educated general public, do we not know the smallest and largest measurement of a length?

To remedy that situation, let us first establish the smallest and the largest lengths. Then, let us multiply the smallest by 2, and each result by 2, until we reach the largest length. It may sound simple, albeit a bit tedious, but the process has many surprises as we begin to explore this simple exponential notation of our seemingly complex universe.

Smallest. We will assume that Max Planck was right in 1900 when he calculated the Planck Length. Certainly it is well-worth the time to study its derivation; however, for this discussion, take it as a given that the Planck length is the smallest unit of measurement of a length.

The number is 1.61619926×10-35 meters. By comparison, a hydrogen atom is approximately 1.06×10−10 meters.

Largest. The largest measurement of length is an on-going research effort; and although there are many different conclusions, there is a general direction and concurrence within the scientific community. Among the more recent calculations, we turned to the Sloan Digital Sky Surveys (SDSS-III), particularly her Baryon Oscillation Spectroscopic Survey (BOSS) measurements from March 2012 to establish a working range from the smallest-to-the-largest lengths. We are simply taking it as a given that the SDSS BOSS measurement is close enough. They have confirmed earlier statements, i.e. “The universe is 13.78 billion years old.” Others are as high as 14.1.

The first calculations for Planck Length

You would think it is a rather straightforward conversion, yet, even those calculations (convert years to a length) are diverse.

With their Scale of the Universe Cary Huang suggests 9.3×1026 meters (93 trillion light years). Paul Halpern, physics professor and author of the book, Edge of the Universe: Voyage to the Cosmic Horizon and Beyond, told me (via email) that a better figure is 4.3×1026 meters.

Yet, that end figure is not as important as the starting figure; it just helps to know when we are getting close.

Multiplying by 2. To create a simple relation between everything in the universe, take the Planck Length and multiply it by two, then continue multiplying each result by 2 until we are out to that largest measurement. It is a natural progression much like the unfolding of life within cellular division.

So, based on their calculations, how many times will we multiply by 2 to go from the smallest to the largest measurement of lengths in the universe?

That process is called base-2 exponential notation and the answers are quite surprising:

NASA physicist, Joe Kolecki, calculated 202.34 notations or doublings. So, let us start with his figure. Others calculate it a bit differently. Halpern’s calculations give us about 204+ notations. Jean-Pierre Luminet of the Observatoire de Paris suggests around 205.1 notations.

Notwithstanding, that range, 202.34 to 205.1, is a very small number of doublings (notations, layers or steps) from the smallest to the largest measurement of a unit of length.

It is a cause for wonder. At the 202nd doubling of the Planck Length, the measurement is 1.03885326×1026 meters. At the 203rd doubling it is 2.07770658×1026 meters. And, at the 204th it is 4.15541315×1026 meters and the 205th doubled to 8.31082608×1026 meters.

Within respectable universities, the number is as high as 2×1028 meters! Wikipedia is: 8.8×1026 m 

201.34 to 205 doublings from the smallest measurement of a length to the largest. It is quite fascinating to find the thickness of a human hair (around 40 microns) at notation 101, the thickness of paper at 102, and diameter of an egg cell at 103.

Step back and look at the board. Here we have the universe, all of it, within a natural ordering sequence and we can begin to see relations between everything. A high school geometry class initially explored this simple model in December 2011 and we have been trying to understand why we have not found it anywhere within the academic community. It seems to be an oversight. We were studying nested geometries, starting with the tetrahedron, discovering the octahedron and four tetrahedrons inside it. Then the students discovered the half-sized objects inside the octahedron — six octahedrons and eight tetrahedrons. It was this progression that opened the smallest-largest question.

The purpose of the many discussions within these pages of the Big Board – little universe is to encourage students to do the following:

There is a lot of work to do.

The Big Board-little universe as a working tool is a bit awkward. It is five feet by 1 foot. On Friday, May 17, 2013, back in front of the high school geometry classes, we were using the Periodic Table of Elements, particularly the wonderfully interactive, online version by Michael Dayah. It is called the Dynamic Periodic Table ( It was during those lessons that it became apparent that the Periodic Table took its form for a reason. Knowing that we would not get it right the first time, it was decided to reduce the Big Board – little universe to a ten-column table so it could be readily opened on a smartphone.

This exercise is important for several reasons. First, we will use it as a simple way to order information. Then, we will see if each notation opens new areas for speculations and analysis. The first sixty steps have never been critically explored. With just a very simple geometry of nested tetrahedrons and octahedron, a simple structure for coherence throughout the universe is created in less than 206 notations. Today we are systematically adding and analyzing layers of complexity using all the five platonic solids.

That is why we have posted these pages here. We invite you to explore the universe in the simplest ways possible, then ask, “What difference does it make?”

Editor’s Note: Initiated in April 2012 —  — was last updated on June 14, 2013 and briefly reviewed for link integrity on December 23, 2021.

  1. Go to an Introduction & Overview
  2. A Wiki-like Overview
  3. Current work: