A Key Insight

History. This website began back in December 2011 with work in a high school geometry class in New Orleans. We had first observed how the tetrahedron is perfectly filled with four “half-sized” tetrahedrons (one in each corner) and an octahedron (revealing one triangular face in the center of each of the four faces of the tetrahedron). We then observed how the octahedron was perfectly filled with six “half-sized” octahedrons (one in each corner) and eight tetrahedrons (one in each face). We observed how the two objects together perfectly fill space, theoretically tiling and tessellating the universe. The question was asked, “How far within can we go?” By filling each tetrahedron and octahedron with smaller and smaller tetrahedrons and octahedrons, we discovered the smallest limits. In about 45 steps within we were in the range of particle physics! Within another 67 steps within we were within the Planck scale!

By multiplying each by 2, and the result by 2, over and over again, we reached the upper limit in just 90 steps or doublings.

Once we defined those boundaries, we began re-examining the processes.

202 notations or doubling or groups or steps. In just three steps going within, we had so many tetrahedrons and octahedrons, we turned to paper! Within those 45 steps, going further and further within, we were at the sizes used by CERN Labs (Geneva) Atlas program. Within another 67 steps within, we were facing what we called the Planck Wall and realized that this is where Zeno had finally reached a limit.

When we multiplied our desktop objects by 2, we were equally surprised that we reached the approximate size of the universe in just over 90 steps. That we mapped the entire universe in 202 doublings with geometries and multiplication by 2 was as satisfying as it was mystifying.

Mystery. First, we couldn’t find any references to it on the web! We did find Kees Boeke’s base-10 and that was some comfort, but where is our more granular base-2? How could something so simple be new?

Then we asked the question, “Does this geometry represent anything in reality?” It is symbolic, of course, but in many different ways, it is also observable. We’ve kept that question open.

A STEM Tool. Now, both surprised and a bit perplexed, we asked, “What can we do with this nascent model?” We decided to share our map of the universe with other schools. For us, it seemed like an excellent STEM tool. There were no silos of information. Everything necessarily grew out of the earlier notations.

To summarize, in this first mapping of the universe, we de facto discovered the Planck Length, base-2 notation, a rather overwhelming continuum of geometries, and a delightful way of ordering information along a scale of the universe.

Of course, we still had a lot of work to do. First, we started to learn about the other Planck base units. How would they track with the Planck Length expansion? We could not even begin to guess. We are told that we live in an expanding universe, so we realized that we would have to keep track of the top numbers and the expansion. We also asked, “What comes before the Planck units?”

We didn’t have a clue.

It took us three years to be somewhat convinced that what we did — a base-2 expansion from the Planck base units — had not been done or yet acknolwedged. We realize that we backed into it as a result of thinking about Zeno’s paradox and simple embedded geometries. That we had a base-2 chart of the universe was a pleasant surprise. So, we then started to learn about the Planck base units.

If you would like to return to the webpage that opened this page, you have choices:

Related Webpages:

December 2011:

January 2012:

2013:

Letters to
Scientific American