Rejected by AAAS Science Magazine

A student and his math & science teachers ask,

“What is wrong with our simple logic and math?”

by Bryce Estes,1 Cathy Boucvalt,2 Steve Curtis,3 and Bruce Camber4

New Orleans, LA 70123, USA

High school is a place to explore the unknown and to build on the basics. It is a time to soak up information, knowledge, and wisdom. Some of those unknown concepts are quite simple but perplexing. For example, look what happened when we divide the edges of a tetrahedron and its internal octahedron by 2. The process creates “half-sized” embedded geometries. In about 45 steps going within, we were at the CERN-scale, and in another 67-steps going within, we were at the Planck scale. We then multiplied our little models by 2 and in about 90 doublings we were somewhere in the range of the Observable Universe. We created a chart of the universe with just 202 notations.

It was remarkable to see the universe all mathematically notated on one chart.

We quickly discovered that a base-2 scale of the universe hadn’t been done and was not part of any formal academic program. We discovered Kees Boeke’s 1957 work in Holland, a base-10 chart with just 40 jumps. It had become quite popular, but it did not start at the Planck units nor did it go to the Age of the Universe. We started to write up our results and we attempted to get others involved. In 2012 our first article for Wikipedia was rejected. (Editor’s note: AAAS rejected this article in April 2016.)

So, now we go back over our logic and math and ask, “What are we doing wrong?”

Since December 2011 a small group of high school students and a few of their teachers have been trying to figure out what to do with an all-encompassing but simple mathematical and geometrical model. Findings to date are presented with the hope that the academic-scientific community can tell us how best to proceed with our very simple charts.

As noted above, the project began by dividing the edges of a tetrahedron by 2. We connected the new vertices and discovered a tetrahedron in each of the four corners and an octahedron in the middle. We then divided the edges of the octahedron by 2 and discovered an octahedron in each of the six corners and a tetrahedron in each of the eight faces. Delighted with the simple complexity, we continued to divide each subsequent object in the same way until we had to resort to paper. By the 45th step within we were in the range of a proton. In another 67 steps we were in the range of the Planck base units. Back up inside the classroom (Notation-112), we  multiplied by 2. Within about 22 steps we were out to the International Space Station. In a total of 90 steps, we were out to the edges of the Observable Universe and the Age of the Universe.

1 Electronic address: Webpage:
2 Electronic address: Webpage:
3 Electronic address: Webpage:
4 Electronic address: Webpage:

Steve Curtis is a mathematics teacher at the Curtis School.  He is also a football coach (defensive backs).

This project officially began on December 19, 2011 in Steve’s classroom with his three geometry classes and two ACT preparation classes. These classes learned about base-2 exponential notation by observing nested geometries using the tetrahedron and octahedron.

The process. These classes went deeper and deeper inside each object by dividing each edge in half and by connecting the  new vertices to create a smaller set of nesting tetrahedrons and octahedrons. By about the 45th step within — on paper — the size of the tetrahedron and octahedron were about the size of a fermion. Within about 67 more steps, that size was in the range of the Planck Length. At that time there were about 112 steps within from the size of our original plastic models.

We then went out into the universe by multiplying each edge by 2. Somewhere between 90 and 98 steps, we were in the area of the Observable Universe. It wasn’t until we followed Planck Time to the Age of the Universe did we finally settle on a total of 202 notations.

This project has been under the watchful eye of Steve Curtis right from its beginning.

About   Home


“What is wrong with our simple logic and math?”     Thursday, June 16, 2016              page 2


It was a delightful process charting the universe in what we quickly learned was base-2 exponential notation. We thought Kees Boeke’s base-10 work was interesting but not granular enough. As we began to study our work, we realized it was an excellent Science-Technology-Engineering-Math (STEM) tool so we began sharing it with others within very preliminary web pages, Cf. the Ref. [1]

The first chart was a 60″ by 11″ board that started with the Planck Length and went to the Observable Universe, Cf. the Ref. [2] so we called our little project, Big Board-little universe. Because we didn’t know where to stop, we got a little help with our calculations, Cf. the Ref. [3]. A year later a desktop version of the chart was started; it was dubbed, Universe Table, Cf. the Ref. [4]. In December 2014 Planck Time was added to the chart, Cf. the Ref. [5]. Those numbers tracked well with the Planck Length. Now we had a better number, the Age of the Universe, to determine when to stop multiplying by 2.

Here we discovered that that Max Planck’s ratio of of the Planck Length to Planck Time within each of the 202 notations was always within .1% of the speed of light. In February 2015, we added the other three Planck base units to the chart, Cf. the Ref. [6]. Here we discovered that each number is a ratio to the others and provides summary data about inherent order within the universe.

The teachers include three of the authors of this paper, Bruce Camber, Cathy Boucvalt, and Steve Curtis and a student, Bryce Estes, who represents the many years of students, including a sixth grade class back in the Spring of 2012 when they were introduced to these charts.

The very first observation was that each chart is a highly efficient way to organize vast amounts of information, but these charts also raised some rather fascinating questions.

Questions and Challenges

First, we wondered why couldn’t we find some vestiges of these charts within our textbooks or someplace on the World Wide Web. In May 2012 Wikipedia rejected an article we submitted because it was “original research,” , Cf. the Ref. [7]. We asked, “Isn’t all this information somewhere within the academic world?

Stepping back from our charts, we asked, ”Isn’t each column of the chart a very basic continuity equation from a Planck base unit to its largest possible measurement? Isn’t continuity the bedrock of order? Should this be the first principle within our work?”

The small numbers were impossibly small and the large numbers were impossibly large, yet the 202 notations were relatively manageable. The nagging question was, “Is there a problem with our logic and math?” It was exponential notation that helped us get comfortable with both extremes. It helped make the numbers more manageable. Yet, it takes time. It has been a steep learning curve for our feelings or intuitions about the very nature of a number. We asked more questions, “What are these numbers? What are they telling us about the universe and ourselves?

The geometries started simple, but became exceedingly complex. We asked, “What is geometry? How is space necessarily defined? Does it require all the Planck base units? Does it require the extended Planck units?”

The human family seems to dominate the middle of this chart yet the time epoch for humanity’s existence is entirely within notation 202. What is the correlation, the working relation, between the current time and the other notations? Are all notations concurrent, active and forever? What does that imply about the nature of space and time?


“What is wrong with our simple logic and math?”    Thursday, June 16, 2016             page 3


When the chart is divided into thirds, the small-scale universe is extremely small. It goes from the Planck Length to about the size of the fermion. This particular view of the small-scale universe is virtually unknown yet it has a substantial amount of data waiting to be properly analyzed. We reached out to many of the finest scholars for their inputs.

Everybody seemed puzzled. The human scale and large scale did not seem to challenge our simple logic until the time column was observed, particularly the figures at one second, Cf. the Ref. [8]. What does it mean that the Planck Length multiple is the distance light travels in a second? Well over two-thirds of all the notations are within one second and within an area defined by the earth to the moon. What does that tell us?

Nobody seems to know what to do with these charts. So, to get some scrutiny, online articles, blogs and emails, Cf. the Ref. [9] were written. Feedback has been limited. How can that change?

Prof. Dr. Freeman Dyson (email, Cf. the Ref. [10]) recommended that we use dimensional analysis and scaling laws to determine the number of possible vertices starting at the Planck base units. The numbers become extremely large rather quickly; nevertheless, because the first 60+ notations were not on anybody’s charts of known things within space and time, we concluded that these vertices must be shared by the entire universe and have something to do with homogeneity, isotropy, the very nature of symmetry and the symmetry of nature, and the cosmological constant. Of course we had self-doubts and asked, “Are we crazy or what?”

Concluding Questions

Are these numbers important? Is this model important? We believe these numbers are trying to tell us something very new and rather special so we will continue writing blogs about our ideas, intuitions, conjectures, and sometimes rather-wild speculations until we learn why our simple logic and simple math have failed us.

Have they?


  1. Big Board-little universe, “Can our entire universe be meaningfully encapsulated within just over 201 base-2 exponential notations?” the basic web page.
  2. Big Board-little universe, “An exploration of 101 steps from the smallest measurement, the Planck length, to the human scale, and then 101 more steps to the Observable Universe” December 2011
  3. Big Board-little universe, “Some Calculations and Thoughts Regarding Measurement by Joe Kolecki, NASA scientist, retired”
  4. Big Board-little universe, “Universe Table – Human Scale”
  5. Big Board-little universe, “UniverseView: Begin at the Planck Time and Planck Length, Use Base-2 Exponential Notation”
  6. Big Board-little universe, “Five Planck Base Units – Vertically scrolled, 204-to-0” February 2015
  7. Big Board-little universe, “The original Wikipedia article as written in March 2012”
  8. Big Board-little universe, “Five Planck Base Units – Horizontally scrolled, 0-204 to the first second” May 2016.
  9. Big Board-little universe, a working “Index” of articles from January 2012
  10. Big Board-little universe, “Freeman Dyson: A Guiding Light” October 22, 2012

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