PERFECTION STUDIES: CONTINUITY•SYMMETRY•HARMONY GOALS.15-16.April.2024
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Science gets confused when we do not study our mistakes
by Bruce E. Camber (IN PROCESS, working first draft)

Introduction. So much of Aristotle’s voluminous writings reflect the spirit and beliefs of his time. To focus on a single mistake might seem a bit picky, yet a mistake is a mistake. Scholars in the 1400s were the first to point out a pivotal error; then, an MIT mathematics professor re-reviewed it 1926, and mathematicians from Ann Arbor and Beijing again reviewed its history in 2012.* Even with that attention, there has not been much scholarly discussion about Aristotle’s mistake over these many years (well over 2300 years). Given that the error applies to how we see the universe, particularly the geometry of the universe and its gaps† (quite possibly quantum fluctuations), it seems that this is one mistake that deserves further study.

Process. We are told early in our education not to go beyond anything that we do not understand. Ask a question until you understand the answer. Aristotle’s seemingly innocuous statement that precipitated those three critical articles is this:

“It is agreed the that there are only three plane figures which can fill a space, the triangle, the square, and the hexagon and there are only two fill solids, the pyramid and the cube…”^[1]

– De Caelo (On the Heavens), Aristotle (384 BCE – 322 BCE), written 350 B.C.E Translated by John Leofric Stocks, Oxford, 1922 Online, page 306 b, 8.5 (page 126 of 250)

Aristotle missed this simple gap. ^[2]

Discussion. Science is a process of successive approximations. We never have it perfectly right. Although some of his writings appear laughable as we read Aristotle today, many of our beliefs today may well appear laughable to those who follow us in a millennium — yet perhaps only decades. It is quite impossible to measure the influence Aristotle’s writings have had on anyone doing geometric modeling in the 1400s, in 1926, or in 2012. Asked about the implications of the gap’s manifestation within spacetime, our 2012 authorities, Lagarias and Zong had no comments. Their focus has been on packing densities. Even asking a great diversity of scholars, including theoretical physicists, chemists, biologists, mathematicians, and philosophers, no one would speculate.

Aristotle’s pyramid, assumed by Aristotelian scholars to be a tetrahedron, does not tile and tessellate space. To fill space perfectly — without gaps — it needs an octahedron and another tetrahedron. There is a necessary symbiotic relation between these two. First within every tetrahedron there is an octahedron. Please look at these images closely. Tetrahedrons are perfectly filled with four smaller tetrahedrons, one in each corner, and an octahedron. The octahedron is perfectly and simply filled with six smaller octahedrons, one in each corner, and eight tetrahedrons, one in each face. The tetrahedron and octahedron are the most-simple structures within the five most-basic solids which are known as the Platonic solids. It seems that this simple geometry is not known to most, including geometers. I have personally asked hundreds, “What is perfectly enclosed within an octahedron? Only John Conway said, “Well, let’s figure it out.” And, of course, he did.

One of the least recognized facts about the octahedron is that there are four hexagons outlined within every octahedron.^[3] That’s not taught in Geometry 101.^[4]

Just the two little-known facts, that gap and the origin of hexagon could have changed the face of Euclidean geometry and the fundamentals of chemistry. As the baseline of all geometries and the evolution of so many new flavors of geometry within a short two hundred years, what have we missed? What have we not seen?

If the gap is a baseline for quantum physics and the perfections of the octahedral-tetrahedral complex is the basis of a physics of perfected states within the infinitesimals of spacetime, the answer is, “Too much!”

Science evolve; there’s no question about it. Yet, often it seems to evolve in spite of us. Yet, philosophies, ethics, and our understanding of ourselves evolve as well and so much of it all is a direct result of our sciences. Today, that is not a pretty picture. Something is fundamentally off. And, it seems that two rather simple errors, what amounts to missing pieces of the puzzle, throws everything off, and then it becomes a platform for mistaken beliefs. Even with all our mistakes, it is a remarkable testimony to human insight, creativity and intelligence that we’ve progressed as far as we have.

Conclusions. Though unwittingly introduced, mistakes are mistakes. Eventually a mistake becomes misinformation; and over centuries, it becomes disinformation. When the error goes on for over 1800 years, it becomes too big an embarrassment for our intellectual community to grasp, so the data gets kind-of, sort-of, “unwittingly” buried and forgotten.

I am ever thankful for the scholarship of Dirk Struik. Yet, although he recognized the potentials within that pivotal mistake, he had written his article in Dutch so its readership was limited. Even after translation^[5] decades later, his article was still rather tenuously received by scholars. I would guess that even today, nobody knows what to make of it. Then finally, over eighty years later in 2012, out of Ann Arbor and Beijing, two scholars, Lagarias and Zong, captured the essence and its rather long history. And, even though that article was recognized in 2015 by the American Mathematical Society^[6] with a most-distinguished award, the issue continues to be virtually ignored. It is too much to take in, a major academic faux pas, a mistake by a master that was missed by the masters and dismissed as trivial because it must be trivial if the academics are not talking about it.

What a dilemma! The very foundations of our academic enterprise are incomplete. It is no wonder why we are so messed up.

First, we have to come to terms with the mistake. Our quick summary of the 2012 work by Lagarias and Zong may be helpful as a start — https://81018.com/biased/#Aristotle As noted, we have asked many scholars for their explanation as to how that gap may appear within space time. While we are waiting, for someone to respond, we took this stab at it:  https://81018.com/geometries/

Science and mathematics are flawed without fully grasping the dimensions of that gap in our knowledge systems and how mistakes beget mistakes. And, it all started with a mistake by a genius who was one of the greatest influencers of Western culture, education and knowledge.

That’s an “oopsie-daisey” if there ever was one.

So, let’s stop ignoring our mistakes. When we acknowledge our mistakes, we grow. Plus, mistakes happen to the very best of us, including Aristotle, Newton, Einstein, and Hawking…. yet, the beat of life goes on.

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References
As these references are studied, key references and resources are be added and updated.

[*] Jeffrey Lagarias and Chuanming Zong, retrieved 8 April 2024: https://81018.com/biased/#Zong
• Dirk Struik, retrieved 8 April 2024, https://81018.com/struik/
• 0ur initial study on the web,retrieved on 8 April 2024, https://81018.com/biased/#Aristotle

[†] Gaps. Tetrahedral gap. retrieved on April 8, 2024, https://81018.com/gap/

[1] De Caelo (On the Heavens), Aristotle (384 BCE – 322 BCE), written 350 B.C.E Translated by John Leofric Stocks, Oxford, 1922 Online, page 306 b, 8.5 (page 126 of 250)

[2] Gaps retrieved 8 April 2024, https://81018.com/gaps/

[3] Octahedron’s four hexagons, retrieved 8 April 2024: https://81018.com/octahedron/

[4] Geometry 101, retrieved 8 April 2024: https://81018.com/geometry-prentice-hall/

[5] Translation retrieved 8 April 2024, https://81018.com/struik-senechal/

[6] American Mathematical Society retrieved 8 April 2024, https://81018.com/ams/ Also: https://www.ams.org/prizes-awards/paview.cgi?parent_id=29

References-resources-footnotes-endnotes are being developed. -BEC, April 2024

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Resources
As the references are studied, additional resources may be suggested.

There are five pages that work together with this homepage:
• From the smallest to largest scales: https://81018.com/reformat/
• On identifying keys to our Universe: https://81018.com/tighter/
• The Qualitative: https://81018.com/qualitative/
• Pi Day: https://81018.com/2024-piday/
• Number Theory: https://81018.com/numbers-numbers-numbers/

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Endnotes & Footnotes
Personal reflections.

1. This homepage was precipitated from an endearing note from Siri Carpenter, one of the founders of “The Open Notebook” for science writers.
2. Does it always come back to a discussion about the finite-infinite? It seems that way. One of our attempts to reopen those discussions within a very different context is based on pi: https://81018.com/csh/
3. Symbiosis and symbiotics. The tetrahedron-octahedron relation necessarily is the most simple and abundant example. From this page and day, we will begin an active and in-depth analysis of symbiotic relations. It seems every year we are challenge by new data that tell us how dynamically interdependent everything is. If we redefine time, we might say, “How dynamically interdependent everything, everywhere-for-all-time is.”

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Emails
There will be emails to many of our scholars about key points.

• Bryna Kra, current president, American Mathematical Society, Providence, RI (and Northwestern)
• Chuanming Zong, Tianjin, China
• Jeffrey Lagarias, University of Michigan, Ann Arbor, MI
• Siri Carpenter, The Open Notebook, Madison, WI
• Robin Hendry and Nancy Cartwright, Durham University, Durham, England
• Ruth Charney, Brandeis, Waltham (MA), former president, American Mathematical Society

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IM
There will also be many instant messages to thought leaders about these key points.

11 April 2024 @ 10:03 AM Come on Berkeley; you’re all smart people. You can’t use instrumentation to measure smaller than the CERN scale for length and the MPI Optical Lab for time. You’ve got to use pure math to get to the Planck scale: https://81018.com/chart/ It seems base-2 notation opens a path.

10 April 2024 @sjmontlake The tenor and flavor of your family’s sojourn to the total eclipse in Vermont was endearing. A simple reality for your children is to embrace the universe from the first moment in time, Planck Time, to this day in just 202 base-2 notations. We did it in a high school in New Orleans in 2011: https://81018.com/big-board/ It took several years to begin to figure it out: https://81018.com/stem/ We are still working on it: https://81018.com/ very, very tricky…

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Critique ____ You are always invited.

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Keys to this page, study

• This page became the homepage on 9 April 2024.
• The last update was 12 April 2024.
• This page was initiated on 8 April 2024.
• The URL for this file is https://81018.com/study/
• The headline for this article: Science easily confused when we do not study our mistakes.
• First teaser* is: Hard to believe but our ultimate principles are wrongly assumed….

*Or, wicket, kicker or eyebrow.

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