Tetrahedral & octahedral and other gaps
Five-Tetrahedral Gap
0.12838822… radians
7.3561031724+ degrees
7.356103172453456846229996699812179815034215504539741440855531+º (DEGREES)
Five-Octahedral gap
Radians: 0.12838822…º radians
Degrees: 7.3561031724+ degrees Believed to be a never-ending, never-repeating number, this has not yet been proven. Just below it is taken out to over 60 digits:
7.356103172453456846229996699812179815034215504539741440855531+º (DEGREES)
Beginning in 2016, it has been a key part of our analysis of numbers, geometries, chaos, quantum fluctuations, and human will. Is it possible that this little gap could be the basis for diversity, creativity, openness, indeterminism, and uniqueness? [1]
In 2022 we also began to consider a pentagonal gap with five octahedrons sharing a centerpoint and each octahedron sharing at least one face. That gap has not yet been discussed in any of the geometry textbooks. It cannot be easily created by the geometry construction sets. Some of computer programs for geometers like Mathematica do render these gaps.
There are also the icosahedral gaps, and the Pentakis dodecahedral gaps that are extensions of the five-tetrahedrons.
History: Sometimes referred to as Aristotle’s Gap [2] because Aristotle was wrong about tiling the universe with tetrahedrons. That mistake stood uncorrected until sometime in the 1400s. Yet, even today it is still a little-known gap. [3] With five perfectly-shaped tetrahedrons, the imperfection is easily observed. That gap is a key. Tetrahedrons have four vertices. Octahedrons have six. Five regular tetrahedrons have seven or eight vertices. Cubes have eight. This progression is important.
The gap appears to be transcendental, non-repeating, and never-ending.
Of course, together with a ratio of two tetrahedrons to one octahedron, you can create a whole, ordered, rational, and perfect object that will perfectly tile and tessellate the entire universe.
The indeterminate and chaotic reside somewhere deep within the structure of the universe. We believe that place just may begin right here with the “Pentastar Gap.”
Here may well be the basis for broken symmetries. Of course, for many readers, this will be quite a stretch. That’s okay. For more, we’ll study chaotic maps, the classification of discontinuities, imperfections, and quantum fluctuations. More…
The Pentastar logo by Chrysler has a fan club! Now that Fiat’s involved, the Pentastar logo is being flushed out of their branding. There is a group of over 13,000+ advocating that they continue using it. I suggested in a note to them, “Chrysler made it an artform! The gap is the beginning of quantum fluctuations. It is the beginning of the geometry of living.”
Please, again note: If you use the Zometool (and other competitive manufactured products) or computer program to generate the five tetrahedrons or five octahedrons, there’ll be no gap. If there is a gap, please-please-please let me know! -BEC
[1] Numbers: The first analysis within this website of the pentastar gap was done in on January 8, 2016 within an articles, “Numbers: Creating Our Universe From Scratch.”
[2] History: The subject was re-engaged on July 4, 2018 within an article on scientific revolutions. The question is asked, squishy geometry anyone?
[3] Further analysis: One of the best analysis is “Mysteries in Packing Regular Tetrahedra” by Jeffrey C. Lagarias and Chuanming Zong. Our analysis.
[4] Octahedral Pentagon: With just octahedrons, a pentagon with the same gap emerges.
[5] Again, please also visit the pentagon, octahedron, tetrahedron, the sphere, and their geometries.
Foundational Questions Institute called for papers about it….
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