Tetrahedral and Octahedral Gaps
Five-Tetrahedral Gap = θ
Degrees: 7.3561031724+ degrees
Geometric or Pentagonal Gap: The first little-known gap that we studied was the 7.356103+ degree gap created by five tetrahedrons. Since 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?  In 2022 we also began to consider the pentagonal gap created by five octahedrons sharing a centerpoint and each octahedron sharing at least one face. There are also the icosahedronal gaps, and the Pentakis dodecahedronal gaps that are extensions of the five-tetrahedrons. 
History: Aristotle had it wrong; he claimed that one could perfectly tile and tessellate the universe with tetrahedrons.  The world believed him for over 1800 years. However, if you have five perfectly- shaped tetrahedrons, a gap is easily observed. That gap is a seminal. The gap appears to be transcendental, non-repeating, and never-ending.
Of course, together with the ratio of two tetrahedrons to one octahedron, you can create a whole, ordered, rational, and perfect object that will tile and tessellate the entire universe with no gaps.
The indeterminate and chaotic reside somewhere deep within the structure of the universe. We believe that place just may begin right here as a 7.35610+ degree gap.
Here may well be the basis for broken symmetries. Of course, for many readers, this will be 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.”
Note: If you use the Zometool (and other manufactured products), there is no gap. -BEC
Endnotes and Footnotes:
 Numbers: The first analysis within this website of the five-tetrahedral gap was in January 8, 2016 within an articles, “Numbers: Creating Our Universe From Scratch.” The subject was re-engaged on July 4, 2018 within an article on scientific revolutions.
 History: The first analysis within this website of the five-octahedral gap was in May 2022. Octahedral pentagon: With just octahedrons, a pentagon with a gap emerges
 Further analysis: One of the best is “Mysteries in Packing Regular Tetrahedra” by Jeffrey C. Lagarias and Chuanming Zong.
Again, please also visit the pentagon, octahedron, tetrahedron, and the sphere.
Foundational Questions Institute called for papers about it.
- Frank, F. C.; Kasper, J. S. (1958), “Complex alloy structures regarded as sphere packings. I. Definitions and basic principles”, Acta Crystall. 11. and Frank, F. C.; Kasper, J. S. (1959), and “Complex alloy structures regarded as sphere packings. II. Analysis and classification of representative structures”, Acta Crystall. 12.
- “A model metal potential exhibiting polytetrahedral clusters” by Jonathan P. K. Doye, University Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, United Kingdom, J. Chem. Phys. 119, 1136 (2003) The compete article is also available at ArXiv.org as a PDF: http://arxiv.org/pdf/cond-mat/0301374
- “Polyclusters” by the India Institute of Science in Bangalore has many helpful illustrations and explanations of crystal structure. PDF: http://met.iisc.ernet.in/~lord/webfiles/clusters/polyclusters.pdf
- “Mysteries in Packing Regular Tetrahedra” Jeffrey C. Lagarias and Chuanming Zong, a focused look at the history.