• Eight vertices can be ordered to make five regular tetrahedrons.

• These five regular tetrahedrons using a common center edge will create one 7.38º (degree) gap which is, to date, unnamed.

• Five irregular tetrahedrons were used to make the Chrysler logo; introduced in 1962,  it was named the Pentastar.

• For the purposes of this article our object is called a Pentastar and the focus is on the 7.38º  gap.

There is a dynamic between the Pentastar and pi.  There is also a dynamic with cubic close packing, bifurcation theory and cellular automaton.

The purpose of this short article is to introduce the Pentastar 7.38º  gap, to invite initial comments, and to invite experts to study those relations and dynamics with pi.  Is it here that we find the foundations for quantum theory, identity, chaos, indeterminacy, and the self?

It is one of the most simple constructions in the universe, and it must in very special ways interact with the non-repeating, never-ending numbers of pi and the other nine fundamental numbers and number groups that are discussed in the article about the importance of numbers, On Constructing The Universe From Scratch.

References:

1. 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.
2. “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
3. “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
4. “Mysteries in Packing Regular Tetrahedra” Jeffrey C. Lagarias and Chuanming Zong, a focused look at the history.
5. http://www.hyperflight.com/pentagon-construct.htm