Geometries of spheres, triangles, tetrahedrons, fluctuations, imperfections and probabilities

Starts with a circle
by Bruce E. Camber, February 2018, Austin, Texas

Sphere to tetrahedron-actahedron couplet
Attribution: I, Jonathunder

Spheres, sphere stacking. Our studies of sphere dynamics (stacking), particularly the internal dynamics such as the Fourier transform, is still quite formative.  When we started we were most interested in Platonic forms, not the circle or sphere. The focus on spheres began to emerge when we discovered this sphere-stacking dynamic gif on the right. There is no known study of the possible dynamics of each sphere within the infinitesimal. The Fourier transform would be the beginning of such a study, yet the specific dynamics of adding one sphere at a time must be studied as well.  Three spheres create an internal triangle, four create the square, five the pentagonal, and six the hexagon.


Tetrahedrons. A geometry of fluctuations, imperfections and probabilities comes out of our studies of the pentastar (five tetrahedrons or five octahedrons sharing a common centerpoint). There are several configurations of the pentastar to develop an icosahedron (twenty tetrahedrons) whereby a flexible five-tetrahedral cluster can be found with any group of five tetrahedrons.


Among our high school students, tetrahedral geometries are known as squishy geometry. Perhaps a better word for it is quantum geometry.

There are many excellent articles and studies that form the basis of our growing understanding of the tetrahedral clusters and the tetrahedral-octahedral clusters.

Introductions to the tetrahedron clusters:
•  Mysteries in Packing Regular Tetrahedra” by Jeffrey C. Lagarias and Chuanming Zong


•  Cluster structure. Within our of the earliest articles, first published within Wikipedia in March 2012, explored the work of  Jonathan Doye who studied at Cambridge University and teaches at Oxford.

•  The tetrahedral-octahedral chain where an octahedron sits on two faces of the upper tetrahedron and two faces of the lower tetrahedron.  It forms a natural double helix and begs the question about its fundamentality as a more basic architecture that generates the helix.

•  The icosahedron (twenty tetrahedrons sharing a common centerpoint). It appears that these same geometries are also responsible for quantum fluctuations.

In the study of each notation we will look to see how each notation is necessarily derivative of the prior. And, we will continue to study the nature of these fluctuations.

Thank you. Your comments and suggestions are always welcomed. -Bruce