D & M Pyramid Geometry Addendix A

Appendix A


The Geomorphology and Geometry of the D&M Pyramid

Erol O. Torun

added: June, 1989

Fivefold Symmetry in Crystals

Section 4.5 states ". . it is impossible for any crystal to possess 5-fold symmetry." This is still true for periodic crystalline structures (structures in which lattice distances repeat themselves at regular intervals).

But there is evidence for the existence of a nonperiodic, or quasiperiodic lattice that exhibits 5-fold symmetry, or, when viewed in 3 dimensions, icosahedral symmetry. Mathematical physicist Roger Penrose, formerly at Oxford, now at Rice Univ. in Houston, developed a 2-D tiling pattern that exhibits a 5-fold non-periodic symmetry. Penrose used two diamond shaped figures, a "thick" and a "thin" rhombus. When extended into 3-D, the structure has icosahedral symmetry.

The discovery of a substance having this symmetry was made by Dan Shechtman of the Israel Institute of Technology in Haifa, while working at the National Bureau of Standards in Gaithersburg, Md. Shechtman produced the substance by depositing a molten mixture of aluminum and manganese, iron or chromium onto a spinning, water cooled copper wheel. If the mixture cools too fast, a metallic glass is formed, and if the mixture cools too slowly, the atoms will assume a periodic pattern consistent with classic crystallographic rules. An intermediate cooling rate will generate the icosahedral "quasicrystal" that remains stable for several hours before rearranging itself into a regular periodic form.

John W. Cahn of the NBS feels that this quasicrystal growth may solve the mystery surrounding the ability of iron pyrite to form a pentagonal dodecahedron. The pyrite may exist as a quasicrystal at higher temperatures and pressures, and, upon excavation, slowly assume a periodic crystalline structure while retaining its outward shape. NBS expects quasicrystals to have unusual properties because they are highly structured and, unlike conventional crystals, isotropic.

Quasicrystal growth is unlikely to explain the existence of the D&M Pyramid, due to its size and the fact that it deviates substantially from true 5-fold or icosahedral symmetry.


Science News, Vol.127, Jan.19, 1985; p.37

Science News, Vol.127, Mar.23, 1985; pp. 188-9