Icosahedral Quasicrystal as a packing of regular tetrahedra

Icosahedral Quasicrystal as a packing of regular tetrahedra , updated 7/21/20, 10:06 PM


Fang Fang, Julio Kovacs, Garrett Sadler, Klee Irwin (2014)

This paper presents the construction of a dense, quasicrystalline packing of regular tetrahedra with icosahedral symmetry. This quasicrystalline packing was achieved through two independent approaches. The first approach originates in the Elser- Sloane 4D quasicrystal – a 3D slice of the quasicrystal contains a few types of prototiles. An initial structure is obtained by decorating these prototiles with tetrahedra. This initial structure is then modified using the Elser- Sloane quasicrystal itself as a guide. The second approach proceeds by decorating the prolate and oblate rhombohedra in a 3-dimensional Ammann tiling. The resulting quasicrystal has a packing density of 59.783%. We also show a variant of the quasicrystal that has just 10 plane classes (compared with the 190 of the original), defined as the total number of distinct orientations of the planes in which the faces of the tetrahedra are contained. The small number of plane classes was achieved by a certain “golden rotation” of the tetrahedra.

About Klee Irwin

Klee Irwin is an author, researcher and entrepreneur who now dedicates the majority of his time to Quantum Gravity Research (QGR), a non-profit research institute that he founded in 2009. The mission of the organization is to discover the geometric first-principles unification of space, time, matter, energy, information, and consciousness.


As the Director of QGR, Klee manages a dedicated team of mathematicians and physicists in developing emergence theory to replace the current disparate and conflicting physics theories. Since 2009, the team has published numerous papers and journal articles analyzing the fundamentals of physics.

Klee is also the founder and owner of Irwin Naturals, an award-winning global natural supplement company providing alternative health and healing products sold in thousands of retailers across the globe including Whole Foods, Vitamin Shoppe, Costco, RiteAid, WalMart, CVS, GNC and many others. Irwin Naturals is a long time supporter of Vitamin Angels, which aims to provide lifesaving vitamins to mothers and children at risk of malnutrition thereby reducing preventable illness, blindness, and death and creating healthier communities.

Outside of his work in physics, Klee is active in supporting students, scientists, educators, and founders in their aim toward discovering solutions to activate positive change in the world.  He has supported and invested in a wide range of people, causes and companies including Change.org, Upworthy, Donors Choose, Moon Express, Mayasil, the X PRIZE Foundation, and Singularity University where he is an Associate Founder.

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