Golden, Quasicrystalline, Chiral Packings of Tetrahedra

Fang Fang, Garrett Sadler, Julio Kovacs and Klee Irwin (2012)

Since antiquity, the packing of convex shapes has been of great interest to many scientists and mathematicians. Recently, particular interest has been given to packings of three-dimensional tetrahedra. Dense packings of both crystalline and semi-quasicrystalline have been reported. It is interesting that a semi-quasicrystalline packing of tetrahedra can emerge naturally within a thermodynamic simulation approach. However, this packing is not perfectly quasicrystalline and the packing density, while dense, is not maximal. Here we suggest that a “golden rotation” between tetrahedral facial junctions can arrange tetrahedra into a perfect quasicrystalline packing. Using this golden rotation, tetrahedra can be organized into “triangular”, “pentagonal”, and “spherical” locally dense aggregates. Additionally, the aperiodic Boerdijk-Coxeter helix (tetrahelix) is transformed into a structure of 3-or 5-fold periodicity—depending on the relative chiralities of the helix and rotation—herein referred to as the “philix”. Further, using this same rotation, we build (1) a shell structure which resembles a Penrose tiling upon projection into two dimensions, and (2) a “tetragrid” structure assembled of golden rhombohedral unit cells. Our results indicate that this rotation is closely associated with Fuller’s “jitterbug transformation” and that the total number of face-plane classes (defined below) is significantly reduced in comparison with general tetrahedral aggregations, suggesting a quasicrystalline packing of tetrahedra which is both dynamic and dense. The golden rotation that we report presents a novel tool for arranging tetrahedra into perfect quasicrystalline, dense packings.

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.

Tetrahedra plane class packed face Golden rotated arccos structure tetrahedral Dense packed

Fang Fang, Garrett Sadler, Julio Kovacs and Klee Irwin (2012)

About Klee Irwin

Tetrahedra plane class packed face Golden rotated arccos structure tetrahedral Dense packed

Golden, Quasicrystalline, Chiral Packings of 
Tetrahedra
Fang Fanga, Garrett Sadlera, Julio Kovacs, Klee Irwina, b
Quantum Gravity Research Group, Topanga, California 90290, USA
Since antiquity, the packing of convex shapes has been of great interest to many scientists 
and  mathematicians  [1-7].  Recently,  particular  interest  has  been  given  to  packings  of 
three-dimensional tetrahedra [8-20]. Dense packings of both crystalline [8, 10, 15, 17, 19] 
and  semi-quasicrystalline  [14]  have  been  reported.  It  is  interesting  that  a  semi-
quasicrystalline  packing  of  tetrahedra  can  emerge  naturally  within  a  thermodynamic 
simulation approach [14]. However, this packing is not perfectly quasicrystalline and the 
packing density, while dense, is not maximal. Here we suggest that a "golden rotation" 
between tetrahedral facial junctions can arrange tetrahedra into a perfect quasicrystalline 
packing.  Using  this  golden  rotation,  tetrahedra  can  be  organized  into  "triangular", 
"pentagonal",  and  "spherical"  locally  dense  aggregates.  Additionally,  the  aperiodic 
Boerdijk-Coxeter helix [23, 24] (tetrahelix) is transformed into a structure of 3- or 5-fold 
periodicity—depending  on  the  relative  chiralities  of  the  helix  and  rotation—herein 
referred to as the "philix". Further, using this same rotation, we build (1) a shell structure 
which  resembles  a  Penrose  tiling  upon  projection  into  two  dimensions,  and  (2)  a 
"tetragrid" structure assembled of golden rhombohedral unit cells. Our results indicate 
that this rotation is closely associated with Fuller's "jitterbug transformation" [21] and 
that  the  total  number of  face-plane  classes  (defined  below)  is  significantly  reduced  in 
comparison with general tetrahedral aggregations, suggesting a quasicrystalline packing of 
tetrahedra which is both dynamic and dense. The golden rotation that we report presents a 
novel tool for arranging tetrahedra into perfect quasicrystalline, dense packings.
Packings  of  spheres  and  the  platonic 
solids  have  been  of  great  interest  to 
mathematicians since ancient times [1-7]. With in 
the past few years, rapid progress has been made 
in the problem of dense packings of tetrahedra [9-
20].  Previous  studies  have  mainly  focus  on 
crystalline packings [8, 10, 15, 17, 19] and have 
since appeared to reach a plateau. Recently, the 
possibility of quasicrystals displaying tetrahedral 
phases has been examined [27], and a subsequent 
report  [14]  has  interestingly  and  unexpectedly 
produced a quasicrystalline packing of tetrahedra, 
motivating  the  present  work.  In  fact,  the  close 
relationship
 between
 
the
 
tetrahedron, 
icosahedron,
 and  pentagonal  bipyramid 
(discussed below) strongly suggests the existence 
of a dense, quasicrystalline packing of tetrahedra, 
as  pentagonal  and  icosahedral  symmetry  are 
forbidden  in  crystals  but  are  common  in 
quasicrystals (QC) [22].
In our analysis of tetrahedral packings and 
assemblages, we have found utility in a geometric 
object's "plane classes". Planes are said to belong 
to the same plane class if and only if their normal 
vectors are parallel. The number of plane classes 
for  a  collection  of  polytopes  is  defined  as  the 
number of distinct plane classes comprising the 
collection's
 
two-dimensional
 
faces.
 A 
characteristic feature of crystals and QCs is that 
they have a finite number of plane classes. For 
example, a three-dimensional cubic lattice has a 
total of three plane classes. A three-dimensional 
a
These authors contributed equally to this work.
b Group director. Correspondence should be directed to: klee@quantumgravityresearch.org
1
generalization of the Penrose tiling [28, 29] may 
be  put  into  correspondence  with  rhombic 
triacontahedron, and consequently comprises 15 
plane  classes.  Finally,  the
 tetrahedral, 
quasicrystalline  approximant  produced  by Haji-
Akbari  et  al.  [14],  in  principle, comprises  an 
infinite  number  of  plane  classes  due  to  the 
underlying  stochastic  nature  of  that  structure's 
effecting Monte Carlo simulation.
The irrationally valued dihedral angle of 
the tetrahedron presents a difficulty towards the 
production  of  tetrahedral  packings  with  long-
range  order  and/or  a  finite  number  of  plane 
classes.  As  an  extreme  example,  consider  the 
Boerdijk-Coxeter  helix  [23,  24]  depicted  in 
Figure  1e.  This  structure  is  well-known  to  be 
aperiodic and, if extended to comprise an infinite 
number  of  tetrahedra,  produces  infinitely  many 
plane classes.
In  the  pursuit  of  a  tetrahedral  structure 
2
Figure 1 | Tetrahedral structures subjected to the golden rotation and associated junction types. Groups 
of  tetrahedra  uniformly  distributed  about  a  common  edge  (a–c),  and  vertex  (d).  A linear  arrangement  of 
tetrahedra  forms  the  Boerdijk-Coxeter  helix  (tetrahelix)  (e).  "Twisting"  these  structures  eliminates  inter-
tetrahedral gaps and reduces the number of plane classes (f–j). The corresponding projections of facial junctions  
are shown in (k–o). The relative angular relationship between faces in k, m, n, and o is related to the golden  
ratio. Hence, this rotational relationship is referred to as the golden rotation. In its canonical form, the Boerdijk-
Coxeter helix is aperiodic, however, following application of the golden rotation, this structure forms the "philix" (j) 
with  a  3-  or  5-tetrahedron  periodicity,  depending  on  the  relative  chiralities  of  the  helix  and  rotation.  Axial  
projections of this helix show the 5-period (p, q) and 3-period (r, s) cases. The significant reduction of plane 
classes obtained using these face junctions vs. a randomly generated set is shown in (t).
with  long-range  order  and  a  small  number  of 
plane  classes,  we  have  found  useful  five 
categories  of  face  junctions—defined  as 
orientational  facial  relationships  between 
coincident  tetrahedra.  These  face  junctions  are 
shown in Figures 1k–o and are referred to here as 
3G,  4G,  5G,  20G,  and  FC  (face  centered), 
respectively. The 3G, 4G, and 5G face junctions 
may be obtained by "twisting" the n tetrahedra of 
a  collection  (n <  6)  arranged  about  a  common 
edge (Figures 1a–c) by an angle αn, expressed as
αn = tan
−1(√cos2( θ2 )− cos2( θn2 )
sin( θ2 )cos( θn2 )
)
(1)
where  θ = arccos(1/3) is the tetrahedral dihedral 
angle  and  θn =  2π/n,  about  an  axis  passing 
between  the  midpoints  of  the  central  and 
peripheral  edge.  (In  the  case  of  the  20G, 
tetrahedra  are  rotated  by  an  angle  of 
arccos((√5+3)/(4√2))  about  an  axis  that  passes 
from  the  icosahedral  center  through  each 
tetrahedron's  exposed  face.)  This  procedure  is 
equivalent  to  twisting  the  tetrahedra  of  Figures 
1a–d uniformly until  all  gaps have been closed 
(Figures 1f–i). The resulting angular relationship 
between faces in a junction pair is determined by
βn = 2 tan
−1(√cos2( θ2 )− cos2( θn2 )
cos( θn2 )
)
(2)
Assembling  tetrahedral  structures  with  these 
facial  junctions  has  the  effect  of  reducing  the 
total  number  of  plane  classes  on  certain 
3
Type Twisting angle, αn
Facial rotation angle, βn Facial center offset
Symmetry Num. 
plane 
classes
3G
arccos( 1
√6 )≈ 65.91̊
arccos(− 14 )≈ 104.48̊
3 τ + 1
2 √6 τ 2
≈ 0.4564
3-fold
9
4G
arccos( 1
√2 )= 45̊
arccos( 12 )= 60̊
1
2 √3
≈ 0.29
4-fold
4
5G
arccos(√ 12 + 1√5 )
= arccos (
τ 2
√2 (τ + 2 ) )≈13.28̊
βg ≡ arccos(1 + 3√5
8
)
= arccos(3 τ − 1
4
)≈15.52̊
1
√6 (√5 + 3)
=
1
2√6 τ 2
≈ 0.078
5-fold
10
20G
arccos(√5 + 3
4 √2 )
= arccos ( τ
2
2√2 )≈ 22.24̊
βg ≡ arccos(1 + 3√5
8
)
= arccos(3 τ − 1
4
)≈15.52̊
2
√6 (√5+ 3)
=
1
√6 τ 2
≈ 0.15
A5
10
FC
n/a
βg ≡ arccos(1 + 3√5
8
)
= arccos(3 τ − 1
4
)≈15.52̊
0
3-fold
9
5-fold
10
Table 1 |  Important parameters for  facial  junction types.  "Twisting"  angles are given by  αn. The relative 
angular relationship between face pairs in a junction are given by βn. The golden rotation angle is denoted by  βn. 
Values for facial center offsets represent the distance between facial centers of a junction pair. Here, A5 denotes 
the orientation-preserving subgroup of the icosahedral symmetry group, the alternating group on 5 letters.
arrangements of tetrahedra (see Figure 1t).
A list  of  relevant  parameters  for  these 
"closed-twisted" structures is provided in Table 1. 
Of note is the fact that the relative face-to-face 
rotations for the 5G, 20G, and FC junctions are 
all  equal  (βg =  arccos((3τ –  1)/4)  ≈  15.5225°, 
where  τ  is  the  golden  proportion).  Hence,  this 
value is referred to as the golden rotation,  βg. In 
these  structures—as  for  the  twisted  4G 
arrangement—the  number  of  plane  classes  is 
reduced  compared  with  
the  uniform 
arrangements. Additionally, the twisting angle of 
the 20G is equal to the rotational angle of Fuller's 
"jitterbug  transformation"  which  transmutes  the 
icosahedron  to  the  cuboctahedron  [21].  Taken 
together,  these  two  features  strongly  suggest  a 
"transformable"  quasicrystalline  packing  of 
tetrahedra. (See Figure 2.)
In  attempting  to  construct  a  dense 
quasicrystalline  packing  of  tetrahedra,  we  have 
employed  both  bottom-up  and  top-down 
approaches.  The  former  consists  of  using  the 
various face junctions of Figures 1k–o either (1) 
to  construct  a  skeleton  of  the  quasicrystal, 
subsequently  filling  gaps  with  appropriately 
shaped tiles (e.g., "the shell model", Figure 3a), 
or (2) to  build construct a primitive cell  of the 
quasicrystal (e.g.,  the "tetragrid model", Figures 
3d,e). These models may be assembled using the 
following procedures.
Shell model construction
1. To an  initial  tetrahedron,  four  tetrahedra 
are appended such that the facial centers 
of  their  incident  faces  coincide.  Each 
appended tetrahedron is then rotated such 
that the relative angle between the faces is 
the golden rotation angle, βg.
2. Tetrahedra  are  then  iteratively  appended 
to  each "exposed"  face (i.e.,  not  already 
affixed  to  another  tetrahedron),  and 
oriented in the same manner as above.
3. Following  each  iteration,  tetrahedra  are 
scaled down (with centers fixed) until all 
collisions are eliminated.
Tetragrid model construction
1. Apply  the  golden  rotation  to  the 
constituent  tetrahedra  of  a  Boerdijk-
Coxeter  helix  to  obtain  a  "philix"  with 
linear periods of 3 or 5 tetrahedra.
2. The periodic nature of the philix along its 
axis  allows  for  the  formation  of  three-
dimensional grids of philices, by spacing 
their  points  of  intersection  every  3 or  5 
4
Figure 2 | Jitterbug transformation associated with handedness of the golden rotation.  8 selected faces 
(red)  on the icosahedron with  octahedral  symmetry  (a).  When rotated through an angle  of  α20 these faces 
coincide with 8 outer faces of the "twisted" 20G (b). When the handedness of rotation is alternated, these faces 
coincide with 8 faces of the cuboctohedron (c).
tetrahedra.
3. At  each  crossing  point,  there  are  three 
directions along which to  place philices, 
each given by one of the opposing faces 
in  a  dipyramid  (two  consecutive 
tetrahedra sharing a face)
4. Since these grid lines form three families 
of  parallel  lines,  rhombohedra  may  be 
formed. In the case of 3-periodic philices, 
golden  rhombohedra  are  produced, 
whereas rhombohedra with a diagonal to 
edge ratio of √2 are formed for 5-periodic 
philices.
These models are shown in Figures 3a, d, 
and  e.  The  two-dimensional  projection  of  the 
shell  model  very  closely  resembles  a  Penrose 
tiling  (Figures  3b,  c).  In  the  tetragrid  case 
employing  3-periodic  philices,  a  golden 
rhombohedral  primitive  cell  is  obtained, 
reminiscent of the golden rhombohedron prototile 
of the three-dimensional Penrose tiling [28, 29]. 
These  are  encouraging  indications  for  the 
feasibility  of  
 generating  quasicrystalline 
packings  using  the  face  junctions  described 
above.
Our  top-down  approach  involves  the 
production of a four-dimensional quasicrystal via 
5
Figure 3 | The shell and tetragrid models. 3D shell structure with 5 iterations and a total of 284 tetrahedra (a)  
and its 2D projection down a 5-fold symmetry axis (b). Coloring this projection reveals similarities to the Penrose 
tiling (c). Tetragrid structure comprised of 5-periodic (d) and 3-periodic (e) philices. A layer of the 3D projection of 
V', viewed from an axis of 10-fold symmetry, is shown in (f).
the cut-and-project method [22, 26] from the E8 
lattice.  Selected  points  of  this  structure  are 
subsequently  projected  into  three-dimensional 
space.  The  details  of  this  procedure  are  as 
follows:
1. Determine  the  set  of  edge  of  minimal 
length l among all points of the projected 
(8D to 4D) set. Call this set E.
2. From this  set,  select  those  vertices  that 
belong to 12 or more of these edges (i.e., 
vertices forming connections with at least 
12 others at a distance of l). Let the set of 
these vertices be denoted by V.
3. Select those edges of E whose end points 
are contained in V. Denote this set by E', 
and let  V' denote the set of endpoints of 
edges in E'.
4. Finally,  project  the  vertices  of  V'  from 
four-dimensions to three-dimensions.
We  expect  this  procedure  to  produce  a 
three-dimensional  quasicrystal  of  irregular 
tetrahedra. Through the application of our golden 
and 4G face junctions, we anticipate the ability to 
regularize  tetrahedral  cells  while  introducing 
inter-tetrahedral  gaps.  We  have  achieved  some 
preliminary  results  in  this  direction.  Figure  3f 
shows  a  layer  of  the  3D  projection  from  V', 
viewed along a five-fold axis of symmetry. This 
layer  is  clearly   a  cartwheel-type  quasicrystal, 
similar to the two-dimensional projection of the 
shell  model.  An  interesting  fact  about  this 
projection  method  is  its  relationship  with  the 
"Sum of Squares" law [25], which states that for 
any  edge-transitive
 polytope  projected 
orthographically  from  m dimensions  to  n 
dimensions, the ratio of the sum of squared edge 
lengths is m/n (m > n > 0).
In  conclusion,  the  golden  rotation 
described  in  this  writing  produces  plane-class 
reduction in tetrahedral packings, and is closely 
related  to  Fuller's  jitterbug  transformation. 
Consequently,  we  suspect  that  a  tetrahedral 
quasicrystal may be constructed using this golden 
rotation, and that a dynamic quasicrystal may be 
achieved  using  jitterbug-like  transformation 
between  consecutive  quasicrystal  frames. 
Towards this goal, both bottom-up and top-down 
approaches are currently under investigation.
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