fractal and fractional
Article
Cabinet of Curiosities: The Interesting Geometry of
the Angle β = arccos ((3φ − 1) /4)
Fang Fang, Klee Irwin *, Julio Kovacs and Garrett Sadler
Quantum Gravity Research, Topanga, CA 90290, USA; Fang@quantumgravityresearch.org (F.F.);
jak3377@gmail.com (J.K.); imgarypenn@gmail.com (G.S.)
* Correspondence: klee@quantumgravityresearch.org
Received: 12 September 2019; Accepted: 27 October 2019; Published: 30 October 2019






Abstract:
In this paper, we present the construction of several aggregates of tetrahedra.
Each construction is obtained by performing rotations on an initial set of tetrahedra that either
(1) contains gaps between adjacent tetrahedra, or (2) exhibits an aperiodic nature. Following this
rotation, gaps of the former case are “closed” (in the sense that faces of adjacent tetrahedra are
brought into contact to form a “face junction”), while translational and rotational symmetries are
obtained in the latter case. In all cases, an angular displacement of β = arccos (3φ− 1) /4 (or a closely
related angle), where φ =
(
1 +
√
5
)
/2 is the golden ratio, is observed between faces of a junction.
Additionally, the overall number of plane classes, defined as the number of distinct facial orientations
in the collection of tetrahedra, is reduced following the transformation. Finally, we present several
“curiosities” involving the structures discussed here with the goal of inspiring the reader’s interest in
constructions of this nature and their attending, interesting properties.
Keywords: tetrahedron; Golden Ratio; rotational transformation
1. Introduction
The present document introduces the reader to the angle β = arccos ((3φ− 1) /4), where φ =
(
1 +
√
5
)
/2 is the golden ratio, and its involvement, most notably, in the construction of several
interesting aggregates of regular tetrahedra. In the sections below, we will perform geometric rotations
on tetrahedra arranged about a common central point, common vertex, common edge, as well as
those of a linear, helical arrangement known as the Boerdijk–Coxeter helix (tetrahelix) [1,2]. In each
of these transformations, the angle β above appears in the projections of coincident tetrahedral faces.
Noteworthy about these transformations is that they have a tendency to bring previously separated
faces of “adjacent” tetrahedra into contact [3] and to impart a periodic nature to previously aperiodic
structures. Additionally, after performing the rotations described below, one observes a reduction in
the total number of plane classes, defined as the total number of distinct facial or planar orientations
in a given aggregation of polyhedra. These aggregates of tetrahedra might give clues to periodic
tetrahedral packing in quasicrystals [4] .
2. Aggregates of Tetrahedra
In this section, we describe the construction of several interesting aggregates of regular tetrahedra.
The aggregates of Sections 2.1 and 2.2 initially contain gaps of various sizes. By performing special
rotations of these tetrahedra, these gaps are “closed” (in the sense that faces of adjacent tetrahedra
are made to touch), and, in each case, the resulting angular displacement between coincident faces
is either identically equal to β or is closely related. In Section 2.3, a rotation by β is imparted to
tetrahedra arranged in a helical fashion in order to introduce a periodic structure and previously
unpossessed symmetries.
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2.1. Aggregates about a Common Edge
Consider aggregates of n regular tetrahedra, 3 ≤ n ≤ 5, arranged about a common edge (so that
an angle of 2π/n is subtended between adjacent tetrahedral centers; see Figure 1a for an example
with five tetrahedra). In each of these structures, gaps exist between tetrahedra that may be “closed”
(i.e., faces are made to touch) by performing a rotation of each tetrahedron about an axis passing
between the midpoints of its central and peripheral edges through an angle given by
αn = arctan

√
cos2 γ2 − cos2
θn
2
sin γ2 cos
θn
2
 ,
(1)
where γ = arccos(1/3) is the tetrahedral dihedral angle and θn = 2π/n. When this is done, an angle,
βn, is established in the “face junction” between coincident pairs of faces such that
βn = 2 arctan

√
cos2 γ2 − cos2
θn
2
cos θn2
 .
(2)
Α5
(a)
Α5
(b)
Β5
(c)
Figure 1. “Twisting” tetrahedra centered about a common central edge to close up gaps between
adjacent tetrahedra. When this operation is performed, the angle β = arccos ((3φ− 1) /4) is produced
in the projection of a “face junction”. (a) Five tetrahedra arranged about a common edge. In this
arrangement, small gaps exist between the faces of adjacent tetrahedra. Each tetrahedron is to be
rotated by α5 about an axis passing between the midpoints of its central and peripheral edges.
(b) The tetrahedra after rotation. In this arrangement, the faces of adjacent tetrahedra have been
brought into contact with one another. (c) A projection of a “face junction” between two coincident
faces in Figure 1b. The angular displacement between the faces is β5 = β.
The present document is focused on the angle β = arccos ((3φ− 1) /4), which is, in fact, the angle
obtained in the “face junction” produced by executing the above procedure for n = 5 tetrahedra
(see Figure 1). It is interesting, however, that a simple relationship may be established between this
angle and β3. By evenly arranging three tetrahedra about an edge and rotating each through the axis
extending between the central and peripheral edge midpoints, the face junction depicted in Figure 2c
is obtained. The angle between faces in this junction, β3, may be related to β in the following way:
2π
3
− β = β3.
(3)
To see this, note that
2π
3
− arccos
(
3φ− 1
4
)
= 2 arctan

√
cos2 γ2 − cos2
θ
2
cos θ2

(4)
gives the solution θ = ± 2π3 , which reduces the right-hand side of Equation (4) to β3.
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Α3
(a)
Α3
(b)
Β3
(c)
Figure 2. Arranging and rotating three tetrahedra (as done in the n = 5 case) to “close up” gaps between
adjacent tetrahedra. When this operation is performed, the angle β3 = 2π3 − β is produced in the
projection of a “face junction”. (a) Three tetrahedra arranged about a common edge. In this arrangement,
large gaps exist between the faces of adjacent tetrahedra. Each tetrahedron is to be rotated by α3 about
an axis passing between the midpoints of its central and peripheral edges. (b) The tetrahedra after
rotation. In this arrangement, the faces of adjacent tetrahedra have been brought into contact with one
another. (c) A projection of a “face junction” between two coincident faces in Figure 2b. The angular
displacement between the faces is β3.
In this section, we have produced two aggregates of tetrahedra whose face junctions bear a
relationship to the angle β = arccos ((3φ− 1) /4). In the section that follows, we will locate this angle
in the face junctions produced through rotations of tetrahedra about a common vertex.
2.2. Aggregates about a Common Vertex
Consider the icosahedral aggregation of 20 tetrahedra depicted in Figure 3a. The face junction of
Figure 3c is obtained when each tetrahedron is rotated by an angle of
α20 = arccos
(
φ2
2
√
2
)
(5)
about an axis extending between the center of its exterior face and the arrangement’s central vertex.
As above, this operation “closes” gaps between tetrahedra by bringing adjacent faces into contact.
Interestingly, the face junction obtained here consists of tetrahedra with a rotational displacement
equal to the one obtained in the case of five tetrahedra arranged about a common edge above,
i.e., β20 = β5 = β. (It should be noted, however, that α20 and β20 are not produced by Equations (1)
and (2), respectively, as those formulæ are only valid for 3 ≤ n ≤ 5.)
In all of the cases described above, gaps are “closed” and “junctions” are produced between
adjacent tetrahedra in such a way that the angle β = arccos ((3φ− 1) /4) appears in some fashion in
the angular displacement between coincident faces. For the cases of 5 tetrahedra about a central edge
and 20 tetrahedra about a common vertex, this angle is observed directly. For the case of 3 tetrahedra
about a central edge, the angular displacement between faces is closely related: β3 = 2π3 − β.
We now turn to an arrangement obtained by directly imparting an angular displacement of β
between adjacent pairs of tetrahedra in a linear, helical fashion known as the Boerdijk–Coxeter helix.
An interesting result of performing this action is that a previously aperiodic structure is transformed
into one with translational and rotational symmetries.
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Α20
(a)
Α20
(b)
Β20
(c)
Figure 3. When 20 tetrahedra are organized into an icosahedral arrangement, gaps between adjacent
tetrahedra may be “closed” by performing a rotation of each tetrahedron by α20 about an axis passing
from the central vertex through each tetrahedron’s exterior face. When this is done, an angle of
β is produced in the projection of faces in a “face junction”. (a) Twenty tetrahedra arranged with
icosahedral symmetry about a common central vertex. In this arrangement, gaps exist between faces of
adjacent tetrahedra. Each tetrahedron is to be rotated by α20 about an axis passing from the central
vertex through its exterior face. (b) The tetrahedra after rotation. Like in the cases above, the faces of
adjacent tetrahedra have been brought into contact. (c) A projection of the “face junction” between two
coincident faces in Figure 3b. The angular displacement between the faces is β20 = β5 = β.
2.3. Periodic, Helical Aggregates
In Sections 2.1 and 2.2, we described a procedure by which initial arrangements of tetrahedra
were transformed so that adjacent pairs of tetrahedra were brought together to touch. In each of these
structures, coincident faces are displaced by an angle equal or closely related to β. Here, we will
construct two periodic, helical chains of tetrahedra by directly inserting an angular offset by β between
each successive member of the chain. For their close relationship with the Boerdijk–Coxeter helix,
we refer to these structures by the term modified BC helices.
The construction of a modified BC helix is depicted in Figure 4. Starting from a tetrahedron
Tk = (vk0, vk1, vk2, vk3), a face fk is selected onto which an interim tetrahedron, T′k, is appended.
The (k + 1)th tetrahedron is obtained by rotating T′k through an angle β about an axis nk normal to fk,
passing through the centroid of T′k. (Note that this automatically produces an angular displacement of
β between two faces in a “junction”, see Figure 5a).
fk
Tk
(a)
Tk
Tk
'
(b)
Tk
Tk+1
nk
(c)
Figure 4. Assembly of a modified BC helix. (a) A segment of a modified BC helix with face fk identified
on tetrahedron Tk. (b) An intermediate tetrahedron, T′k (shown in blue), is appended (face-to-face) to fk
on Tk. (c) Finally, Tk+1 is obtained by rotating T′k through the angle β about the axis nk.
The structure that results from this process depends on the sequence of faces F = ( f0, f1, . . . , fk)
selected in order to construct the helical chain. This sequence determines an underlying chirality of
the helix—i.e., the chirality of the helix formed by the tetrahedral centroids—and plays a pivotal role
in the determination of the structure’s eventual symmetry. (However, it should be noted, of course,
that some sequences of faces do not result in helical structures. Faces cannot be chosen arbitrarily or
randomly; they must be selected so as to build a helix).
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By performing the procedure depicted in Figure 4, using an angular displacement of β between
successive tetrahedra, periodic structures are obtained with 3- or 5-fold symmetry (upon their
projections, see Figures 6b,c), depending on the relative chiralities between the rotational displacement
and the underlying helix: when like chiralities are used, one obtains 5-fold symmetry; when unlike
chiralities are used, one obtains 3-fold symmetry. In addition to rotational symmetry, these structures
are given a linear period, which we quantify here as the number of appended tetrahedra necessary to
return to an initial angular position on the helix. For a modified BC helix with a period of m tetrahedra,
we use the term m-BC helix. Accordingly, the procedure described above produces 3- and 5-BC helices,
which are shown in Figure 6. (See [5] for a proof of these structures’ symmetries and periodicities).
Β
(a)
(b)
(c)
Figure 5. BC helix projections and face junction. (a) A “face junction” between tetrahedra of a 3-
or 5-BC helix. (b) A projection of the 5-BC helix along its central axis, showing five-fold symmetry.
(c) A projection of the 3-BC helix along its central axis, showing three-fold symmetry.
(a)
(b)
(c)
Figure 6. Canonical and modified Boerdijk–Coxeter helices. (a) A right-handed BC helix. (b) A “5-BC
helix” may be obtained by appending and rotating tetrahedra by β using the same chirality of the
underlying helix. (c) A “3-BC helix” may be obtained by appending and rotating tetrahedra by β using
the opposite chirality of the underlying helix.
3. Curiosities
We have seen the construction of several aggregates of tetrahedra. Each of these structures contains
tetrahedra with coincident faces, offset angularly by β or a closely related angle ( 2π3 − β). We will
now explore some of the interesting features of these structures. As already noted, it is interesting
that β appears in the face junctions of structures generated by “closing” gaps between tetrahedral
aggregates. It is additionally interesting that when this angle is employed in the construction of
a helical chain of tetrahedra, a periodic structure emerges (whereas the canonical BC helix has no
non-trivial translational or rotational symmetries).
The features we will highlight in this section involve the reduction of the overall number of
“plane classes” and the linear displacements between facial centers in a face function. Here, we say
that two planes belong to the same plane class if and only if their normal vectors are parallel. The
number of plane classes for a collection of tetrahedra, then, is defined as the number of distinct plane
classes comprising the collection’s two-dimensional faces. By rotating tetrahedra so as to bring faces
into contact (as in Sections 2.1 and 2.2), or rotating tetrahedra to obtain periodicity (as in Section 2.3),
the overall number of plane classes for an aggregate is reduced. Clearly, as the values of α3, α5, and α20
are such that they bring faces of adjacent tetrahedra into contact, we would expect to see a reduction in
the number of plane classes in the corresponding aggregations of tetrahedra featured in Sections 2.1
and 2.2. It is interesting, however, that rotation of the tetrahedra in a BC helix by β (observed in the
Fractal Fract. 2019, 3, 48
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face junctions of Figures 1c, 2c and 3c) obtains a reduction from an arbitrarily large number of plane
classes (3n + 1, where n is the number of tetrahedra in the helix) to relatively small numbers: 9 plane
classes in the case of the 3-BC helix, 10 plane classes in the case of the 5-BC helix. Table 1 provides the
numbers of plane classes for the tetrahedral aggregates described in Section 2 before and after their
transformations.
Table 1. Plane class numbers for aggregates described in Section 2.
Structure
αn
βn
Plane Classes
Before After
3 tetrahedra, common edge
arccos
(
1√
6
)
2π
3 − arccos
(
3φ−1
4
)
12
9
4 tetrahedra, common edge
π
4
π
3
16
4
5 tetrahedra, common edge
arccos
(
φ2
√
2(φ+2)
)
arccos
(
3φ−1
4
)
20
10
20 tetrahedra, common vertex
arccos
(
φ2
2
√
2
)
arccos
(
3φ−1
4
)
60
10
n-tetrahedron 3-BC helix
n/a
arccos
(
3φ−1
4
)
3n + 1
9
n-tetrahedron 5-BC helix
n/a
arccos
(
3φ−1
4
)
3n + 1
10
Finally, an appealing feature is observed in the face junction projections of the tetrahedral
aggregates discussed in this paper. Figure 7 provides a side-by-side comparison of these face junctions.
As the angular displacement between tetrahedra in all face junctions is related to β, we can see
that translation of a tetrahedron in one junction can produce any of the other junctions. Let the
displacement between tetrahedra in the face junction of 5 tetrahedra about a common edge be denoted
by δ =
〈
a
2φ2
√
6
, 0
〉
, where a is the tetrahedron edge length. Starting from the face junction of a 3- or
5-BC helix, the remaining face junctions corresponding to 20 tetrahedra about a vertex, 5 tetrahedra
about an edge, and 3 tetrahedra about an edge may be obtained by translating a tetrahedron of the
junction by −2δ, δ, and (3φ + 1) δ, respectively.
-2 ∆
∆
∆ H3 Φ + 1L
Figure 7. Side-by-side comparison of face junctions. All face junctions may be obtained by
translation of the (projected) tetrahedra of the 3- and 5-BC helix (pictured center left) by integer-
and golden-ratio-based multiples of δ =
〈
a
2φ2
√
6
, 0
〉
, where a is the tetrahedron edge length, the
displacement between tetrahedra of a face junction for 5 tetrahedra about a common edge (pictured
center right). From left to right, the displacements between the tetrahedra are −2δ, 0, δ, and (3φ + 1) δ.
4. Conclusions
In this paper, we have presented the construction of several aggregates of tetrahedra. In each case,
the construction process involved rotations of tetrahedra by a value related to β = arccos (3φ− 1) /4.
The structures produced here have several notable features: faces of tetrahedra are made to touch
(“closing” previously existing gaps between tetrahedra), aperiodic structures are imparted with
periodicity, and the total number of plane classes is reduced, as shown in Figures 1–4. The purpose of
the present document, however, is not merely descriptive; it is hoped that these notable features have
generated interest in the reader of rotational transformations of tetrahedra involving the angle β. In
particular, it is desired that further observations may be found that transform aggregates of tetrahedra
Fractal Fract. 2019, 3, 48
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such that faces are brought into contact and the number of plane classes is reduced, as shown in Table 1.
These aggregates of tetrahedra could link tetrahedral packing to quasicrystals.
Author Contributions: Conceptualization, F.F. and K.I.; Investigation, F.F., J.K. and G.S.; Methodology and
Writing Manuscript, G.S. and F.F.; Software, G.S. and F.F.; Supervision, K.I.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
Coxeter, H.S.M. Regular Complex Polytopes; Cambridge University: Cambridge, UK, 1974.
2.
Boerdijk, A.H. Some remarks concerning close-packing of equal spheres. Philips Res. Rep. 1952, 7, 30.
3.
Fang, F.; Clawson, R.; Irwin, K. Closing Gaps in Geometrically Frustrated Symmetric Clusters: Local
Equivalence between Discrete Curvature and Twist Transformations. Crystals 2018, 6, 89. [CrossRef]
4.
Fang, F.; Irwin, K. An Icosahedral Quasicrystal as a Golden Modification of the Icosagrid and its Connection
to the E8 Lattice. arViv 2015, arViv:1511.07786. [CrossRef]
5.
Sadler, G.; Fang, F.; Clawson, R.; Irwin, K. Periodic modification of the Boerdijk-Coxeter helix (tetrahelix).
Mathematics 2019, 7, 1001. [CrossRef]
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