By experimenting with
Escher's tessellations I stumbled on an elegant extension to 3D space. By rotating
the elements in the two-dimensional tilings I obtained spiraling bodies that
close pack to fill space. I named them Tordated Sinusoids of order N: [N]TS.
It appears that only the bodies based on triangles (TTS), squares (QTS) and
the combination of triangles and hexagons (HTS) are valid close packers. The
[N]TS are limiting cases of the k-armed Spirallohedra discovered by Russell
Towle.
Check out his cool YouTube movies here. And particularly this one:
The paper below gives
a full description. PDF version here.
This paper presents
a new class of three-dimensional mathematical objects that close-pack to
fill space. These so-called Tordated Sinusoids ([N]TS) are based on rotational
variations of two-dimensional tessellations of equilateral triangles (which
forms TTS) and of squares (QTS). The combination of sixfold Tordated Sinusoids
(HTS) and TTS gives rise to a composite 3D lattice. It appears that the
ruled surfaces of an [N]TS are linearly tordated and are circumscribed by
N circular spirals. It also appears that the TTS, QTS and HTS can be described
as a cube, Kepler's dodecahedron and a triacontahedron respectively with
their rhombic surfaces parametrized in curvilinear coordinates.
1. From 2D
tessellations to 3D close-packing
Two-dimensional tessellations
of a plane are well-studied by both mathematicians, crystallographers and
artists alike. The basic regular tessellations are ascribed to Plato [1].
There are only three regular polygons which can be used to tile a surface
using only likes: triangles, squares and hexagons. Tilings using combinations
of different regular polygons are called Archimedean [1].
The Dutch graphic artist M.C. Escher (1898-1972) used such tessellations to
bring his parallel universe with all its bizarre inhabitants come to life
[2]. The present and earlier work [3]
shows how Platonic and Archimedean tessellations can be elegantly extended
into the third dimension to form spiraling close-packers.
Figure
1.
Animation of rotating
squares with two different phases α.
1.1.
Consider the regular tessellation
of white squares in the starting frame of figure 1. By
rotating each square around its center whilst decreasing its size, a second
set of (black) squares appears in the animation at the locations of the vertices
of the original white squares. By further rotating and scaling around their
centers, the white and black squares generate a complete sequence of tessellations
until the black squares form a regular Platonic tessellation. By switching
the colors here, we are basically back at the starting point, only with a
spatial shift. But in fact the cycle is only halfway through here and it can
be completed by rotating and scaling the black squares according to the same
recipe until you are back at the beginning.
This procedure gives us
a clue that the evolution of a white square is identical to that of a black
square, only with a 180º phase shift. By transforming the parametrization
α of the sequence of figure 1 into a third spatial coordinate
as in figure 2, it indeed appears that the white and black
squares make up the same spiraling three-dimensional shape. Therefore this
shape is an intrinsic close-packer of 3D space, just like a square is a close-packer
of 2D space. For reasons that are revealed in the next sections, these bodies
are named "Tordated Sinusoids" of order N ([N]TS). The body with fourfold
rotational symmetry based on squares is called "QuatTorSin" or just QTS. Of
course these Tordated Sinusoids shape have twofold degeneracy: it can spiral
both clockwise and counter-clockwise with equal packing properties. In crystallography
the lattice structure of the QTS is called "body-centered cubic".
Figure 2.
Three-dimensional close-packing
of QTS.
1.2.
The same trick is also
possible with Platonic tilings of equilateral triangles as indicated in figure
3. Here, a third color (grey) is necessary to distinguish between the
different sets of triangles. It is obvious that the three different sets have
a phase shift of 120º.
Figure
3.
Animation of rotating
equilateral triangles with three different phases α.
Again making the step
to 3D space, these sequences result in spiraling close-packers with threefold
rotational symmetry, called "TriTorSins" (TTS). They are to be stacked in
a "cubic" fashion as shown in figure 4.
Figure 4.
Three-dimensional close-packing
of TTS.
1.3.
By trying out the exact
same procedure of rotating and scaling for the Platonic tessellation of hexagons,
the gaps turn out to be triangular instead of hexagonal like the orginal tessellations
(figure 5). Apparently we need to resort to Archimedean
tessellations for the intermediate tilings. Note that the triangles evolve
just like in figure 3, but here there is only one single
phase. Yet, there are two sets of triangles, rotated at 60º around their axes
with respect to each other.
Figure
5.
Animation of rotating
hexagons in combination with equilateral triangles.
The combination of spiraling
hexagons and the bodies of figure 4 give rise to a composite
close-packing of "HexaTorSins" (HTS) and TTS in a 1:2 ratio as depicted in
figure 6.
Figure 6.
Three-dimensional close-packing
of HTS and TTS.
1.4.
The shapes and close-packing
properties of the [N]TS are very similar to the Spirallohedra discovered and
described by Towle[4]. These Spirallohedra are obtained
by transformations of polar zonohedra and are therefore covered by many rhombic
faces, whereas the N surfaces of an [N]TS are fully double-curved. In fact,
the [N]TS are the limiting cases of the k-armed Spirallohedra when the number
of faces goes to infinity. Therefore the horizontal cross-sections of a Spirallohedron
generally will not deliver regular polygons. Nevertheless it is intriguing
to see how two totally different approaches deliver similar objects.
2. Mathematical
description of the ruled surfaces
Several methods can be
employed to mathematically describe the ruled surfaces of an [N]TS.
2.1.
The most intuitive way
to describe the surface of an [N]TS is just to look at the tilings of figures
1, 3 and 5 as
horizonal cross-sections of close-packing QTS, TTS and HTS/TTS respectively.
By following the evolution of one square as in figure 7a
it is possible to identify a limited number of cardinal grid-points that govern
the rotation and scale. In three dimensions these points form vertical axes
that describe the entire close-packing.
Figure 7a-c.
Cardinal axes of a
QTS, TTS and HTS.
First of all, it is clear
that the squares rotate around their center axes Ab,w. To
enable continuity of the pattern, the edges (or the continuation thereof)
have to cross fixed points B. These points B therefore form vertical straight
lines on the tordated surface of a QTS. After a closer study of the patterns
of figure 1 it appears that the vertices of the squares are all located on
perfect circles after completion of a cycle. The centers of these circles
are tagged with C. The circles intersect at A and B points.
These circles are parametrized linearly (completing double the circular angle
as compared to the rotation of the squares) and form either right- or left-handed
screwing spirals in 3D space. Figure 7b-c shows that also
a TTS and a HTS are described with these A, B and C grid-points.
It is concluded that all
[N]TS only principally differ in the number of ruled surfaces N. But a study
of Platonic and Archimedean tilings shows that only the bodies based on triangles
(N=3), squares (N=4) and hexagons (N=6) are suitable for 3D close-packing.
Figure 8.
The surface of a TTS
as a horizontally ruled surface.
Figure
8 shows how the surface of an [N]TS can be built up from horizontal rules
governed by the vertical axes A, B and C.
2.2.
If the parametrization
of the rotating polygons of figures 1, 3
and 5 is called α, which cycles through 360º
for creating one [N]TS, it can be proven that the angle of one edge of the
polygon with respect to the angle at α ↓ 0 is proportional
to ½α. In other words, the tordation of the surface ruling around
the B axis is linear and completes 180º over the full length. Furthermore,
with the description of the circular orbits of the vertices the length of
the edge of such a polygon can be shown to be proportional to sin(½α).
This statement can easily be proven by taking the total surface of a white
and a black square in figure 1, which of course needs
to be constant. With implementing their phase relation of 180º, that surface
is proportional to sin²(½α) + sin²(½(α+180º)) =
sin²(½α) + cos²(½α) = 1. Equivalently for the triangles
of figure 3, sin²(½(α-120º)) + sin²(½α) + sin²(½(α+120º))
= 3/2.
The range of mathematical
bodies [N]TS as described above form a surprising set of spiraling close-packers
for N=3, 4 and 3/6. Almost in contradiction to their very organic shapes they
can be used as “bricks” to build intricate three-dimensional structures. This
creates many new possibilities for sculptors, computer artists and architects.
With the recipes of section 2 the shapes are relatively easily created. The
spiraling shape is fascinating in itself with its mathematical elegance. Of
course it reflects nature on all scales, from DNA to spiral galaxies. With
the space-filling properties decribed in this paper they even encompass infinity.
Examples of my sculptures
and sketches based on [N]TS are to be found here
and here respectively.
Figure
1. 
Figure
3. 
Figure
5. 
