The following is a webified
version of:
George W. Hart, "Icosahedral
Constructions"
Bridges: Mathematical Connections
in Art, Music, and Science
Reza Sarghani (editor),1998,
pp. 195202.
Icosahedral Constructions
Abstract
A mathematical concept has inspired me to create a series
of sculptures. While the media, surface forms, and general impact of these
constructions may vary considerably, there is an ancient underlying structure
common to the series. The mathematical basis behind these sculptures is
the chiral icosahedral symmetry group. This underlying form binds these
pieces in a way that may be obvious to a mathematician accustomed to the
study of patterns, yet invisible to a casual observer. For centuries, geometry
has been considered fundamental to an educated mind and has been both tool
and inspiration to many artists, yet it has lost this status in our current
culture. One purpose of my artwork is to show that geometry still has a
power and a relevance. I hope to prod the viewer into seeing the type of
deeper connection which is the subject of this series of sculptures. In
addition, there is a natural aesthetic which many artists have found in
polyhedral symmetry.
Icosahedral Symmetry

Fig. 1 Dodecahedron and icosahedron,
by Leonardo Da Vinci for Luca Pacioli’s The Divine Proportion.

The ancient Greeks who wrote of the five Platonic solids
certainly appreciated the geometric beauty which they embody. When any
of us now turns a dodecahedron in our hand we see the same quintessential
form which inspired Pythagoras and many others through the ages. Leonardo
DaVinci’s drawings of the icosahedron and the dodecahedron, published in
1509, are shown as Fig 1. Although the dodecahedron may appear superficially
very different from the icosahedron, there is a level of analysis at which
they are identical.
We are all accustomed to geometric abstraction in which
we think of ideal versions of physical objects. For example we may hold
a handful of marbles and imagine an ideal Platonic sphere more perfect
than any marble, free of even the atomicscale imperfections present in
any physical sphere. Similarly, we can imagine the ideal dodecahedron and
the ideal icosahedron as distinct objects inhabiting a Platonic domain
of polyhedra, free from the imperfections of any cardboard polyhedron model.
Yet there is a deeper level — one floor down in Platonic structure — at
which the ideal dodecahedron and ideal icosahedron are but two models of
a single, more basic form which they share.

Fig. 2. The 31 symmetry axes of the
icosahedron and dodecahedron, superimposed so they align (six 5fold, ten
3fold, fifteen 2fold).

The deep structure common to the dodecahedron and icosahedron
is termed their symmetry group. It is the pattern of rotations and
reflections inherent in their geometric form. For example, a line passing
through the centers of two opposite faces of a dodecahedron is a 5fold
axis of the dodecahedron. Rotating the dodecahedron about this axis through
an angle of 72 degrees (or any multiple of 72 degrees) leaves it appearing
unchanged. Altogether, there are six such axes for a dodecahedron (one
axis for each pair of opposite faces), arranged at very specific angles
to each other in space. The icosahedron, though made of triangles instead
of pentagons, also has six 5fold axes of symmetry arranged at the same
angles. In the icosahedron, the 5fold axes pass through pairs of opposite
vertices. Similarly, both solids have ten 3fold axes. They pass through
opposite vertices of the dodecahedron and through the centers of opposite
faces of the icosahedron. They also each have fifteen 2fold axes which
pass through the opposite edgemidpoints in either case. The total arrangement
of symmetry axes is identical in the two cases as Fig. 2 illustrates. If
one starts with a dodecahedron or icosahedron, constructs all of its symmetry
axes, and then erases the polyhedron, one can not tell whether it was the
dodecahedron or icosahedron that one started with.

Fig. 3 The 15 icosahedral symmetry planes,
each represented by a disk.

The symmetry group common to the icosahedron and dodecahedron
is usually called icosahedral, though it lies behind other polyhedra
as well. It is different from the symmetry of many other polyhedra; for
example, the cube has three 4fold axes, four 3fold axes, and six 2fold
axes. The sculptures shown below all have icosahedral symmetry. To fully
understand them, one must see through not only the irregularities of technique
which any physical object must manifest, but deeper—past their form—to
the common symmetry group which binds them.
There is another aspect of polyhedral symmetry beyond
axes of rotation. One can also look at the mirror planes of a polyhedron.
The planes of symmetry are the imaginary planes passing through the center
of the object which reflect one side to exactly match the other side. Again,
the dodecahedron and icosahedron exactly agree in their symmetry planes.
Each has fifteen planes of symmetry in exactly the same arrangement. Fig.
3 shows this arrangement, with one circle for each symmetry plane. The
planes divide the surface of a sphere into 120 right triangles called Mobius
triangles, which are alternately lefthanded and righthanded. The
symmetry axes are located where the planes intersect.
When one studies the possible sets of symmetries, what
mathematicians call group theory, it turns out that only a relatively
small number of polyhedral symmetry structures are possible. In the case
of icosahedral symmetry, there are two possibilities: an object either
has fifteen mirror planes or it has none. An object with no mirror planes
is called chiral (Latin for hand). A chiral object appears
different in a mirror; it comes in lefthand and righthand forms. The
opposite, e.g., the icosahedron which has one or more planes of reflection,
is called reflexible.
Examples

Fig. 4 Radiolarian, by Haeckel.

In nature, icosahedral symmetry does not show up at human
scales, only at microscopic scales. The icosahedral symmetry of certain
species of radiolaria are well known from the drawings of Ernst Haeckel.
I surmise that he found this symmetry particularly attractive since he
chose an icosahedral form for the first figure of the first plate of his
beautiful 1904 collection, Artforms in Nature. It is shown here
as Fig. 4. Other natural examples, all discovered relatively recently,
include the forms of certain viruses, microscopic quasicrystals, and "fullerene"
molecules. (Incidentally, crystals of iron pyrite do form as dodecahedra
with equal 5sided faces, but the faces are not regular pentagons, and
so the crystals do not have any 5fold axes of symmetry; they are not of
the icosahedral symmetry group.)

Fig. 5 Snub icosidodecahedron, discovered
by Archimedes, rediscovered by Kepler.

Although the ancients discovered and created objects with
icosahedral symmetry, I know of no surviving description of any chiral
icosahedral object from that period. It is likely that Archimedes discovered
the snub icosidodecahedron, and that this was the first example of chiral
icosahedral symmetry known to humankind, but his writing on this topic
is lost and only inferred from later commentaries. Kepler independently
rediscovered Archimedes' lost snub icosidodecahedron, and published it
in his 1625 book Harmonies of the World; see Fig. 5. One pentagon
and four triangles meet at each vertex.
However, before Kepler, there was an independent discovery
of this symmetry. The earliest surviving example I know of an object with
chiral icosahedral symmetry is a 1568 drawing by Jamnitzer. His Perspectiva
Corporum Regularium contains many beautiful imagined objects with icosahedral
symmetry, e.g., his monument shown as Fig 6. However, only one, shown here
as Fig 7, has chiral icosahedral symmetry. To come up on his own with something
of this class is an enormous artistic and intellectual achievement in my
view.


Fig. 6 An icosahedral monument by Wentzel
Jamnitzer.

Fig. 7 Detail of a drawing by Jamnitzer.

These developments were part of a surge of interest in
geometry and polyhedra in Renaissance Europe. Polyhedra were valued not
just for their classical associations, but were considered worthy objects
of an artist's study and important tests of an artist's mastery of the
newly developed techniques of perspective. In Albrecht Durer's 1525 Painter's
Manual, he not only teaches techniques of perspective, but he is the
first to show polyhedra in print as an unfolded net. His net of the icosahedron
is shown as Fig 8.

Fig. 8 The eariest known net of an icosahedron,
by Albrecht Durer.

Although icosahedral forms have been created through the
centuries since then, we now skip ahead to our century's most notable artist
of polyhedra. M.C. Escher is best known for his tessellations, but he also
had a great affinity to polyhedra. A chiral icosahedral object is featured
in his lithograph Gravity, which shows a small stellated dodecahedron
perforated in a chiral manner and then populated with colorful creatures.
Even more relevant is that Escher made a number of sculptures with chiral
icosahedral symmetry, including a puzzle of twelve intertwined starfish,
and an icosahedral candy tin design. Fig. 9 shows Escher’s carved maple
flower, made in 1958.

Fig. 9 Carved maple sculpture in the
form of twelve flowers arranged as a dodecahedron, by M.C. Escher.

Method
A simple general method for seeing a construction with icosahedral
symmetry is available, if the object is to be reflexible. The icosahedral
kaleidoscope, invented by Mobius in 1852, is a set of three mirrors,
each a thin circular wedge, with angles of 20.91, 31.72, and 37.37 degrees,
respectively. They are arranged in the form one of the small triangles
shown in Fig. 3, i.e., the points of the wedges meet at the center of the
sphere, with the reflective side facing into the triangular region. If
any object is placed in the triangle and viewed from inside the triangle,
the "hall of mirrors effect" causes it to be replicated with the symmetry
of an icosahedron. Details can be found in Coxeter [1963], but this method
is not suitable when one is interested in chiral objects, which have no
planes of symmetry.
To physically construct an object with chiral icosahedral
symmetry, one first imagines the symmetry axes, then places components
about them in the proper positions. For example, if one starts with sixty
identical copies of anything, they can be placed on the surface of a sphere,
five around each of the twelve points where the six 5fold axes penetrate
the sphere. However, if one starts with objects that have 5fold symmetry,
then only twelve are needed, as each can be centered over an axis. Similarly,
twenty components with 3fold symmetry can be placed at the 3fold points,
or thirty components with 2fold symmetry can be placed on the 2fold points.
One could analogously assemble 120 copies of a component
(60 lefthand pieces and 60 righthand pieces) to construct a reflexible
icosahedral model, but that symmetry group was not the subject of this
series.
Constructions
In these constructions, the components are carefully designed
both for the overall form which the assemblage is to achieve and for the
necessary lengths and angles required to make the components meet properly.
In some cases I have selected massproduced elements to use as components,
but in most cases I fabricate the elements to be assembled. However, the
most challenging part of the construction of the following sculptures is
the construction of the necessary jigs to hold the components together
at the proper relative positions for assembly. I usually spend far more
time constructing jigs than actually using the completed jigs to assemble
components.
The sculptures shown here as Figs. 1018 were all constructed
by the author in 1997, as part of a series. Each displays the identical
chiral icosahedral symmetry, but with very different forms and media. I
leave it to the reader to locate the thirtyone symmetry axes in each case,
and to decide if the symmetry carries a natural aesthetic within each piece.


Fig. 10. I’d like to make one thing perfectly
clear. (18”) This is a construction of 30 identical pieces of clear
acrylic plastic (plexiglass). Each piece started as a plastic parallelegram
with two polished edges, then was heated (baked in an oven at 300 F for
3 minutes) and bent into a form with 2fold symmetry. Cooled in a
jig which maintains two edges at the proper relative angle, each piece
is centered on a 2fold axis of symmetry, and exactly spans the 63.4 degree
angle between two 5fold axes. These edges were beveled to a 72 degree
angle, so five fit around an axis, and glued with a solvent cement. 
Fig. 11. The Plastic Tableware of Damocles.
(25”) This is a construction of 180 plastic dinner knives, in black,
white, and grey. It can be seen as sixty equilateral triangles, each
containing one knife of each color. The triangles are arranged in
the form of a dodecahedron in which the pentagons are replaced by concave
dimples of five equilateral triangles. Knives of a color can be seen
to spiral around the axes of symmetry. 


Fig. 12. Battered Moonlight. (21”)
This is an icosahedral topological surface with one side and six edges,
inspired by the work of Brent Collins. It is constructed of paper
machee over a steel (hardware cloth) framework, and painted in white latex/acrylic.
The twelve holes contain the 5fold symmetry axes, and the six edges are
each 5fold symmetric “equators” about one 5fold axis. 
Fig. 13. Gazmogenesis. (12”)
This is a construction of 30 identical pieces of copper. Each piece
was cut into the shape of an elongated ellipse, hand formed into a curve
with 2fold symmetry, and the assemblage was soft soldered. The intention
was to create a form both organic and geometric, reminiscent of a radiolarian
with a spiked spherical body containing internal structure. Each
copper ellipse spans one edge of an icosahedron, along a detoured route
which passes through the interior. 


Fig. 14. Fire and Ice. (24”)
Sixty identical pieces of oak and ten strips of brass are interwoven.
The wooden components were cut on a computercontrolled router table, then
hand beveled to the proper dihedral angles. After the wood was glued,
the brass strips were woven through the wood and each other, and soldered
into loops. Each brass loop has 3fold symmetry and circles a 3fold
axis of the whole. 
Fig. 15. No Picnic. This is 16’ mobile
with three major spherical assemblages of plastic tableware: one ball consists
of 150 knives in three colors, one of 180 spoons in six colors, and one
of 240 forks in six colors. The plastic utensils were jigged on frameworks
built from a commercial plastic construction toy, glued with solvent cement,
and then the jigs were disassembled and removed through the spaces between
the utensils. The supporting booms are steel tubes. 


Fig. 16. Fork component of Fig. 15 (23”).

Fig. 17. Knife component of Fig. 15
(29”).


Fig. 18. Spoon component of Fig. 15
(32”).

References

H.S.M. Coxeter, Regular Polytopes, Macmillan, 1963,
(Dover reprint, 1973).

Peter R. Cromwell, Polyhedra, Cambridge, 1997.

Michele Emmer (editor), The Visual Mind: Art and Mathematics,
MIT, 1993.

George W. Hart, Virtual Polyhedra, http://www.georgehart.com/virtualpolyhedra/vp.html

George W. Hart, Geometric Sculpture, http://www.georgehart.com/sculpture/sculpture.html

Jay Kappraff, Connections: The Geometric Bridge Between
Art and Science, McGraw Hill, 1990.

Doris Schattschneider, M.C. Escher: Visions of Symmetry,
Freeman, 1990.

Marjorie Senechal, George Fleck (eds.), Shaping Space:
A Polyhedral Approach, Birkhauser/MAA, 1988.

Hermann Weyl, Symmetry, Princeton U. Pr., 1952.
copyright 1998, George W. Hart