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The Mathematical Tourist
Dirk Huylebrouck, Editor
Fullerenes,
Polyhedra, and
Chinese Guardian
Lions
EUGENE A. KATZ AND BIH-YAW JIN
Does your hometown have any mathematical tourist
attractions such as statues, plaques, graves, the cafe
´
where the famous conjecture was made, the desk where
the famous initials are scratched, birthplaces, houses, or
memorials? Have you encountered a mathematical sight
on your travels? If so, we invite you to submit an essay to
this column. Be sure to include a picture, a description
of its mathematical significance, and either a map or
directions so that others may follow in your tracks.
âSubmissions should be uploaded to http://tmin.edmgr.com
or sent directly to Dirk Huylebrouck,
huylebrouck@gmail.com
W
We, the coauthors of this paper, met for the first
time in 2014 at the 2nd International Conference
“Science, Technology, and Art Relations” in Tel
Aviv. In his lecture “Fullerene-like Architecture in Nano-,
Micro-, and Macro-worlds” Eugene Katz showed his pho-
tographs of two statues of guardian lions at the Forbidden
City in Beijing and mentioned that the lions’ paws rest on
balls, which, combinatorially, are fullerene polyhedra and
their duals. Bih-Yaw Jin challenged him on several points.
This short discussion spurred our collaboration on this
article, and it became a report on a mathematical tourist
detective investigation.
Statues of Chinese guardian lions, known as Shishi for
stone lions or Tongshi for bronze lions, stand in front of
Chinese Imperial palaces, Imperial tombs, government
offices, temples, and the homes of government officials and
the wealthy; the tradition dates back to the Han Dynasty
(206 BC–AD 220). The lions are believed to have powerful
mystical protective powers. They are always presented in
pairs, a manifestation of yin and yang, the female repre-
senting yin and the male yang. The female has a cub under
the left paw, representing the cycle of life. The male lion is
essentially identical, but has its right front paw on an
embroidered ball. Sometimes the ball is carved with a
spherical network that is similar to that of a fullerene
polyhedra or a fullerene dual. (A fullerene polyhedron is
one with only pentagonal and hexagonal faces. It is easy to
show that no matter how many hexagonal faces such a
polyhedron has, the number of pentagonal faces must be
twelve. For more about fullerene polyhedra and their duals,
see the Appendix to this paper.)
One of us (E.K.) first learned about the fullerene-like
decorated sphere under the paw of the guardian lion in
front of the Gate of Heavenly Purity (the Qianqing Gate) at
the Forbidden City in Beijing from Istvan Hargittai’s article
[1]. Hargittai seems to have been the first scientist to pay
attention to this pattern. On his first visit to Beijing, E.K.
wanted to find and photograph it. He did (Fig. 1a–c), and
he also found, surprisingly, the paw of a male lion in front
of the Hall of Spiritual Cultivation (Yangxing Dian) on a
ball with the fullerene dual structure (Fig. 1b–d). E.K. rashly
decided that these statues were erected in the same period
as the Forbidden City was built (Early Ming Dynasty,
*1420).
B.-Y.J. questioned this dating. The buildings in the
Forbidden City are made of wood, and many of them have
been burned down and rebuilt. The palace behind the
bronze lion has been rebuilt several times. Even though the
bronze lion could not be burned down so easily, the statues
should be dated according to historical records.
B.-Y.J. also queried the fullerene structure, having ear-
lier compared it to a picture he found on the Internet
©2016 Springer Science+Business Media New York, Volume 38, Number 3, 2016 61
DOI 10.1007/s00283-016-9663-0
(Fig. 2). If the structure had icosahedral symmetry, the
centers of the two pentagons (marked in red) would be the
vertices of an equilateral triangle whose third vertex would
be the center of a third pentagon at the location of the
white dot. However, there is a hexagon at that position.
Analysis of the “dual fullerene” (Fig. 1b–d) showed three
pentagonal arrangements of triangular faces (or three ver-
tices of degree five) on its front side; see the red pentagons
in Figure 3. (Goldberg vectors (2,1) and (4,0) could be
identified; see the Appendix).
We agreed that these structures do not have icosahedral
symmetry. But do they have a structure of any other full-
erene (or fullerene dual)? In other words, do they have 12
pentagonal faces (or 12 pentagonal connections)? To
answer this question one should examine the entire surface
of the ball.
B.-Y.J. studied the Chinese literature [2] and learned that
near or on each sculpture in the Forbidden City there
should be information about its dating. He asked his former
postdocs, Yao Shen and Qing Ai, now working in Beijing,
to visit the Forbidden City to check this information and to
photograph the balls from various perspectives. Later, he
also visited the Forbidden City and carefully examined the
three Bronze lion statues with claws stepping on “fullerene-
like” balls in front of Qianqing Gate, Yangxing Dian, and
Ningshou Gate (Gate of Tranquil Longevity).
The Forbidden City was built in the early fifteenth cen-
tury (*1420–1644), but all these lion balls decorated with
“fullerene-like” structures were constructed much later,
during the Emperor Qianlong’s reign in the Qing dynasty
(1740–1800), an era of prosperity in China. Looking at the
ball in front of the Qianqing Gate from various sides, one
finds only four pentagonal faces. Although it is not possible
to look at the ball from its backside, it is unlikely that this
small part contains the eight missing pentagons. Closer to
the bottom of the ball, the tiling pattern becomes quite
irregular (Fig. 4a). Furthermore, one can identify two edge-
sharing triangles and also a distorted hexagon sharing a
Figure 1. Male gilded guardian lions at the Forbidden City (a, b) and the balls under their right front paws (c, d). The balls are
carved with a pattern that resembles a fullerene (a, c) and a fullerene dual (b, d). Beijing, China. Photograph by E. A. Katz.
62 THE MATHEMATICAL INTELLIGENCER
single edge with two other polygons. Thus, we can safely
say that this is not a fullerene structure.
The ball with triangular tiling (Fig. 1b–d) located at
Yangxin Dian certainly does not have the structure of a
fullerene dual, since Yao Shen and Qing Ai found some
vertices of degree seven (marked by blue heptagons in
Fig. 4b).
The name “Flower of Life” is given to a plane tiling
composed of multiple evenly spaced, overlapping circles
(Fig 5a). The center of each circle is on the circumference
of six surrounding circles of the same diameter. This flat
figure forms a flowerlike pattern with so-called Seed of Life
as basic component of such tiling. The latter consists of
seven circles being placed with sixfold symmetry (Fig. 5b).
It is an easy consequence of Euler’s formula
FVþE¼2
however that this pattern cannot be inscribed on a sphere.
The Flower of Life pattern has symbolic meaning in
many cultures and religions worldwide. For example in the
Judeo-Christian tradition, the Seed of Life is supposed to
symbolize the seven days of creation. Examples of these
patterns can be found in the ancient Egyptian, Phoenician,
Assyrian, Indian, Asian, Middle Eastern temples, and in
medieval art. Figure 6depicts ancient artifacts with the
Flower of Life and the Seed of Life found in Israel.
In the Forbidden City, at the Gate of Supreme Harmony
(the Taihe Gate), Yao Shen and Qing Ai found another
sculpture of a lion with a ball tiled in a “Flower of Life-like”
pattern (Fig. 7). This was also constructed in the Qianlong
period. One cannot find any pentagonal connection on the
ball surface; evidently the anonymous sculptor did not
perform a proper geometrical analysis.
Figure 2. (a) The decorated ball shown in Figures 1a–c; (b)
fullerene C
140
with the Goldberg vector (2,1). (For definition of
Goldberg vector, see the Appendix to this article.)
Figure 3. Analysis of the “dual fullerene” shown Figure 1d.
Red pentagons mark the locations of the pentagonal connec-
tions of the triangular faces (or vertices of degree five). The
white dot marks the six-degree vertex where the five-degree
should occur.
Figure 4. (a) Molecular “graph” on the “Fullerene-like” structure. (b) Blue heptagons indicate locations of the heptagonal
connection of the triangular faces on the surface of the ball with triangular tiling. Photograph by Yao Shen and Qing Ai.
Figure 5. Flower of Life (a) and Seed of Life (b).
©2016 Springer Science+Business Media New York, Volume 38, Number 3, 2016 63
Finally, we note that a guardian lion statue with the
“fullerene-dual-like” decorated sphere under its paw has
already been mentioned in scientific literature [3]. This one
is located in front of the East Gate, the main entrance of the
Summer Palace (Yiheyuan garden) in Beijing. One can find
three vertices with five connections and a few Goldberg
vectors (see the Appendix) connecting them ((2,1) and
(3,2)). Another pair of bronze lion statues in the Summer
Palace is located in front of Paiyun Gate. The decoration on
the ball under the male lion statue’s paw also has a pattern
of “Flower of Life” with no pentagons. According to the
historical records, these two pairs of bronze lion statues
were both built in Qianlong’s reign in China (1736–1795).
However, the pair in front of Paiyun Gate was originally
located in the main entrance of the Old Summer Palace
(Yuanmingyuan Garden) [4].
The truncated icosahedron is one of the 13 Archime-
dean bodies referred to by Pappus of Alexandria (AD 290–
350), as well as a member of the fullerene family. The
oldest published image of the truncated icosahedron can
be found in the book “De Divina Proportione” (printed in
1509) [5], written by Luca Pacioli (1445–1514) and illus-
trated by Leonardo Da Vinci. However, Pacioli is often
blamed for plagiarism of unpublished manuscripts of his
mathematical teacher, the great artist Piero della Francesca
(*1415–1492) [6], who nowadays is given credit for the
rediscovery in the West of the truncated icosahedron and
some other Archimedean bodies [7]. His manuscript
Libellus de Quinque Corporibus Regularibus (Short Book
on the Five Regular Solids), written around 1480, probably
has the oldest convincing image of the truncated polyhe-
dron [8].
Anyway, one can conclude that, the “fullerene-like” balls
shown in Figure 1are much younger than Piero’s drawing
and they are not mathematically correct. It seems that their
creator never built the entire sphere. However, a number of
questions for historians of science and art remain open and
still require additional investigation. Two of the most
important are: (1) Were these beautiful “fullerene-like”
artefacts subjected to mathematical analysis at the time of
their creation? Is there any connection of such artistic cre-
ations with the scientific investigation of geometry of
polyhedra at corresponding historical periods? What is the
earliest historical period when such analysis could be
performed in China? (2) Is it possible that some older
“fullerene-like” artefacts can be found in medieval Chinese
sculptures? What is the earliest historical period when such
artefacts could have been created?
We know the answer to the first question: the mathe-
matical analysis of such complicated polygonal structures
could have been performed in China only after the scien-
tific and pedagogical activity of the Jesuit priest Matteo
Ricci (1552 –1610) and his followers at the Imperial Court in
Beijing. They introduced the geometry of polyhedra into
China.
Figure 6. Ancient artifacts with the Flower of Life (a) and the
Seed of Life (b) in Israel: (a) Stone mosaic floor from the
Herod’s Palace at Herodium, 1st century BCE, Israel Museum,
Jerusalem; (b) Decorated altar with basin on top from the
Palace of Agrippa, Banias, 1st century CE. Photographs by E. A.
Katz.
Figure 7. Ball tiled with a structure resembling the “Flower of
Life” under the paw of the Lion located at the Taihe Gate,
Forbidden City, Beijing. Photograph by Yao Shen and Qing Ai.
64 THE MATHEMATICAL INTELLIGENCER
Wending Mei was the first Chinese mathematician who,
in the early Qing period, did a thorough study on the
Platonic solids in his book, “The Jihe bubian” (JHBB,
Supplementary Notes on Geometry, 1692) [10], in which he
intended to make a more complete study to extend the
short descriptions of these five solids given previously by
Jacobus Rho in “Celiang Quanyi”[11]. In addition to the
extensive studies on geometric properties of the five Pla-
tonic solids in JHBB, Mei also described two Archimedean
solids, cuboctahedron and icosidodecahedron (Figure 8a–
b), quite carefully [12,13]. There is a short supplementary
note in JHBB that indicates four other Archimedean solids,
truncated tetrahedron, truncated cube, truncated octahe-
dron, rhombicuboctahedron, and one nonconvex stellated
solid, stella octangula, proposed by his friend, Linzong
Kong. It is worth noting that only a small part of the
icosidodecahedron (Figure 8b) was shown in JHBB [12,13].
The complete drawing of icosidodecahedron, particularly
the superimposed figure with dodecahedron or icosahe-
dron simultaneously, which would look exactly like
pentakis-dodecahedron (C
60
-dual, C
60
*
) (see Appendix), is
not shown in JHBB. Thus, it is hard to tell whether Mei
knew the C
60
*
structure or not.
Appendix: Fullerenes and Their Duals
The C
60
molecule, with carbon atoms at the 60 vertices of a
truncated icosahedron (an Archimedean solid in the shape
of a modern soccer ball) (Fig. 9a), discovered in 1985 [14],
was named buckminsterfullerene, or buckyball, because of
the connection between the C
60
molecular structure and
Buckminster Fuller’s geodesic domes. The discovery of C
60
was followed by the hypothesis of the existence of an
entire family of carbon molecules, the fullerenes, in the
shape of convex polyhedra with vertices of degree three
and only pentagonal and hexagonal faces [15] that was
afterward experimentally confirmed. Mathematicians often
give their own definition [16]: a fullerene (C
V
) is a poly-
hedron with Vvertices of degree three and only pentagonal
and hexagonal faces [17].
For any convex polyhedron with Ffaces, Eedges, and V
vertices, the Euler relation holds [18,19]:
VEþF¼2ð1Þ
It is easy to show that the faces cannot all be hexagons.
For fullerenes, where f
6
and f
5
are the numbers of hexag-
onal and pentagonal faces, respectively, it is almost as easy
to show that f
5
=12 and V=2(10 +f
6
). Thus the number
of pentagonal faces is always 12. The value of f
6
can be any
number but 1 [20]. Accordingly, the smallest fullerene, C
20
,
has the shape of a Platonic dodecahedron, formed only
by pentagons (Fig. 9b). The next fullerenes are C
24
,C
26
,
C
28
,…,C
60
,C
70
,C
2(10+h)
…
According to group theory, the C
60
structure (truncated
icosahedron) belongs to the same point group of symmetry
as the Platonic icosahedron and dodecahedron, I
h
. This
Figure 8. Two Archimedean solids described in Wending
Mei’s “Jihe bubian”: (a) Cuboctahedron, (b) Icosidodecahe-
dron [10]. Figure 9. (a) C
60
, (b) C
20
.
Figure 10. Axes of rotational symmetry in truncated icosahedron: (a) twofold axes, (b) threefold axes, (c) fivefold axes.
©2016 Springer Science+Business Media New York, Volume 38, Number 3, 2016 65
group of icosahedral symmetry includes the highest num-
ber of symmetry elements (120 elements) of all types:
center of symmetry (inversion center), planes of mirror
reflection (bilateral symmetry), and axes of two-, three-,
and fivefold rotational symmetry (Fig. 10).
Some other fullerenes, although not all, belong to the I
h
group as well as to another icosahedral group I(without
the center of symmetry). No such symmetrical molecule
was known before the discovery of C
60
.
The number of vertices (carbon atoms in a molecule C
V
)
Vin all icosahedral fullerenes can be described by very
simple formulas:
V¼20Tð2Þ
and
T¼a2þab þb2
;ð3Þ
where aand bare integer numbers.
For fullerenes with I
h
symmetry: a=b=0orb=0.
For example, the smallest fullerene C
20
, Platonic
dodecahedron, is characterized by (a,b)=(1, 0) and T=1;
C
60
, truncated icosahedron by (a,b)=(1, 1) and T=3; C
80
with I
h
symmetry by (a,b)=(2, 0) and T=4.
Icosahedral fullerene polyhedra are commonly refer-
red to as Goldberg polyhedra [21] and parameters (a,b)
are considered as coordinates of the Goldberg vector that
connects the two closest pentagons. The credit for the
formal definition of this polyhedral family and the anal-
ysis with Eqs. (2–3) goes to American mathematician
Michael Goldberg who published a pioneering paper in
1937 [22]. This paper received a lot of credit in the
context of pioneering research of the structure of viruses
[23].
In geometry, polyhedra are associated into pairs called
duals, in which the vertices of one correspond to the faces
of the other. This correspondence can be simply explained
by the example of the five regular Platonic solids, four of
which form two pairs of such duals (cube-octahedron and
icosahedron-dodecahedron) (Table 1).
As one can see from Table 1, the numbers Fand Vare
interchangeable for cube and octahedron (6,8) – (8,6) and
for icosahedron and dodecahedron (20,12) – (12,20),
respectively. The parameters mand nare also inter-
changeable for these pairs of polyhedra. Cube and
octahedron are dual to each other; icosahedron is dual to
dodecahedron (and vice versa), and tetrahedron is dual to
itself.
An important property of a pair of dual polyhedra is that
both possess the same symmetry and they must have the
same number of edges.
The dual of an isogonal polyhedron, having equivalent
vertices, is one that is isohedral, having equivalent faces.
Because fullerene polyhedra have 12 pentagonal and a
various number of hexagonal faces and all their vertices
have the degree of 3, fullerene duals have only triangular
faces, 12 vertices with degree of 5 and various number of
vertices with degree of 6. In the sequel, we denote a dual of
fullerene C
V
by C
V
*
.
The smallest fullerene dual is the Platonic icosahedron
(dual of the dodecahedron) with 20 triangular faces and 12
five-valent vertices. The dual of truncated icosahedron, C
60
,
is pentakis-dodecahedron C
60
*
with 60 triangular faces, 12
five-degree vertices, and 20 six-degree vertices (Fig. 11).
Figure 11. Pentakis-dodecahedron, dual of truncated icosa-
hedron.
Figure 12. Duals of icosahedral fullerenes: (a) C
80
*
[I
h
;(a,b)=
(2, 0)]; (b) C
140
*
[I; (a,b) = (2, 1)].
Table 1. Characteristics of Platonic Solids
Polyhedron Number of Edges
per Each Face,
m
Vertex Degree (Number of Edges
that Connect in Each Vertex),
n
Number
of Faces,
F
Number
of Edges,
E
Number
of Vertices,
V
Tetrahedron 3 3 4 6 4
Cube 4 3 6 12 8
Octahedron 3 4 8 12 6
Icosahedron 3 5 20 30 12
Dodecahedron 5 3 12 30 20
66 THE MATHEMATICAL INTELLIGENCER
Since every fullerene polyhedron and its dual share the
same symmetry group, duals of icosahedral fullerenes also
have icosahedral symmetry. The only difference is that
Eq. (2) describes the number of faces in duals C
20T
*
, where T
again is equal to a
2
+ab +b
2
. Now, the parameters aand
brepresent the coordinates of the vector joining the two
closest vertices of degree five (Fig. 12).
Many of the famous geodesic domes and most of
icosahedral viruses have the structure of dual icosahedral
fullerenes [24].
ACKNOWLEDGMENTS
The authors acknowledge Drs. Yao Shen and Qing Ai for
their crucial help in this exploration and some photos, as
well as Professors Ioannis Vandoulakis, Chikara Sasaki,
Dun Liu, Ko-Wei Lih, and Dr. Jos Janssen for fruitful dis-
cussions. Bih-Yaw Jin thanks the Ministry of Science and
Technology, Taiwan, for partial financial support and
Yuan-Jia Fan for preparing the figures of fullerenes.
Eugene A. Katz
Department of Solar Energy and Environmental Physics
J. Blaustein Institutes for Desert Research
Ben-Gurion University of the Negev
Sede Boqer Campus
Midreshet Ben-Gurion 84990
Israel
and
Ilse-Katz Institute for Nanoscale Science and Technology
Ben-Gurion University of the Negev
Beer Sheva 84105
Israel
e-mail: keugene@bgu.ac.il
Bih-Yaw Jin
Department of Chemistry and Center for
Quantum Science and Engineering
National Taiwan University
Taipei 10617
Taiwan
e-mail: byjin@ntu.edu.tw
REFERENCES AND COMMENTS
1. I. Hargittai, Fullerene geometry under the lion’s paw, Mathematical
Intelligencer, Vol. 17, no. 3, pp. 34–36, 1995.
2. According to Jun Cheng’s “Traditional Lion Art in China (中國傳統
獅子工藝),” published by Beijing Arts and Crafts Publishing House
(北京工藝美術出版社, 2013, Beijing), there are six pairs of bronze
lion statues in the forbidden city, which are located in front of Taihe
Gate (太和門), Qianqing Gate (乾清門), Yangxin Dian (養心殿),
Changchun Palace (長春宮), Ningshou Gate (寧壽門), and
Yangxing Dian (養性殿), respectively. Three male lions in front of
Qianqing Gate, Ningshou Gate, and Yangxing Dian step on balls
decorated with fullerene-like patterns; whereas the two balls under
male lions in front of Yangxin Dian and Changchun Palace have
triangular tiling patterns.
3. T. Tarnai. Geodesic Dome under the Paw of an Oriental Lion.
http://www.mi.sanu.ac.rs/vismath/visbook/tarnai/.
4. The Qing Dynasty inherited the Forbidden City directly from the
Ming Dynasty. Without the need to build a new palace, the focus
thus moved to the construction of the Imperial Gardens. In the
peak of the Qing dynasty (Qianlong’s reign), more than 90 imperial
gardens were built in the great Beijing area. The most famous
gardens among them are Yuanmingyuan (Old Summer Palace,
圓明園) and Qingyiyuan (清漪園). Both Yuanmingyuan and Qin-
gyiyuan were destroyed by Anglo-French armies during the
Second Opium War in 1860. Between 1884 and 1895, during the
reign of the Guanxu Emperor, Qingyiyuan was reconstructed and
given its present-day name, “Yiheyuan” (New Summer Palace).
There were at least two pairs of bronze lion statues in the Yuan-
mingyuan and one pair of bronze lion statues in the Qingyiyuan. The
British army took one pair of bronze lion statues located in Yuan-
minyuan during the Second Opium War and returned them back to
China more than 100 years later, in 1984. This pair is now located in
the Diaoyutai State Guest House, Beijing. The other pair from
Yuanmingyuan is now located in front of Paiyun Gate of Yiheyuan.
5. L. Pacioli, De Divine Proportione, 1509 (Ambrosiana facsimile
reproduction, 1956; Silvana facsimile reproduction, 1982).
6. K. Williams, Plagiary in the Renaissance, Mathematical Intelli-
gencer, Vol. 24, pp. 45–57, 2002.
7. M. D. Davis, Pierro della Francesca’s mathematical treatises,
Ravenna, Italy: Longo Editore (1977).
8. Some artefacts with a fullerene-like form, and even with that of
C
60
, dated from Antiquity (Hellenistic/Roman times). For instance,
Jos Janssen found in Kunsthistorisches Museum in Vienna a Late-
Roman sprinkler from the Eastern Mediterranean area, the
spherical part of which is decorated by pentagons and hexagons
presumably in the form of C
60
(see Ref. 9).
9. J. W. A. M. Janssen, A soccer ball in the Kunsthistorisches
Museum? Forum Archaeologiae 64/IX/2012 (http://farch.net).
10. Wending Mei, Jihe Bubian (幾何補編, Complements of Geometry),
1692, in Meishi Congshu Jiyao (梅氏叢書輯要), Zuangao Mei, ed.,
1871 (Taipei Yiwen Publisher reproduction, 1971).
11. Jacobus Rho, Celiang Quanyi (測量全義,The Complete Meaning of
Measurement), 1631, in Chongzhen Lishu (崇禎曆書, Calendar
Compendium of the Chongzhen Reign), Guangqi Xu, ed., 1644
(Shanghai Classic Publishing House reproduction, 2009).
12. D. Liu, Platonic solids and Archimedean solids in the late 17th
century China. In: Symmetry: Art and Science, Special issue for
the 9th Congress-Festival of the International Society for the
Interdisciplinary Study of Symmetry, 2013, pp. 216–220.
13. D. Liu, Archimedes in China: Archimedes and His Works in Chi-
nese Literature of the Ming and Qing Dynasties. In: Eberhard
Knobloch, Hikosaburo Komatsu, Dun Liu, Seki, Founder of
Modern Mathematics in Japan, Springer, 2013.
14. H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, R. E.
Smalley, C
60
: Buckminsterfullerene, Nature,Vol.318,pp.162–
163, 1985.
15. A. D. J. Haymet, Footballene—A theoretical prediction for the
stable, truncated icosahedral molecule C
60
,J. Am. Chem. Soc.,
Vol. 108, pp. 319–321, 1986.
16. M. Deza, P. W. Fowler, A. Rassat, K. M. Rogers, Fullerenes as
tilings of surfaces, J. Chem. Inf. Comput. Sci. 2000, 40, 550–558.
©2016 Springer Science+Business Media New York, Volume 38, Number 3, 2016 67
17. There is an even more general mathematical definition: a fullerene
is a finite trivalent map with only pentagonal and hexagonal faces
embedded in any surface (not only in the spherelike one) [3].
18. L. Euler, Elementa Doctrinae Solidorum, Novi Commentarii Aca-
demie Petropolitanae, Vol. 4, pp. 109–140, 1758.
19. L. Euler, Demonstratio Nonnullarum Insignium Proprietatum,
Quibus Solida Hedris Planis Inclusa Sunt Praedita, Novi Com-
mentarii Academie Petropolitanae, Vol. 4, pp. 140–160, 1758.
20. B. Gru
¨nbaum, T. S. Motzkin, The number of hexagons and the
simplicity of geodesics on certain polyhedral, Can. J. Math., Vol.
15, pp. 744–751, 1963.
21. G. Hart, Goldberg Polyhedra. in Shaping Space: Exploring Poly-
hedra in Nature, Art, and the Geometrical Imagination,M.
Senechal, ed., Springer, 2013, pp. 125–138.
22. M. Goldberg, A class of multi-symmetric polyhedra, To
ˆhoku Math.
J., Vol. 43, pp. 104–108, 1937.
23. D. L. D. Caspar, A. Klug, Physical principles in the construction of
regular viruses. In: Cold Spring Harbor Symp. Quant. Biol. Vol. 27,
pp. 1–24, 1962.
24. H. S. M. Coxeter, Virus Macromolecules and Geodesic Domes,
in A Spectrum of Mathematics, J. C. Butcher, ed., Auckland,
1971.
68 THE MATHEMATICAL INTELLIGENCER
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