A Fullerene in a Lion’s Paw
Topology, Nanotube Belts, and a C120 Hidden in a Bronze Ornament
Abstract. At the East Gate of Beijing’s Fragrant Hills Park, a bronze lion rests one paw on an ornamental ball whose surface forms a trivalent network of pentagons and hexagons. The pattern can be identified combinatorially with a C120 fullerene: 120 vertices, 180 edges, and 62 faces, of which twelve are pentagons and fifty are hexagons. A closer examination reveals something subtler. Two closely related placements of the pentagons produce two legitimate C120 fullerene graphs, but with markedly different global shapes. The distinction originates in a one-step shift in an alternating third shell around a coronene-like cap. One choice produces an elongated cage with an armchair-type belt; the other produces the shorter, fuller structure actually found on the bronze ornament, whose middle region is a two-layer segment of a zigzag (12,0) carbon nanotube. The example illustrates, in unusually tangible form, the distinction between molecular formula, topology, and geometry—and how a small local change can determine the shape of an entire cage.
A Ball Beneath a Lion’s Paw
At the East Gate of Beijing’s Fragrant Hills Park, a bronze lion rests one paw on an ornamental ball. Such xiùqiú—literally “embroidered balls”—are familiar motifs in Chinese decorative art. Their surfaces may carry flowers, ribbons, interlacing bands, or polygonal patterns.
This one is different.
Its surface can be read as a trivalent network made entirely of pentagons and hexagons. Once noticed, the resemblance is difficult to unsee: the ball looks remarkably like the skeleton of a fullerene.
I first encountered it in 2018, after years of examining similar ornaments on Chinese lions in temples, gardens, monuments, and architectural decoration. This search grew out of earlier work with Eugene A. Katz on fullerene-like polyhedral patterns associated with Chinese guardian lions [1]. Many examples appeared tantalizingly close to fullerene patterns but failed under closer inspection. Some were incomplete; others contained incompatible polygons or abandoned the pattern on the hidden side.
The search had begun to resemble the celebrated line of the Song-dynasty poet Xin Qiji: one searches “a thousand times among the crowd,” only to find the object unexpectedly close at hand.
At Fragrant Hills, I finally seemed to have found what I had been looking for.
That was the end of one story—the story of finding the object. It was the beginning of another.
What, mathematically, was the object I had found?
Figure 1. The bronze lion at the East Gate of Fragrant Hills Park, Beijing, with the ornamental ball beneath its paw.
Counting the Cage
Ignore chemistry for the moment and regard the pattern simply as a trivalent graph embedded on the sphere. If it has V = 120 vertices, trivalence gives
E = 3V/2 = 180.
Euler’s formula, V − E + F = 2, then gives
F = 62.
For a fullerene, all faces are pentagons or hexagons. The usual Euler count then forces exactly twelve pentagons, leaving fifty hexagons:
V = 120, E = 180, F = 62, 12P + 50H.
Thus the notation C120 refers to 120 vertices—carbon atoms in the molecular interpretation—while the visible surface contains only 62 polygonal faces.
So far everything is determined by counting. But the counting does not determine the fullerene. There are many ways to distribute twelve pentagons among fifty hexagons, and different distributions can produce different graphs.
Which C120 is it?
Twelve Pentagons, Twelve Sources of Curvature
The fixed number of pentagons is one of the most attractive features of fullerene geometry. An infinite hexagonal lattice is locally flat. Introducing a pentagon into that lattice creates positive discrete curvature. Twelve such curvature defects are required to close a trivalent pentagon–hexagon network into a sphere.
Fullerenes can therefore grow larger by inserting more hexagons while retaining the same twelve pentagons. What changes is their location. And because the pentagons carry curvature, where they are placed matters greatly.
Begin with Coronene
Rather than examining the complete C120 at once, begin at one end.
The first two shells have the topology of coronene: a central hexagon surrounded by six additional hexagons. If the growth is continued in the most straightforward benzenoid fashion by adding a third shell of twelve hexagons, one obtains circumcoronene.
benzene → coronene → circumcoronene
The Fragrant Hills structure follows this sequence through coronene and then makes a different choice.
Its third shell is not twelve hexagons. Instead, it consists of twelve polygons alternating between pentagons and hexagons.
5–6–5–6–… or 6–5–6–5–…
Combinatorially, the difference is only a shift by one place. Geometrically, it is anything but trivial.
Figure 2. From coronene to two C120 topologies. The first two shells form a coronene-like patch. A third shell of twelve hexagons would give circumcoronene. Replacing that shell by alternating pentagons and hexagons gives two phase choices: 5–6–5–6–… and 6–5–6–5–…. The corresponding partial Schlegel diagrams show the shift of pentagons from faces 8,10,12,14,16,18 to 9,11,13,15,17,19, leading respectively to armchair- and zigzag-type belts.
Two Alternating Shells
In my first reading of the ornament, I chose
5–6–5–6–…
In the corresponding spiral ordering, the first pentagon in the third shell is face 8, giving the pentagon positions
[8,10,12,14,16,18,45,47,49,51,53,55].
This reconstruction produced no contradiction. It gave a perfectly legitimate fullerene with 120 vertices, 180 edges, 62 faces, and exactly 12P + 50H.
Yet the resulting object looked wrong. It was too slender. More revealingly, its central band had an armchair arrangement.
The actual bronze ornament showed a different pattern. Its middle belt was zigzag.
That small visual discrepancy forced a return to the original photographs. The correct third shell begins not with a pentagon, but with a hexagon:
6–5–6–5–…
The first pentagon is therefore face 9 rather than face 8, and the spiral code becomes
[9,11,13,15,17,19,44,46,48,50,52,54].
The shift is only one face. The resulting global shape is quite different.
The Surprise Was That Both Were C120
I had expected the incorrect reading to fail. Perhaps the cage would not close; perhaps the Euler count would be wrong; perhaps the graph would cease to be trivalent.
None of these things happened.
Both structures are legitimate C120 fullerenes. Both have 120 vertices, 180 edges, 62 faces, twelve pentagons, and fifty hexagons.
What differs is connectivity.
The count tells us that it is C120; the connectivity tells us which C120.
From an Armchair Belt to a Zigzag Belt
The difference becomes especially transparent when the middle portion of each cage is viewed through the geometry of carbon nanotubes.
A single-wall carbon nanotube may be described by its chiral indices (n,m). Tubes of type (n,0) are zigzag; those of type (n,n) are armchair.
The middle of the corrected Fragrant Hills cage is not merely vaguely “nanotube-like.” Its hexagonal band can be identified as a very short, two-layer segment of a zigzag (12,0) carbon nanotube.
cap + two-layer (12,0) zigzag CNT belt + cap
The initially misread structure has the complementary armchair pattern. Its middle belt is consistent with a very short (6,6) armchair nanotube segment.
(6,6) armchair-like belt vs. (12,0) zigzag belt
Why One Cage Is Fatter
The nanotube interpretation also helps explain a feature that is obvious by eye.
The corrected zigzag cage is shorter and fatter; the armchair alternative is narrower and more elongated.
For a nanotube with graphene lattice constant a, the circumference associated with chiral indices (n,m) is proportional to
|Ch| = a√(n² + nm + m²).
For the two candidate belts,
|Ch(12,0)| = 12a,
|Ch(6,6)| = 6√3 a.
Their ratio is
2/√3 ≈ 1.155.
Thus, in ideal nanotube geometry, the (12,0) belt has a circumference about 15% larger than the (6,6) belt. This is precisely the direction suggested by the reconstructed cages: the zigzag version is broader, while the armchair version is slimmer.
The complete fullerene shapes also depend on their caps and on geometric relaxation, so this simple ratio is not a prediction of the full cage aspect ratio. But it gives a satisfying geometric explanation for the contrast.
A Local Shift with a Global Consequence
The two structures share the same coronene-like core. At the third shell,
5–6–5–6–…
places the pentagons in one set of alternating positions and leads to an armchair-type continuation. Shifting the pattern by one face,
6–5–6–5–…
changes which sites carry curvature and produces a zigzag continuation.
pentagon positions → cap boundary geometry → nanotube chirality → global aspect ratio
A local combinatorial choice becomes a global geometric one.
Figure 3. Same formula, different shape. Two legitimate C120 fullerene cages. The pentagon-first construction produces the more elongated structure with an armchair-type belt, consistent with a very short (6,6) nanotube segment. The hexagon-first construction produces the shorter, broader cage with the two-layer zigzag (12,0) belt observed in the Fragrant Hills ornament.
Formula, Topology, Geometry
The example illustrates a useful hierarchy:
formula → face census → connectivity → geometry
The formula C120 determines the number of vertices. The census 12P + 50H determines the numbers of the two face types. The spiral code contains far more information about connectivity. The three-dimensional realization finally reveals global shape.
Twelve Integers
There is a pleasing compression in the spiral representation. The corrected cage is encoded by the twelve integers
[9,11,13,15,17,19,44,46,48,50,52,54].
They identify the pentagons within a sequence of 62 faces.
bronze ornament → polygonal pattern → spiral code → graph → three-dimensional cage
An Echo of Goldberg
There is also an older mathematical resonance.
Michael Goldberg’s investigations of polyhedra in the 1930s began in part from an isoperimetric question: among polyhedra of prescribed combinatorial complexity, which ones best approximate a sphere?
The two C120 structures here are not candidates in Goldberg’s precise optimization problem. Their comparison nevertheless offers a miniature visual version of the same theme.
They possess exactly the same numbers of vertices, edges, faces, pentagons, and hexagons. Yet their shapes are clearly different.
Counting the polygons is not enough. Their arrangement matters.
The Artisan’s Choice
Why did the Fragrant Hills ornament use the zigzag version?
We cannot know what reasoning guided its maker. Perhaps there was no explicit geometric reasoning at all. The choice may have arisen from symmetry, fabrication, decorative convention, or simply an experienced eye.
Still, the result is intriguing. The armchair version is mathematically legitimate, but elongated. The zigzag (12,0) version is broader and more nearly ball-like.
For an ornamental sphere under the paw of a bronze lion, the latter seems an aesthetically natural choice.
One might say that the artisan selected the better aspect ratio without ever formulating an optimization problem.
Geometry can be felt before it is formalized.
Does the Lion Hold a Molecule?
Of course not. The ball is bronze, not carbon.
Calling its pattern a C120 fullerene means that its polygonal network is combinatorially equivalent to a fullerene graph with 120 vertices. Likewise, describing its middle belt as a (12,0) nanotube segment is a statement about the topology and geometry of the hexagonal network—not a claim about its material composition.
A real C120 carbon molecule would face additional constraints: bond lengths, bond angles, electronic structure, strain, and energetic stability.
The historical claim should therefore remain modest. The artisan did not “discover carbon nanotubes,” nor did the ornament anticipate fullerene chemistry in any chemical sense.
Something more interesting, and less sensational, has happened. The same abstract structural language appears independently in decorative art, polyhedral geometry, graph theory, fullerene chemistry, and carbon nanotubes.
Seeing the Ball Twice
When I first encountered the bronze lion in 2018, the pleasure was one of recognition. After a long search, I had found what appeared to be a complete fullerene pattern.
Returning to it years later produced a different experience. The ball could now be counted, encoded, and compared with neighboring possibilities.
An incorrect interpretation did not collapse into nonsense. It produced another perfectly legitimate C120.
Only a small geometric detail exposed the mistake: the middle belt was armchair when the bronze ornament was zigzag.
That observation led back to a single alternating shell, and eventually to an unexpectedly precise description of the central band:
a two-layer (12,0) zigzag nanotube belt.
At first I had found a shape. Later I began to see the structure inside the shape.
number ≠ arrangement
formula ≠ topology
topology ≠ geometry
Two cages may both be C120. A single shift in an alternating shell may separate them.
And sometimes a small zigzag on a bronze ball is enough to tell us which one we are looking at.
References
1. E. A. Katz and B.-Y. Jin, “Fullerenes, Polyhedra, and Chinese Guardian Lions,” The Mathematical Intelligencer 38(3) (2016), 61–68. DOI: 10.1007/s00283-016-9663-0.
2. M. Goldberg, “A Class of Multi-Symmetric Polyhedra,” Tôhoku Mathematical Journal 43 (1937), 104–108.
3. N. Hamada, S. Sawada, and A. Oshiyama, “New One-Dimensional Conductors: Graphitic Microtubules,” Physical Review Letters 68 (1992), 1579–1581.
4. S. Reich, L. Li, and J. Robertson, “Structure and Formation Energy of Carbon Nanotube Caps,” Physical Review B 72 (2005), 165423.
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