Searching a Thousand Times: The C120 Fullerene Beneath a Lion’s Paw
I have a rather unusual habit.
Whenever I see a male Chinese guardian lion in Beijing, I walk over and inspect the embroidered ball beneath its paw.
I am not looking first at the craftsmanship, nor at whether the decoration is beautiful. I am looking for pentagons.
Because if the ball is truly a fullerene, twelve pentagons must be there. Not eleven. Not thirteen. Exactly twelve.
1. April 15, 2018: The East Gate of Xiangshan Park
On April 15, 2018, I visited the East Gate of Xiangshan Park in Beijing.
As at many Chinese palaces, temples, and gardens, a pair of guardian lions stood at the entrance. The female lion was accompanied by a cub, while the male rested one paw on an ornamental ball.
By then I had already inspected many such lions in Beijing: in the Forbidden City, the Lama Temple, Fayuan Temple, the Summer Palace, and the Old Summer Palace. I had gradually developed the habit of checking the ball under every male lion I encountered.
At first, the Xiangshan lion seemed no different.
I walked closer and began looking.
One pentagon.
Then another.
From a different angle, another one.
I moved around the lion, taking photographs and trying mentally to reconstruct the parts partly hidden by the paw.
The longer I looked, the more surprised I became.
I stood beside the lion for a long time.
After examining so many guardian-lion balls in Beijing, I had finally found a genuine fullerene.
But the story had actually begun four years earlier.
2. Eugene Katz Showed Me a Chinese Lion
In 2014, I attended the Second Conference on Science, Technology and Art Relations in Tel Aviv, where I first met Eugene Katz.
Eugene had long been fascinated by fullerene-like structures appearing across very different scales, from the nano world to architecture. In his lecture he showed a photograph of a Chinese guardian lion in the Forbidden City in Beijing.
Under the paw of the male lion was an ornamental ball. Its surface was covered with pentagons, hexagons, and related polygonal patterns. It certainly looked fullerene-like.
My first reaction, however, was not:
“Ancient Chinese craftsmen had already made fullerenes!”
Instead I asked:
“Wait a moment. Is this really a fullerene?”
At that time I had never been to Beijing. I could only study photographs found online and whatever other images I could obtain. The more carefully I looked, the less simple the question became.
That question became the starting point of my collaboration with Eugene. Together we investigated the unusual polyhedral balls beneath Chinese guardian lions and eventually wrote an article connecting fullerenes, icosahedral symmetry, geometry, and Chinese guardian lions.
Looking back, it was the beginning of a mathematical tourist detective story.
3. Twelve Pentagons, No More and No Less
Only a little topology is needed to understand the central issue.
A conventional fullerene is a closed trivalent network: each vertex has degree three, and every face is either a pentagon or a hexagon.
From Euler’s formula,
together with the trivalence condition, one obtains one of the most elegant facts in fullerene topology:
Every closed pentagon–hexagon fullerene contains exactly twelve pentagons.
The number of hexagons may vary. The number of pentagons may not.
So when I inspect a guardian-lion ball, the important question is not whether it resembles a soccer ball.
4. Carbon Cannot Escape Euler; a Craftsman Can
After examining more and more guardian-lion balls in Beijing, I began to notice a recurring pattern.
The craftsmen seemed to favor hexagons. On the visible parts of the sphere, they often extended regular hexagonal patterns as far as possible, introducing pentagons or other polygons only when needed.
But a ball beneath a lion’s paw has two very convenient places where the pattern does not have to be completed.
One is hidden beneath the paw.
The other is where the ball meets the pedestal or ground.
These regions are difficult to see, and in many examples the polygonal network simply does not form a fully closed surface there.
From the viewpoint of craftsmanship, this is perfectly reasonable.
From the viewpoint of topology, however, it changes everything.
The theorem requiring twelve pentagons applies to a closed spherical surface. Once a boundary is introduced, the topological bookkeeping changes.
These ornamental balls therefore should not necessarily be regarded as “incorrect fullerenes.”
A better way to say it is:
The craftsmen were never obliged to make a fullerene in the first place.
They only needed to complete what the viewer could see.
A real carbon cage has no such freedom. If carbon atoms form a closed cage, every part of the network must close.
5. 2016: Go and Look at the Object Itself
I first visited Beijing in 2016.
For the first time, I could examine the famous bronze guardian lions of the Forbidden City directly, rather than through photographs found on the Internet.
I carefully photographed the gilded bronze lions near the Palace of Heavenly Purity, including angles that ordinary visitors might not find especially important.
That visit also brought up another question: dating.
Many online sources described the lions simply as Ming-dynasty objects, apparently because the Forbidden City itself was built during the Ming dynasty.
But the age of a palace does not determine the age of every object standing in front of it.
The answer was actually written on the lion itself.
There is an inscription on the body of the bronze lion recording information about its casting. It is not easy to read casually at the site, but careful photographs can be enlarged and examined. The evidence points to the Qianlong period of the Qing dynasty.
That experience left a strong impression on me.
Do not rely only on the label.
Go and look at the object itself.
The same principle applies to fullerene-like decorations.
Do not call a ball a fullerene simply because it resembles a soccer ball.
Count the pentagons.
6. 2018: Beijing Becomes My Fullerene Field Trip
I returned to Beijing in the spring of 2018, this time at the invitation of Professor Yingxia Wang of Peking University.
Over approximately six weeks, I gave more than ten classes related to bead-string models. It was the first time I organized bead-string model building into a sustained course. After returning to Taiwan, I gradually developed these ideas into my general-education course, Molecular Aesthetics.
Those six weeks also gave me something that a short conference visit rarely provides:
Outside the classroom, Beijing became my fullerene field trip.
The Lama Temple — inspect the lions.
Fayuan Temple — inspect the lions.
The Summer Palace — inspect the lions.
The Old Summer Palace — inspect the lions again.
Whenever I saw a male guardian lion, I looked at the ball beneath its paw.
And again and again, the result was disappointing.
The balls looked fullerene-like, but they were not complete closed fullerenes.
Eventually I stopped expecting to find one.
Then, on April 15, I decided to visit Xiangshan Park.
7. Xiangshan: This Time, It Really Closed
At the East Gate stood another pair of bronze guardian lions.
As usual, I walked over to the male lion.
My first thought was:
Probably another incomplete one.
But the longer I looked, the more surprised I became.
This time, the pentagons did not disappear beneath the paw.
I began to locate them one by one.
Change angle.
Take a photograph.
Count again.
Eventually there was no doubt.
This was not merely an ornament that imitated a fullerene over its visible surface.
It was a truly closed trivalent pentagon–hexagon network.
It really was a fullerene.
I remained there for a long time.
Eventually I entered the park and spent several hours walking in the hills. But on my way out, I stopped at the same lion again.
I looked again.
And took a few more photographs.
8. Not C60, but C120
The structure was not the familiar C60.
Nor was it one of the standard highly symmetric icosahedral Goldberg fullerenes.
My reconstruction of the network gave:
with a
rotational symmetry axis.
Visually, the craftsman made a sphere.
Topologically, however, the network is easier to understand as a capped cylinder.
The two ends can be viewed as coronene-like caps, connected through alternating pentagon–hexagon regions,
to a zigzag cyclic belt in the middle.
The craftsman then shaped or projected the entire network onto an approximately spherical ornamental ball.
The eye sees a sphere;
topology reveals a capped cylinder.
We do not know who designed this ball.
We certainly should not say that the craftsman “knew fullerene chemistry,” nor that he “knew Euler’s theorem.”
A more precise statement is:
An unknown Chinese craftsman produced a closed pentagon–hexagon network topologically equivalent to a C120 fullerene, without the conceptual language of modern fullerene chemistry.
Was this an accident?
Was it inherited from an ornamental tradition?
Did craftsmen develop some empirical rule for closing polygonal patterns on a sphere?
I do not know.
That uncertainty is part of what makes the object so fascinating.
9. Thirty-One Days Later: “Searching for Him a Thousand Times”
Soon after finding the Xiangshan fullerene, I posted several photographs on Facebook and explained why I believed it to be a genuine fullerene.
Eugene Katz saw the post and was immediately interested. So did my friend Dirk Huylebrouck. Eugene encouraged me to write a short piece for The Mathematical Intelligencer.
I did not.
At the time I thought:
It is only a small observation.
There were always other things to do.
Then, only thirty-one days later, the ball appeared in a formal academic lecture.
On May 16, 2018, the historian of science Liu Dun gave a lecture at Tsinghua University’s Institute for the History of Science. His title was:
From Timaeus to Fullerenes — A Glimpse of Geometry in Ancient Art and Artifacts
When he came to the Xiangshan lion, one of his slides carried six Chinese characters:
The line comes from a famous poem by Xin Qiji and can be translated approximately as:
“Searching for him a thousand times in the crowd.”
The slide showed an old photograph of Liu himself as a boy beside that very lion.
Young Liu had already been there. He had even climbed onto the lion and had his photograph taken.
Of course, as a child, he had no idea what a fullerene was.
Decades later he became a distinguished historian of science and had looked at many Chinese artifacts and many Beijing guardian lions. Yet he had never noticed that this particular ball was topologically different from the others.
Then came the next slide.
It showed my photograph beside the Xiangshan ball.
Liu explained to the audience:
He had learned from me that the ball beneath the Xiangshan guardian lion was a C120 fullerene.
Fortunately, I was sitting in the audience that day.
I photographed the two slides, and also photographed Liu at the podium with the “Searching for Him a Thousand Times” slide behind him.
10. Seeing Is Not Recognizing
Liu’s childhood photograph made me think about a question that is even more interesting than the fullerene itself:
When do we truly “see” something?
A child sees a lion large enough to climb on.
A tourist sees a traditional Chinese guardian lion.
An art historian may see period, style, craftsmanship, symbolism, and casting technique.
Someone trained in fullerene topology may immediately ask a completely different set of questions:
Where are the twelve pentagons?
Is every vertex trivalent?
Is the surface closed?
What symmetry does it have?
The object has not changed.
What changes is the conceptual framework in the observer’s mind.
This has become increasingly important to me through years of teaching Molecular Aesthetics and working with bead-string models.
We do not teach Euler’s theorem, polyhedra, or Goldberg constructions merely so that students can memorize a few more formulas.
Learning a structure is also learning a new way to see the world.
Something that was once merely ornament becomes topology.
Something that was once merely an embroidered ball can suddenly be read as a molecular graph.
11. 2024: Visiting an Old Friend
Six years later, in the spring of 2024, I returned to Peking University and once again taught a course related to bead-string models.
By then I had passed sixty.
This brought an unexpected practical advantage in Beijing: admission to Xiangshan Park, the Summer Palace, and the Old Summer Palace was free for me.
During that stay I went hiking at Xiangshan once or twice almost every week.
And whenever I passed the East Gate, I would look again at the bronze lion and the C120 beneath its paw.
In 2018, I stood there for a long time because I could hardly believe that all twelve pentagons were truly present.
In 2024, I no longer needed to count them.
The lion was still there.
The C120 was still there.
What had once been an unexpected discovery had become an old friend.
12. A Letter Eight Years Later
The story might have ended there.
Then, recently, Eugene wrote to me again.
He is preparing a book about fullerene structures and architecture. He remembered the Xiangshan guardian lion I had posted in 2018 and asked:
Did you ever publish that ball?
I had to answer:
No.
I found some of my old photographs and sent them to him, telling him that he was very welcome to include the Xiangshan lion in his book.
But his question also reopened a story that had been quietly sitting in my memory for years.
2014 — Tel Aviv: Eugene shows me a Forbidden City guardian lion.
2016 — My first field investigation in Beijing.
2018.4.15 — Xiangshan: discovery of the closed C120.
2018.5.16 — Liu Dun at Tsinghua: “Searching for Him a Thousand Times.”
2024 — Return to Peking University and repeated visits to Xiangshan.
2026 — Eugene asks: “Did you ever publish it?”
More than a decade had passed.
In 2018, I thought finding a C120 beneath a lion’s paw was probably too small a discovery to deserve its own article.
Looking back now, I think what is worth recording is not merely the answer.
Epilogue: It Had Always Been There
The line Liu Dun used on his slide in 2018 now seems almost perfect:
眾裡尋他千百度
Searching for him a thousand times in the crowd.
As a child, Liu had already climbed onto that lion.
I, meanwhile, had looked at lion after lion in Beijing, counting polygon after polygon on incomplete ornamental balls.
And in the end, the real fullerene was not hidden in a museum storeroom, nor buried in some forgotten manuscript.
It stood at the East Gate of Xiangshan Park.
Beneath the paw of a bronze lion.
Countless visitors must have walked past it.
Some things are difficult to discover not because they are well hidden.
They have been there all along.
We simply have not yet learned how to see them.
眾裡尋他千百度
The phrase on Liu Dun’s slide comes from Xin Qiji (辛棄疾, 1140–1207), one of the great poets of the Southern Song dynasty. It appears near the end of his celebrated lyric Qingyu'an · Lantern Festival (《青玉案・元夕》).
眾裡尋他千百度，
驀然回首，
那人卻在，燈火闌珊處。
A thousand times I searched for him in the crowd.
Suddenly I turned my head—
and there he was,
where the lantern lights were fading.
The line is especially fitting for the Xiangshan story. Liu Dun had stood beside — and even climbed upon — this very guardian lion as a child. Decades later, after both of us had seen many guardian lions in Beijing, the mathematically remarkable object turned out to have been there all along: beneath the lion’s paw.
The fullerene was not hidden.
We simply had not yet learned how to recognize it.
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