## Hoàng Xuân Sính

20 June, 2022

During the Vietnam war, Grothendieck taught math to the Hanoi University mathematics department staff, out in the countryside. Hoàng Xuân Sính took notes and later did a PhD with him — by correspondence! She mailed him her hand-written thesis. She is the woman in this picture:

As you might guess, there’s a very interesting story behind this. I’ve looked into it, but what I found raises even more questions. Hoàng Xuân Sính’s life really deserves a good biography.

Hoàng Xuân Sính was born in 1933 in a village called Cót, one of seven children of a fabric merchant. Her mother died when she was eight years old, and she was raised by a stepmother.

She spent a lot of time sewing and designing clothes. However, this image of a magazine from 1981 shows a picture of her holding a book—and on the website where I found this, a caption in Vietnamese says “Cót village girl is passionate about math”.

She would have been 18 at the time. In this year she completed a bachelor’s degree in Hanoi, studying English and French, and then traveled to Paris for a second baccalaureate in mathematics. She stayed in France to study for the agrégation (the competitive examination for civil service) at the University of Toulouse, which she completed in 1959, before returning to Vietnam and teaching mathematics at the Hanoi National University of Education.

Grothendieck visited North Vietnam in late 1967, during the Vietnam War, and spent a month teaching mathematics to the Hanoi University mathematics department staff, including Hoàng Xuân Sính, who took the notes for the lectures. Because of the war, Grothendieck’s lectures were held away from Hanoi, first in the nearby countryside and later in Đại Từ.

After Grothendieck returned to France, he continued to teach Hoàng Xuân Sính in an exchange of letters. According to the web page of the university she founded, Thang Long University, Hoàng Xuân Sính remembers two main impressions from her contacts with Alexander Grothendieck:

1) A good teacher is a teacher who turns something difficult into something easy.

2) We should always avoid anything that is fictitious, live in accordance to our own feelings and value simple people.

She finished her thesis in 1972. Around Christmas that year, the United States dropped over 20,000 tons of bombs on North Vietnam, mainly Hanoi. So, it’s not surprising that she only defended her thesis three yearslater, when the North had almost won. But she mentions another reason. She later wrote:

I was a doctorate student during wartime. Back then, I was teaching at Hanoi Pedagogical University, there was not a mode to take leave to study for the doctorate. I taught during the day and worked on my thesis during the night under the kerosene lamp light. I wrote in French under my distant teacher’s guidance. When I got the approval from France to come over to defend, there were disagreeable talks about not letting me because they was afraid I wasn’t coming back. The most supportive person during the time was Lady Ha Thi Que—President of the Vietnamese Women Coalescent organization. Madame Que was a guerilla, without the conditions to get much education, but gave very convincing reasons to support me. She said, firstly, I was 40 years old, it is very difficult to get a job abroad at 40 years old, and without a job, how can I live? Second, my child is at home, no woman would ever leave her child… so comrades, let’s not be worried, let her go. I finished my thesis in 1972, and 3 years later with the help and struggle of the women’s organization, I was able to travel over to defend in 1975….

She went to France to defend her thesis at Paris Diderot University (also called Paris 7). Her thesis committee included not only Alexander Grothendieck but also Henri Cartan, Laurent Schwartz, Michel Zisman, and Jacques Deny.

Her thesis defense lasted two and a half hours. And soon thereafter she defended a second thesis, entitled “The embedding of a one-dimensional complex in a two-dimensional differential manifold”. I don’t know who, if anyone, directed this second thesis.

She later became the first woman mathematics professor in Vietnam — and the second came 35 years later.

In 1988, she started the first private university in Vietnam, Thang Long University in Hanoi. For a while she was not only the head, but also the janitor, bringing water to the school and sweeping floors. Later she said “When I look back at it, I thought it was the most romantic idea I’ve had.”

In 2003 she was awarded France’s Ordre des Palmes Académiques. She is still alive! I hope someone has interviewed her, or does it now. Her stories must be very interesting.

Her thesis classified Gr-categories, which are now called ‘2-groups’ for short. A 2-group is the categorified version of a group: it’s a monoidal category where every object and morphism is invertible. (An object $x$ is invertible if there’s an object $y$ with $x \otimes y \cong y \otimes x \cong I$, where $I$ is the unit for the tensor product.)

From a 2-group you can get two groups:

• the group $G$ of isomorphism classes of objects, and

• the group $A$ of automorphisms of the unit object $I$.

The group $A$ is abelian, and $G$ acts on $A$. But there’s one more thing! The associator can be used to get a map

$a \colon G^3 \to A$

The pentagon identity for the associator implies that this map obeys an equation. And this equation is familiar in the subject of group cohomology: it says $a$ is a ‘3-cocycle’ on the group $G$ with coefficients in $A$.

Even better, she showed that cohomologous 3-cocycles give equivalent 2-groups. (Equivalent as monoidal categories, that is.)

So, we can classify 2-groups using cohomology! The most exciting, least obvious part of this is the cohomology class $[a] \in H^3(G,A)$. This is often called the ‘Sính invariant’, though I believe Hoàng is her surname, not Sính.

This connection between 2-groups and cohomology is no coincidence. It’s best understood using a bit more topology.

Any connected space with a basepoint, say $X$, has a fundamental group. But it also has a fundamental 2-group! This 2-group has $G = \pi_1(X)$ and $A = \pi_2(X)$. And if all the higher homotopy groups of $X$ vanish, this 2-group knows everything about the
homotopy type of $X$, at least if $X$ is reasonably nice, like a CW complex.

So, Hoàng Xuân Sính’s thesis sheds light on ‘homotopy 2-types’: that is, homotopy types of nice spaces with $\pi_n(X) = 0$ for $n > 2$. They are just 2-groups!

Thus, her thesis illuminated one of the simplest — yet still important — special cases of Grothendieck’s ‘homotopy hypothesis’, namely that homotopy n-types correspond to n-groupoids.

You can see Hoàng Xuân Sính’s thesis along with a handwritten
summary in English here:

That website also has three nice photos of Grothendieck in Vietnam. I showed a colorized version of one at the top of this article, and here is another, with Hoàng Xuân Sính at far left:

Here are the original uncolorized versions of all three:

## Civilizational Collapse (Part 2)

24 January, 2013

Last time I told you a story of the American Southwest, starting with the arrival of small bands of hunters around 10,000 BC. I focused on the Anasazi, or ‘ancient Pueblo people’, and I led up to the Late Basketmaker III Era, from 500 to 750 AD.

The big invention during this time was the bow and arrow. Before then, large animals were killed by darts thrown from slings, which required a lot more skill and luck. But even more important was the continuing growth of agriculture: the cultivation of corn, beans and squash. This was fueled a period of dramatic population growth.

But this was just the start!

### The Pueblo I and II Eras

The Pueblo I Era began around 750 AD. At this time people started living in ‘pueblos’: houses with flat roofs held up by wooden poles. Towns became bigger, holding up to 600 people. But these towns typically lasted only 30 years or so. It seems people needed to move when conditions changed.

Starting around 800 AD, the ancient Pueblo people started building ‘great houses’: multi-storied buildings with high ceilings, rooms much larger than those in domestic dwellings, and elaborate subterranean rooms called ‘kivas’. And around 900 AD, people started building houses with stone roofs. We call this the start of the Pueblo II Era.

The center of these developments was the Chaco Canyon area in New Mexico:

Chaco Canyon is 125 kilometers east of Canyon de Chelly.
Unfortunately, I didn’t see it on my trip—I wanted to, but we didn’t have time.

By 950 AD, there were pueblos on every ridge and hilltop of the Chaco Canyon area. Due to the high population density and unpredictable rainfall, this area could no longer provide enough meat to sustain the needs of the local population. Apparently they couldn’t get enough fat, salt and minerals from a purely vegan diet—a shortcoming we have now overcome!

Yet the population continued to grow until 1000 AD. In his book Anasazi America, David Stuart wrote:

Millions of us buy mutual funds, believing the risk is spread among millions of investors and a large “basket” of fund stocks. Millions divert a portion of each hard-earned paycheck to purchase such funds for retirement. “Get in! Get in!” hawk the TV ads. “The market is going up. Historically, it always goes up in the long haul. The average rate of return this century is 9 percent per year!” Every one of us who does that is a Californian at heart, believing in growth, risk, power. It works—until an episode of too-rapid expansion in the market, combined with brutal business competition, threatens to undo it.

That is about what it was like, economically, at Chaco Canyon in the year 1000—rapid agricultural expansion, no more land to be gotten, and deepening competition. Don’t think of it as “romantic” or “primitive”. Think of it as just like 1999 in the United States, when the Dow Jones Industrial Average hit 11,000 and 30 million investors held their breath to see what would happen next.

### The Chaco phenomenon

In 1020 the rainfall became more predictable. There wasn’t more rain, it was simply less erratic. This was good for the ancient Pueblo people. At this point the ‘Chaco phenomenon’ began: an amazing flowering of civilization.

We see this in places like Pueblo Bonito, the largest great house in Chaco Canyon:

Pueblo Bonito was founded in the 800s. But starting in 1020 it grew immensely, and it kept growing until 1120. By this time it had 700 rooms, nearly half devoted to grain storage. It also had 33 kivas, which are the round structures you see here.

But Pueblo Bonito is just one of a dozen great houses built in Chaco Canyon by 1120. About 215 thousand ponderosa pine trees were cut down in this building spree! Stuart estimates that building these houses took over 2 million man-hours of work. They also built about 650 kilometers of roads! Most of these connect one great house to another… but some mysteriously seem to go to ‘nowhere’.

By 1080, however, the summer rainfall had started to decline. And by 1090 there were serious summer drought lasting for five years. We know this sort of thing from tree rings: there are enough ponderosa logs and the like that archaeologists have built up a detailed year-by-year record.

Thanks to overpopulation and these droughts, Chaco Canyon civilization was in serious trouble at this point, but it charged ahead:

Part of Chacoan society were already in deep trouble after AD 1050 as health and living conditions progressively eroded in the southern districts’ open farming communities. The small farmers in the south had first created reliable surpluses to be stored in the great houses. Ultimately, it was the increasingly terrible conditions of those farmers, the people who grew the corn, that had made Chacoan society so fatally vulnerable. They simply got back too little from their efforts to carry on.

[….]

Still, the great-house dwellers didn’t merely sit on their hands. As some farms failed, they used farm labor to expand roads, rituals, and great houses. This prehistoric version of a Keynesian growth model apparently alleviated enough of the stresses and strains to sustain growth through the 1070s. Then came the waning rainfall of the 1080s, followed by drought in the 1090s.

Circumstances in farming communities worsened quickly and dramatically with this drought; the very survival of many was at stake. The great-house elites at Chaco Canyon apparently responded with even more roads, rituals, and great houses. This was actually a period of great-house and road infrastructure “in-fill”, both in and near established open communities. In a few years, the rains returned. This could not help but powerfully reinforce the elites’ now well-established, formulaic response to problems.

But roads, rituals, and great houses simply did not do enough for the hungry farmers who produced corn and pottery. As the eleventh century drew to a close, even though the rains had come again, they walked away, further eroding the surpluses that had fueled the system. Imagine it: the elites must have believe the situation was saved, even as more farmers gave up in despair. Inexplicably, they never “exported” the modest irrigation system that had caught and diverted midsummer runoff from the mesa tops at Chaco Canyon and made local fields more productive. Instead, once again the elites responded with the sacred formula—more roads, more rituals, more great houses.

So, Stuart argues that the last of the Chaco Canyon building projects were “the desperate economic reactions of a fragile and frightened society”.

Regardless of whether this is true, we know that starting around 1100 AD, many of the ancient Pueblo people left the Chaco Canyon area. Many moved upland, to places with more rain and snow. Instead of great houses, many returned to building the simpler pit houses of old.

Tribes descending from the ancient Pueblo people still have myths about the decline of the Chaco civilization. While such tales should be taken with a huge grain of salt, these are too fascinating not to repeat. Here are two quotes:

In our history we talk of things that occurred a long time ago, of people who had enormous amounts of power, spiritual power and power over people. I think that those kinds of people lived here in Chaco…. Here at Chaco there were very powerful people who had a lot of spiritual power, and these people probably used their power in ways that caused things to change, and that may have been one of the reasons why the migrations were set to start again, because these these people were causing changes that were never meant to occur.

My response to the canyon was that some sensibility other than my Pueblo ancestors had worked on the Chaco great houses. There were the familiar elements such as the nansipu (the symbolic opening into the underworld), kivas, plazas and earth materials, but they were overlain by a strictness and precision of design that was unfamiliar…. It was clear that the purpose of these great villages was not to restate their oneness with the earth but to show the power and specialness of humans… a desire to control human and natural resources… These were men who embraced a social-political-religious hierarchy and envisioned control and power over places, resources and people.

These quotes are from an excellent book on the changing techniques and theories of archaeologists of the American Southwest:

• Stephen H. Lekson, A History of the Ancient Southwest, School for Advanced Research, Santa Fe, New Mexico, 2008.

What these quotes show, I think, is that the sensibility of current-day Pueblo people is very different from that of the people who built the great houses of Chaco Canyon. According to David Stuart, the Chaco civilization was a ‘powerful’ culture, while their descendants became an ‘efficient’ culture:

… a powerful society (or organism) captures more energy and expends (metabolizes) it more rapidly than an efficient one. Such societies tend to be structurally more complex, more wasteful of energy, more competitive, and faster paced than an efficient one. Think of modern urban America as powerful, and you will get the picture. In contrast, an efficient society “metabolizes” its energy more slowly, and so it is structurally less complex, less wasteful, less competitive, and slower. Think of Amish farmers in Pennsylvania or contemporary Pueblo farms in the American Southwest.

In competitive terms, the powerful society has an enormous short-term advantage over the efficient one if enough energy is naturally available to “feed” it, or if its technology and trade can bring in energy rapidly enough to sustain it. But when energy (food, fuel and resources) becomes scarce, or when trade and technology fail, an efficient society is advantageous because it simpler, less wasteful structure is more easily sustained in times of scarcity.

### The Pueblo III Era, and collapse

By 1150 AD, some of the ancient Pueblo people began building cliff dwellings at higher elevations—like Mesa Verde in Colorado, shown above. This marks the start of the Pueblo III Era. But this era lasted a short time. By 1280, Mesa Verde was deserted!

Some of the ruins in Canyon de Chelly also date to the Pueblo III Era. For example, the White House Ruins were built around 1200. Here are some of my pictures of this marvelous place. Click to enlarge:

But again, they were deserted by the end of the Pueblo III Era.

Why did the ancient Pueblo people move to cliff dwellings? And why did they move out so soon?

Nobody is sure. Cliff dwellings are easy to defend against attack. Built into the south face of a cliff, they catch the sun in winter to stay warm—it gets cold here in winter!—but they stay cool when the sun is straight overhead in summer. These are good reasons to build cliff dwellings. But these reasons don’t explain why cliff dwellings were so popular from 1150 to 1280, and then were abandoned!

One important factor seems to be this: there was a series of severe droughts starting around 1275. There were also raids from other tribes: speakers of Na-Dené languages, who eventually became the current-day Navajo inhabitants of this area.

But drought alone may be unable to explain what happened. There have been some fascinating attempts to model the collapse of the Anasazi culture. One is called the Artificial Anasazi Project. It used ‘agent-based modeling’ to study what the ancient Pueblo people did in Long House Valley, Arizona, from 200 to 1300. The Villages Project, a collaboration of Washington State University and the Crow Canyon Archaeological Center, focused on the region near Mesa Verde.

Quoting Stephen Lekson’s book:

Both projects mirrored actual settlement patterns from 800 to 1250 with admirable accuracy. Problems rose, however, with the abandonments of the regions, in both cases after 1250. There were unexplained exceptions, misfits between the models and reality.

Those misfits were not minor. Neither model predicted complete abandonment. Yet it happened. That’s perplexing. In the Scientific American summary of the Long House Valley model, Kohler, Gummerman, and Reynolds write, “We can only conclude that sociopolitical, ideological or environmental factors not included in our model must have contributed to the total depopulation of the valley.” Similar conundrums best the Villages Project: “None of our simulations terminated with a population decline as dramatic as what actually happened in the Mesa Verde region in the late 1200.”

These simulation projects look interesting! Of course they leave out many factors, but that’s okay: it suggests that one of those factors could be important in understanding the collapse.

For more info, click on the links. Also try this short review by the author of a famous book on why civilizations collapse:

• Jared Diamond, Life with the artificial Anasazi, Nature 419 (2002), 567–569.

From this article, here are the simulated versus ‘actual’ populations of the ancient Pueblo people in Long House Valley, Arizona, from 800 to 1350 AD:

The so-called ‘actual’ population is estimated using the number of house sites that were active at a given time, assuming five people per house.

This graph gives a shocking and dramatic ending to our tale! Lets hope our current-day tale doesn’t end so abruptly, because in abrupt transitions much gets lost. But of course the ancient Pueblo people didn’t disappear. They didn’t all die. They became an ‘efficient’ society: they learned to make do with diminished resources.

## Civilizational Collapse (Part 1)

20 January, 2013

A few weeks ago I visited Canyon de Chelly, which is home to some amazing cliff dwellings. I took a bunch of photos, like this picture of the so-called ‘First Ruin’. You can see them and read about my adventures starting here:

• John Baez, Diary, 21 December 2012.

Here I’d like to talk about what happened to the civilization that built these cliff dwellings! It’s a fascinating tale full of mystery… and it’s full of lessons for the problems we face today, involving climate change, agriculture, energy production, and advances in technology.

First let me set the stage! Canyon de Chelly is in the Navajo Nation, a huge region with its own laws and government, not exactly part of the United States, located at the corners of Arizona, New Mexico, and Utah:

The hole in the middle is the Hopi Reservation. The Hopi are descended from,the people who built the cliff dwellings in Canyon de Chelly. Those people are often called the Anasazi, but these days the favored term is ancient Pueblo peoples.

The Hopi speak a Uto-Aztecan language, and so presumably did the Anasazi. Uto-Aztecan speakers were spread out like this shortly before the Europeans invaded:

with a bunch more down in what’s now Mexico. The Navajo are part of a different group, the Na-Dené language group:

So, the Navajo aren’t a big part of the story in this fascinating book:

• David E. Stuart, Anasazi America, U. of New Mexico Press, Albuquerque, New Mexico, 2000.

Let me summarize this story here!

### After the ice

The last Ice Age, called the Wisconsin glaciation, began around 70,000 BC. The glaciers reached their maximum extent about 18,000 BC, with ice sheets down to what are now the Great Lakes. In places the ice was over 1.6 kilometers thick!

Then it started warming up. By 16,000 BC people started cultivating plants and herding animals. Around 12,000 BC, before the land bridge connecting Siberia and Canada melted, people from the so-called Clovis culture came to the Americas.

It seems likely that other people got to America earlier, moving down the Pacific coast before the inland glaciers melted. But even if the Clovis culture didn’t get there first, their arrival was a big deal. They be traced by their distinctive and elegant spear tips, called Clovis points:

After they arrived, the Clovis people broke into several local cultures, roughly around the time of the Younger Dryas cold spell beginning around 10,800 BC. By 10,000 BC, small bands of hunters roamed the Southwest, first hunting mammoths, huge bison, camels, horses and elk, and later—perhaps because they killed off the really big animals—the more familiar bison, deer, elk and antelopes we see today.

For about 5000 years the population of current-day New Mexico probably fluctuated between 2 and 6 thousand people—a density of just one person per 50 to 150 square kilometers! Changes in culture and climate were slow.

### The Altithermal

Around 5,000 BC, the climate near Canyon de Chelly began to warm up, dry out, and become more strongly seasonal. This epoch is called the ‘Altithermal’. The lush grasslands that once supported huge herds of bison began to disappear in New Mexico, and those bison moved north. By 4,000 BC, the area near Canyon de Chelly became very hot, with summers often reaching 45°C, and sometimes 57° at the ground’s surface.

The people in this area responded in an interesting way: by focusing much more on gathering, and less on hunting. We know this from their improved tools for processing plants, especially yucca roots. The yucca is now the state flower of New Mexico. Here’s a picture taken by Stan Shebs:

David Stuart writes:

At first this might seem an unlikely response to unremitting heat and aridity. One could argue that the deteriorating climate might first have forced people to reduce their numbers by restricting sex, marriage, and child-bearing so that survivors would have enough game. That might well have been the short-term solution [….] When once-plentiful game becomes scarce, hunter-gatherers typically become extremely conservative about sex and reproduction. […] But by early Archaic times, the change in focus to plant resources—undoubtedly by necessity—had actually produced a marginally growing population in the San Juan Basin and its margins in spite of climatic adversity.

[….]

Ecologically, these Archaic hunters and gatherers had moved one entire link down the food chain, thereby eliminating the approximately 90-percent loss in food value that occurs when one feeds on an animal that is a plant-eater.

[….]

This is sound ecological behavior—they could not have found a better basic strategy even if they had the advantage of a contemporary university education. Do I attribute this to their genius? No. It is simply that those who stubbornly clung to the traditional big game hunting of their Paleo-Indian ancestors could not prosper, so they left fewer descendents. Those more willing to experiment, or more desperate, fared better, so their behavior eventually became traditional among their more numerous descendents.

### The San Jose Period

By 3,000 BC the Altithermal was ending, big game was returning to the Southwest, yet the people retained their new-found agricultural skills. They also developed a new kind of dart for hunting, the ‘San Jose point’. So, this epoch is called the ‘San Jose period’. Populations rose to maybe about 15 to 30 thousand people in New Mexico, a vast increase over the earlier level of 2-6 thousand. But still, that’s just one person per 10 or 20 square kilometers!

The population increased until around 2,000 BC. At this point population pressures became acute… but two lucky things happened. First, the weather got wetter. Second, corn was introduced from Mexico. The first varieties had very small cobs, but gradually they were improved.

The wet weather lasted until around 500 BC. And at just about this time, beans were introduced, also from Mexico.

Their addition was critical. Corn alone is a costly food to metabolize. Its proteins are incomplete and hard to synthesize. Beans contain large amounts of lysine, the amino acid missing from corn and squash. In reasonable balance, corn, beans and squash together provide complimentary amino acids and form the basis of a nearly complete diet. This diet lacks only the salt, fat and mineral nutrients found in most meats to be healthy and complete.

By 500 BC, nearly all the elements for accelerating cultural and economic changes were finally in place—a fairly complete diet that could, if rainfall cooperated, largely replace the traditional foraging one; several additional, modestly larger-cobbed varieties of corn that not only prospered under varying growing conditions but also provided a bigger harvest; a population large enough to invest the labor necessary to plant and harvest; nearly 10 centuries of increasing familiarity with cultigens; and enhanced food-processing and storage techniques. Lacking were compelling reasons to transform an Archaic society accustomed to earning a living with approximately 500 hours of labor a year into one willing to invest the 1,000 to 2,000 yours coming to contemporary hand-tool horticulturalists.

Nature then stepped in with one persuasive, though not compelling, reason for people to make the shift.

Namely, droughts! Precipitation became very erratic for about 500 years. People responded in various ways. Some went back to the old foraging techniques. Others improved their agricultural skills, developing better breeds of corn, and tricks for storing water. The latter are the ones whose populations grew.

This led to the Basketmaker culture, where people started living in dugout ‘pit houses’ in small villages. More precisely, the Late Basketmaker II Era lasted from about 50 AD to 500 AD. New technologies included the baskets that gave this culture its name:

Pottery entered the scene around 300 AD. Have you ever thought about how important this is? Before pots, people had to cook corn and beans by putting rocks in fires and then transferring them to holes containing water!

Now, porridge and stews could be put to boil in a pot set directly into a central fire pit. The amount of heat lost and fuel used in the old cooking process—an endless cycle of collecting, heating, transferring, removing and replacing hot stones just to boil a few quarts of water—had always been enormous. By comparison, cooking with pots became quick, easy, and far more efficient. In a world more densely populated, firewood had to be gathered from greater distances. Now, less of it was needed. And there was newer fuel to supplement it—dried corncobs.

Not all the changes were good. Most adult skeletons from this period show damage from long periods spend stooping—either using a stone hoe to tend garden plots, or grinding corn while kneeling. And as they ate more corn and beans and fewer other vegetables, mineral deficiencies became common. Extreme osteoporosis afflicted many of these people: we find skulls that are porous, and broken bones. It reminds me a little of the plague of obesity, with its many side-affects, afflicting modern Americans as we move to a culture where most people work sitting down.

On the other hand, there was a massive growth in population. The number of pit-house villages grew nine-fold from 200 AD to 700 AD!

It must have been an exciting time. In only some 25 generations, these folks had transformed themselves from forager and hunters with a small economic sideline in corn, beans and squash into semisedentary villagers who farmed and kept up their foraging to fill in the economic gaps.

But this was just the beginning. By 1020, the ancient Pueblo people would begin to build housing complexes that would remain the biggest in North America until the 1880s! This happened in Chaco Canyon, 125 kilometers east of Canyon de Chelly.

Next time I’ll tell you the story of how that happened, and how later, around 1200, these people left Chaco Canyon and started to build cliff dwellings.

For now, I’ll leave you with some pictures I took of the most famous cliff dwelling in Canyon de Chelly: the ‘White House Ruins’. Click to enlarge: