Topology Rocks

In the high frontier country of California, stands a range of rock formations. [1] It’s weathered granite forms are absolutely alluring. The whole place pulls you in with a kind of mythical intrigue.

This is the Alabama Hills, on the outskirts of the town of Lone Pine, California.[2] Savina and I have traveled here to explore the country and admire the rocks.

Math on our minds

The evening before, we arrived to nearby Tuttle Creek campground to spend the night. Along the way we stopped to inspect a roadside tourist map and informational sign. We found indicated on this map a feature called the “Mobius Arch”. We had never heard of this before. Could it be related to the famous mathematical object, the Möbius Strip? The map had a thumbnail image of the Mobius Arch to show what it looked like. Yes, I could tell right away, from its distinct “twist” partway along the arch, it was indeed named after the Möbius strip. What a pleasant surprise for a Möbius enthusiast such as myself! It’s lovely enough that this natural sculpture exists out there in the form of a Möbius strip. But what is even lovelier is the thought that there was one or more other Möbius enthusiasts out there of sufficient influence to give this arch its mathematically inspired name.

The next morning we break camp just before dawn, in order to see this place in a magical early morning light. We head straight for the Möbius arch, and reach it easily:

In all the world, could there be a greater monument to the Möbius strip than this? While it is actually impossible to have a Möbius strip, a two-dimensional object, exist materially in our three-dimensional world, this mighty arch does a fine job of representing one in the hearts and minds of math-minded humans. To add to its prestige, as you can see, this arch forms a perfectly framed view of the majestic Mt. Whitney, the highest summit in the contiguous United States.

In this moment, I can’t help but think of Carlo Séquin, the greatest Möbius enthusiast that I know. Carlo is a Möbius enthusiast to such a degree, that he has a whole line of research and invention on the topic of Möbius bridges. Here, you can see him explaining all about it:

Mobius Bridges and Buildings with Carlo Sequin
Video by Brady Haran for Numberphile - April 9, 2014

I had met Carlo for the first time a few weeks ago, while attending a talk at the Arts, Technology and Culture Colloquium at the University of California, Berkeley. I found a kindred spirit in him and we got acquainted. I really enjoyed learning about his passions in math, art, and computer science. So, naturally, I was contemplating all this, in the midst of the marvelous Möbius Arch, perhaps the greatest nature-made Möbius bridge in all the land.

The difference between spheres and arches

Savina and I share a picnic breakfast beside the Möbius Arch, and we observe it carefully. But there is more to see in this place. We set out on foot to roam around further. We walk, and look, and ponder the combination of ruggedness and smoothness all around us. This otherworldly place has a lot to offer. Suddenly, we come across a hill of spheres:

It is spheroidal weathering of course. Ok, I didn’t know it at the time, but I learned about it only after returning home and reading about this place. How does spheroidal weathering work? Well, it is an elaborate, four part process:

  1. A large, unbroken mass of granite is formed from magma and lies underground.
  2. Somehow (through mechanical stresses I presume) it develops a three dimensional pattern of cracks which segment the granite into large blocks. This is known as jointed granite.
  3. Water from above ground enters these cracks and causes chemical decomposition, starting at the surface of each block and gradually advancing inwards. Due to the geometry of the arrangement, the decomposition happens fastest at the corners, intermediate at the edges, and then slowest at the faces of each block.[3]
  4. Sometime later, all this gets exposed at the surface, where conventional erosion goes to work, removing the decomposed rock with ease and leaving the rounded solid cores undamaged.

So that’s how you end up with a hill of spheres. But look back again at the photo above, some of these spheres have little caverns on the sides. Ah ha! These ones seem to be on their way to becoming arches. Do arches form like the spheres do? Do they too form underground through the same chemical decay process? My guess would be so. And I have a hypothesis as to how:

My hypothesis is that arches tend to form where there’s a T-joint in the granite, as illustrated above. Are you a geologist reading? If so, can you speak to this?

If this isn’t the case, then how else could arches form? I guess you could simply have a cavity on one side of a sphere that starts to decay, and thus more water enters and it decays more and more, and maybe it meets up with another such cavity on the other side and thus you have a connected tunnel.

And so we return to the original question: what’s the difference between spheres and arches? Well… an arch is just a sphere with a hole through it. In other words, an arch is a donut.

Topological Safari

Topography: the study of the arrangement of features in a landscape.

Topology: the study of shape and space, related to geometry, but somehow more abstract.

We continue to hike, exploring the topography of this landscape, exposed spheres and arches all around. I still have math on my mind, and I get to thinking about topology.

To a topologist, all objects are made of infinitely stretchable rubber. Objects are categorized not by their size, nor curvature, nor volume, nor surface area, nor any of those traditional geometric concepts. No, topological objects can be categorized by just three particular properties:

1. How many borders?
2. How many sides?
3. Connectivity (genus)

This knowledge is fresh in my mind (wouldn’t you know?) thanks to just a few days ago having watched another of Carlo’s videos, recently published, in which he explains all this and more:

Super Bottle with Carlo Sequin
Video by Brady Haran for Numberphile - Dec 13, 2016

Here’s few topological objects to consider:

Look through each one and try and see for yourself how each of these objects can be defined by these three properties.

  1. The number of borders – This might be the easiest to understand. From any point on an objects surface, can you travel freely without running out of surface? On a sphere and a donut you can. On a disk you can’t. You’d come up against an edge.
  2. The number of sides – This is a bit trickier. Familiar surfaces have two sides, a disk has a front and a back, a sphere has an inside and an outside. But some special surfaces have only a single side, like the famous Möbius strip, and Klein bottle. My understanding is that no surface can have three or more sides. Are you a topologist reading? If so, can you speak to this?
  3. The genus – this is the most perplexing, I think. Genus tells you the number of “handles” the object has. I put “handles” in quotes, because these are handles in the special, topological sense. Sometimes, when trying to identify an object’s topological identity, parts which seem like a handle are not, or a part which is actually handle might not seem like one at all. A more effective way of determining genus is this: the maximum number of cuts through the object that you can make without the object coming apart at all.
EDIT:
Thanks to Daniel (via his comment below on this post) for pointing out a flaw in my original explanation. I was incorrectly applying genus to the disk and the perforated disks. Daniel points out that the topological cuts associated with genus need to be closed curves, as seen in the two images above of the sphere and donut. I checked a few topological sources, and they confirm this. Therefore, the genus of the disk, and perforated disks (of any number of holes) is zero. Think of taking a cookie cutter to a disk, you'll end up with the cut piece being separated. See, I told you genus was tricky!

Using these three properties, you can find the true identities of objects. You can determine if two seemingly different objects are actually the same. For example, what about this: an object in the form of a straw?

Answer: it has two borders, two sides and genus 1. Therefore it is topologically equivalent to the disk with one hole above.
Is this equivalent to any of those above?

How many borders does this straw have? How many sides? What genus? Once you’ve got this, does it correspond to any of the above objects? If so, which one? Then, if you imagine freely stretching and distorting it, can you convince yourself that it is the topological equivalent?

Here’s another puzzler, how about this object, which resembles a T-shirt?

Or try this one, which is like a vaulted ceiling with four pillars. How many handles does it have?

As we walk, I think about genus. What rock formations can we find of different genus?  We come across a monumental spheroid (genus 0):

Sometime later, we come across another beautiful arch, the “Eye of Alabama” (genus 1):

What other topological species are out there waiting to be found?

Next, we come upon this intriguing specimen. It doesn’t quite look like an arch. Rather a cavern with a pillar at its entrance. Or you could see it as a tunnel whose entrance and exit are right next to each other.

But, of course, with a little topological know-how, you can identify it as genus 1, therefore equivalent to a simple arch, or a donut. I am reminded of the interconnected relationship between an arch and a tube, or between a bridge and a tunnel. Neither can exist in isolation. Where there is one there must be another. Like inverse pairs, like yin and yang.

By now I’ve become eager to search for higher genus formations. Could we hope to find a double-arch (genus 2), even – dare I say it – a triple arch (genus 3)? How rare would these be?

We walk and walk, and we see rock after rock. We decide we’ve gone far enough. We turn back and return to our car. Despite my high hopes, we haven’t even found a single formation greater than genus-1. Too bad.

When suddenly:

Oh my word! This thing is off the charts. It’s beyond a double arch, or a triple arch. It’s even greater than a quadruple arch (genus 4)…

Look at this detail, a lovely double-arch, and this is just one part of this whole topological complex.

Now, the question of the day: what is the genus of this outstanding rock? Let’s take a look. There’s two main chambers. The first (Chamber A) is too small to enter, but it clearly has three openings.

The second (Chamber B) is cramped, but it’s easy enough to crawl inside. Here’s a panoramic view from within:

So we have found a formation with seven openings. Three in Chamber A and four in Chamber B. What genus does it have? How many handles? Look at how many places you could theoretically make a cut through without anything falling apart. Those are the “arches”, or “pillars”. Chamber A has two pillars, and Chamber B has three. So then, what we have here is a quintuple arch (genus 5) Wow!

Question: what would the genus of this formation be if we were to cut a hole that would connect the two chambers directly?

We admire this rare quintuple arch for a while more. And then, content with our discovery, Savina and I make our way back. It would have been enough to find a double arch, even a triple, even a quadruple. But today nature graced us with a quintuple arch. How about that?

We are nearly back to the trailhead. And, just when I thought we’d had enough topology for one day, we find this, beside the dusty trail:

An old can, shot full of holes for target practice. It been shot five times, and so it has ten holes (five entry wounds and five exit wounds). What then, is the genus of this can?


More from Carlo Séquin:
- Check out his full playlist on Numberphile
- If you like topological analysis of natural formations, then you'll probably like his paper 2-Manifold Sculptures on the topology of mathematical sculpture.

Footnotes

  1. From possible designations of urban, rural, and frontier, the United States Census Bureau identifies this area as “frontier”.
  2. Why does this place have the name it does? We learn it was named after a famous Civil War battleship the CSS Alabama, by some local prospectors in 1864 who were sympathetic to the Confederacy. To us, the name feels ill-suited. Why should this remote and magnificent natural wonderland take its name from a reference to this distant war, whose fighters probably had no connection whatsoever to this place? Like it or not, you can read more about the naming of this place here.
  3. The process of rounding off the corners and edges of a block into a spheroid has a lovely intuition. For a process of decomposition, what needs to happen is the water needs to penetrate into the block. Think of a bit of rock somewhere just inside the surface of the block. If that bit of rock is just beneath the center of one of the block faces, then water can only attack it from one direction. If that bit of rock is just beneath an edge, then water can attack it from two directions. And if that bit of rock is just beneath a corner, then water can attack it from three directions. I’d like to model this programmatically and play with simulating it. Can it be said then that the decomposition happens twice as fast at the edges and three times as fast at the corners? If you modeled it as a 3x, 2x, 1x process, would you end up with a sphere?

Journey in the Tufa Kingdom

In a distant land, across mountains and deserts and upon a high plain, lies a lake of great peculiarity. Water flows in, but no water flows out. The only way water leaves is by evaporation. And when the water evaporates, it leaves salt and other goodies behind. So the lake is salty, and alkaline, and rich in carbonates. What’s more, there are subterranean springs which bring calcium-rich water. When this spring water meets the lake water, presto! You have a reaction, and calcium carbonate, a.k.a limestone is precipitated. Get a spring like this going for many decades or centuries and amazing limestone towers known as tufas will form. Savina and I journeyed to see these famous tufas at Mono Lake, and here is what happened:

Continue reading Journey in the Tufa Kingdom

Weightless Wreath

This sculpture is a modern take on a traditional ornament. It is both familiar and unfamiliar at the same time.

It can be called a wreath in the sense that it is a circular arrangement of plant matter made for aesthetic purposes. It can also be called a tensegrity, or floating compression, in the sense that it is a mathematical arrangement of sticks and strings which holds itself together using a counterbalancing network of tension and compression.

Continue reading Weightless Wreath

Real Make Believe Space Programs

or: How I learned to love the Tom Sachs Space Program by reading Nicholas de Monchaux and learning the history of the Apollo Missions.

Did you know there is more than one NASA? There is of course, the NASA which is a United States government agency established by President Eisenhower in 1958. Then there is another NASA, which is an independent agency established by the artist Tom Sachs in 2007.

This article is for Tom Sachs fans like myself, who might not be so well schooled on the historical context for Space Programs. My argument is that by increased awareness of one NASA, the reader can have a deeper appreciation for the other NASA. And along the way, may come to see that the real and the make-believe aren’t so far apart as we might think.

Continue reading Real Make Believe Space Programs

Thanksgiving in the snow

The following trip report is written by my wife Savina. I think it is really great, so I asked to post it here on my website, and she agreed. I hope you enjoy reading, and I hope Savina keeps writing. -Jeremy

Thursday, November 26th

The night before, we prepared most of the stuff. Jeremy got the camping kitchen from the closet. We woke up early saying Happy Thanksgiving. We were the most excited to go to the snow. The previous days we checked the weather twenty times and yes, there would be snow.

Continue reading Thanksgiving in the snow

Seamless Mobius Weave

Seamless Möbius Weave (2015)
Hemp twine, two pieces (dyed & natural colors)
3" x 3" x 2"

This summer I studied weaving. I had a delightful time learning from the great Lou Grantham who is owner/operator/instructor at San Francisco Fiber.

After making a handful of traditional pieces in Lou’s studio and at home, I then couldn’t resist the desire to make a weaving of an unconventional and special topology, the Möbius Strip. But simply weaving a strip and then joining it together at the ends with a single twist in the middle wouldn’t do. I wanted it to be seamless, with the warp and the weft running uninterrupted throughout the whole band.

So I needed a special loom, to support the weaving process. My first attempt at loom construction was a single piece of wire, like so:

But this was too flimsy, it wouldn’t hold its width. I needed some structural cross-members, so I added the following with careful wire-plier work:

As you can see, each cross-member acts as a bridge to keep the two rails apart at the proper distance, while each bridge remains out of the way of the weaving plane.

From here, the warping of the loom was easy. It consisted of simply wrapping the hemp fiber around the rails and keeping a more or less consistent spacing the whole way round.

Then, the main weaving activity could begin.

But wait!

It is not immediately clear how to go about this. I can’t merely start weaving from one edge to the other. Due to the counter-intuitive topology of the Möbius strip, every time you complete a pass around, along one edge, you end up back where you started but along the other edge. (Well, technically, a Möbius strip has only one edge, but if you just look at your local spot on the strip, then you can talk about it having two edges)

So, what can we do about that? Start from the middle and work outwards? With each pass proceeding outwards from the middle along alternating edges?

Consider the following path for the single weft fiber, starting from the middle:

This didn’t satisfy me very much, too asymmetric I thought. So I came up with the following alternative:

What I’ve done is to start the weft in the midline of the strip (indicated by the thick line in the above drawing), as two fibers doubled over and proceeding together. Then the weaving process proceeds symmetrically outwards, with two weft fibers advancing together, each new time around “switching sides” (but not really,  because a Möbius strip has only one side, remember?)

I settled on the second, more symmetric option, but was this the right choice? I traded off bilateral symmetry for a double-thickness of warp along the midline. Is this a fair trade? What does bilateral even mean in the context of Möbius topology? Is the lack of obvious symmetry option here related to what happens when you cut a Möbius strip in half?

Back to weaving. In practice, the weaving was more like embroidery, as I was passing the weft through with a needle, over, under, over, under. There is no convenience of heddles to form a shed and pass a shuttle of weft through here. No, each over and under was done manually with needle technique.

Here, you can see the weaving coming along nicely:

Once I was satisfied with the number of weft passes, it was finally time to remove the work from the loom. Only, in this case, the only way to accomplish this is to destroy the loom, turning it into pieces with my wire cutter. In other words, I removed the loom from the weaving!

Once the piece was free from the loom, I noticed it had quite a lot of slack. So I set out to remove the slack, first concentrating all of the warp passes together like so:

And then carefully working the weft through, one bit at a time, until the density gap was closed:

And, finally, I tied the two ends of the weft to the two ends of the warp, to complete the weave, and trimmed the excess down to the knot.

As promised, a seamless Möbius weave.