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?

2 thoughts on “Topology Rocks”

  1. Hey Jeremy,

    Great work. I particularly liked your drawings to go with the discussion. However there is something that I found puzzling: when you examine the genus of disks with holes, you show that a disk with one hole has genus 1—how could this be? If I make a closed curve cut on this surface it should be impossible to avoid immediately separating it; there are two types of closed loop-cuts: a loop-cut that does not circumnavigate the hole (which cuts out a disk and leaves a disk with two holes), and a loop-cut that does circumnavigate the hole (which cuts out an annulus and leaves another annulus). By this analysis, I would think that the disk with one hole (or any number of holes, for that matter) has genus 0.

    I see that you define genus without the requirement that the cuts be closed curves. I worry about this way of defining it, because one could make a bunch of minute slits in the surface, thereby making the “genus” infinite. Reading online, sources seem to avoid talking about the genus of surfaces with boundary, and they add the condition of closed cuts, which I assume is because of this subtlety.

    http://mathworld.wolfram.com/Genus.html

    1. Daniel,

      Thank you 🙂 This was fun to write and draw.

      I am reluctant to admit it, but I looked into it and, you are right. I was improperly attributing genus to the “flat” objects in my educational gallery. I searched around the web and found that indeed cuts are required to be along closed loops. I was intuitively casting the genus behavior of the donut (torus) onto the perforated disk (annulus) as if it was a simple cast from 3d to 2d. And I was envisioning the cut from one border to the next.

      FYI, when searching around, I did find I wasn’t the only one with this misconception, for example, I came across a debate as to the genus of a pair of pants, (equivalent to a disk with two holes, genus 0).

      So, I’ve gone and revised my article, making corrections to the genus values. I’ve also gotten rid of my use of the word, “hole” which in light of your revelation, I find to be misleading. I’ve gone for simply explaining genus with the word “handle” instead. I’d say “handle” is intuitively associated with wrapping your thumb and fingers around something to form a closed loop.

      I go back to the notion of genus as “connectivity” and have to admit, there’s not much connectivity in a perforated disk, at least not in the same sense as a torus, or multi-torus. Though, I’m still fond of my intuitive conception of genus extending to the perforated disks. Maybe in some other context a definition as I originally (erroneously) stated would be appropriate. Such as, a cut must be a closed loop, OR it must extend from one border to a different border. In that case a pair of pants would have genus 2.

      Thanks for your constructive criticism. This article has been a learning process for me. Next time, I’ll run my math-related posts by you first for a math check before publishing 😉

Leave a Reply

Your email address will not be published. Required fields are marked *