Hanging Lake (Colorado II)

Last week’s post was a bit of a cliff hanger, and so is Hanging Lake.

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It is precariously sitting on top of a cliff, with waterfalls in the back as a bonus.


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The emerald green water creates an eery play between underwater world and the reflections of the upper world behind.

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What more could one wish for? Well, there is more. A very short hike up above is Spouting Rock, a single, taller waterfall that by itself is worth a visit.

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Long time exposure doesn’t do it much good.

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In this case, I like the dramatic spattering or the quiet drip-dropping much better.

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It is a wondrous place. Remember, come early.


The Translation Invariant Costa Surface

Out of the flurry of minimal surfaces that was inspired by the Costa surface, a particularly fundamental new surface is the Translation Invariant Costa Surface, discovered by Michael Callahan, David Hoffman, and Bill Meeks around 1989.Chm11

Like Riemann’s minimal surface, its ends are asymptotic to horizontal planes, but it is invariant under a purely vertical translation, and the connections between consecutive planes are  borrowed from the Costa surface. Surprisingly, in a few ways this surface is even simpler than Costa’s surface. To see this, let’s look at a quarter of a translational fundamental piece from the top:


It is bounded by curves that lie in reflectional symmetry planes, and cut off with an almost perfect quarter circular arc. Hence the conjugate minimal surface will have an infinite polygonal contour, like so:


It is not too hard to solve the Plateau problem for such contours, and adjust the edge length parameter so that the conjugate piece is the one used for the Translation Invariant Costa Surface. It is also possible to argue that the Plateau solution is embedded, and conclude the same for the Translation Invariant Costa Surface. All this is not so easy for the Costa surface itself.


Above is a variation with one handle added at each layer. Surprisingly, the corresponding finite surface does not exist. One can add deliberately more handles. Below is a rather complicated version that I called CHM(2,3), with a wood texture rendered in PoVRay in 1999, when I had figured out how to export Mathematica generated surface data to PoVRay.

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Hanging Lake Trail (Colorado I)

Hanging Lake is one of the most popular hikes in Colorado. In the summer, the daily 1000+ visitors don’t hike the trail anymore, but stand in line all the way up and down.DSC 8523

I avoided all this by getting there at 7am, which gave me time to enjoy the trail itself.

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It climbs up steeply among trees and rock cliffs. On a crowded day, it would be impossible not to overlook perfect sceneries like this one.

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The semi-vandalized hut below hints at how much work it must be to maintain the trail and keep it in its almost pristine condition.

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What to do with an incessant stream of visitors? Let it grow, or cut?

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It all depends, of course, what is at the end of the hike. We’ll learn that next week…

Boundary Considerations, Part II

As promised, today we will look at a close cousin of last week’s surface. A good starting point is the CLP surface of Hermann Amandus Schwarz, about which I have written before.



Up above are four copies of a translational fundamental piece. There are horizontal straight lines meeting in a square pattern, vertical symmetry planes intersecting the squares diagonally, vertical lines through edge midpoints of the squares and horizontal symmetry planes half way between squares at different heights. What more could one want? Well, CLP has genus 3, and we wouldn’t mind another handle.





There are various ways of doing that, and one of them leads to today’s surface, shown above. For adding a handle we had to sacrifice the vertical straight lines, but all other symmetries are retained. These are, in fact,  essentially the same symmetries we had in last week’s surface, except that there, the squares in consecutive layers were shifted against each other. The similarities go further.


Again we can ask how things look at the boundary. Pushing the one free parameter the the other limit, gives us again doubly  periodic Scherk surfaces and Karcher-Scherk surfaces. There is a subtle difference (called a Dehn twist), however, how the two types of Scherk surfaces are attached to each other in both cases.


Finally, as usual, the cryptic rainbow polygons that encode everything. Today, the two fit together along their fractured edges, which has to do with the period condition these surfaces have to satisfy.


Trillium Grandiflorum

Spring this year was short, and so was the wildflower season.  

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This post is dedicated to the largest trillium in the state, the trillium grandiflorum. It is not particularly rare, but the large white flower petals whither quickly, and is a favorite food of the abundant deer population.  DSC 1292

Both the petals and leaves are deeply veined. The flower sits on a stem above the leaves, in contrast to the drooping trillium where it drops down below the leaves.

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One of the woodland trails in Turkey Run State Park has them usually in abundance, but only this year I saw them in their prime time.

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How Does This Look Like At The Boundary?

A common recommendation to the layperson who is stranded among a group of mathematicians and doesn’t know what to say is to ask the question above. It will almost always trigger a lengthy and incomprehensible response.




For example, let’s look at the surface below. It constitutes a building block that can be translated around to make larger pieces of the surface. That this works has to do with the small and large horizontal squares. It is similar to Alan Schoen’s Figure 8 surface, but a bit simpler (it only has genus 4)



This surface belongs to a 5-dimensional family about which little is known. The only simple thing I can do with it is to move the squares closer or farther apart. So, how does this look at the boundary? On one hand, when the squares get close, we see little Costa surfaces emerging, as one might expect:


At the other end of infinity, things look complicated, but depending what we focus on, there is a doubly periodic Scherk surface or a doubly periodic Karcher-Scherk surface:


Below are, for the sake of their beauty, the two translation structures associated to two of the Weierstrass 1-forms defining this surface. Next week we will study a close cousin of this surface.




Little Things in Yellow and White and Green

The Turkey Run State Park has not only some of the most interesting rock formations in Indiana, but also an exciting vegetation.  Today we focus on the little things. Let begin with the liverworts (marchantiophyta), belonging to the bryophytes family. These bryophytes neither use roots nor make flowers, but interesting leave patterns. This stuff is what covers the rock formations, unless the rocks have been abused as slides. People should (i) look and (ii) think before they do their thing.DSC 1231

Here is another very little one I don’t know the name of. 

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Below, I think, is a buttercup flower, getting ready.DSC 1257

Unfortunately, searching for yellow with white hair is not very helpful. So I also don’t know what this one is called. But it does know about right angles and 5-fold symmetry.DSC 1258

Finally, another flowerless plant with pretty hair:DSC 1286