Enough of minimal surfaces, at least here, for a while. Time for a relaxing little game. Let’s put six tokens in a row, three showing a 1, the other three a 2. There are evidently 6 choose 3 = 20 possibilities for that. A move consists of swapping two neighboring tokens, which also evidently only makes sense if they are different. 


Above is the graph that shows all possible states, the connections indicating the possible moves. What is interesting here? Not much. The graph is connected, which is easy to explain, and I mention all this only because it will serve as the 1-dimensional example of what we’ll do in two dimensions next week and in even higher dimensions later. So, to make things more complicated we are allowing the six tokens to show three values 1, 2, 3. We begin with two tokens of each kind.


The graph is more complicated but offers no interesting insights. Let’s forbid to swap neighbors 12 or 21. The game graph will have six identical components. This usually makes for hideous puzzles, asking the innocent victim to find moves that change say 112233 to 332211. But here this is too obvious, as we clearly cannot change the order of the 1s and 2s while the 3s can move freely among them.

To fix this, we make one more change: We arrange the tokens in a circle, still use 112233, and disallow a 12-21 swap.


Two different components. Why can’t we move from 112233 to say 132213? Again, order plays a role: There is just no way to get from 1133 to 1313, even if we allow cyclic permutations.

Finally, an example with just four tokens, having values 1 through 4, arranged in a square. We allow the swaps 12-21, 23-32, 34-43, and 41-14, but forbid 13-31 and 24-42.


This time we get two components as well. Why can’t we move from 1234 to 4123, i.e. rotate all tokens clockwise by 90 degrees? This time, the order still plays a role, but involves the orders of both pairs 13 and 24 simultaneously, which is hard to see. So this makes, in all its simplicity, a pretty nice puzzle. 




The Overstory

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Some views and places, like the arching tree above (still there!), about which I have written before in one of the more often revisited blog posts of this remote little blog, lend themselves easily to metaphors. Others, like the one below, are more hermetic.

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The spot is the same as in one of my earliest posts. Back then, the composition off fallen tree on rock before a steep canyon wall kept intriguing me over several seasons, triggering something I can’t put into words. After a seasonal flood swept away the tree, the place became less fascinating, until I noticed that something was happening. Two little Sycamore trees had taken possession, visible already above.

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If they survive a few more years rock fall, floods, and human mischief, they will have outgrown the canyon and dominate the scene for years beyond me and you.

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By the way, read that book in the title. It is better than I think.

Vertical Flux (The pleasures of Failure IV)

One of the standard examples of a failed minimal surface construction is the Finite Riemann surface, named in reference to the rather successful infinite Riemann minimal surface




The attempt was to make an embedded minimal sphere with 3 ends. Two ends is easy: the catenoid will do. If you want more, the ends can only be catenoidal or planar, and the catenoidal ends need to point in the same direction, i.e. have the same limiting normal. Such a sphere will always have what is called vertical flux, which is disastrous in its usefulness.


Whenever you have vertical flux, you can deform the minimal surface in a very simple way, keeping the catenoidal ends vertical but tilting non-horizontal planar ends like above. This is nice but also disastrous because it allows to prove that embedded surface of this sort cannot exist.


This is a famous theorem by Francisco José Lópe and Antonio Ros. The proof consists of three steps: If the surface is embedded, its stays embedded for all parameter values of the deformation. Secondly, if such a surface has either a planar end or a finite point with vertical normal, it cannot stay embedded. These two steps imply that such a surface can only have catenoidal ends and no points with vertical normal. Being a sphere then implies that there can only two ends, and we have the catenoid.



It is amusing to make more examples of surfaces with vertical flux. Above are single periodic versions with vertical catenoidal and annular ends. The first one feels a little bit embedded because the only intersections that happen are those of a catenoidal end with a translational copy, and I am inclined not to count that.




There is curious little town near Bloomington I haven’t written about that fits into my series of occasional posts about towns with a past (Hollbrook, Jerome, Leadville).

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This is Nashville, Indiana. Getting there early in the morning is smart, because then you get free parking and avoid the crowds.

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The attraction? Nashville used to be an artist colony. Locals doing their thing and having a good time. Their good sense of taste brought others with taste, and the town flourished.

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It still does, but with the money people arrived with more sense for money than taste.

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Occasionally there is still a trace of the former times left, like below in the inconspicuously elegant choice of a color pallette.

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From H to P and beyond (H-3)

In March we had a look at the Schwarz H surface, and it is time to revisit it. We begin by turning it on its side, for comfort:


Then horizontal lines and vertical symmetry planes cut the surface still into simply connected pieces like this one


The H-surfaces form a natural 1-parameter family with hexagonal symmetry. It turns out that in this representation one gains another parameter at the cost of  losing the hexagonal symmetry. This allows to deform the H-surface into new minimal surfaces, and the question arises what these look like. To get used to this view, below is yet another version of the classical H-surfaces near one of its two limits. 


The new deformation allows to shift the catenoidal necks up and down, until they line up like so:


This surface is a member of the so-called orthorhombic deformation of the P-surface of Schwarz so that we can deform any H-surface into the P-surface, and from there into any other member of the 5-dimensional Meeks family.

This is remarkable because the H-surface does not belong to the Meeks family, but to another 5-dimensional family of triply periodic minimal surfaces that is much less understood. The final image is another extreme case of the newly deformed H-surfaces:


Encapsulation (Mimosas II)

After drinking mimosa tea, the mind opens up to the beauty of the tree.DSC 9752

The flowers sit like alien insects above the fern shaped leaves, an odd contrast both in shape and color.DSC 9758

When in late summer the seed pods join in, we get  in a single picture the story of an entire life en miniature.

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Mimosas are fascinating for many reasons, and animals know about it. I am not patient enough for wildlife photography, so the image below says nothing about my skills but everything about the abundant presence of the hummingbirds.DSC 9769

What else could one possibly desire but sit back and watch all this.




The Catenoid With a Handle (The Pleasures of Failure III)

In the Beginning, Leonhard Euler created the Calculus of Variations and gave many examples. In one of them he proved that if you want to minimize the area of a surface of revolution (among surfaces of revolution), you will get a piece of a catenoid or plane.TwoCatenoids 3

The story is not as simple as it seems, because catenoids stop being minimal at a certain size. Above, for instance, two catenoids that have the same two circles as boundaries. Clearly only one of them has minimal area.

One of the early stories of failure was the attempt to add a handle to the catenoid. Maybe one could even save area this way?Cathandle

Not so, as the handle doesn’t even extend far enough to close up. Rick Schoen proved more generally that catenoids don’t come with handles of any sort and number. But one can try other things, like making the catenoid more symmetric. This is of course silly, but Luquesio Jorge and Bill Meeks did just that by turning the reflectional symmetry at a horizontal plane into a dihedral symmetry, thus creating the k-Noids. 3 noid

This works for any order and gets a little boring after a while.7 noid

But, somewhat surprisingly, one can add a handle to these k-Noids when k is at least 3:

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Like last week, k does not need to be an integer, and one can see clearly what goes wrong when one pushes k below 3: Here we have a broken catenoid with a handle.

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