Category: Mathematics

Some Caterpillars Stay Caterpillars

This blog post is (as will be several in the next weeks) inspired by the art of Matthew Shlian. Below you see ragged squares (or Aztec diamonds, if rotated by 45º), tiled by P-pentominos. On the right, we get by without using mirrored copies, and on the left we see all eight rotations and reflections of the P-pentomino.

The embossment on the left comes from Matt’s instagram pages where you can find a paper sculpture called Some Caterpillars Stay Caterpillars 42. He realizes this tiling using folded paper, and my different color are his different shades of grey, coming from natural light, and much more subtle in his sculpture than here.

The colors used in the embossment above add different esthetic and structural flavors. For instance, on the right the parallel slopes of the region are colored in different shades of the same color, emphasizing striped bands that completely eliminate the original tiling by P-pentominoes.

Matt’s original sculpture maintains a poetic balance between the mathematical rigor of the tiling and the more organic nature of a landscape of valleys and ridges that appear in his embossment.

Above is a tiling of the next larger Aztec diamond that can be tiled by P-pentominoes. It turns out that here you need the P-pentomino together with its mirror, but I don’t know a reason for this.

Elusiveness (Fischer-Koch II)

By varying angles and edge lengths of the fundamental piece, one can repeat last week’s construction of the 6-ended Fischer-Koch surfaces and make surfaces with more ends. Above and below you see images of the 10-ended version, together with their twisted friends.

Again one obtains parking garage structures as limits, and the position of the helicoidal axes is indicated below: The case k=3 corresponds to last week’s surfaces, the case k=5 to the ones above. The colored disks represent helicoidal axes, with the color showing the different spins.

What about the case k=4, which should lead to 8-ended surfaces? Daniel Freese has shown that one can untwist the parking garage structures to screw motion invariant minimal surfaces:

and

But these surfaces can not be obtained using the Fischer-Koch construction. Below you see the completely untwisted version with annular ends.

One key difference to the Fischer-Koch surfaces is that opposing ends have opposite normals in Daniel’s surfaces (or differently colored sides, as visible above). If the vertical line was really a straight line, it would be a rotational symmetry line, and opposing ends had the same color.

So things are not always quite what they seem.

Lost in Translation (Fischer-Koch I)

In 1990, Werner Fischer and Elke Koch classified embedded triply periodic minimal surfaces that can be obtained by extending Plateau solutions for Euclidean polygons.

Above to the left you see a minimal 8-gon, extended to a twisted minimal annulus to the right. The horizontal lines make a 60 degree angle, and if the height of the shorter vertical segments is one half of the gap size between two of these segments, further rotations will deliver an embedded surface.

It is hard to believe that something like this is possible, isn’t it?

Hermann Karcher describes a variation of this construction that creates 6-ended singly periodic minimal surfaces. He also mentions that these surfaces can be twisted, i.e. deformed into screw-motion invariant minimal surfaces with helicoidal ends.

Above you can see what happens when the surface is twisted clockwise. On the right, we approach a parking garage structure with five helicoidal columns, four of them with positive spin and axes at the corners of a square, and one with negative spin at the center of the square. Not getting lost in a parking garage like this would be very difficult…

As these surfaces are chiral, twisting them counterclockwise leads to essentially different surfaces. In this case, the limit parking garage structure consists of three helicoidal columns with axes placed on a straight line. So during this entire deformation, two of the helicoids have magically cancelled.

Next week we will see a very recent and surprising variation of this construction.

History Lessons (Berlin X)

If you search the internet for Cemeteries at Hallesches Tor, you will find blog posts about this cemetery, and some of them mention an encounter with a friendly middle aged man.

Saying a greeting led to a polite exchange, which in turn led to a conversation about the cemetery in general, which in turn led to an in-depth discussion of many specific graves.

This is another one of these blog posts. I, too, met this person, and received the best history lesson I had in my entire life.

For instance, I learned about the strange markings on many graves that look like gun shot holes. They are from gun shots, inflicted in one of the many utterly senseless battles of the Second Word War, when the Nazis forced teenage boys to confront the Soviet army, with only a handful of ammunition and no hope but death.

Or about the bunker that the Nazis build on this cemetery after making room by eliminating all traces of the Jewish graves, a bunker that was never used as it filled instantly with ground water, a bunker that has resisted demolition ever since, a bunker that couldn’t be more meaningless.

Or about the grave of Archduke Leopold Ferdinand of Austria, one of the last hopes of the Austrian Monarchy after the assassination of Archduke Franz Ferdinand in Sarajevo, one of the escalations that led to the First World War, but who happily renounced his title in order to marry the sex worker Wilhelmine Adamovicz. His grave (together with this third wife Clara Hedwig Pawlowski) is reportedly still visited yearly by mourning monarchists.

Other visitors commemorate E.T.A. Hoffmann’s death by following his request not to bring flowers but champagne…

And of course there are graves of mathematicians, like the one of the immortal Carl Gustav Jacob Jacobi, and less famous ones, like the one of Horst Kirchmeier, whose brave attempts to liberalize the German law governing sexual offenses went rather far.

I could continue my history lesson for a while. Or write about the symbolic of the cast iron fences.

Or about the massive thefts of tomb decorations that apparently sell world wide for enormous sums. Or what happened to the churches to which this cemetery belonged, and what buildings are there now… Another day.

Broken Circles

Here are two circles with an eighth removed. As we can see, they can move together. What about adding more such broken circles? How densely can we slide them together?

Circle45 01

Here is an example with three circles, each one sixth missing:

Circle60 01

This is maybe a little bit remarkable: If you take two such broken circles and rotate them by 60º against each other, you can slide them along each other so that one end point of one circle moves on the other circle, and vice versa. The first question (which must have an easy answer, of course) is: why does this work for these angles?

Circle60b 01

As shown above, this allows us to pack three of these broken circles together, creating a mild form of prettiness.

Now let’s use four broken circles, with a quarter removed each, rotated by 90º, and colored appropriately:

 

 

 

Circle90 01

Again, one can slide these circles along each other. Can you do this with six broken circles? Does this work with other angles?

Hopefully continued.

 

 

Remains to be Seen (Berlin VIII)

Mona Hatoum’s impressive installation of this name was shown in the Diversity United exhibit in Berlin this year, and the catalogue speculates that what we see here are the ghost-like fragments of a house.

I saw these hanging concrete pieces as probes into space, an attempt to make visible what has disappeared.

In this reconstruction, I am using PoVRay to probe textured space. A texture in PoVRay is function of the three spatial coordinates whose values is used in a color map to determines the color value of an object at the point given by the three coordinates. Above the function is sqrt(x2+y2+z2), and the color map a simple grayscale gradient, so that spheres centered at the origin have the same color value. Objects placed into the scene appear to be carved out of this space.

Above is a more complete reconstruction of Haroum’s installation, using the same spatula texture with added reflection. And below are the same probes, using an entirely different texture based on the function sin(x)+sin(y)+sin(z).

The Library of Babel II

Today we are taking the 2-dimensional floor plans of the Library of Babel  to the third dimension. The simplest way is to use a single floor plan and copy it for every level of the library. For instance, last time’s finite hexagon becomes a daunting infinite tower, at least in our imagination.

Babel hex 01

One could also do the same with several separate hexagons, but the resulting library would consist of several buildings, which, while not explicitly prohibited by Borges, seems unacceptable.

Doublyplus 01

But there is another possibility. For instance, using the arrangement of hexagons in horizontal lines, and repeating them vertically (as on the left above), but then, on the second floor, using the same floor plan albeit rotated by 90º (as in the middle), and then repeating this periodically, we arrive at a single building where it is sometimes necessary to climb up, walk across, and then down again to reach a different room on the same floor.

Nearby places can sometimes be terribly far away.

In the double floor plan up above on the right we see that this can be done by aligning the vestibules in a square pattern, leaving star-shaped Voids. Below is a partial view of this magnificent library:

Babel3The condition that the staircases in the vestibules extend infinitely in both directions is quite limiting, even if one doesn’t require that each staircase can be reached at every floor. A good strategy for designing even more complicated libraries is to begin with a floor plan that includes hexagons and squares, and use on each floor a different subset of these squares and hexagons as vestibules and galleries. For instance, we can start with this Archimedean tiling that has large dodecagonal Voids:Archi2 01

 

One individual floor then could look like this, seemingly giving each gallery three exits to vestibules with staircases, one of which, however, will be blocked off by a bookshelf on two of its sides:

Third 01

This floor plan will be rotated by 120º on each subsequent floor (about the center of one of the dodecagonal Voids), creating a single labyrinthian library.

Here is a deliciously maddening view into the resulting skeleton of this library. I haven’t closed off the inaccessible staircases yet, so please watch your step.

Babel5

Happy are those of us who can get lost in a single book like in this Borgesian library. 

The Library of Babel I

In the story The Library of Babel, Jorge Luis Borges describes a library whose design follows near axiomatic principles:

It is composed of an indefinite, perhaps infinite number of hexagonal galleries. In the center of each gallery is a ventilation shaft, bounded by a low railing. From any hexagon one can see the floors above and below—one after another, endlessly. The arrangement of the galleries is always the same.Babel2

One of the hexagon’s free sides opens onto a narrow sort of vestibule, which in turn opens onto another gallery, identical to the first—identical in fact to all. […] Through this space, too, there passes a spiral staircase, which winds upward and downward into the remotest distance.

If we remove all the cosmetics, we might end up with a design like the above, clearly unsatisfactory. Nothing is said about the underlying geometry of the library, Euclidean, spherical, hyperbolic, or even more esoteric. We will assume that the universe is Euclidean, for now, because this is still interesting enough. In this first post I will discuss the floor plans of a single floor. The combination of hexagonal galleries and square vestibules suggests that we are looking at floor plans that can be derived from this Archimedean tiling of the plane:

Babel 1

Of course the triangles will be Voids, and we have too many square vestibules. We get one more clue (or axiom) from Borges: Twenty bookshelves, five to each side, line four of the hexagon’s six sides […]

This means that each gallery has just two vestibules where one can enter or exit. As all galleries are identical, this leaves us with three distinct possibilities how a gallery can look like.

Babel straight 01

If we assume that the vestibules are placed at opposite sides of a gallery, our floor plan will necessarily look like the one above (which is used in the top image, too), representing a favorite labyrinth of Borges, the line (!). In the other extreme case, when the vestibules are at adjacent sides of each hexagon, there are two possible floor plans:

Babel 60

As a single floor plan, neither looks exciting, but we’ll see. There is one more option when between the two exits to the vestibules there is just one wall with shelves, like so:

Babel 120a 01

And fascinatingly, this last options allows for much more intricate floor plans, like this infinite double spiral:

Babel spiral 01

Next time we will investigate how the connections between different floors makes the life of the librarians even more exciting.

Tripods III

After realizing that while choices allow for free will, too many choices make everything possible and create only an illusion of free will. So let’s allow very few choices in our tripod-universe:

Two 1

Above are the tripods we are allowed to use, and below the first three generations of our expanding universe if we pick the first of the three above and place it at the center:

 

Two 2 01

We see that at each step, there are only two choices that occur along precisely one branch (marked red). So while the number of possible universe histories grows exponentially, the overwhelming majority of its inhabitants (i.e. the leaves at the end of the trees) don’t have a choice, their future is predetermined and can’t even be affected by the single monarch who can only determine their own future. More choice is needed.

Two 3

So let’s allow all six tripods that use three colors with just one color occurring twice as leaves. We choose one of them for the Big Bang at time 0. At time 1 we already have 27 different possible histories, because at each leaf there are always 3 choices that can be made:

Two 4

In the next generation, we will have already 19683 different possibilities. This looks promising, so let’s see how much these tripods can control their future. Below are two universes at time 2 that use only the colors yellow/green and yellow/blue at the leaves. Can you find a universe where all leaves are yellow? Or blue?

Two 5

Some more questions:

  • Suppose you succeeded in making all leaves yellow. How many more generations does it take you to make all leaves blue?
  • Can you have a universe where no two neighboring leaves have the same color?
  • Below is a universe where in generations 1,2, and 3 each we have an equal number of leaves of each color. Eg, in the current generation 3, there are 8 green, blue, and yellow leaves. Can you continue like this? Is there a recipe for it?

 

Two 6

HOF+ (Tetrasticks I)

A polystick is a connected finite subgraph of the grid graph, and a tetrastick is a polystick with four edges. There are 25 of them, counting mirror copies.

Tetrasticks 25 01

In other words, these are squiggles you can make with four strokes. They’d make a nice alphabet for people who are addicted to abstraction.

Hof 01

 

Today we are focussing on six of them, fattened and colored above. They are denoted by the letters H, O, F, +, and the mirrors of H and F. For reasons to become clear later we consider O and + also as mirrors of each other in a certain sense. The goal is to tile rectangles with them, like in the 3×7 and 2×12 rectangles below.

Example 01

There are many constraints on what tiles one can use, and how many. For instance, an a x b rectangular grid has a(b+1) vertical and (a+1)b horizontal edges, for a total of 2ab+a+b edges. This number is divisible by 4 only if a-b is divisible by 4, so squares are good for tiling, as are the two rectangles above. They both consist of 52 segments and thus require 13 tetrasticks. Below is a different example.

Example2 01

Note that all our 6 letter except for H and its mirror use two horizontal and two vertical segments. As the 3×7 rectangle has 4 more horizontal edges than vertical ones, we need at least four H-tetrasticks (or its mirrors) to tile this rectangle. We can use more, but then only an even number of them. Likewise, we need at least two H-tetrasticks to tile the 2×12 rectangle.

Example3 01

This brings us to today’s puzzle: Tile the 3×7 rectangle with your choice of 13 tetrasticks from our selection of six, and then use the same set to tile the 2×12 rectangle. The examples on this page are attempts that require to flip an H or an F into its mirror (or an O into +). Can you find a perfect solution that doesn’t require flipping a tetrastick over?