Month: October 2016

Pillow Puzzles (From the Pillowbook V)

After admitting a few pillows with straight edges, there is no end to it. Here are all 24 pillows based on a square that either have a straight, concave, or convex edge. We disregard rotational copies but keep mirrors.

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Usually, polyformists try to tile simple shapes using each polyform exactly once. The archetypical example is to tile a 6×10 rectangle with all 12 pentominoes. This is in most cases a tedious exercise that doesn’t teach you much more than backtracking. On the other hand, nothing is worse than not knowing, so here you go: Three puzzles that ask to tile the outlined region by using each of the 24 pillows exactly once.

Puzzles 01

The grid is there to help placing the pillows. These puzzles are actually not so bad. The first one for instance requires to make economic use of the pillows with straight edges. I post the solutions below, mainly because nobody would do them anyway and to prevent future waste of time.

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Note in the solution above the second column consisting entirely of pillows with parallel straight edges. I think this has to appear in any solution of this puzzle.

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The one above is my favorite. Unfortunately, one could go ahead and ask to find solutions of similar puzzles where the shape of the hole in the center is any of the remaining 23 pillows. No.

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Asleep and Awake

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After visiting Brown County State Park on a very foggy earl Fall day, revisiting the same location two days later on a very sunny day shows a different landscape.

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Just before sunrise, the lake is still partially covered with morning fog, but within an hour, the appearance changes completely.

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That one hour of snoozing gives plenty of time to walk around Strahl Lake,

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slowly separating dream and reality.

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Cubes, Cylinders and Triangles

If you don’t have the bricks available that I used as substitutes for a rhombic dodecahedron, you can still make simple models jut using cubes: Take an ordinary cube, and choose three edges, one in each coordinate direction, and so that they don’t share a vertex. There are, up to rotations, two ways of doing so. Let’s call them blue and red. Make a few dozen of the blue cubes.

Bluecube

Now comes the tricky part: You are only allowed to attach two cubes so that they share one of their blue edges. This is fairly easy in zero gravity, or in your favorite computer software, like Minecraft. The structure you get this way is yet another version of the Laves graph. This looks clumsy, but it is useful for prototyping things. It also gave me the idea of a further reduction that is even harder to hold together but much more elegant: Replace each marked cube by the equilateral triangle that has its vertices at the midpoints of the marked edges.

Triangles

Now one even has plenty of room to show the two intertwining Laves graphs simultaneously. What one cannot see very well in the above ethereal image is that if one orthogonally pierces a cylinder through the midpoint of any triangle, the cylinder will periodically hit other triangles in the same way, without interfering with any other triangles or cylinders.

Cylinders

Out of the sudden, there is structure. And it gets better: Because the cylinders don’t interfere, we can make their radii so big that they reach the vertices of the triangles. This way the cylinders will touch precisely at the vertices of the triangles. This means that the cylinder packing that uses cylinders in all four directions of the diagonals of a cube can be used to construct the Laves graphs: Determine where the cylinders touch. Each of these points belongs to two equilateral triangles equitorially inscribed in the two touching cylinders. Use the triangles centers as vertices of the Laves graph, and connect them by an edge if the triangles meet at a vertex.

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Landscape in the Mist

Fall has arrived.

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As you can see, the Indiana landscape does have opportunities for outlooks, at least in theory.

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I used the first rainy fall day to revisit Brown County State Park with its two lakes.

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My favorite lake front at Strahl Lake has changed only little since my first post about this place, even though some trees are dying.

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There is also Ogle Lake below, which is larger and not as intense.

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The far side of it is more interesting, with groups of trees guarding the secrets of the place.

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Walking through a landscape in the mist has become a ritual since I first watched the film by Theo Angelopoulos with the same title. Fog, light, and borders will never mean the same again.

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More Choices

Last week we saw that using just the left handed of the two bricks that I based on the rhombic dodecahedron produces nothing but the Laves graph. Using the right handed brick makes the mirror image of the Laves graph, and one can see

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that they intertwine nicely. Of course it would be better to have real bricks, and with help from Martha and the friendly people at MadLab of our Fine Arts School, I could play with a few dozen left and right bricks.

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In the above picture left and right bricks are color coded, and the sculpture starts with a hexagonal ring and then grows tentacles in a single color. These will come together and close, but leaving gaps looked more interesting.

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Here (above and below) you can see that I cheated, because I am also using a brick with four sides. It is geometrically much simpler, but of course still based on the rhombic dodecahedron, replacing four of its sided by their inscribed ellipses, and then taking their convex hull. This allows for tighter loops as in the image above, and allows for more design options.

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Another Brick in the Wall

When Apple announced in July this year they had sold 1 billion iPhones, I started wondering about another brick maker: How many blocks has Lego made? Their friendly customer service couldn’t tell me how many elements they have made in total, but the yearly production is 19 billion. Scary. Unfortunately, the shape of the standard lego brick is too limited for my needs. For a long time, I had wanted a lego brick in the shape of a rhombic dodecahedron (better would be a four dimensional lego hypercube of which the rhombic dodecahedron is a mere shadow, but let’s not be delusional). As you can see, this polyhedron tiles space as well if not better than the cube.

RhombicDodecahedronTiling

Various companies have produced shapes with more or less cleverly embedded magnets, but keeping track of the polarity on all faces of a 12 sided object is tricky. And this would be a lot of magnets. The actual problem, however, is the enormous amount of choices one has: 12 faces to attach to is just too much. I strongly believe that Lego’s success stems from the fact that they have reduced the number of possible ways how you can attach two lego pieces dramatically. No choice means dictatorship, two choices US capitalism, but more choices sounds like European liberalism or even anarchism, and we see where that leads.

This gave me the idea to replace the complicated rhombic dodecahedron by a simple object that is less attachable. Here is the new brick.

Brick

To make it, take three faces of the rhombic dodecahedron that are symmetrically positioned, and replace each of the three rhombi by its inscribed ellipse. Then take the convex hull of the ellipses. The resulting shape consists of the ellipses, two equilateral triangles in parallel planes, and three intrinsically flat mantel pieces.

You will notice that there are two versions of this brick, a left and a right handed one. This leaves just the right amount of choices.

Hexring

If you alternatingly attach a left to a right brick, you get a hexagonal annulus. Remember that we are still tiling space using slimmed down versions of the rhombic dodecahedron. Due to our imposed limitation of choice, nor every place can be reached anymore. The hexagonal annulus is a little simplistic. What do we get if we just use the left handed brick?

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Let’s start with a red central brick, attach a brick on all three sides, and another six at the free faces of the new bricks. We notice that the bricks can occur in four different rotated positions. I have distinguished them by color. Add another 12 bricks:

Prestar

And another 24. No worry, no intersections can occur, because, I insist, we just tile a portion of space with rhombic dodecahedra.

Star

Now we see that the tree like structure we have produced so far does not persist. In the next generation, we obtain closed cycles of length 10, and we finally recognize the Laves graph.

Ball

In the very near future you will see what else one can make with these bricks.

Scherk in Clay

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This innocent minimal surface, which can be obtained from Heinrich Scherk’s traditional surface by adding two wings and bending them towards each other, poses interesting challenges when printed (vertically, i.e. rotated by 90 degrees) in clay. First of all, there are three horizontal cross sections which look like branches of hyperbolas (but aren’t, not even for the original Scherk surface, in contrast what Wikipedia currently claims).

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When printing this layer by layer, the nozzle has to move from branch to branch, and as the printer can’t stop printing while it skips across, it leaves hairy artifacts.

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They clearly have their own charme.

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Another problem arises from the saddle points that are printed without support. This leads to other imperfections and sometimes structural complications that might take away from the elegance of the original surface but contribute to wild interior landscapes.

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Watching the printer work for two hours is dramatic, because failure in the form of collapsing walls can happen any minute.

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Up?

Let’s continue the delightful examination of the green-white-black pre-Fall landscape of Indiana.

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Once in a while it overcomes me and I want to be able to look around. Unfortunately, most of the Midwest of the USA has no mountain peaks to climb, so one is pretty much limited to a horizontal, 2-dimensional perspective.

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This gets worse in Clifty Falls State Park, where the main (and most exciting) option is down. At the bottom, one can only walk along the creek, and is reduced to a 1-dimensional perspective. There are traces from a still existing outside world, mostly in the form of large boulders.

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We do not ask what’s underneath. And yes, there is life, if given time.

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Thoughts about a way out seem preposterous. How dare we think about an up when there is only forward?

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So, instead of the longed for outlook I had another look inward, reducing everything to the very next step. There are many ways rinse oneself.

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