June 2020 – The Inner Frame

Darkness and Light

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Nino Haratischwili’s book The Cat and the General is a difficult book. It talks about guilt, and the unhealthy death wish that can come with it. It’s also a long book, and might not satisfy the reader expecting satisfying exterior context. This books is about minds.

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The novel often appears to be talkative, giving too many irrelevant details, but these are just part of an undercurrent of themes that connect victims with perpetrators. One such pattern is that of Darkness and Light.

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After Sonja’s Death, Ada had begun to be afraid of the dark. She only wanted to sleep in bright light, holding a pillow in front of her eyes.

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“Why is there Darkness and Light”, he heard his daughter ask, then just five years old.

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— Because we couldn’t see the light without darkness, and the darkness not without light, he answered, and felt doubtful.

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— But why do I have to see darkness at all?

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Darkness is nothing but a disguise for the light!

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This dialogue between the general and his young daughter replicates a dialogue between Nura and her father, and is one of many parallels that live in the subtext of the book.

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The pictures here were taken during a recent visit to Turkey Run State Park.

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Besides Light and Darkness, there is also the theme of wood and rock in these images, of growth and strength.

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Alan Schoen’s Cubons (Solitaire XIV)

Alan Schoen is best known for the discovery of the gyroid, but he has also invented an enormous number of puzzles. The one I will discuss today he named cubons.

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Above you see his model of the 24 cubons which I currently have on loan for exploration.

Cubons of order n are obtained from a regular cube by dividing each edge into n+1 equal segments, and choosing on each edge one of the n subdivision points. These are then joined with the vertices on the same edge, the face centers on the adjacent faces, and the center of the cube. Adding faces results in eight polyhedra that can be assembled into a cube, obviously.

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There are n^2+n(n-1)(n-2)/2 different cubons of order n, which gives the sequence 1, 4, 11, 24, 45, 76, …

The 24 cubons for n=4 are particularly interesting because they might be used to assemble three cubes, using every cubon just once. Finding a single solution is not so easy, because there are 735471 ways to select 8 from the 24 cubons, but only 18844 will allow themselves to be assembled into a cube, many of them in several different ways.

Still, there are a mind-blowing 1050759 different solutions to partition the 24 cubons into three groups of 8, each of which can be put together into a cube. One might want to put additional restrictions on the solution. For instance, one could ask that each cube has an equator, i.e. four consecutive unbroken segments.

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The picture shows front and back side of each cube on top of each other, the back is obtained by rotating the fron by 180º about a horizonta axis parallel the screen. There are still 1887 solutions of this case. One can also aim for the opposite, namely insisting that there is no straight dividing edge on any face. This allows 6361 many solutions.

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An esthetically pleasing limitation asks to have parallel edges being subdivided the same way. There are only two different solutions, using 16 cubons. Unfortunately, the remaining 8 cubons do not fit together.

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One can also us the color of the faces to impose restrictions. I am using here Alan’s color scheme (which employs 5 colors, one for two of the twelve possible cubon faces that are visible in a cube), but I am sure there are many possibilities.

For instance, below is a solution so that each face uses two different colors. There are 2544 of those…

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Stereo Moss (Moss III)

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Continuing humidity has led to the prospering of micro-jungles also called moss.

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I have posted two sets of extreme macro images earlier, and today we continue with macro stereo images of moss.

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The images are all for cross-eyed viewing. Ideally, view them on a large monitor, sit back, and focus your eyes at a point half way between you and monitor. Then relax the focus to the monitor. If you have difficulties, try the last one first, then slowly scroll up.

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Hand made stereo images of still life-sized scenes are easy to make by taking two pictures, moving the camera approximately eye distance in between. Our brains are tolerant…

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This doesn’t work for macro images, one needs to move the camera by mere millimeters, using a focussing rail sideways. Enjoy.

Quadrons (Solitaire XIV – From the Pillowbook XI)

After discussing trions and hexons, it’s time for the simplest variety, the quadrons. They are obtained by cutting from a square a quadrilateral that has as its vertices the square center, a vertex, and two points on the edges adjacent to the chosen vertex.

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The points on the edges are chosen from the n possible points that divide the edge into n+1 equal segments. I call the number n the generation of the quadron. Above are the four generation 2 quadrons, which fit nicely into a single square, and can be regrouped to form the six square pillows I introduced a long time ago.

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For generation 3, there are 9 quadrons, so they don’t fit together into squares. But we can leave one out, and try to assemble the remaining ones into two squares. There are 9 pairs of such squares, but not all quadrons can be left behind. Which ones can? The solution has to do something with the area (which color codes the quadrons above).

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The 16 generation 4 quadrons fit nicely into four squares as shown above. There are 48 individual squares, which will make nice cards for another puzzle…

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Then there are 75 eye-straining ways to select four of these 48 squares to obtain a complete set of generation 4 quadrons.

Higher generation quadrons become very tedious. For generation 6, there are 36 quadrons, and 2139255 many ways to fit them into a set of 9 squares.

I promised a solution for last week’s puzzle, here it is. Now you can guess the second one yourself.

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Mushrooms

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Now that all the pretty spring wildflowers are gone, it’s time to pay attention to some of the other vegetation that prospers in the humid Midwest. As you can see, they still have to learn about social distancing.

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I know next to nothing about mushrooms. The beauty above is probably a coral fungus. But don’t trust me, and in particular don’t eat one just because you have seen it here.

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They grow with an astonishing speed, and take on shapes that range from gracile to monstrous. And they are usually gone after a few days.

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Some look so strange that I don’t even know whether they are mushrooms.

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The ones above were photographed with a macro lens. For the one below I didn’t have one, but I also didn’t need one.

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Araneae 2 (Solitaire XIII)

Last week I introduced the six balanced spiders which reappear below with a slightly improved design, and asked to use six of each kind to create a 6×6 spiderweb so that adjacent cards color match at the legs, and each row and column contains exactly one balanced spider of each kind.

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I got hooked on these little critters and wrote some code to hunt down solutions for me.

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The solution above is perfect in the sense that also left and right (resp. top and bottom) edges match, so that we can get a toroidal spiderweb (or an infinite periodic one). There are still several hundred solutions. Not every possible first row can be extended to a perfect spiderweb, but the one below extends to a unique one. Can you find it?

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There are other subsets of the 20 possible spiders that make nice puzzle sets. For instance the following are all possible spiders that have four different side patterns, i.e. one edge has to have green legs, another one pink, the remaining two mixed colors in different order. Let’s call them mixed spiders.

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We can again ask for perfect 6×6 spiderwebs, now using mixed spiders. In this case, there are only two different solutions. Below is the first row of one of them:

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I have no idea how hard it is to find the solution by hand. When I saw it, I was a bit upset. I will post it next week…

Tradescantia (Spiderwort)

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The spiderwort is a late spring wild flower. The local kind has blossoms that close a night and dangle down, suggesting spider legs.

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But maybe it’s the long leaves that are often bent like spider legs. In either case, each individual plant is an architectural master piece.

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The flowers have three petals. Unlike trilliums, I have never seen a four leaf variation . Another explanation for the spider name are the hairs around the anthers.

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Maybe we should consult an expert. They seem to like these flowers, too.

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Araneae (Solitaire XII)

Spiders are, as we know, square shapes with eight legs extending in pairs to the four sides. The legs come in two colors, and each color occurs four times. There are 20 different suborders (up to rotations), shown below. They are related to quadrons, to be discussed at a later time. Araneae 01

Today we will focus on balanced spiders that extend legs of different colors to each face. There are just six of them: Six 01

Spiders build spiderwebs by holding together along equally colored legs, as shown above. They prefer it if no two identical spiders occur in the same row, also as shown above. This single row of all six balanced spiders was easy.

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The second row is a little harder, consisting again of all six different balanced spiders. We had to rotate some of them, they don’t mind that.

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Adding more rows seems to be equally hard. Is this always possible? Then it should be possible to arrive at a 6×6 square of balanced spiders, each row a complete set of the six.

The dream of the spiders is to build a 6×6 spiderweb so that also all columns contain a complete set of balanced spiders. Please help them.