The Inner Frame

Infidelity (Mushroom 3)

After harvesting, cleaning, processing, eating, freezing, photographing my beloved chanterelles for over a month, it is maybe time to elope a little and have at least a careful look at what else is out there in the local woods.

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Let’s keep it anonymous. I don’t know enough about mushrooms to tell you their names.

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Nor do I know how safe they are. After all, I am just looking…

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Their beauty is admittedly tempting, though.

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Well, some of them look tastier than others.

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And this one’s best days are over. To be continued…

Colored Pillows (Solitaire XXI – From the Pillowbook XVI)

Another lonely day gave me the idea to spice up the six standard pillows into domino type puzzle pieces by coloring them like so:

ColorPillows deluxe

Below is a less pretty but easier to cut version, assembled in a 5×2 rectangle,

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which solves what a mathematician would call a boundary value problem:

Boundary 1

Here is another puzzlable boundary contour for your solitary enjoyment. The rule is to place the puzzle pieces so that they fit & match in color when they meet:

Boundary 2

These are nice little puzzles, not too easy, not too hard, but 10 is an awkward number (why don’t we have 16 fingers, like everybody else out there?), and it is somewhat annoying to have to turn around some of the pieces by 180º to see whether they finally fit, so I decided to modify the coloring a bit, like so:

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The 16 puzzle tiles above must not be rotated anymore. Equivalently, the horizontal and vertical color gradients need to match with adjacent fitting pieces. The pieces above all fit together to form a 4×4 square which periodically tiles the plane, moreover, this square is symmetric across one of its diagonals. Can you find other such square, tiled by using each of the 16 colored pillows exactly once? There are a few, but not too many.

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These 16 puzzle pieces would make great 2-person games, for horizontal and vertical players… Their time will come. For now, enjoy the two boundary value problems above. One of them is very very very hard.

Housekeeping

The quotes are from Marilynne Robinson’s first novel Housekeeping. It is rare that the imagery of a book resonates so strongly with how I experience nature. The book revolves around three complex metaphors, the Lake, the House, and the Bridge. It would be a bit simplistic to view them as tokens for past, presence, and future, because they each hold much more.

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It seems there was a time when the dimensions of things modified themselves, leaving a number of puzzling margins, as between the mountains as they must have been and the mountains as they are now, or between the lake as it once was and the lake as it is now.

Indeed, where we were we could feel the reach of the lake far behind us, and far beyond us on either side, in a spacious silence that seemed to ring like glass.

He had grown up in the Middle West, in a house dug out of the ground, with windows just at earth level and just at eye level, so that from without, the house was a mere mound, no more a human stronghold than a grave, and from within, the perfect horizontality of the world in that place foreshortened the view so severely that the horizon seemed to circumscribe the sod house and nothing more.

Imagine that Noah knocked his house apart and used the planks to build an ark, while his neighbors looked on, full of doubt. A house, he must have told them, should be daubed with pitch and built to float cloud high, if need be. A lettuce patch was of no use at all, and a good foundation was worse than useless. A house should have a compass and a keel. The neighbors would have put their hands in their pockets and chewed their lips and strolled home to houses they now found wanting in ways they could not understand.

I believe it was the crossing of the bridge that changed me finally. The terrors of the crossing were considerable.

Something happened, something so memorable that when I think back to the crossing of the bridge, one moment bulges like the belly of a lens and all the others are at the peripheries and diminished.

Two (Prairies II)

Today’s pictures are from an early morning visit to the prairies of the Columbia Mine Preserve in the middle of summer.

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The light is extraordinary, as is the variety of life around you.

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In a few weeks, all the insects will become quiet, and the migratory birds will get ready to leave.

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The grass will become brown,

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and the trees will explode in color.

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This has been a strange year, and rarely have I longed more for it to be over.

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But there are still choices to be made.

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Chanterelle (Mushroom 2)

There are, more or less, two kinds of mushrooms: Those you can eat safely, and the other ones. Among the edible, I can identify morels and chanterelles. The latter ones have become abundant lately:

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Over the last few weeks, I probably have collected about 10 pounds.

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They are not only beautyful, they also taste extremely well. So here are two micro-recipes:

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Step 1: Clean. A toothbrush will do. Don’t be squeamish, a bit dirt won’t harm you. This will also help to sort out the fake chanterelles, in case you accidentally got some. The fake chanterelles have true grilles, while the true chanterelles have fake grilles. Easy.

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Fry at medium heat. For immediate consumption, I use a bit of olive oil, garlic, pepper, sprinkle with cilantro. In that form, they freeze well, or better are eaten right away. I like them with tagliatelle noodles.

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A second option is to use lots of butter. When almost done, add a bit tomato paste and vinegar, and blend it carefully with more butter until smooth. Spice it up, if you like, but make sure not to overpower the delicate taste of your chanterelle butter.

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This will go well with the right kind of bread, for instance a decent loaf of baguette (one doesn’t always need to have a thick crust…)

The butter will also freeze well. If there is anything left.

Inflation (Solitaire XX – From the Pillowbook XV)

In order to have a planar realization of Alan Schoen’s tetrons and cubons, we need to add a few coins to our currency. Here are the 24 coins you’ll need:

Cubarrows

Below is an example how to decorate three cube skeletons with them. This doesn’t look as pretty as the 3D-cubons, but one can much more easily play with them. Coins need to be placed on the vertices of a graph so that the two arrows that share an edge have the same color and point in consistent directions.

Cubons plane 01

What other pretty cubic graphs are out there? The Foster census lists one cubic symmetric graph with 24 vertices (the Nauru-graph), and it is a challenge quite in Alan’s spirit to try to decorate it with his 24 cubon-coins. This is indeed possible:

Naura

The above representation of the Nauru graph lives in a hexagonal torus. It is also the dual graph of the Octahedral 3¹² polyhedral surface, which is the genus 4 quotient of a triply periodic polyhedron by its period lattice. Here is a picture of it. Note that this is twice a fundamental piece, as the period lattice is spanned by the half main diagonals of the cube.

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Octahedral 3¹² is intimately related to Alan Schoen’s I-WP surface. Everything we do at a certain depth is connected to everything else, it seems.

Skeletons (Prairies I)

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Now that the summer here has (temporarily) cooled down a bit, I took the chances and visited the Columbia Mine Preserve, 90 minutes away, in order to see the prairies in full glory. I had to pay a price, but it was worth it.

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They are teeming with life, with many insects, and fortunately for me, also many insect eating insects.

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Most of us humans deny them consciousness, so maybe they don’t know how beautiful they are.

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They are beautiful to us, but what gives us the right to think that they are only beautiful to us?

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Maybe we shouldn’t be so arrogant, but instead be thankful for what we are allowed to witness, for a limited time.

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Decorated Maps (Solitaire XIX – From the Pillowbook XIV)

We continue decorating cubic graphs with our four coins:

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Below is the complete bipartite graph K₃₃, realized as the edge graph of the Heawood map of the torus. Remember that opposite edges are identified. Heawood 01

Here is a little puzzle to warm up. If we order the four coins as up above, we can for each decoration of the Heawood map make a tally like 0330 which lists for each coin how often it occurs. The 0330 is the tally for the simple solution to the the left. Besides that, the following tallies are possible: 1221, 1140, 0411, 1302, 2112, 2031, 3003. Find one decoration in each case.

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A more interesting map is the Möbius-Kantor map on a genus 2 surface, represented by an octagon with opposite edges identified. The map consist of 6 octagonal regions. Can you decorate it so that the boundary of each octagon uses each type of coin exactly twice? Here is a hint: This map is the double cover of the cube map, branched over the centers of the faces. So you if you can first decorate the cube such that each square uses each coin once, you can lift this decoration…

Pappus 01

Finally for today, here is the Pappus map on a torus. Again identify opposite edges of the diamond, matching all three coins on one side with the three corresponding three coins on the other side. Can you decorate this map using one blue coin, and for the rest only use purple and brown coins? It will help to remember what we learned about deficits for pillows a long time ago.

Archilochus colubris

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It’s almost time to say good-bye to the mimosas this year, but at the moment, they are still attracting visitors.

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This is the ruby-throated hummingbird. Annoyingly, I haven’t seen any adult males this year, who are responsible for the name. The adult females have, if my information is correct, a clear white throat, while the young males have a speckled throat, as the ones here.

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They are hyperactive all day round, chasing each other as well as the larger birds, and chattering intensely. They must have a lot to talk about, unlike me. The one above is poking his tongue out a little. I think it can extend it quite a bit further. Amazing possibilities…

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I never expected to take wildlife photos, I am not that patient. But sitting in the garden with a good cup of tea and just waiting for the birds to get into position is a surprisingly pleasant way to spend a Sunday afternoon.

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The 2-dimensional Nature of Time

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I must have said this before, the DePauw Nature Park challenges me to new views. Today I am trying to combine a wide format (3:1) with a shallow depth of field.

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These are per se contradictory, and to be effective, the shallowness has to be extreme.

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This sliver of sharpness acquires a strong horizontal nature, like a line of text in a book that we read, oblivious of the past and future lines. Time becomes horizontal. There is only one way to move, everything seems to be determined.

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Sometimes, it also becomes discrete, when there doesn’t seem to be a before or an after. There just is the singular moment, evidently still full of potential.

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This becomes less effective for things far away (or in the distant future) when it still seems possible to move forward and not just sideways, giving us the hope that there is free will.

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Is this all just perception? Can we think the barriers out of the way, by looking at them properly?

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