The 2-dimensional Nature of Time

DSC 3589

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.

DSC 3571

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

DSC 3554

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.

DSC 3511

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.

DSC 3507

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.

DSC 3488

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

DSC 3469

Plane and Simple (Solitaire XVIII – From the Pillowbook XIII)

Alan Schoen’s Cubons and Tetrons make beautiful and interesting puzzles, but few people will have the patience to build them. So here is a workaround. I begin with simplified tetrons where the edges are divided either 1:2 or 2:1. There are just 4 of them, and they nicely fit together into a single tetrahedron. Here are three views of the same tetrahedron. Smalltet1111

We now represent the tetrahedron by its edge graph K₄, and each cubons becomes a disk with three arrows placed on the vertices of this graph. The graph on the left represents the tetrahedron above.Tetrahedron 01

An arrow pointing away from the center of the disk means that the corresponding edge of the cubon is long, and short otherwise. So instead of elaborately assembling tetrahedra, we can just place one of the four types of coins on the vertices of the graph so that the two arrows at the end points of an edge point in the same direction. As an exercise, try to find the tetrahedron below among the four graphs above:


Here is a little worksheet so that you can cut out coins in our new currency. You will have realized that these four coins correspond to the four rounded trillows. In essence, we are doing nothing but decorating the vertices of cubic graphs with trillows.

Coins 01

The same procedure works for simplified cubons. There are of course again just four of them, represented by the same set of coins.


Instead of trying to parse a 3D image, we decorate the edge graph of the cube with our coins. Below you see what the cube on the left above looks like. Try to find decorations of the graph that correspond to the other solutions.

Cube2222 01

Next time we will look into decorating other cubic graphs.

Above and Below (Solitaire XVII – From the Pillowbook XII)

I wrote the first Solitaire post in March, exactly four months ago, being almost certain that after maybe two months I could safely move on to two person games. Now it looks like this will have to continue for a while. At least I can assure you that by the time I run out of topics, the pandemic will be over, one way or the other…

HeawoodMap 01

Today’s puzzle is concerned with the Heawood map. This is a map consisting of seven hexagons arranged as up above in the Heawood tile to the left, with edge-zigzags matching in pairs of equal color. This matching can be used to periodically tile the plane as to the right, or to interpret this map as a map on a torus, thus showing that one needs at least 7 shades of gray to shade a general map on a torus. (7 is indeed optimal).

Hexpillows 01

After the square pillows and triangular pillows, it is now finally time to introduce the 14 hexagonal pillows above. That’s all there is with curvy edges only — if you allow for straight edges, you get (too) many more. It is (for some of us) tempting to replace the hexagons of the Heawood tile by seven pillow tiles, so that the entire Heawood tile can be used to periodically tile the plane. If you only use one type or pillow, there are only two possibilities:

One 01

With two different pillows, it gets more interesting (and prettier). Below are two (slightly different) solutions using the same two pillows.

Two 2 01

And here are two more, again using the same two pillows, which are less similar. You can find four more by reflecting all these, but that’s it with two pillows.

Two 1 01


Now let’s jump ahead and try to use seven different pillows. Here is a simple example:

Simplex 01

How hard is this? There are 3432 ways to select 7 different pillows from the 14, but only 380 of these choices allow you to form a Heawood tile. That’s maybe not too hard. But, of course, you (I) would want  to assemble the remaining 7 pillows also into a Heawood tile. That’s today’s challenge, and I think it’s rather difficult (there are still many different solutions). The hint below may not be that useful. It merely shows the contours of the Heawood tiles for one particular solution to this problem. But at least it now becomes a very concrete puzzle: Just tile the two regions using all of the 14 pillows exactly once.

Hint 01


Food is art.

DSC 3005

Local Indiana goat cheese (which can be very good, as visitors from France have admitted) has become more local since the Goat Conspiracy from Bloomington added artisan goat cheese to their product line.

DSC 3012

Their (very limited) highlight (so far) is Phantasia, coated with a thin layer of charcoal to encourage the growth of the characteristic mold.

DSC 4042

At current temperatures, the cheese quickly starts to flow, calling back memories of the lava flows of Iceland.

DSC 3022

It goes exceedingly well the excellent local bread from Muddy Fork Bakery. What better is there to enjoy in this weather?

DSC 3079

Alan Schoen’s Tetrons (Cubons IV and Solitaire XVII)

After Alan’s cubons, now his tetrons. There is no end to it…

DSC 3027

They are constructed like the cubons: Take a tetrahedron, divide each edge into fifths, pick one of the four inner subdivision points on each edge, connect them to face centers and tetrahedron center to decompose the tetrahedron into four tetrons. Up to motions, each tetron is determined by the choice of three numbers from 1 to 4 (up to cyclic permutation), and, as for the cubons, there are 24 possible choices. In fact, the tetrons are just squished cubons.


The natural question for anybody obsessed with puzzles is whether these 24 tetrons can be used to assemble six tetrahedra. The answer is yes, you can group them in 11417 different ways into six sets of four so that this is possible. Above is one of them, and it is clear that this image is lacking, so below is the same solution, unfolded into nets. You will recognize what I discussed earlier as trions (albeit there with fewer subdivision points):

Sampletetronnet The tetrons are computationally much simpler than the cubons. For instance, we can again separate the 24 tetrons into 8 chiral and 16 achiral ones. Surprisingly, the 16 achiral ones can be assembled into four tetrahedra in exactly five different ways (up to rotations). Here they are, unfolded:


For the 8 chiral ones, the situation is a bit more complicated. There are two ways they can be grouped into two sets of 4, and in each case, there are two ways to assemble each set into tetrahedra. If we denote a single tetron by a list if three numbers that give the number of the chosen subdivision point as seen from the tetrahedron vertex of the tetron, then the partition of the 8 tetrons for the first solution can be denoted like so: {(1, 2, 3), (4, 1, 3), (2, 1, 4), (3, 2, 4)} and {(1, 2, 4), (4, 2, 3), (1, 4, 3), (3, 2, 1)}. Prettier are the nets. The first way to assemble them into tetrahedra is on the left, the second on the right.


And here is the other partition: {(1, 2, 3), (4, 2, 3), (2, 1, 3), (3, 2, 4)} and {(1, 2, 4), (4, 2, 1), (1, 4, 3), (3, 4, 1)}.

Again there are two different ways to assemble each quartet of tetrons into tetrahedra. I’ll show the solution next week.

From North by Hill, From South by Lake, From West by Paths, From East by River

So the title of a little book by László Krasznahorkai, better known for Sátántangó, and responsible for the stories behind a few of Béla Tarr’s films.

DSC 3353

Main protagonist is the grandson of Prince Genji, who is visiting a monastery near Kyoto.

DSC 3227

In 49 short chapters, we get a tour, both of the monastery, and of what the grandson perceives. Everything is treacherous.

DSC 3229

It is as if the visitor and the place are resisting their fictionality: Their possibility is enough to contemplate how place and visitor react to each other.

DSC 3276

Physical reality becomes secondary, what counts is the permanence of the imagination.

DSC 3300

9 is less than 10

DSC 3102

As hinted at earlier, Laowa has made a 9mm full frame lens, and I couldn’t resist.

DSC 3129

The lens is small, much smaller than the Samyang 10mm lens, which may come at a price. Its widest aperture is only 5.6, and there is a bit of distortion going on, but I can’t complain about sharpness.

DSC 3137

I think I’ll call this lens my black hole, it sucks everything in.

DSC 3147

No human eye is capable of a perspective like this. 

DSC 3153

The pictures here were taken a McCormick’s Creek State park, following the creek from its over-photographed waterfall  until it merges into the White River, a tributary to the Wabash, itself flowing into the Ohio, then flowing into the Mississipi. That’s a long trip for a little water.

DSC 3159

My trip was shorter, and a nice contrast to the loop of the Pate Hollows trail from a day ago.

DSC 3166

When you reach the end, that’s it. The only option is to turn back.

DSC 3172

Does one need this? A lens with extreme perspective? Or, to follow a stream until it ends?

DSC 3190

The reason is simple, sometimes: Turning back is not that different from changing the perspective.

DSC 3214

But learning is hard.

Seven Crossings

DSC 2825

Pate Hollow Trail is a 6 mile loop, with a few shortcuts as variations. In its longest version, it requires seven stream crossings.

DSC 2854

We follow them here counterclockwise, so to speak, if you look at the trail from above on a map.

DSC 2866

We also follow them in spring, when the streams carry water, for additional challenge.

DSC 2873

The crossings are small obstacles and excellent landmarks, telling how far we have progressed.

DSC 2876

The actual obstacles, however, are the in-betweens, where one has to climb up a few hundred feet just to descend again to the next crossing.

DSC 2882

So we can learn a lesson: progress is measured by landmarks, but achieved by what we do in-between.

DSC 2886

Care to join me for another round?

Pillars (Cubons III and Solitaire XVI)

This (for now) last past on Alan Schoen’s Cubons is dedicated to what Alan calls pillars.

DSC 3026

Above you see a page from one of several notebooks of Alan, introducing the pillars. A cubon solution has a pillar structure if all four horizontal faces are cut by unbroken lines. The pictures should make clear what this means. There are 456 ways to partition the 24 cubons into 3 groups of 8 so that one can assemble 3 pillar cubes.



If we restrict our attention to those that in addition have top and bottom face each cut into unbroken lines, there is only one such pair, consisting of all 16 symmetric cubons. They are shown above, in front and back view, and below as nets.


The remaining 8 chiral pillars can also be assembled into pillar cubes, in 8 different ways (not counting symmetries):


Last week I asked about polarity, which divides the set of 24 cubons into polar pairs, which use a complementary subdivision of the cube edges. It turns out that there is no solution to the problem to divide the 24 cubons into three sets of 8 so that each set can be assembled into a cube and consist of four pairs of polar cubons.

SymapolarOn the other hand, the eight achiral cubons obviously form four polar pairs (and can be assembled into a single cube). The remaining 16 symmetrical cubons can then be divided in four different ways into two sets of eight that are polar to each other, and that can both be assembled (in several different ways) into cube. Above are 3D solutions (one pair each column), and the nets are below.