Bucket of Blood Street (Arizona II)

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The little town Holbrook in Arizona offers convenient accommodation after visiting the Petrified Forest National Park. This is not a wealthy town, but the downtown area has its own nostalgic charm. You wonder what life was like here a hundred years ago.

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Then you come across this street sign. Choosing a name is a delicate thing. Apparently, in the good old times a saloon shooting ended in such a way that the establishment was renamed the Bucket of Blood Saloon. In the long run, this didn’t help much, and after the building fell apart, the name survived as the street name, to this day.

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Other local attractions allude to that bit of the town’s history in appropriate color.

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The moral? Appearances change, names stay. But it seems the town hasn’t quite figured out whether that name is a curse or an opportunity.

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More about Decorated Squares (Five Squares III)

In order to classify surfaces that have five coordinate squares around each vertex, we were led to consider planar tilings with six different colored squares. Today we will discuss a special case of this, namely tilings that use just two of these squares. The only rule to follow is that colors of tiles need to match along edges. Here is an example:

Sample

To classify all tilings by these two squares (and their rotations), we first simplify by solely focussing on the gray color (making it dark green), and considering the blue, orange, green as a single color, namely light green. This way we get away with just one tile. Of course we hope that understanding how this single tile can fill the plane will help us with the two tiles above.

One

We first note that placing the tile determines three of its neighbors around the dark green square. So instead of tiling the plane with copies of this squares, we can as well place dark green squares on the intersections of a line grid so that for each cell of the grid, precisely one corner is covered by a dark green square, like so:

Shiftedrows

We first claim that if we do this to the complete grid, we must have a complete row of squares or a complete column of squares. Below is a complete row (given the limitations of images). The red dots indicate where we cannot place green squares anymore, because the grid squares have all their green needs covered.

Row

If we do not have such a row, there must be a square without left or right neighbor. Let’s say a square is missing its right neighbor, as indicated in the left figure below by the rightmost red dot.

Existence
Notice how the two grid squares to the right of the right dark square have only one free corner. We are forced to fill these with dark squares, as shown in the middle. This argument repeats, and we are forced to place consecutively more squares above and below, completing eventually two columns.

As soon as we know that we have (say) a complete horizontal row, directly above and below that row we will need to have again complete rows of squares, as in the example above. These rows can be shifted against each other, but that’s it. So any tiling of the plane by the dark/light green tile consists of complete rows or columns with arbitrary horizontal or vertical shifts, respectively.

Finally we have to address the question whether this tells us everything about tilings with the two tiles above. This is easy: Each dark green square represents a light gray square that is necessarily either surrounded by blue or orange tiles. So we can just replace each dark green square by an arbitrary choice of such a blue or orange cluster. The final image shows such a choice for the example above.

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It is now easy to stack several such tiled planes on top of each other, thus creating infinite polyhedral surfaces that have five coordinate squares at each corner.

Moonscape (Arizona I)

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My obsession (?) with taking pictures in moonlight is not so much due to a romantic trait of mine, but rather because of my more general fascination with alternate lighting.

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The Blue Mesa of the Petrified Forest National Park is part of the Painted Desert. The eeriness of the landscape increases in the moonlight, which brings out more blue than is really there.

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The pictures here were taken shortly after sunset with rapidly decreasing light and increasing exposure times.

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These last two pictures were taken in almost complete darkness. They show the landscape as we would see them with more sensitive eyes.

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My thanks go to the friendly park rangers who didn’t fine us despite staying after sunset.

More Examples (Five Squares II)

To review, let’s start with the following tiling

New 4 tile

Now use the dictionary below to replace each tile by the corresponding 3-dimensional shape. Each tile from the bottom row is an abstraction of an idealized top view (top row) of a rotated version of five coordinate squares that meet around a vertex (middle row).

Dictionary

By using the top left quarter, we get the top layer of the polygonal surface below. The bottom layer uses the same pattern as above with blue and orange exchanged. This is a fundamental piece under translations, and we can see that the quotient has genus 4. This also follows from the Gauss-Bonnet formula, which says that a surface of genus g uses 8(g-1) of our tiles (12 for the top and bottom each in this case.

Genus 4

Similarly, this tiling

Genus5 tile

encodes one layer of the following surface of genus 5:

Genus 5

To make things more complicated, the next surface (of genus 4 as well)

Plus 1

needs four layers until it repeats itself. Two of them are shown below.

Pluslayers

These tilings exhibit holes bordered by gray edges which complicates matters, as we will now also have to understand partial tilings (with gray borders).

Verticality

Near where McCormick’s Creek merges into the White River, the area becomes quite swampy and is often flooded. There are two views from a boardwalk trail through this swamp that have caught my attention. The first is a quadruplet of sycamore trees in the foreground.

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Clearly the weeds are about to conquer the world, you might think. Of course, the sycamores know better.

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The other spot is a hundred yards further down the board walk, where the view opens up into a stage like space where we wait in vain for a performance to begin.

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But it is us who are lacking the patience: The performance is happening, all the time, mostly without us.

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Columns

After looking at the intersections of symmetrically placed cylinders and obtaining curved polyhedra, it is tempting to straighten these intersections by looking at intersections of columns instead.

The simplest case is that of three perpendicular columns. The intersection is a cube. Fair enough. But what happens if we rotate all columns by 45 degrees about their axes?

Col 3 union

Before we look, let’s make it more interesting. In both cases, we can shift the columns so that their cross sections tile a plane with squares. Surely, every point of space will then be in the intersection of a triplet of perpendicular columns. In other words, the intersection shapes will tile space.

Col 3 shifts

Yes, right, we knew that in the first case. I find the second case infinitely harder to visualize. Fortunately, I have seen enough symmetrical shape to guess what the intersection of the three twisted columns looks like it is a rhombic dodecahedron.

But not all triplets of columns that meet do this in such a simple way, there is a second possibility, in which case the intersection is just a twelfth, namely a pyramid over the face of the rhombic dodecahedron.

Col 3 twist

Together with the center rhombic dodecahedron they form a stellation of the rhombic dodecahedron, or the Escher Solid, of which you have made a paper model using my slidables.

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Above you can see a first few of Escher’s solids busy tiling space.

The Spider

When I was in third grade, my father brought home a beautiful 2 volume edition of Edgar Allan Poe’s short stories, illustrated by Alfred Kubin.

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The genre name Horror Story describes very unsatisfactorily what Poe accomplishes. The conventional horror story utilizes a simple scheme: It wins our trust by first presenting a plausible scenario, and then abuses this trust in order to get away with less plausible events.

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In Poe’s best stories, this is not the case. The horror story is happening in the protagonist’s mind, and we become afraid that this same horror might as well infest our own brains.

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There are a few European stories that achieve the same effect, and one of them is Hanns Heinz Ewers’ story The Spider, from 1915. In it, the tenant of a small apartment starts to play a game with a woman in a window across the street: They make movements with their hands, which the other is supposed to copy. The narrator, whose diary we read, is at first surprised how quickly his neighbor can repeat his own movements, until he realizes that he is in fact, against his own will, only repeating the movements of the neighbor.

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This realization comes too late, obviously. No good horror story can end well. The same is true for Hanns Heinz Ewers himself, unfortunately. Despite having understood the machinations of manipulation, he fell under the spell of a much larger spider, even though he didn’t share their racial ideology, had conflicting sexual preferences, and his books were banned.

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Alchemy (From the Pillowbook X)

Here is a variation of the pillow theme. This time, the tiles are not based on squares as the regular pillows or on triangles as in an older post, but on 60 degree rhombi. I only use pieces with convex or concave edges, so there are seven different rhombic pillows up to symmetry, this time also not distinguishing between mirror symmetric pieces. The main diagonals of the original rhombi are marked white. For the purpose of the Alchemy game below, I call them elements.

Elements

These elements can be used to tile curvy shapes like the curvy hexagon below. Again, for the purpose of the game, I call such a tiled hexagon a Philosopher’s Stone.

Stone1

I leave going through the brain yoga to discuss tileability questions to the dear reader. Instead, here is the game I designed these pieces for.

Alchemy

A Game for 2-6 Players

Purpose

To complete the Magnum Opus by crafting a Philosopher’s Stone.

Material

The seven elements above in seven colors, colored on both sides, at least 4 of each kind for each player;
One transmutation card for each player;
One Philosopher’s Stone outline for each player;
Pencils and glue sticks.

Below is a template for the transmutation card. It shows a heptagon with the elements at its vertices, and all possible connections (transmutations, that is).

Transmutations

Preparation

All elements are separated into resource piles according to color/shape. Each players takes a transfiguration card and an outline of the Philosopher’s Stone.

Outline

Above is an outline of the Philosophers stone, with little notches to indicate where the corners of the elements have to go. The elements are shown next to it to scale so that you get the elements in the right size.

Completing the Magnum OpusGoals

The goal of the game is to accomplish the Opus Magnum by filling the outline of the Philosopher’s Stone with elements using as few transmutations as possible. Elements must be placed so that

at least one corner matches a notch or a corner of another element that has already been placed;
elements don’t overlap and don’t leave gaps;
no two equal elements may share a curved edge (but they may share a vertex).

Scoring

When a player has completed a Philosopher’s Stone, he or she determins the used transmutations:
A transmutation occurs in the Philosopher’s Stone when two elements share a curved edge.

The players record a transmutation on their transmutation card by drawing a straight red edge between two elements that share a curved edge in their completed Philosopher’s Stone.

The unused edges are then drawn black. The player with the largest number of black edges becomes the master alchemist.

Below is the completed transmutation card for the Philosopher’s Stone at the top. This was a pretty poor job, the player used all but three of all possible transmutations.

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One can turn this game also into a puzzle. Can you tile the Philosopher’s Stone with the seven elements that its transmutation card is the one below?

Transmutations puzzle

Pate Hollow Trail

Here are some early morning impressions from hiking the Pate Hollow Trail north of the Painetown State Recreation Area. The mood for the hike is set right at the beginning, thanks to the Little People who built this tree house.

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Nothing can go wrong after seeing it. The trail takes about three hours to hike and mostly follows the ridges. You get the usual fare of broken wood, that I always find photogenic.

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The other highlights of the hike are the views of Lake Monroe.

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Nothing dramatic, but a little blue is very welcome. When the trail touches down to the lake front, you can meditate either about structural simplicity of the opposite shore

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or the irritating complexity of simultaneous reflections of frozen weeds and preposterous trees. It is March, after all.

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