From Space to Plane (Five Squares I)

Euclid allows us to place four squares around a vertex. If we are not satisfied with that, we can either move into the hyperbolic plane, or into space. A neglected configuration is that of five squares parallel to the coordinate planes that meet at a common vertex, like so:


This is, as you convince yourself quickly, the only way to to this, up to 24 symmetric cases. The squares trace out a polygonal arc on the faces of the cube above, which we can interpret as a marking of the faces of the cube that contains the five squares. Two faces are marked by a straight segment (front and  left), three by an L-shaped segment (back, right, top), and one face is unmarked (bottom). As we did with the simple six color marking, we can centrally project the 24 cubes into the plane.


The image above shows six of these projections. The remaining ones can be obtained by 90 degree rotations. I have colored the faces of the cube to indicate what the path is doing within that face (green=go straight, blue=turn one way, orange = turn the other way, gray=don’t be there). Convince yourself that the colors suffice to reconstruct the path.

Thus we obtain a set of six tiles that allow us to explore layers of polygonal surfaces that have five squares around each vertex. For prettification, I have dropped the path and filled in the hole. No information has been lost. We are allowed to place tiles next to each other if the colors match. This is it:


Here is a simple example of such a surface.


It is triply periodic and incidentally related to Schwarz P minimal surface. Two consecutive horizontal layers are represented by the two tilings below:


So, we have the burning questions: Are there more polyhedra like these, do the tilings help us, and can we understand the tilings? More about it in a week or two.

Brown (Charles Deam Wilderness II)

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Indiana has a fifth season: A good winter brings snow, harsh light and the contrasts that blind your eyes. A good spring brings mild air, green buds, tree blossoms, and wild flowers. But in between, there is at least a month of nothingness.

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It is the season of decay, and its color is brown. The contrasts of winter light disorient us because they provide information conflicting with the physical landscape. It is almost as if a fourth dimension has been added which we cannot parse. Today, the low contrast of an overcast sky and the muted colors make the contours disappear and appear 2-dimensional.

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Still, the monotonicity has its appeal, in particular when it is interrupted by an alien intrusion.

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Why should we hate what we are attracted to?

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We can play domino with identical copes of a single cube by insisting that the cubes have matching colors at the faces where they touch. This is hard to convey in a perspective image like the one above that shows a 4x4x1 cubomino tiling, so I will switch to a 2D representation that shows each cube in central perspective from above. Here is the flattened version:


As you can see, the single 3D cubomino can be represented by six 2D cubomino squares, which may be rotated:


These are a subset of the tiles I used for the Compass game a while ago.

This new Cubomino puzzle is a simple example that teaches to analyze tilings by understanding them as boundary value problems. To see how this works, we first notice that to some extent the color of the lower and the left edge of a tile determines that tile and its rotation.


It works for most color combinations, but there are exceptions. These occur precisely when the two chosen colors for bottom and left edge are antipodal, i.e. are either equal or are one of the three pairs of colors of two opposite faces of our cube:


This observation extends to larger rectangles: No antipodal colors can occur both in the left edge and the bottom edge of a tiled rectangle. To see this, assume the contrary. The vertical edge on the left of the rectangle that has one of the antipodal colors (pale yellow below) determines a horizontal strip of tiles that have the antipodal colors as vertical edges, and the horizontal edge on the bottom of the rectangle with the second color from an antipodal pair (dark purple below) determines a vertical strip of tiles that have the antipodal colors as horizontal edges. These two strips meet in a tile that must have a boundary consisting just of antipodal colors, which is impossible.


On the other hand, if the left and bottom edge have no antipodal colors in common, like in the example below,
there is always a unique tiling of a rectangle that has the two edges as its lower and left edge.


The reconstruction process is easy: Each choice of a left and bottom edge color determines a tile and its rotation uniquely. We begin by placing the only possible square into the bottom left corner of our boundary, and work our way to the right and up.

Winter Light (Charles Deam Wilderness I)

The history of the settlement of Indiana has been a history of forest destruction.
So the first sentence of the preface to the second edition of Charles Clemon Deam’s book Trees of Indiana, from 1919.

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Deam was Indiana’s first state forrester, and the state’s only designated wilderness area is named after him. The Charles Deam Wilderness is located in Karst hills bordering Lake Monroe. Deforestation here is pointless, as the ground is not suitable for farming. That there are still plenty of trees in southern Indiana does not contradict the above sentence.

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In today’s images I have been trying to catch some of the harsh light that one can experience on cold winter mornings (This year, this has not been easy).

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When colors retreat and contrast is everything, even simple landscapes can become disorienting.

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In truth, there is nothing here but water, ice, wood, and sky.

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Lakes should be horizontal, and trees should grow vertical.

The Advantages of Different Viewpoints (From the Pillowbook IX)

It all started with a question about polysticks: I wanted to see how to tile parts of the square grid with 3-sticks so that there are always three 3-sticks touching. A 3-stick is nothing but a T with all segments the same length: it has 3 arms that end at the hands, and are joined together at a head. By tiling I mean that the arms align with the edges of the standard square grid and don’t overlap. And, as I said, I want always three of them to hold hands. Here is an example (that you should imagine continued periodically):

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The condition about holding hands in threes means that each such tiling has a dual tiling where the new 3-sticks have their heads wherever three hands come together, the same arms, and the previous heads are replaced by three hands. This also implies that a tiling and its dual will tile the same portion of the square grid, as below to the left and right.

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We can also combine a tiling and its dual into a single figure by centrally scaling each 3-stick by 50%, and taking the union: The gaps created by the scaling makes room for the 3-sticks from the other tiling. You can examine the result up above in the middle. The new skeleton will have all 3-sticks hold hands in pairs instead of in triples. Here is another, more complicated example. The original periodic 3-stick tiling:

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The inflated version that combines the original with its dual:

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In the inflated version we can replace each 3-stick by a square so that the sides touch when the 3-sticks hold hands. The unused edge of the square is pushed inwards, turning the square into the familiar 3/4 pillow we admired the last time (the next image shows only a quarter of the previous piece. It repeats itself using horizontal and vertical translations).

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As a final simplification, we can fill in the holes as follows: We replace the 3/4 pillows that border a hole by the polyomino they cover  (thereby filling in the hole). Below are the two simplest polyominoes that surround a hole:

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These polyominoes will tile the plane as before, because each 3/4 pillow must belong to exactly one hole. Our 3/4-pillow tiling now becomes a very simple polyomino tiling:

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Thus periodic 3-stick-tilings with triple hand holding, 3-stick tilings with hand holding in pairs, 3/4-pillow tilings with holes, and  tilings by polyominoes that surround holes are all the same thing.

You can use this to design  much more intricate patterns, with holes of any size, for instance.

Sources of Healing

Last October I went on an early morning to McCormick’s State park, not expecting to see anybody.

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But there was this guy, sitting next to the little waterfall in the dark. We started chatting. He was from Florida, evacuating for the week because of Hurricane Matthew. Friends had told him to check out this place, and he was quite impressed.

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They had also told him of a Spring of Healing that could be found here, and he wanted to know about it.

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I couldn’t help.

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But it is true that some places are special. My virtual substitute here can only be a reminder that they still exist.

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They need our protection.

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Intersecting Cylinders

After learning how to intersect two cylinders, let’s try it with more. The next simple case are three cylinders, and the most symmetric case has them aligned along the coordinate axes. The union is on the left, and the (rescaled) intersection on the right.

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We obtain an inflated cube that has twelve cylindrical rhombi as faces, one for each edge of the cube. This is therefore a version of the rhombic dodecahedron. Using four cylinders, the most symmetric way is to align them with the diagonals of a cube. We see here (again) that each cylinder contributes a chain of cylindrical faces that touch at opposite vertices.

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For six cylinders, there are two notable choices for the axes. The first is to use the six edges of a tetrahedron, recentered so that they all pass through the origin. Alternatively, we could use lines that connect midpoints of opposite edges of a cube.

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The other, even more symmetric choice is to use the diagonals of an icosahedron. Using axes related to the platonic solids creates simple patterns that can all be understood in terms of finite reflection groups acting on the sphere.

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By now, I have probably created the impression that the appearance of the chains of equally colored cylindrical faces has something to do with the symmetry of the chosen axes. Not so. The next image shows a random choice of 10 cylinders of equal radius through the origin, and their intersection to the right.

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The black curves are the waists of the cylinders, i.e. their intersections with the sphere of equal radius centered at the origin. The waist of a cylinder must belong to the intersection, because any other cylinder will properly contain it. Thus our chains appear along these waists, and they are pinched whenever to waists meet.