Triangles and Squares

There are two Archimedean tiling using triangles and squares.

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Both of them use twice as many triangles than squares. I find the first one is more interesting, maybe because it is chiral. There are still many other ways to tile the plane say periodically with just triangles and squares. There are three different ways to assemble two triangles and a square, and all of them give polyforms that can be used as a single subtile for the first Archimedean tiling:

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Among these three polyforms I like the middle one best, maybe because it cannot be used to subtile the second Archimedean tiling. It is an amusing exercise to doodle around and find other tilings of the plane with this tile. Here, for instance, are two small turtles and a giant caterpillar, all part of a big creation.

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I find it amusing how this simple polyform lends it self easily to organic shapes and abstract designs.

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There are (I think) 10 ways to combine two of them into a single polyform, not counting mirror images. At least two look like cats.

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Confusing as they look, almost all of them tile the plane. The two exceptions are shown below. It is not difficult to find an argument why these two do not tile.

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More interestingly the other eight tile, even though they look much more complicated. Typically one needs for each tile its mirror, suitably rotated. Here are two pretty examples. Homework is to find the others.

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The Loop

Once again I am returning to the fascinating Pine Hills Nature Preserve in Shades State Park, walking the loop trail there.

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I have done this several times, at different seasons, and both the fact that I keep repeating this hike and that it itself is a loop (returning to its beginning) makes be wonder about the purpose of this.

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Return and repeat: Aren’t these early signs of failure? Wouldn’t it be better to give up and move on?
After being exposed to Iceland’s permeating Black, Green, and White last summer, I was surprised to find the same monochromaticity here, in late summer.

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Green is a difficult color, and doesn’t pair well with a single other color I think, but it does exceedingly well in combination with black and white.

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When we return, we are different, and view things differently, and possibly even the completion of a loop teaches us something new. That what makes us repeat is maybe the feeling that there is unfinished business, that the circle has been left open, in the way the ensō brush stroke is often left open.

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So the loop, as a pattern, is nothing but a sophisticated mechanism to move on.

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Pictures are Better than Words

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With Fall around the corner, it is time to revisit a few friends. One of the less traveled trails in Shades State Park is the loop #2 in the eastern part of the park (but still west of Pine Hills Nature Preserve).

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It begins with a steep descend to Sugar Creek (using stairs).

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One can wade through the creek westwards about 100 yards to get a view of Silver Cascades Falls

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and then turn back

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in order to continue upwards into Pearl Ravine.

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This is again steep and sometimes very wet. After some minor obstacles

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one reaches the Maidenhair Falls.

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They are small but pretty.

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From there, it goes up and out.

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Arrows (From the Pillowbook IV)

So far, we have looked only at pillows with concave and convex edges. Today, we begin also to allow straight edges. To keep it simple, let’s look at the three different pillows that have two straight edges, one concave, and one convex edge. Here they are. I call them the arrow pillows.

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Because they have straight edges, we can finally tile rectangles that have straight edges, too, like so:

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There are a few immediate questions: Is this always possible? Can we say something about the number of arrows of each type we need? The key to the answers is indicated in the right image. The convex edge of one arrow pillow (the predecessor) fits snugly into the concave edge of a second arrow pillow (the successor), thus providing us with a recipe to move from one pillow to a neighbor. If we have a tiling of a rectangle just by arrow pillows, this sequence of consecutive successors must form a closed cycle. Therefore, the entire rectangle will be covered by possibly several such closed cycles, so we have what is called a Hamiltonian circuit. Readers of my blog have seen these before.

Vice versa, given any Hamiltonian circuit and a direction for each component, we can lay out the arrow pillows along each path to obtain a tiling. Below are two more examples with two components each that use only right and straight arrow pillows.

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Can you tile a 5×5 square with arrow pillows? If you checkerboard color the rectangle black and white, any path alternates between black and white squares, so a closed path will cover the same number of white and black squares. Thus in particular Hamiltonian circuits must have an even length on rectangles.

Let’s look at a single closed cycle, and let’s assume we follow it clockwise. Then there must be four more right turns than left turns. We have seen examples with no left turn arrow pillow, and with two left turns. Below are examples with just one and just three left turns.

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These little insights not only help to show that some tiling is impossible, they also give hints to design tilings. For instance, suppose you want to tile a square using the same number of straight, right, and left arrow pillows. Then the smallest square for which this could work is the 6×6 square. We also see that we need an even number of cycles in our Hamiltonian circuit in order to balance the left and right arrow pillows. The simple solution below uses two mirror symmetric tilings of 3×6 rectangles.

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Just Two (From the Pillowbook III)

A while ago we learned how to tile curvy 3×3 squares with pillows. Most of the possible tilings need at least three different kinds of pillows. The only way to tile a curvy 3×3 square was using the Blue and Yellow. This changes when we look at tilings of larger curvy rectangles. For instance, below is a tiling of a 5×7 rectangle with Red and Blue:

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I have overlaid the curvy rectangle with a 3×4 ragged rectangle, tiled by L-trominoes. Each L-tromino is replaced with a cluster of 3 Reds, and the T-junctions of the L-tromino tiling are filled with Blues. As we have seen that every ragged rectangle whose area is divisible by 3 can be tiled with L-trominos. This gives plenty of examples. In fact, every
tiling of a curvy rectangle with just Blue and Red comes from an L-tromino tiling of a ragged rectangle.

To see this, one can look at the possible ways a red pillow can be surrounded by blue and red pillows, and one almost finds that each red pillow belongs to a unique L-tromino. There is one exception that causes a little bit of headache that leads to circular clusters as in the example below (dark red and pink).

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But one can show anyway that such clusters can be tiled (in multiple ways) with L-trominoes.

Another challenge is to find a curvy square that can be tiled with just Blue and Red. That this is impossible follows from the deficit formula: We need to have the area r+b to be a square and r-2b =2 for r Reds and b Blues. But this implies that -1 is a square modulo 3, which is false.

Tiled squares are possible with other two color combinations. The example of a tiling of curvy 5×5 square tiled by Yellow and Green is deceivingly simple. The next case of the 7×7 square below is more complicated. Can you find a pattern?

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The only really simple case is tilings by Yellow and Blue. All curvy rectangles can be tiled, and in only just one way.

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A mean little exercise is to ask somebody to tile any curvy rectangle with Green and Red. There is no solution, because the deficit formula tells us that r-g=2, but r+g needs to be odd, because curvy rectangles have odd area.

There are a few more color combinations to consider. For instance using either orange or purple pillows together with a second color is impossible. By the deficit formula, this would require to be either a single yellow pillow or precisely two red pillows. For purple this means that there would be a line of purple pillows through the rectangle. But such lines always end at one concave and one convex segment, which can’t be. For orange this would require al least two orange corner pillows, which also doesn’t work.

Columbus out of Focus

A while back I confessed that I had acquired a Velvet 56 from Lensbaby. Yesterday I decided to try this lens with street photography and architecture, abusing charming Columbus (Indiana) for that purpose. Of course, all images were taken wide open. Brace yourself.

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Above is the entrance of the Cummins Headquarter building. This lens has clearly difficulties here. The overall softness distracts from the graphical elements. If you don’t know what Cummins is making, you can see one of their products below. It is not a space ship, nor a gun.

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While Cummins would probably not use this image for marketing, the Velvet 56 does a much better job when there is an obvious foreground. With the rental bikes lined up below, one can see nicely how the lens progresses into unsharpness and how it deals with highlights.

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I still like that image, and even more so the image below. Beautiful couple on beautiful bikes.

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My favorite is the last one, however. The unreal mini-halos about all the highlights on the chairs complement the mural as if the lens just came out of the bar…

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Dissect and Conquer

Many basic mathematical concepts are easy to convey to the layperson. For instance, most people are ok with numbers, distances, and right angles. An example of a concept that I found very hard to explain is that of a group action, and the related concept of a fundamental domain. Equivalence classes in general seem to be completely out of this world.

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Periodic tilings give many examples. The colored square tiling above for instance is periodic with respect to a group of (color respecting) translations, all of which can be written as a combination of the two orange arrows at the bottom left, or their reversed arrows. The collection of all these translations is called the lattice of the tiling.

More complicated looking tilings can have simpler lattices. For instance, the tiling by the differently sized yellow and blue squares below has the same lattice as the tiling by the outlined orange squares.

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The not so simple consequence of this simple observation is the following dissection of a large square into two smaller squares:

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The reason why this works is that both the large square and the union of the two smaller squares are a fundamental domain for the common lattice of the two tilings. You can think about the orange grille as a cookie cutter, and the yellow and blue squares as periodic dough. Cutting a blue and a yellow dough square with that cutter gives you five pieces that just fill one larger square of the cutter. There are many different ways to place the cutter over the dough, and all are allowed, as long as cutter and dough have the same lattice. This means that you can translate the cutter, but not rotate.

This method is well known among dissectionists. My favorite example is the dissection of a regular octagon into a square.

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To explain how to find it, we tile the plane with octagons and yellow squares. This tiling has the same lattice as a tiling by two unequal squares, where we choose the smaller purple squares to be exactly the same size as the yellow squares.

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The Economy Bender

If you want to build a column that has two elliptical cross sections at the top and bottom with different major and minor axis and that can roll, you can just take the ruled surface whose lines connect points with parallel tangents on the two ellipses.


When placed horizontally on a sheet of paper, the column will touch the paper in these lines, and you can wrap the paper around the column, making evident that this is a developable (or flat) surface. You can try it out yourself with the template below.

This simple trick has dramatic economic applications. Suppose you want to transform a boring, stagnating economy into a vibrant, growing economy:

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To do so, you just need to join points with parallel tangents on the two economy graphs by straight lines. Here is Martha with a wooden prototype of the Economy Bender.

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You can now wrap some expensive looking material over it. To explain to your CEO how you will be able to transform your failing company into a successful one, just take a printout of last year’s dire company report, put it onto the lower part of the bender next to the stagnating economy curve, and slowly move it upward towards the growing economy curve. You can do this by keeping the report tight on the surface, neither tearing not stretching it. This should convince anybody that a smooth transition into a brave new world is always possible. Here is the template:

Anybody buying it?

Ragged Rectangles (From the Pillowbook II)

In a ragged rectangle, the sides zigzag diagonally as in the left figure below, which shows a ragged rectangle of dimensions 6⨉7, and within a ragged 3⨉3 square. Note that the boundary changes directions at every unit step. These shapes make interesting candidates for regions to be tiled with polyominoes. The example in this post illustrates nicely how the interplay between making examples and generalization leads to a miniature theory.

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To tile a shape like this with polyominoes, it will help to know its area in terms of unit squares. This is easy: If you color the squares in a ragged a⨉b rectangle beige and brown, you will get a⨉b squares of one color, and (a-1)⨉(b-1) squares of the other color.

This right away shows that it is hopeless to tile a ragged rectangle with dominoes. The first really interesting case is to use L-trominoes. The area formula implies that we need one dimension of the rectangle to be divisible by 3, and the other to leave remainder 1 after division by 3. Thus the shortest edge that can occur has length 3, and the other them must have length 3n+1. The figure below shows how to tile any ragged rectangle of dimensions 3x(3n+1) with L-trominoes:


The next shortest edge possible has length 4, and then the other edge must have length 3n. Again, a few experiments lead to a general pattern which shows that any 4x(3n) ragged rectangle can be tiled with L-trominoes:


This covers the two basic kinds of thin and arbitrarily long rectangles. What about larger dimensions? If we already have a ragged rectangle tiled with L-trominoes, we can put a frame around it that is also tiled with L-trominoes:


These three constructions together show that a ragged rectangle can be tiled with L-trominoes if and only if its area is divisible by 3. Next time we will see how this helps us to tile curvy rectangles with pillows.