Kratzke: Flying Monohedral Tessellating Monkeys

PREFACE: THE WIZARD AND I

I told my baby sister that it would be neat if she cut a Wizard of Oz puzzle in which all the shapes were monohedral (same shape and size) tessellating (no spaces or gaps) monkeys. I even wrote to a lady who had happened to create some of her own monohedral tessellating monkeys, and asked if she would share her shape with us. She did not respond, and naturally, I assumed that the tessellating monkey affair was over. What a silly, silly goose I had been. Liên came to Austin and presented me with her own design of tessellating monkeys, and they were much better than the ones from the unresponsive lady because they were flying tessellating monkeys. She's so smart.

The Wizard & Dorothy

Flying Monohedral Tessellating Monkeys

Ladies and Gentlemen, Liên Kratzke

Liên's friend, upon witnessing the planning of such a project, commented in reference to me, "she knows not what she asks." True enough! And so it must be that this post is dedicated to that friend. (Who knows not what they ask now?) Liên's friend, I don't know your name, so for the purpose of this post, I will refer to you warmly as "Nikko," the leader of the flying monkeys. We all know that I'm the Wicked Witch of the West, which means you and I are already as thick as thieves.

CHAPTER 1: THE SHAPES

Any 3 or 4 sided shape with straight lines can tessellate.

The sum of the interior angles of an n-sided polygon is (n-2)(180°). So in the case of quadrilaterals, the sum of the interior angles is 360°. If you line up one of each interior angle, pinwheeling them together, you'll always reach 360° and you'll leave no gaps.

I can't, or maybe won't, prove that what's left always leaves the option to continue creating pinwheels of 360°; I am sure you can imagine this. Or rather, it's difficult to imagine its falsehood. Triangles sum up to 180°, and you can always create a pinwheel out of two straight lines.

Nikko, if you don't believe me, all you have to do is fold a piece of paper into eighths or so, cut a 3 or 4 sided shape with straight lines, and puzzle it out for yourself. Mind you, I make no promises regarding the border of your new jigsaw.

At the time of this writing, it has been discovered that there are 15 pentagons and 3 hexagons that can also tessellate. Go ahead and click on those links and scroll down to "Monohedral Convex Pentagonal Tilings" and "Monohedral Convex Hexagonal Tilings" to check them out.

Regular (equal angle and equal side) shapes that tessellate are the triangle, the square, and the hexagon.

CHAPTER 2: THE GRID

At this junction, I have a choice of shapes in which to begin our explorations, but one is clearly superior, and that is the 120°/60° rhombus. This is the only shape which can pinwheel with either 4 angles (creating a checkerboard) or 3 (creating a hexagon). My first step was to make a grid, which is harder than it sounds because pixels are squares, and in terms of horizontal and vertical axes, my units for equilateral triangles are 1 and √3. Nikko showed me how to verify that with the Pythagorean theorem.

Graph Paper

Rhombus

Checkerboard

Now that we have our graph paper and our shape, it's time to design an "innie," which we will use to generate a host of "outie" possibilities. I'll confine our innie to one quadrant and have it originate on one side. I'll also label a distinct number of positions to try placing an outie.

Pink Thumb

Kentucky, Virginia, North Carolina, Georgia, Alabama, Mississippi, Arkansas, Missouri

The rules of the game are that we will always start with the pink thumb innie on MO. From this position, we will explore what happens to the grid when we place an outie on KY, VA, NC, GA, AL, MS, and AR. This is what happens to the grid with KY. Isn't that wild?

Notice that this hexagon contains all of its thumbs. When all the thumbs are "confined to the hexagon," we can create variations of the pattern by rotating adjacent hexagons however we like. Rotating that pattern by increments of 120° would yield the same results, as Nikko pointed out, but that's not true if we use colors. Once you imagine 3 different thumb colors, or rather 3 differently colored puzzle pieces, you could start rotating adjacent hexagons willy nilly and enjoy the chaos that ensued.

CHAPTER 3: THE PINK THUMBS

Notice that this pattern is "confined to a double row."

That means we can shift each couple of linked rows left or right in indiscrete ways.

This pattern is "confined to the horizontal" OR "confined to the vertical."

We can shift rows OR columns in any number of indiscrete ways.

Of course, we only have such a luxury because our shape is so boring.

Alabama does not work. It's beautiful, but requires a second shape to fill in the gaps.

Tessellations with two or more shapes are called semi-regular tessellations. Fail.

Mississippi is like Alabama, but "a little bit louder and a little bit worse." Fail.

Arkansas is "confined to the horizontal" only.

Slide those rows like an abacus!

CHAPTER 4: THE PURPLE THUMBS

But what if we want to add reflections to the mix? I'll reflect my pink thumb into a purple thumb.

Purple Thumb

We now need a new set of rules. Henceforth, I shall only put thumbs on obtuse angles, as our innie was originally designed that way. In the world of squares, we would have no such guidelines, and we would run amok our equiangular grid with abandon. One thing you will notice immediately is that you can generate an image by replacing "every other" pink thumb from Chapter 3. Most of these will be non-monohedral, but they're still worth a look. I'll label those with "-NM," and any new grids we stumble upon with the prime symbol.

KY-NM

VA-NM

Another VA-NM

NC-NM

GA-NM

AL-NM

Another AL-NM (Interesting!)

MS-NM

MS'

AR'

CHAPTER 5: THE MONKEYS

Five Little Monkeys

Flying on a Grid

Liên Chose KY

(We know this because MO went to KY)

From the West:

MO outie - KY innie

AR innie - VA outie

From the South:

MS outie - NC innie

AL innie - GA outie

Monkey Pinwheel

Fly! Fly! Fly!

On the Grid

Off the Grid

EPILOGUE: THE MASTER

The last time I checked, the world of monohedral tessellations was dominated by the work of Dutch artist of Maurits Cornelis Escher (1898-1972). Nikko and I, hand in hand, have flown us through Oz, exploring the concepts of innies and outies, the flexibility of placement, the importance of symmetry, the use of rotation, and the possibility of reflection. We're at last ready to gaze upon the works of the master; we close this post simply by admiring a few of his pieces.