Kratzke

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°). 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 (and by this, I mean lining up all the angles so that they meet at an origin), you'll always reach 360° and 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 in the world that can pinwheel with either 4 angles at an origin (creating a checkerboard) or 3 (creating a hexagon). My first step was to make equilateral triangular graph paper upon which to draw, which is harder than it sounds because pixels are squares, and in terms of horizontal and vertical axes, my units are of the ratio 1:√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 inward protruding element, which I will henceforth call an "innie." I'll confine our innie to one quadrant and have it originate on one side.

Pink Thumb

Next, we'll use our innie to generate a host of "outie" possibilities. Here's a map with the positions we'll try placing our outie.

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

Here's how this works: we will always begin with a pink thumb innie on MO. From this position, we'll explore what happens when we place an outie on KY, VA, NC, GA, AL, MS, and AR.

Before you read on, picture this in your mind: try placing a pink thumb in Missouri and another pink thumb protruding from anywhere else. North Carolina is probably the easiest.

This is our KY piece. Can you imagine what happens when we duplicate it and start puzzling the pieces together?

KY Piece

This generates our hexagon!

Notice that all of the thumbs are confined within the border of this hexagon. When all the thumbs are "confined to the hexagon," we can create variations of the pattern by rotating adjacent hexagons by increments of 60° however we like. This is easier to picture if you imagine differently colored puzzle pieces. Once you have that in your mind, you can start rotating adjacent hexagons willy nilly to get irregular patterns.

CHAPTER 3: THE PINK THUMBS

Confined to the Hexagon

VA-NT (non-tessellating)

This doesn't work because our grid doesn't allow for tessellations!

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

Confined to the Row

You can slide those rows like an abacus and your shapes will still be monohedral.

AL-NT

MS-NT

Confined to the Row

CHAPTER 4: THE PURPLE THUMBS

But what if we want to add reflections to the mix? Nikko reflected my pink thumb into a purple thumb! Time to play with purple thumbs.

Purple Thumb

We need a new set of rules. Henceforth, I shall only put thumbs of any color 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.

In plugging an outie purple thumb in KY, VA, NC, GA, AL, MS, and AR, we get all sorts of results. Sometimes, we generate something that's non-monohedral. These would still make great puzzle cuts! Sometimes, we stumble upon something a little closer - a tessellating grid in which the shapes would be monohedral if we could pick them up, flip them around, and put them back down. In other words, the entire pieces themselves would be reflections of one another. And sometimes, we get a brand new monohedral tessellating pattern. Nikko says, "Buckle up!"

Key:

-NM = pieces are neither monohedral nor reflections of one another

These patterns have a purple thumb pointing into an acute angle

-NT = grid does not allow for tessellating without the addition of new shapes

As seen in the previous chapter

-RF = some pieces are reflections (and of course rotations) of other pieces

These are actually my favorite

-II = second monohedral tessellation

Because we are working with purple thumbs, the outie is a reflection of the innie

KY-NM

VA-RF

NC-NM

GA-NM

AL-NM-NT

AL-RF-NT

MS-NM-NT

MS-RF

AR-II

CHAPTER 5: THE FINDINGS

Sorting through the rubble, Nikko and I present 10 findings. We have 5 monohedral tessellating puzzle piece patterns, 2 reflecting tessellating puzzle piece patterns, and 3 non-monohedral tessellating puzzle piece patterns. Liên, I hear the scroll saw calling for you.

MONOHEDRAL TESSELLATING SHAPES & THEIR GRIDS

AR-II

REFLECTING PIECE TESSELLATIONS (2)

VA-RF

(Double-heighted rhombuses [parallelograms] are now monohedral tessellating shapes.)

MS-RF

(Double-heighted rhombuses [parallelograms] are now monohedral tessellating shapes.)

NON-MONOHEDRAL PIECES IN LARGER TESSELLATING SHAPES (3)

KY-NM

(Hexagons are now monohedral tessellating shapes.)

NC-NM

(Rows are now monohedral tessellating shapes.)

GA-NM

(Diagonals are now monohedral tessellating shapes.)

CHAPTER 6: THE MONKEYS

(This is the moment we've all been waiting for.)

Five Little Monkeys

Flying on a Grid

Liên Chose KY

We know this because MO went to KY

From the West (rotates by hinging counterclockwise on the NW corner up to the North):

MO outie - KY innie

AR innie - VA outie

From the South (rotates by hinging clockwise on the SE corner over to the East):

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; let us close this post simply by admiring a few of his pieces.