But nobody else does, which means they're not made commercially, and that means we're going to have to make them ourselves at Austin's Pottery Parlor. Little does Austin's Pottery Parlor understand what a steal this is on our end - we'll purchase a piece of reasonably-priced unglazed pottery, and an additional $6 fee will cover all the time we need, unlimited access to 58 colors, and the firing of our pieces. They have no idea how much more than $26 a Heawood Conjecture mug is worth. We double that margin at $52 for two. The heist won't work if we show up for amateur hour.
Let's start at the beginning. My cousin Emily created a solitaire game called Chutes and Levers - it's a lot of cleverness wrapped into a little pack of cards. Each card is a maze of sorts, where the top and bottom edges are considered to be the same edge, and the left and right edges are also considered to be the same edge.
Dad explained to me that this is a 2D representation of a torus, which is the math word for the shape of a donut. If you take a standard sheet of paper and imagine it to be flexible, joining the top and bottom edges turns that rectangle into a tube, and joining the left and right edges turns that tube into a donut.
Emily's game also uses the backs of the cards, which are also tori, so the bounds of her one-card mazes are the exterior and interior surfaces of a torus. In other words, you should get the game, and probably before your next airplane trip.
After Dad explained that Chutes and Levers was a torus maze, I was accidentally interested enough for him to launch into the Heawood conjecture. The Heawood conjecture answers the question, "If I draw any number of adjacent (the right word is actually "contiguous") shapes on the surface of something, what's the fewest number of different crayons I'll need to color my shapes in such a way that no two shapes of the same color are touching?
Ooh - I'll admit this much - it's a good question.
The Heawood conjecture says that if the "something" is a piece of paper, the answer is 4.
You know how the mainland of the United States can be represented as a set of contiguous shapes on a piece of paper? And how some of the shapes (Oklahoma, Kentucky, Maryland) are extremely ugly and irregular? Still 4.
When I was very little, I had a wooden puzzle of the United Staes. (Always do Alaska and Hawaii first because a 48-piece puzzle is easier than a 50-piece puzzle.) Even back then, I remember Dad commenting that the puzzle used more than 4 colors, which was not necessary, so the Heawood conjecture is practically in the water this man drinks.
Then Dad said that if the "something" is a torus, the answer is 7.
And before I had the chance to utter, "That sounds interesting bye," he had tasked me with drawing shapes on a torus in such a way that they would require seven colors, and then he walked off to the South End of Snake Mountain with nary a care in the world.
I tried for a couple of days and could not complete my task. The key point - the one that I didn't understand - is that all 4 corners on the rectangle represent the same point on a torus, so they can, and in fact should, be part of the same region. That changes everything. With this in mind, I'll walk you through the solution in two dimensions.
To restate this quest, can we draw 7 shapes on a rectangle representing a torus so that each of the 7 shapes is in contact with the other 6? Remember, the top and bottom edges of the rectangle are the same, as are the right and left edges. And all 4 corners are the same point.
It makes sense to start off with just one shape that's touching six other shapes. Like a beehive.
And fill seven of them with colors.
One of our six non-central colors (for that one is already "done") must become all 4 corners of our torus-representing rectangle. Let's choose yellow since we just saw a beehive. Yellows must first touch all 6 colors. Then they must also open up a pattern that can be colored in a systematic and regular way.
This is one of two acceptable ways to do that. (The other is to have one yellow nestled into both blues, and the other yellow sitting on top of red.)
To double check that this placement has the potential to make a regular coloring pattern, imagine the fourth yellow corner.
Now the only thing left to do is connect four yellow corners in such a way that the top and bottom edges have the same colors and right and left edges have the same colors.
This ought to do it.
And there it is - a pattern that necessitates 7 colors on a torus.
Normally when you see it, it's skewed into a rectangle like this.
That's the most standard and uncreative example we can make, but you can easily imagine bending the black lines without breaking color-adjacent parameters, and that wouldn't disrupt anything. For example, the red hexagon might as well be shaped like a cosmic crisp apple.
You've all done very well, and now we're half way through. Don't blame me; blame my dad.
In order to explain what this has to do with Austin's Pottery Parlor, we now have to talk about what a "genus" is. The genus of a shape is the number of "holes" or "handles" in the shape. A donut has one hole, so it has a genus of 1. A sphere has no holes, so it has a genus of 0.
The Heawood Conjecture says a lot of things; here are some of them:
surface of genus 2 requires up to 8 colors
surface of genus 3 requires up to 9 colors
surface of genus 4 requires up to 10 colors
surface of genus 5 requires up to 11 colors
surface of genus 6 requires up to 12 colors
surface of genus 7 requires up to 12 colors
When two shapes have the same genus number, they're said to be "homeomorphic." And when two shapes are homeomorphic, you can imagine one of the shapes being made of an infinitely flexible material and morphing into the other.
Let's put this to the test with some easy examples first. Many things are homeomorphic to a sphere. An easy comparison is a cube. A slightly harder comparison is a bowling ball. A harder comparison still is a single volume encyclopedia. Do you agree?
A sphere and my Mad Hatter's hat are homeomorphic with genuses of 0. They are homeomorphic to a tea saucer, but not homeomorphic to a teapot (genus of 2). And they are not homeomorphic to a teacup (genus of 1). But a torus IS!! And not only is a torus homeomorphic to a teacup, but the comparison of these two shapes is probably the most famous example of homeomorphism.
And now you can understand what the following tutorial is really about. We'll paint mugs with 7 shapes in different colors in such a way that all 7 shapes are touching the other 6.
It's time to start taking the painting of pottery into very serious consideration, for we are preparing ourselves to move out of the theoretical realm to bring magic into the real world.
The first item of business is to recall from the beehive solution that our purple and red hexagons did not wrap around anything. The other hexagons bent over backwards (rather literally) to reach them.
The most practical way to handle the first of these two regions is to have it include the bottom of the mug, or the piece of the mug that will remain unglazed. That piece is at least as big as the mug's base. It's white, and we have no say in that. We shall henceforth call it Antarctica. It can be bigger than the bottom of the mug, reaching upwards if you like.
The second of these regions has nearly no parameters other than its job to touch Antarctica. But a practical concept would be to push its bounds up and over the mug into the interior, where there's lots of space for other shapes to reach it. It could consume the entire interior, and all the other colors could meet it at the rim. Or it could merely reach down into the mug, forcing the other six to follow it. Or it could overflow the top of the mug and drip down onto all the other colors as a favor to them. If we're pushing it to a space that's far from Antarctica, we might as well call it our Arctic Ocean.
The second item of business is to recall that we don't paint perfectly, and some of us have tremors. Our projects will turn out more impressive if we start the glazing process with the lightest colors first, and make our way towards the darkest, overlapping along the way. The nice man at the Pottery Parlor explained to me that as long as we allow our lightest glaze to dry before applying a darker one, the lighter glaze will not show through the darker glazes once fired. I was explaining why I had to know this, and he was very sorry to tell me that he needed to help the other customers now.
Because Antarctica is white, we only have six colors to choose.
For the purposes of this tutorial, we'll use red, orange, yellow, green, blue, and purple, but this is not a claim on your artistic license. It's most helpful to think of them as two sets of three.
Team Light - yellow, orange, green
Team Dark - blue, red, purple
One team has the task of wrapping around the handle of our mug in a vertical stretch.
Let there be Team Light. And God saw that the light was good.
The other team has the task of wrapping around the cylinder of the mug in a horizontal stretch.
That would be Team Dark.
The Arctic Ocean belongs to Team Dark.
If our Arctic Ocean is feeling polite, it will stretch upwards to make itself easily available.
It's time to let these players onto the field.
Before we begin, imagine your mug is a cylinder, closed on one side, sporting a typical handle.
Rip off the handle and set it aside for later.
Cut directly down through the mug on the opposite side of the handle.
Continue cutting into the circular base.
Uncurl your mug and stretch it into a perfectly flat rectangle.
Make sure the exterior is facing you.
Reshape your handle holes into triangles, for these holes must host at least 3 colors.
Our blank canvas now looks like this, and the interior of the mug is not visible.
I made it that way so that it was easy to understand and easy to paint.
Our blank canvas now looks like this, and the interior of the mug is not visible.
Antarctica & Uncharted Territories
For our first task, we want to use the lightest member of Team Light.
Before we do anything, however, we need to know where our Arctic Ocean will be.
For the purposes of this tutorial, the Arctic Ocean will stretch up into the interior of the mug.
The top and bottom edges are NOT the same like on a torus.
Antarctica is the bottom white region and the Arctic Ocean will be around the top edge.
Paint one shape from one side of the bottom handle hole down to Antarctica.
Paint a second shape from one side of the top handle hole up to the Arctic Ocean.
Notice that these two shapes will be connected later, after we stick the handle back on.
Color 1 cannot engulf the borders of Antarctica or the Arctic Ocean.
Color 1
Our next lightest color must be placed adjacent to the first color, leaving no space in between.
It must also reach Antarctica and the Arctic Ocean.
Notice that I placed my second color to the left of my first in Antarctica.
That means I must also place my second color to the left of my first in The Arctic Ocean.
I'm sure you can see that I could have chosen the right sides instead.
But I could not have chosen one left and one right; more on that in our next step.
Colors 1 and 2 cannot engulf the borders of Antarctica or the Arctic Ocean.
Color 2
Color 3 must be adjacent to the rest of Team Light at Antarctica and the Arctic Ocean.
It can cover the rest of the handle holes.
It doesn't actually have to - you could leave space traveling into the handle for Team Dark.
But to reiterate, Team Light must be adjacent at Antarctica and at the Arctic Ocean.
Of extreme importance - it's necessary to choose opposite sides in ordering the colors.
In other words, this time, you must choose one left and one right.
These colors move up the handle like stripes on a candy cane, so they stay in order.
If they start at (2, 1, 3), then they end at (2, 1, 3) or (1, 3, 2) or (3, 2, 1).
Notice in this example that Antarctica colors are (2, 1, 3).
If we had also colored the Arctic Ocean (2, 1, 3), Color 1 would be trapped from Team Dark!
The Arctic Ocean could not possibly read (1, 3, 2) because we placed Color 3 last.
Our only option is (3, 2, 1).
Team Light must perform a twist.
Team Light cannot engulf the borders of Antarctica or the Arctic Ocean.
Team Light
Let's go ahead and take care of that Arctic Ocean.
Arctic Oceans are blue, right?
Notice that I didn't make this Arctic Ocean big, if you don't count the interior of the mug.
It's long and skinny, reaching up from Antarctica.
It then floods the interior, which is convenient since the interior isn't represented.
And it extends itself down over the rim just a little so we can see how it's connected to itself.
The Arctic Ocean must not be adjacent to Team Light at Antarctica.
There are now two Antarctic holes available - one for each remaining member of Team Dark.
The Arctic Ocean
The final two shapes happen simultaneously once an outline is chosen.
Remember that Team Dark has a horizontal mission.
The shape of Color 5 must touch Antarctica, the Arctic Ocean, and all of Team Light.
But also, the shape of Color 5 cannot hog any of Team Light's colors.
For Color 6 much be able to touch Antarctica, the Arctic Ocean, and all of Team Light as well.
You can see here why the twisting of Team Light was so important!
Antarctica, the Arctic Ocean, and each member of Team Light must remain exposed for Color 6.
Fully Charted Territories
The negative space is reserved for the last and darkest member of Team Dark.
Cylindrical Section Complete
And all that remains is to go back and paint the rest of Team Light into the handle.
Wrap your Team Light colors around once or seventy-one times for all I care; just make sure you start and end where you're supposed to.
Handle
Cut vertically along inner seam & flattened
Before we review everything, I wanted to show you our mug's interior.
Mug Interior
But the Arctic Ocean neither had to reach into the interior, nor did it have to consume it.
The only requirement for the Arctic Ocean was to touch Antarctica.
For example, the Arctic Ocean could have started at Antarctica and flowed into the interior.
Team Light and the other two colors of Team Dark would have had to follow it.
In that case, the interior might have looked something like this.
(Rip off the handle, cut the mug down the handle line, and pull the interior into a rectangle.)
Alternative Mug Interior
TUTORIAL REVIEW IN TEN EASY STEPS
1. Find a Paint-Your-Own Pottery Studio in your hometown.
These are very cool and worth supporting anyway.
Austin's Pottery Parlor even allows you to bring your own snacks.
2. Select a mug and 6 colors.
Oooooo!
Please ignore the fact that teapots might be there, because teapots have a genus of 2.
And those can have contiguous shapes drawn in such a way that 8 colors are necessary.
Nobody wants to paint in the interior of a teapot spout.
And nobody wants to figure out what a Heawood Conjecture Teapot is either.
3. Arrange your colors from lightest to darkest and divide them into two teams.
Your first team has a vertical duty and your second has a horizontal duty.
4. Imagine your Antarctica.
The unfired base will look very nice, but won't show off exciting shapes.
You could choose to extend Antarctica upwards into the body of the mug, or not.
5. Imagine your Arctic Ocean.
Stretching the shape upwards makes the project easier but is not necessary.
Painting the entire interior of the mug in one shade is a little boring but practical.
6. Set Team Light onto "handle-twist-order-at-Antarctica-and-the-Arctic-Ocean" duty.
You can leave space for Team Dark to encroach into the handle if desired.
That's a move reserved for the dextrous and arrogant.
Team Light MUST touch itself directly at Antarctica and the Arctic Ocean.
And of course, the twist is extremely important.
Go ahead and paint the entire handle since you know what you're doing.
7. Paint your Arctic Ocean.
This is very easy because you've already imagined it.
Don't let your Arctic Ocean touch Team Light at Antarctica.
8. Paint Color 5 so that it touches Antarctica, the Arctic Ocean, and all of Team Light.
Remember, it must allow Color 6 the same privileges.
9. Finish painting with your darkest glaze.
Home stretch!
10. Come back the following week to get your very own Heawood Conjecture Coffee Mug!
Heist completed!!
REVIEW SHEET
1. Antarctica has no job, other than not gluing the mug to the kiln.
2. The Arctic Ocean has no job, other than touching Antarctica and not engulfing its borders.
3. Team Light has the mission to be in contact with itself and perform a vertical twist.
It also cannot engulf the borders of Antarctica or the Arctic Ocean.
4. Color 5's job is pretty easy - it contacts everything and leaves room everywhere for Color 6.
5. Color 6 is a coloring book job.
REVIEW TEST
1. How many colors can flow into your handle?
Theoretically, 7.
Each member of Team Light must be touching the other two members of Team Light.
That means:
They're adjacent at Antarctica
They twist on the handle
They're adjacent at the Arctic Ocean.
As long as that's so, other colors (even Antarctica) can swirl into the handle.
Antarctica or the Arctic Ocean could even nearly consume the handle.
2. Can Team Light move in an ascending counter-clockwise spiral?
Sure.
3. Can you make a Heawood Conjecture Coffee Mug in greyscale?
What else are your goth teens going to want for their birthdays?
4. Can the Arctic Ocean touch Team Light at Antarctica?
No.
The Arctic Ocean must leave a space on each side; one for Color 5 and one for Color 6.
5. Can the Arctic Ocean be a dot?
Sure, if you have the brushes and dexterity to deal with that.
PRACTICAL ADVICE
Be patient with yourself and enjoy the process.
Go in early and take your time.
It's more important to be creative than ambitious.
And it's not even important to be creative.
Shapes can outline other well-known shapes, letters, and numbers if you want them to.
If you'd like to make a draft, draw on a paper cup with markers.
Have fun! Your mug should be microwave and dishwasher safe, but not oven-safe.
REVIEW CHARTS
Review Mug Exterior
Review Mug Interior
Review Mug Handle//Matches Review Mug Review Mug Handle//Matches Original
ONE MORE EXAMPLE
We've been using a lot of glaze. But what if we didn't want to?
What if we put our Arctic Ocean between our handle holes?
Our mug would start off like this.
Arctic Ocean Inside Handle
We'd start off with Team Light like we did before.
Team Light
And we'd twist it.
Team Light Twisted
Handle Cut Along Inner Seam
Color 5 has to begin at the Arctic Ocean and contact all of Team Light.
But before we finish that, we have to make sure Color 6 can do the same.
Color 5 (No Yellow Contact)
Subway Map (Just Kidding)
NOW we can finish Color 5.
Arctic Ocean Dot Mug Design
But what if we wanted to use even less glaze?
Well, all the shapes could be smaller and skinnier.
The Arctic Ocean could be more like a line than a dot.
Team Light doesn't even need to travel the entire length of the handle!
Colors 5 and 6 would just have to migrate upwards to meet Team Light.
All of this could all be done with very skinny lines.
In fact, one could even select a mug with a tiny handle to require even less glaze.
Pretty fun to think about.
Does "71 years of age" sound like "7 colors 1 torus" to you?
I don't collect numerical coincidences on purpose; they just fall into my lap, and I occasionally notice them as I brush them off onto the floor.
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