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 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 (without a floating binding). 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 an unlidded 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 contiguous shapes in different colors in such a way that all 7 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 might remain unglazed. Some pottery parlors require that you leave this part unglazed, and some let you glaze the bottoms by firing pieces on stilts in the kiln.
For the purposes of this tutorial, we'll assume that we must leave that part unglazed, and that piece is therefore at least as big as the mug's base. If left unglazed, it will be white, and 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 the 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 impressively 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.
Update: that very nice man turned out not to be a perfectionist.
Underglazes, even lighter ones, do peek through.
It is still a good policy to move from light to dark.
Still, adjacent borders are preferable to overlapping borders.
In other words, if you must overlap, do so with the darker choice.
Too many coats of glaze will flake off, and too few will show brush strokes.
You must be vigilant out there!
Aim for 4 thin layers; it may take 8 hours to do this.
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 Twist - yellow, orange, green
Team Tide - blue, red, purple
One team has the task of wrapping around the handle of our mug in a vertical stretch.
We'll call that Team Twist.
The other team has the task of wrapping around the cylinder of the mug in a horizontal stretch.
It includes the Arctice Ocean.
We'll call that Team Tide.
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.
Our blank canvas now looks like this, and the interior of the mug is not visible.
Antarctica & Uncharted Territories
Before we do anything, 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 of this diagram are NOT the same as they are 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 color must be placed adjacent to the first color, leaving no space in between.
It must also reach Antarctica and the Arctic Ocean.
Colors 1 and 2 cannot engulf the borders of Antarctica or the Arctic Ocean.
Color 2
Now for the hardest part of the entire design: completing Team Twist.
Color 3 must be adjacent to the rest of Team Twist at Antarctica and the Arctic Ocean.
It can engulf the handle holes, but it doesn't actually have to.
It can travel the entire length of the handle, but it doesn't actually have to.
Team Twist has an actual mission, and that is is to twist.
Notice in this example that at Antarctica, Team Twist is in the order: orange, yellow, green.
In other words, yellow is "trapped" in the middle at Antarctica.
If yellow is also trapped in the middle at the Arctic Ocean, the entire design fails.
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As long as the same color is not trapped on both ends,
where Team Twist touches Antarctica and the Arctic Ocean,
then Team Twist has successfully performed a twist.
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In the illustrated example, Team Twist has wrapped around the handle like a barber's pole.But it doesn't actually have to, and here's how.
The first two colors of Team Twist could go straight up the handle.
In that case, Color 3 would also go straight up the handle, too.
I made it that way so that it was easy to understand and easy to paint.
The first two colors of Team Twist could go straight up the handle.
In that case, Color 3 would also go straight up the handle, too.
Color 3 would begin on one side of Team Twist at Antarctica.
But before Color 3 reached the Arctic Ocean, it would have to:
Break off from the "ribbon" of Team Twist,
Loop under the handle independently,
Rejoin Team Twist on the other side, and
Ensure that the "twisted ribbon" joined Antarctica.
If Team Twist engulfs the entire handle hole and travels the entire length of the handle,
then this breaking off happens off the handle.
In other words, Color 3 spins around the handle hole.
Team Twist must perform a twist.
Team Twist cannot engulf the borders of Antarctica or the Arctic Ocean.
Team Twist
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.
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 Twist at Antarctica.
There are now two distinct Antarctic border spaces still available:
One for each remaining member of Team Tide.
The Arctic Ocean
The final two shapes happen simultaneously once an outline is chosen.
Remember that Team Tide has a horizontal mission.
The shape of Color 5 must touch Antarctica, the Arctic Ocean, and all of Team Twist.
But also, the shape of Color 5 cannot hog any of Team Twist's colors.
Antarctica, the Arctic Ocean, and each member of Team Twist must remain exposed for Color 6.
Fully Charted Territories
The negative space is reserved for the last member of Team Tide.
Cylindrical Section Complete
And all that remains is to go back and paint the rest of Team Twist into the handle.
Wrap the 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 Twist and the other two colors of Team Tide 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.
8 hour romantic date night, anyone?
2. Select a mug and 6 colors.
Oooooo!
Please ignore the fact that teapots might be there, because teapots have a genus of 2.
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. Divide your colors into two teams.
Your first team has a twisting duty and your second has a tidal wave duty.
4. Paint Antarctica.
You can sketch any of your shapes in pencil first, if you'd like.
5. Plan the interior of your mug. Plan the Arctic Ocean.
Do you want the interior of your mug to be the Arctic Ocean?
It doesn't have to be, but get a good visualization of each.
Do you want the interior of your mug to be one color?
6. Design and paint Team Twist.
You can leave space for Team Tide to encroach into the handle.
You can leave space for Antarctica and/or the Arctic Ocean in the handle.
Or Team Twist can take it all.
Team Twist must be adjacent to itself at Antarctica and the Arctic Ocean.
Most importantly of all: Team Twist must twist.
It's impossible to mess up the twist by designing Colors 1 and 2.
You have absolute freedom in how you design Color 3.
You have absolute freedom in how you design the Arctic Ocean.
Remember: the twist is a comparison of color order at Antarctica and the Arctic Ocean.
A twist is successful when that order is not the same.
Paint all of Team Twist; after this, the rest is downhill.
7. Paint your Arctic Ocean.
This should be easy because it was in your mind all along.
It must touch Antarctica.
It must touch all of Team Twist.
It must reveal two distinct Antarctic border spaces for the rest of Team Tide.
8. Paint Color 5.
It must touch Antarctica, the Arctic Ocean, and each color of Team Twist.
It must allow Color 6 to do the same.
9. Paint Color 6.
Home stretch - paint everything that is not painted.
10. Come back the following week to get your very own Heawood Conjecture Coffee Mug!
REVIEW SHEET
1. Antarctica has no job, other than not gluing the mug to the kiln.
In the case of pottery parlors that use stilts, Antarctica is not confined to the base.
2. The Arctic Ocean has no job, other than touching Antarctica and not engulfing its borders.
3. Team Twist has the mission of performing a vertical twist.
It must to be in contact with itself at Antarctica and the Arctic Ocean.
It cannot engulf the borders of Antarctica or the Arctic Ocean.
4. Color 5's job is pretty easy - it contacts everything and leaves room for Color 6.
5. Color 6 is nothing more than a coloring book job.
REVIEW TEST
1. How many colors can flow into your handle?
Theoretically, 7.
2. Can Team Twist 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 Twist at the borders of 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 tiny dot?
Sure, if you have the brushes and dexterity to deal with that.
PRACTICAL ADVICE
Be patient with yourself and enjoy the process.
Give yourself many hours, go in early, and take your time.
It's more important to be creative than ambitious.
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 before you go, draw on a paper cup with markers.
REVIEW CHARTS
Review Mug Exterior
Review Mug Interior
Review Mug Handle//Matches Review Mug Review Mug Handle//Matches Original
ONE MORE EXAMPLE
Can we put our Arctic Ocean between our handle holes?
Our mug would start off like this.
Arctic Ocean Dot
We'd start off with Team Twist like we did before.
Team Twist
And we'd twist it.
Twisted
Handle Cut Along Inner Seam
Color 5 has to begin at the Arctic Ocean and contact all of Team Twist.
But before we finish that, we have to make sure Color 6 can do the same.
Color 5 (No Yellow Contact Yet)
Subway Map (Just Kidding)
NOW we can finish Color 5.
Minimalist Mug Design
But what if we wanted to condense the design further?
All the shapes could be smaller and skinnier.
The Arctic Ocean could be more like a line than a dot.
Team Twist doesn't even need to travel the entire length of the handle!
Colors 5 and 6 would just have to migrate upwards to meet it.
All of this could all be done with very skinny lines.
In fact, one could even select a mug with a tiny handle to aid in condensing the design further.
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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My First Heawood Conjecture Mug (My Dad's New Coffee Mug)
Plenty of room for improvement.
But as my dad would say,
"The seven contiguous regions establish the lower bound;
that is, there are maps that require seven colors."
So in that regard, it was a success.
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The Red Dot is the South Pole
The intersection of my imagined red lines kind of shifted
as I stretched that blue circle into linear border,
so it's further into the circle than on the original map.
All of the mapping was done by eye, so it's not that accurate.
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