Coloring a PlayDome (mathematics first)
I wanted to color my PlayDome so that adjacent tires had different colors.
How many different colors of paint did I need to buy? By looking at the 5
large tires that ring the topmost tire, it is clear that I would need at
least 3 different colors. But then the topmost tire will require a fourth
color so that it has a color distinct from the other three used to color
the 5 adjacent tires. So, I need to buy at least four different colors of
paint. Do I need more, 5 colors?
A famous theorem from graph theory, the Four Color Theorem, allows us to
conclude that we should be able to color the PlayDome using only four
colors. By making the tires the vertices of a graph, and then adding edges
where bolts go, we obtain a planar graph on 21 vertices with 50 edges.
Since this graph is planar, the Four Color Theorem asserts that we will not
require more than 4 colors.
Now, how to achieve such a coloring? I started with a greedy algorithm -
order the colors, and use the first possible color that will "work."
However, this leads to a lot of tires using the first color, and not too
many using the last color. What is probably desired is what I might call
an "equitable" coloring - about the same number of tires of each color. In
this case, that would be 3 sets of 5 tires and one set of 6 tires, where
each set uses the same color.
So to achieve an equitable coloring, I did some Kempe interchanges. Now,
if none of this makes much sense, do not despair - this is good fodder for
an undergraduate honors thesis in mathematics (and it has been!).
Coloring Plan
So, if you have lasted this long, here's how I colored my dome:
Start with Curt's detailed schematic (page 2 of
the plans) and choose any one of the buried tires to be tire number 1.
Work your way around the periphery, alternating buried and non-buried large
tires, finally marking tire number 10. Now continue clockwise to the small
tire that will end up above tire number 1, and call this tire number 11.
Continue clockwise, alternating small and large tires until you count up to
tire 20. Finally, call the central, capstone tire, number 21.
Now color as follows:
- Red - 4, 9, 11, 17, 21
- Yellow - 3, 7, 12, 15, 19
- Green - 2, 6, 10, 14, 18
- Blue - 1, 5, 8, 13, 16, 20
You can of course substitute or rearrange the colors to suit your
preferences.
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Created: June 25, 1996, Updated: June 25, 1996.