Tessellation Question

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geekily

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Jan 24, 2007
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I have to do a tessellation project. The directions state that "vertices must meet" and the first problem is "use the triangular grid to complete the star-and-hexagon tessellation that has been started." It shows a grid with three stars shaded, creating a hexagon in the middle of them. Then: "Use different colors to clearly show the stars and hexagons. Do no have shapes of the same color touch each other."

I've been going crazy staring at this thing for about 2 hours. I started coloring - top star blue, bottom star green, hexagon in the middle yellow, hexagon opposite the yellow one red, all 4 coming to a nice vertex point of blue, yellow, green, red. Then I realized that to continue the tessellation, the next star would have to be another color because it touches both the blue and green stars at one point each. However, if you add another color star, it takes the common vertex away. So, when it says not to have shapes of the same color touch each other, does it just mean sides, then? Or, if touching at points really isn't okay, where do I find my common vertex?

Thanks so much for your help.
 
geekily said:
I have to do a tessellation project. The directions state that "vertices must meet" and the first problem is "use the triangular grid to complete the star-and-hexagon tessellation that has been started." It shows a grid with three stars shaded, creating a hexagon in the middle of them. Then: "Use different colors to clearly show the stars and hexagons. Do no have shapes of the same color touch each other."

I've been going crazy staring at this thing for about 2 hours. I started coloring - top star blue, bottom star green, hexagon in the middle yellow, hexagon opposite the yellow one red, all 4 coming to a nice vertex point of blue, yellow, green, red. Then I realized that to continue the tessellation, the next star would have to be another color because it touches both the blue and green stars at one point each. However, if you add another color star, it takes the common vertex away. So, when it says not to have shapes of the same color touch each other, does it just mean sides, then? Or, if touching at points really isn't okay, where do I find my common vertex?

Thanks so much for your help.
Click to expand...

In general, common vertex between same colored shapes are allowed - since those are easily distinguishable.
 
Okay, thank you. Let's say I make that third star blue, though - on one side, the pattern will be correct - blue, red, green, yellow - but on the other side, it will be blue, red, blue, yellow. I just don't see any way around that.

Thanks for your help!
 
Please look at this website: http://www.math.gatech.edu/~thomas/FC/fourcolor.html

This may help you see that you can color ANY tessellation using no more than 4 colors.

You can have vertices that share the same color, but your problem does not prohibit that...you just cannot have two SIDES that share the same color.
 
Thank you. That theorem definitely makes sense. If I can have vertices of the same color, then, I only need 2 different colors, one for the star and one for the polygon, right? I guess what confuses me, then, is when it says "Use different colors to clearly show the stars and hexagons. Do no have shapes of the same color touch each other" because it seemed I would have to have at least 2 colors of at least 1 of the shapes, if not both. Thanks for your help!
 
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