Is this a tesselation question ???

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Hypatia001

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Hello

So this is my first post !!!

The pattern attached is made of regular yellow haxagons and green equilateral triangles. If the pattern carries on forever in every direction what percentage of the area would be green ??

My thoughts are the answer is zero percent but please help.

Thank you
 

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Hypatia001 said:
The pattern attached is made of regular yellow hexagons and green equilateral triangles. If the pattern carries on forever in every direction what percentage of the area would be green ??

My thoughts are the answer is zero percent but please help.
Click to expand...

Please explain your thinking. How do you come to an answer of zero?

What I would do is to look for a single cell (that is, some set of pieces) that is repeated so that you could build an infinite pattern from many of that same cell, treated as a single glued-together unit. (I can see several different ways you could do that.)
 
Well, tell us what you've learned about tessellations, and we can see what hint, beyond what I gave, might help you think about this. Have you seen any examples of tessellating the plane?

But here's an example of what I was talking about: Suppose you glued together one hexagon and one triangle, then made a lot of those. Could you recreate the pattern using those new pieces? If so, then the ratio of colors over the entire plane would be the same as in this one glued unit. If not, try a different unit.
 
Dr.Peterson said:
Well, tell us what you've learned about tessellations, and we can see what hint, beyond what I gave, might help you think about this. Have you seen any examples of tessellating the plane?

But here's an example of what I was talking about: Suppose you glued together one hexagon and one triangle, then made a lot of those. Could you recreate the pattern using those new pieces? If so, then the ratio of colors over the entire plane would be the same as in this one glued unit. If not, try a different unit.
Click to expand...


I have learnt that tesselations in an infinite pattern do not have a fixed starting point. What i am looking for is the actuap repeating pattern. I worked out the area of the tile (24% green triangles) but cannot work out how that would work in an infinite pattern. Thats my dilemma and no matter how i think of it or what i draw or try and use as a start point i cannot get that out of my mind

I was thinking of perhaps just using 1 hexagon and the 6 equilateral triangles aroud this but that would mean a 50% area ... so that has confused me even more !
 
Hypatia001 said:
I have learnt that tessellations in an infinite pattern do not have a fixed starting point. What i am looking for is the actual repeating pattern. I worked out the area of the tile (24% green triangles) but cannot work out how that would work in an infinite pattern. That's my dilemma and no matter how i think of it or what i draw or try and use as a start point i cannot get that out of my mind

I was thinking of perhaps just using 1 hexagon and the 6 equilateral triangles around this but that would mean a 50% area ... so that has confused me even more !
Click to expand...
Can you explain the 24% calculation?

If you use 1 hexagon and 6 triangles as a unit, that will not be able to tile the plane while replicating the pattern:

FMH126227.jpg

What I'm suggesting you do is the same sort of thing I did here, but with a different set of triangles, resulting in a unit that you can repeat forever to make the pattern shown.
 
Maybe I'm being naïve here, but I can't help but notice that the object in the pattern is a hexagon unto itself, which can tessellate infinitely. Am I mistaken to suggest that this isolated portion of the pattern will have the same relative distribution of colors as an infinitely repeating version of it?
 
Mr. Bland said:
Maybe I'm being naïve here, but I can't help but notice that the object in the pattern is a hexagon unto itself, which can tessellate infinitely. Am I mistaken to suggest that this isolated portion of the pattern will have the same relative distribution of colors as an infinitely repeating version of it?
Click to expand...
No, if you tessellate the big hexagon, the colors will not repeat in the same pattern. For example, at each corner you would have yellow hexagons touching each other.

Hint: take off triangles from the star unit I outlined, one at a time, until you get a unit that will tessellate in the right pattern. There are several ways to do that.
 
Dr.Peterson said:
Can you explain the 24% calculation?

If you use 1 hexagon and 6 triangles as a unit, that will not be able to tile the plane while replicating the pattern:

View attachment 23096

What I'm suggesting you do is the same sort of thing I did here, but with a different set of triangles, resulting in a unit that you can repeat forever to make the pattern shown.
Click to expand...


I have taken off a star at a time and tried to to rotate the resulting shape. Perhaps a larger triangle with the hexagon at the middle of it ?? I just cannot picture the infinite pattern. Sorry, just being a little slow.
 
Dr.Peterson said:
Can you explain the 24% calculation?

If you use 1 hexagon and 6 triangles as a unit, that will not be able to tile the plane while replicating the pattern:

View attachment 23096

What I'm suggesting you do is the same sort of thing I did here, but with a different set of triangles, resulting in a unit that you can repeat forever to make the pattern shown.
Click to expand...

The 24% calculation was for the one tile.
 
Hypatia001 said:
I have taken off a star at a time and tried to to rotate the resulting shape. Perhaps a larger triangle with the hexagon at the middle of it ?? I just cannot picture the infinite pattern. Sorry, just being a little slow.
Click to expand...
No rotation will be needed (though some possibilities may rotate or reflect). The units I saw first tessellate by translation only, something like a checkerboard.

Also, don't get bogged down on one shape. If you don't quickly see how to tessellate with one, move on to the next. And you may find it helpful to actually draw them, as I did.
 
Dr.Peterson said:
No rotation will be needed (though some possibilities may rotate or reflect). The units I saw first tessellate by translation only, something like a checkerboard.

Also, don't get bogged down on one shape. If you don't quickly see how to tessellate with one, move on to the next. And you may find it helpful to actually draw them, as I did.
Click to expand...


The only shape i can see that may create an infinite tesselation is where the shape you outlined is halved. I am really stuck on this one :(
 
Hypatia001 said:
The only shape i can see that may create an infinite tesselation is where the shape you outlined is halved. I am really stuck on this one :(
Click to expand...
Well, you can't really halve it, so maybe you don't mean exactly that.

What if you chop off 3 or 4 of the triangles? Give it a try, and show us your drawing. We need to see what you are thinking in detail in order to help more.
 
Do you see how two of those will work? And how to get the percentage from them?
 
I think the last 2 would possibly work but again finding it hard to explain to myself and i can see how to work out the percentages.
 
Hypatia001 said:
I think the last 2 would possibly work but again finding it hard to explain to myself and i can see how to work out the percentages.
Click to expand...
The last one doesn't work; they can tile, but not in the pattern we want:
FMH126227 no.jpg
The third one does:
FMH126227 yes.jpg
One of the others also does. Did you try them to see, or are you just guessing?

Now, what percentage is green?
 
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