help needed Brian Chan's grasshoppers

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shehab hazem
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help needed Brian Chan's grasshoppers

Post by shehab hazem »

I have a problem in finding the reference points. he says the reference point for the perched grasshoppers square root 2+4,For The walking grasshopper square root 2+5,And for the flying grasshopper square root 2+7.So my problem is how could I find these reference points. And please explain how did you find it, And it and it would be much better if you explain it without weird mathematical terms like explain what are those terms are and what are they used for. Too also see that I am a new member so maybe somethings that I don't no. Thanks.
Baltorigamist
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Re: help needed Brian Chan's grasshoppers

Post by Baltorigamist »

There are a few key strategies that you can use to find reference points, actually. The most helpful--at least when you know the ratios involved--is that of crossing diagonals. The idea is that you can divide the reference into a ratio (two numbers that add up to the number required) and then find opposing diagonal lines that fit that ratio. We can assign a ratio to any diagonal line by noting how far it moves up compared to how far it moves out--this is commonly known as "slope".

For sqrt2+n (where n is any whole number), the first step is to realize that an angle bisector has a ratio of 1 : (1+sqrt2). This means that for any rectangle whose diagonal is formed by a 22.5-degree line, the respective proportions of that rectangle are 1 : (1+sqrt2). The height is 1, and the width is (1+sqrt2).
The second step is to subtract that (1+sqrt2) from the ratio you need. For the walking grasshopper, this is [(sqrt2 + 5) - (1 + sqrt2)] = 4--this is the width of the second rectangle. Since the height of the angle bisector is 1, this means that our second rectangle also needs a height of 1.
Finally, you need to place those two rectangles next to each other on the square and find a way to construct them by folding. Depending on your experience, this can be difficult, but luckily this example is fairly straightforward.

https://imgur.com/y3Gx6nt

The image linked above is the reference point for the walking grasshopper. The red lines are your diagonal folds, and the vertical blue lines are in a sqrt2 + 5 proportion. Let me know if anything needs clarification.
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shehab hazem
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Re: help needed Brian Chan's grasshoppers

Post by shehab hazem »

you mean by diagonal line any line coming from the corner ,
Can you also explain the slope definition more clearly. Also please show me a picture of the angle bisector ratio(rectangle ratio 1:(sqrt2+1) )
with the angle bisector drawn red and the rectangle blue just like you did in the sqrt+5
,And Thank you so so much you helped me a lot.
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Re: help needed Brian Chan's grasshoppers

Post by Baltorigamist »

The lines don't necessarily need to come from the corner of the square, but that's the easiest way to picture it (at least for me). Things get a lot more complicated when the lines emanate from somewhere else on the edge.

And slope, as I said, is just a measure of how far up/down a line goes compared to how far out it goes from a given point. For example, you could think of things this way: For every inch (or centimeter, millimeter--whatever unit you'd like to use) the line travels outward, it will travel upward a distance equal to the slope. An angle bisector from the top left corner (as I drew previously) will travel one unit down for every (1+sqrt2) units across, and the numbers would be reversed if the line was below the main diagonal of the square.
The same principle holds for any slope--though, of course, the numbers change.
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shehab hazem
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Re: help needed Brian Chan's grasshoppers

Post by shehab hazem »

I got the meaning of the slope but how do you put line from the edge you said and angle bisector, and angle bisector from the edge? and why do I need to make it from the edge,And I am kind of a visual person so it's hard to understand things in the air so I w be very grateful if you can make this with photos or a power point.
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