construction of a golden-proportion-rectangle

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construction of a golden-proportion-rectangle

Postby Lazur URH » Tue Feb 12, 2013 10:54 am

The idea came from Orimdam here.

Mathematical background

The golden ratio means the following:
a:b=b:(a+b)

which is equal with this:
a*(a+b)=b^2

furthermore
a^2+ab=b^2

b^2-ab-a^2=0


which results as b=(a+-(a^2+4a^2)^0.5)/2=a*(1+5^0.5)/2

thus the golden ratio can be written in a more direct form:
a:b=1:((5^0.5+1)/2),
and 1:((5^0.5+1)/2)=((5^0.5-1)/2):(((5^0.5-1)/2)*(5^0.5+1)/2)=(5^0.5-1)/2:1
-which makes the connection to Fibonacci.

To construct the golden ratio, we will have to draw this (5^0.5-1)/2 ration.

We will use a right-angled triangle which has a leg double times longer than the other.
The ratio between the sides will be then 1:5^0.5:2 by Pytahgoras theorem, this is what the construction uses.
If the shorter leg's length is subtracted from the hypotenuse, it will make the golden ratio with the longer leg.


Here are the steps how it looks like all in inkscape:

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It might not be simpe or fast if you doing it for the first time -or if you making a tutorial of it...-, but
it doesn't take much more time than opening the calculator from the accessories, typing in ((5^0.5)+1)/2 and multiply it with the smaller size of a rectangle, to get the longer size's correct width, that you type in to have the same result as with the construction method.
microUgly wrote:People who figure things out for themselves will be far better at their craft than people who are told how or have someone else do the work for them.
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Re: construction of a golden-proportion-rectangle

Postby druban » Tue Feb 12, 2013 9:27 pm

It seems like a lot more work than necessary! ... In the thread you referred to, I had a much simpler method, did you see it? In the interests of anyone who might actually be looking at this thread for instructions on how to construct a rectangle of these proportions, I am posting the method again here, with annotations in case my other diagram was too difficult to follow.
Off topic:
Incidentally, this can be done with nothing but a straightedge and compass, the way geometry was done originally.
I. Draw a square.
2. Duplicate it.
4. Resize the duplicate from one side until the edge snaps to any midpoint of the original square.
5. Rotate this rectangle until the diagonal is horizontal (use a grid or guide).
6. Resize the original square to snap the corner to the farthest corner of the rotated rectangle.

goldenrect.png
goldenrect.png (74.48 KiB) Viewed 2242 times


Off topic:
brynn wrote:WOW!!
Looking forwards to druban's comments.
Image And look, he managed to post them before I could even post!

Not even sure how that happens when it takes me so long to write the post!
BTW Ragnar, have I made any errors?
Last edited by druban on Tue Feb 12, 2013 9:46 pm, edited 2 times in total.
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Re: construction of a golden-proportion-rectangle

Postby brynn » Tue Feb 12, 2013 9:35 pm

WOW!!

Looking forwards to druban's comments.

Image And look, he managed to post them before I could even post!
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Re: construction of a golden-proportion-rectangle

Postby ragstian » Wed Feb 13, 2013 1:22 am

Hi druban

Off topic:

brynn wrote:WOW!!
Looking forwards to druban's comments.
Image And look, he managed to post them before I could even post!



Not even sure how that happens when it takes me so long to write the post!
BTW Ragnar, have I made any errors?


I managed to draw a golden ratio box using your method - I could not find any error's!! :)

It would be nice to have a extension in Inkscape to make a golden ratio gauge or ruler - an electronic version of what draftsmen use;
http://familywoodworking.org/forums/showthread.php?14892-Fibonacci-Gauge-project
http://www.scrollsaws.com/WoodLathe/woodlathefibonacc.htm
http://www.scrollsaws.com/images/Lathe/RicksGuage.pdf

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Re: construction of a golden-proportion-rectangle

Postby Lazur URH » Wed Feb 13, 2013 12:53 pm

druban wrote:It seems like a lot more work than necessary! ... In the thread you referred to, I had a much simpler method, did you see it? In the interests of anyone who might actually be looking at this thread for instructions on how to construct a rectangle of these proportions, I am posting the method again here, with annotations in case my other diagram was too difficult to follow.

I. Draw a square.
2. Duplicate it.
4. Resize the duplicate from one side until the edge snaps to any midpoint of the original square.
5. Rotate this rectangle until the diagonal is horizontal (use a grid or guide).
6. Resize the original square to snap the corner to the farthest corner of the rotated rectangle.


Image




It was a basic construction for those who don't use the snap tool -like me.
I should have searched for other methods then just applying a basic golden ratio constructioning to inkscape.
Now browsing some things that I never used in inkscape before, the problem looks even easier to solve than your method:

1. set guidelines around the background
2. set guidelines in golden ratio in extensions/guideline maker
3. draw a rectangle-square from the edge guideline of the document (holding down Ctrl) until you snap the other corner on the next guideline paralell to the edge guideline
4. scale the square by holding the same corner horizontally (by holding down Ctrl again) until it snaps to the next guideline.

Image
microUgly wrote:People who figure things out for themselves will be far better at their craft than people who are told how or have someone else do the work for them.
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