Formula for golden rectangle

[nick walsh]

I'm making screens for my backyard and figured a pleasing shape would be the golden rectangle but I forgot the formula and my connection to the net via IE seems to be down at the moment.

[Steven and Gail Peterson]

The ratio of the sides is a little over 1.6

Steve

[gandalf]

------------- the golden ratio = 1.61803399

[Wood Butcher]

There are many formulas but the simplest is

1/x = x-1 Which leads to the quadratic x^2 -x -1 = 0 (ignore the negative root).

Art

I'm making screens for my backyard and figured a pleasing shape would be the golden rectangle but I forgot the formula and my connection to the net via IE seems to be down at the moment.

[alexy]

i.e., (sqrt(5)+1)/2 But the estimates of 1.6, 1.62, or 1.618 are likely about as close as you want.

[John Thomas]

"gandalf" wrote in news: [e-mail]:

Can't you just round it up to 2?

Regards, JT

[phildcrowNOSPAM]

Quick and dirty:

8 to 5.

-Phil Crow

[Another Phil]

Nick:

This is not what you asked for, but here it goes anyway:

The problem is there is a square root of 5 in the equation, thus not rational. Traditionally, only a framing square, drafting compass, and plum-bob were the measuring tools to make a story stick.

The diagonal of a golden rectangle (golden mean) has a angle of 31.7 degrees. Thus if you can construct, with some accuracy, a line (let's call it the baseline) with another line at 31.7 degrees (and call it the golden mean), you could measure off any length on the original line, and drop down to the diagonal line to find the second dimension. Marking both lengths on the story stick. You should be OK. (actual angle rounded off:

31.717474411.... but trust me, measuring 31 degrees is hard, not to mention 31.7.)

The following is to construct the baseline and golden mean, is long and involved.

1) construct a temporary right triangle such that the base is twice the length of the height. Accuracy of 90 degree angle is critical.

--other angles should be about 26.565 and 63.43 degrees. The legs are one unit and two units, while the hypotenuse is square root of 5 units.

2) construct an arc centered on the apex of the 63.4 degree angle, from the right angle to the hypotenuse (this arc is one unit long)

3) Now construct an arc centered on the apex of the 26.5 degree angle, starting at the previous intersection on the hypotenuse and arc it down to the base of the temp triangle. The radius of this second arc is (square root of 5) - 1.

4) You now have the two sides of one golden rectangle, the baseline of the temp triangle, and the length from the apex of the smaller angle to the mark on the base line. (2 units and ((square root of 5)-1 units) (do the math for ratio Phi (capital phi) and phi (lower case).)

5) Use the two dimensions, to construct a golden rectangle, making sure the angles are accurate 90.00 degrees.

6) construct the diagonal mean.

7) extend one the long sides of the constructed rectangle, and extend the diagonal mean.

8) mark on story stick the two lengths you will be using.

I doubt I could make a golden triangle from what I just wrote if I had not done it several times before. Good luck.

I hope this helps, but it is so confusing, it may not.

Phil

[Patriarch]

Use the aproximation of 1.6 something, and make a prototype out of scrap or cheap materials. Ask the significant other in your household if she likes it that way. Adjust it to fit her sense of proportion, and your ability to make it suit the task. What looks 'right' will depend on the surroundings, the contrasts and the materials used.

'Golden' can have multiple, correct meanings. Sociology trumps geometry every time.

Patriarch

[alexy]

Or actually, the ratio of any two successive Fibonacci (sp?) numbers, depending on the accuracy you want. The F numbers is the sequence where each number is the sum of the previous two:

0,1,1,2,3,5,8,13,21,34,55,89,...

8:5 gives 1.6, probably good enough for any practical purpose in design, IMHO. Using it to proportion a door 78" tall gives a width of

48-3/4". Using 55 and 34 gives you a width of 48.218", which is within 1/64" of "true". 8:5 sounds like a darned good rule of thumb to me!

[Oldun]

Could someone please explain what on earth the "Golden Triangle" is, and how is it used in designing a screen?

A clear explaination or simple sketch would help my ignorance.

Thanks.

Oldun

[Han]

"Oldun" wrote in news:4268d054$1 [e-mail]:

There is a whole (almost) cult that regards the Fibonacci series as holy. It turns out that the ratio of ~8:5 is very pleasing to the eye. That's all in a single sentence.

Look up Fibonacci, golden rule, golden ratio on you favorite search engine. Then go to Amazon and do the same. Finally go to your local library and read some .

[Oldun]

Thanks Han, I understand clearly now!!

Oldun

[Guess who]

I could go into a long explanation about Greeks liking a cerain rectangle shape, but try Google. Here's an authoritative explanation:

's as if they looked at some rectangles and decided like Goldilocks ....too thin, too fat ...just right. There was a lot of mysticism in math those days. Some think there is still now.

[WillR]

Oldun:

You could always fool with the Mystic Pentagram - adopted by the Fibonacci Society as it its own...

The Golden Ratio appears in the 5 pointed star -- the Pentagram.

First Rectangles... The Golden Section seems to be 13/8 (phi is the usual symbol an o with a vertical slash). A pleasing rectangle is said to be 13 by 8 . And if you draw a box, of same on the right side partition off a square of 8 X

1. Then put two 1X1 squares along the bottom left. On top of them put a

2X2 and look at the shapes -- after filling in the obvious lines. (Forming a 3X3 to the right and a 5X5 above. Do it in the seq. given and it is amusing and pleasing (and mysterious) if you like to design. :-)

Now Triangles and Stars... Now -- golden triangles... and the pentagram. Have a look at a pentagram

-- ignore all the criss-crossing lines -- and note that you can draw a isosceles triangle (36 deg at the apex) -- rotate it to three different positions and voila! A pentagram! (note the three :-) )

Each Isosceles Triangle forming a point in the star has 72 deg in the base and 36 deg. in the apex. (Isosceles -- Two equal sides -- remember? :-) )

This means that if you draw a right angle from one of the equal length sides to to a base point that it forms another similar triangle with 72 at the base angles and 36 at the apex. It looks rather pleasing and may be what you are thinking of. You could indeed make an interesting Mosaic...

A little more work and you can be a cryptographer... You could always try some ellipses using these numbers and inscribe some triangles and Pentagrams... Write a book about it and you too could be rich.

That should allow you to claim all sorts of mysterious things... :-)

Of course during the full moon... shudder... I can't talk about that part... My Junior Mathematicians oath forbids it under pain of arrggghhhhh nooohh the pain...

[Cyrille de Brébisson]

hello,

if you have an A4 (or any A series) peice of paper, you got it...

regards, cyrille

[nick walsh]

My significant other is a fat spaniel.:) If he pees on it do you think that means he likes it?

[Another Phil]

Oldun:

Take any rectangle (which is not a square), there is one side longer than the other.

A very long time ago someone discovered that there is / was one, and only one, combination of dimensions of a rectangle that:

if you take the smaller side length and make that the long side of another rectangle and you take the difference in length of the smaller from the larger and use that difference as the smaller length in the new rectangle, you end up with the same ratio of sides as before. Thus the next smaller rectangle can be constructed. And repeat, and repeat,......

This very specific ratio of sides that have an infinite repeatable number of smaller rectangles of same ratio of sides is called the golden ratio and a rectangle of this golden ratio is called the golden rectangle, and the diagonal (mean) of this rectangle is called the Golden Mean.

A golden rectangle in and of itself is not that great. It is the rectangle within the rectangle within the rectangle that counts:

classic Greek statue: the whole statue can be boxed by golden rectangle the head can be boxed by golden rectangle the eyes also can be boxed by golden rectangle.

Engineers, and other calculator (slide rule) types are fascinated by Golden Rectangle, Golden Mean, and so forth. As in AH-HA, a math formula for creating ART!! (** Art does not work well with Math formulas in real life, trust me. **)

Phil

>

[Oldun]

Thanks everybody who replied to my question about these golden triangles. I now know what you were talking about. A Google search was also a great help, damn should have asked Google first instead of broadcasting my ignorance.

Cheers. Oldun

[alexy]

My head is spinning! That strikes me as a very convoluted way of saying "take the rectangle and eliminate a square with side equal to the shorter side of the rectangle. What is left is another rectangle with the same proportion.

I have never heard of this definition of Golden Mean. Every place I have seen it, it is used as a synonym for golden ratio. The diagonal of a golden rectangle with sides 1 and 1.618 would be about 1.902. are you saying that this is the Golden mean, different from Phi or phi?