Formula for golden rectangle

[Another Phil]

Alexy:

AFAIK----

Phi (capital) is the ratio of large side to small side, or co-tangent of angle of large side to diagonal of the golden rectangle.

phi (lower case) is the ratio of small side to large side, or the tangent of the angle of the large side to the diagonal of the golden rectangle.

I saw one reference where this angel is called tau (lower case) and the complementary angle is Tau (capital)

I guess I was sloppy in my wording. I should have always stated golden mean ratio and not just golden mean, which could imply I was talking about the length of the diagonal of the golden rectangle. I am going to presume you got the 1.906 as a length of the hypotenuse (mean) which AFAIK, is not used or noted.

Do you have other information?

As far as being convoluted, using words to describe a very simple graphical technique, yes it does become convoluted because of the words. But you just may have to take my word that USING the graphical technique is fast and can be fairly accurate. The graphical technique is really simple to use in which to make marks on a story stick for all the dimensions needed for a series of golden rectangles. Which is how this thread got started; the need to make marks on wood which would, when cut and assembled, end up with a golden rectangle door.

BTW: source of some of my information: FWW September 1987 pages: 66:76-81 (sidebar to main article on Wall Paneling.) Article republished in Best of Fine Woodworking Traditional Woodworking Techniques by Tauton Press. Aside: there is a typo in the reprint sidebar. You will need to re-work the math for it to match the numbers found at web sites by google Golden Rectangle. step 3 should read: 0.61803.... not 0.01803...

In constructing the great churches and cathedrals in Europe, the carpenters, stone masons, and the like are not going to calculate no Fibonacci number or the like.

Phil

[alexy]

Looks good. But I find it easier to follow the process shown in the animation at the top of this page:

words (which are a poor substitute for the picture):

1) Draw a square. 2) Place one point of divider or compass on midpoint of base, and other upper right corner of square. 3) Mark this distance from the center of the base, along the extended baseline. 4) The length of the base of the square plus this "extension" is Phi times the length of the base of the square, and the extension itself is phi times the length of the square.

[alexy]

That's my understanding (although the trig representation seems superfluous--then again, trig was never my favorite math subject) And of course, the interesting (and defining) relationship is that phi=1/Phi=Phi-1

BTDT.

Well, you did say "and the diagonal (mean) of this rectangle is called the Golden Mean", after having already correctly defined the golden ratio. Since a diagonal is a line segment (which I have never heard referred to as a mean), I assumed (always dangerous) that you meant the length of the diagonal. Especially since you were describing it as something distinct from the golden ratio.

(or diagonal). Yep

I agree, although I have never heard the length of a hypotenuse referred to as a mean either.

No, that was a separate post, which I have saved to try out and compare to another method that I posted in response to that post. I was talking about the description in this post, that appeared to be the feature of removing a square from a golden rectangle, leaving the same shape.

[alexy]

[Another Phil]

I have to concede that point--

From my dictionary, there are three derivatives of the American English word "mean"

1- common heritage with German word meinem, to have in mind 2- old English derived from /for common, common place 3- a derivative from Latin similar to median.

And from the 3ed dictionary entry, there is currently a seldom usage of the word "mean" or "median" to be a straight line with longest length that can be drawn inside any 2 dimensional geometric figure. The "mean" is the line, not its length dimension. Thus in a rectangle, the diagonal is the longest line, the diagonal is also the "mean" of the rectangle. And since I use the word interchangeably with diagonal, yes, I am showing my age. It seems to have more common usage as in Mean Distance in orbital mechanics of planets and satellites.

Phil

[Another Phil]

YES!!!

IMHO: that method is the easiest, with the lest errors when just using: framing square, drafting compass, plum bob, string, and marking tool.

Phil

[alexy]

Well, I sure learned something! I guess my statistics background blinded me to other meanings (just as it raises my hackles to see it equated with "median")

[alexy]

Where does the plumb bob come in? Are you also using a level, and just using the plumb bob with the level as a kind of extended square?

Everything here can be done with the classic geometrical construction tools of straight-edge and compass, although the square does allow you to shortcut the process a little.

[WillR]

See "Excursions in Number Theory" by Ogilvy and Anderson (Oxford Press) for confirmation of the notation.

See page 138 -- the section on Fibonacci Numbers.

Although Golden ratio is more commonly used -- 13/8 or 1.625 -- fwiw.

It is truly entertaining for those who enjoy developing visually appealing structures of rectangular or triangular forms -- including stars and pentagrams.

See my other post...

[Guess who]

Let's get it right:

The term "Mean" refers to the fact of it being a "Mean Proportional". The Mean proportional value, x, between two other values, a and b, is such that a/x = x/b.

Writing the proportion [equal ratios] 1/x = x/(x+1) defines "x" as the mean proprtional between 1 and x+1. There is no advantage except as consistent terminology, and this proportion gives the value needed for "x".

An "image" of a mean proplrtional is found by dawing a right triangle, then the distance from the right angle to the hypoteneuse is a mean proportional of the divisions it makes with the hypotenuse of that tgriangle.

The "median" of a triangle joins a vertex to the mid-point of the opposite side. It is easy to see that you can draw a line longer than that in general. The longest line in a polygon is a "diagonal".

[alexy]

Also (more commonly?) known as the geometric mean.

Interesting. I had to draw a picture, but then it is obvious, because of the similar triangles (is that the right term for identical angles, but not identical sides? 8th grade was a LONG time ago!) created.

[Guess who]

It's more general... the [geometric] mean of several [n] quantities is the nth root of their product. This is a special, simplest case of that, and is called the mean proportional.

Yes. It's all sort of intertwined: angle in a semi-circle [right angle], similar triangles, basic trigonometry [ratio of sides], .... Geometry ...always worth a second look with a more mature [than when in high school] outlook.

[Rob Hall]

I need some aspirin.