Formula for golden rectangle

The ratio of the sides is a little over 1.6

Steve

"nick walsh" <[email protected]> wrote in message
news:[email protected]...
>
> 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.

nick walsh wrote:
> 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.

Quick and dirty:

8 to 5.

-Phil Crow

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

>

"nick walsh" <[email protected]> wrote in message
news:[email protected]...
>
> 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.
-------------
the golden ratio = 1.61803399

"Another Phil" <NoSpamming@one two three four five.com> wrote:

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

>I guess I was sloppy in my wording.
BTDT<g>.

> 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.
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<g>) that you meant
the length of the diagonal. Especially since you were describing it as
something distinct from the golden ratio.

>I am going to presume you
>got the 1.906 as a length of the hypotenuse
(or diagonal). Yep

>(mean) which AFAIK, is not used
>or noted.
I agree, although I have never heard the length of a hypotenuse
referred to as a mean either.

>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.

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.

--
Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.

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" <[email protected]> wrote in message
news:[email protected]...
> "Oldun" <[email protected]> wrote in
> news:[email protected]:
>
>> 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
>>
> 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 <grin>.
>
> --
> Best regards
> Han
> email address is invalid

Thanks Han, I understand clearly now!!

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

On Fri, 22 Apr 2005 01:50:32 -0500, Patriarch <[email protected]>
wrote:

><snip of a really tortuous geometry exercise>
>>
>> I hope this helps, but it is so confusing, it may not.
>>
>
>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.

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

"Another Phil" <NoSpamming@one two three four five.com> wrote:

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

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.
--
Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.

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

> Is the method you are describing the bottom animation on this page? If
> so, It looks like a great and easy method. (Guess I am just too much
> of a visual learner--the picture does it for me, but the words were a
> strain.)
> --
> Alex -- Replace "nospam" with "mail" to reply by email. Checked
> infrequently.

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

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

Oldun wrote:
> 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
>
>
>
>

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
8. 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...




--
Will
Occasional Techno-geek

hello,

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

regards, cyrille

"nick walsh" <[email protected]> wrote in message
news:[email protected]...
>
> 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.

"gandalf" <[email protected]> wrote in
news:[email protected]:

>> seems to be down at the moment.
> -------------
> the golden ratio = 1.61803399
>
>

Can't you just round it up to 2?

Regards,
JT

<snip of a really tortuous geometry exercise>
>
> I hope this helps, but it is so confusing, it may not.
>

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

"Oldun" <[email protected]> wrote in
news:[email protected]:

> 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
>
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 <grin>.

--
Best regards
Han
email address is invalid

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

"nick walsh" <[email protected]> wrote in message
news:[email protected]...

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:

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

"Another Phil" <NoSpamming@one two three four five.com> wrote:


>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,......

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.

>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.

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?



--
Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.

On Fri, 22 Apr 2005 11:22:10 +0100, "Oldun"
<[email protected]> wrote:

>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.

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

http://mathworld.wolfram.com/GoldenTriangle.html

It'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.

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

[email protected] wrote:

>
>nick walsh wrote:
>> 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.
>
>Quick and dirty:
>
>8 to 5.
>
>-Phil Crow
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!
--
Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.

Forgot the site.

http://goldennumber.net/geometry.htm

alexy <[email protected]> wrote:

>"Another Phil" <NoSpamming@one two three four five.com> wrote:
>
>
>>The following is to construct the baseline and golden mean, is long and
>>involved.
><major snippage)
>
>Is the method you are describing the bottom animation on this page? If
>so, It looks like a great and easy method. (Guess I am just too much
>of a visual learner--the picture does it for me, but the words were a
>strain.)

--
Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.

"Another Phil" <NoSpamming@one two three four five.com> wrote:


>The following is to construct the baseline and golden mean, is long and
>involved.
<major snippage)

Is the method you are describing the bottom animation on this page? If
so, It looks like a great and easy method. (Guess I am just too much
of a visual learner--the picture does it for me, but the words were a
strain.)
--
Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.

"Wood Butcher" <[email protected]> wrote:
>"nick walsh" <[email protected]> wrote in message
>news:[email protected]...
>>
>>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.
>>

>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).

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.
--
Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.

"Another Phil" <NoSpamming@one two three four five.com> wrote:

>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.
<snip detailed explanation>

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

In 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.

--
Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.

Another Phil wrote:
> 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
>


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...

--
Will
Occasional Techno-geek