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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.

I need some aspirin.

When the Lord calls me home, whenever that may be, I will leave with the greatest love for this country of ours and eternal optimism for its future. ~Ronald Reagan

two

the

can

<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.

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.

#### Site Timeline

- posted on April 23, 2005, 11:48 pm

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.

- posted on April 25, 2005, 12:18 am

When the Lord calls me home, whenever that may be, I will leave with the greatest love for this country of ours and eternal optimism for its future. ~Ronald Reagan

two

the

can

- posted on April 23, 2005, 3:19 pm

Another Phil wrote:

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...

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

Will

Occasional Techno-geek

Click to see the full signature.

- posted on April 22, 2005, 12:21 pm

On Fri, 22 Apr 2005 11:22:10 +0100, "Oldun"

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.

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.

- posted on April 22, 2005, 12:41 pm

Oldun wrote:

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...

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

Will

Occasional Techno-geek

Click to see the full signature.

- posted on April 22, 2005, 1:47 am

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

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

- posted on April 22, 2005, 6:50 am

<snip of a really tortuous geometry exercise>

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

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

- posted on April 22, 2005, 2:51 pm

wrote:

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

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

- posted on April 23, 2005, 4:46 am

<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.

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

- posted on April 23, 2005, 5:19 am

Forgot the site.

http://goldennumber.net/geometry.htm

http://goldennumber.net/geometry.htm

--

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

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

- posted on April 23, 2005, 1:46 pm

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

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

Phil

- posted on April 23, 2005, 2:43 pm

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.

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

- posted on April 22, 2005, 2:18 pm

hello,

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

regards, cyrille

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

regards, cyrille

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