Thread: Basic geometry instead of measuring?

Originally Posted by Auld Bassoon

Isn't that an engineering diagram for an 'A' frame hoist to help the wee lassie back onto her feet - all that weight :eek:

I'm almost sorry I didn't ask about formulae for cantilevering. (Not her again! )

For interactive constructive geometry, the Cinderella tool is pretty sophisticated:

http://cinderella.de/tiki-index.php

Google Sketchup also has an attractor to golden rectangles.

Hey Skew...have you tried proportional dividers? I posted them as an answer only half in jest. If you are serious about no math layout, and you do a lot of work with this ratio, perhaps you could make a pair?

Greg

Thought about 'em? I use an old set of 'em for 2:1 and 3:1 Not adjustable though.

I can't say I've ever seen any of the "non-adjustable" ones for the Golden Mean, but I haven't really looked. Hmmm... I wonder how hard it'd be to make an adjustable pair? :confused:

Dammit! I can already guess what's going to keep me awake tonight...

Originally Posted by Skew ChiDAMN!!

Dammit! I can already guess what's going to keep me awake tonight...

My work here is done.

Yet another fabulous thread.
I have added it to my favourites, too.
The article on the Golden Mean refers to a Duckworth from the 1920's. Is that the same person responsible for what was the original basis for calculating how many runs the team batting second needed to get in shortened one day cricket matches? Or was he the fellow responsible for the semi final system used in Aussie Rules and now league?

BTW, I think I can simplify the language used in the calculations.
Chapter One
Draw a rectangle, ABCD , hereinafter called "the first rectangle".
Add a further line, EF thereby creating a rectangle ABEF, herein after called "the second rectangle",and creating another rectangle CDEF, hereinafter called "the third rectangle,unless of course the line EF is drawn differently whereby we have rectangles AEFD, "the fourth rectangle" and EBCF, "the fifth rectangle".
For ease, we are ignoring at this stage the possibility of the rectangle being better described as ADBC with EF then creating AEFC, "the sixth rectangle" and EDBF, "the seventh rectangle" or possibly ADFE, "the eighth rectangle" and FBCE, "the ninth rectangle".
Take either of the diagonals in the second rectangle, AE or BF, respectively named "the first diagonal" and "the second diagonal" of the second rectangle
Clear so far?
To be continued? No, I think I will rap my nuckles with the ruler and jab myself with the compass and allow sanity to return. There is no place for "Plain English" in geometry.
CJ

Put the lid back on the glue CJ.

OK now you got me thinking so I HAVE to work out the answer.

1.use a compass to draw a square ABCD. AB is the predetermined width.
2.extend AD and BC (2 red lines)
3.use the compass to mark point H, E and F
4.if B, E and F are on the same line then AF is the length. If not, back to pt 3 and try a different length.

This time AB is the predetermined length

1.use a compass to draw a red line square to AB
2.use the compass to mark point I and C
3.draw square ACJI
4.draw line BG
5.use the compass to mark H (copy from IB)
6.if A, H and G are on the same line then AC is the width. If not, back to pt 2 and try a different length.

Hmmm... Dunno, Wongo. Here's the methods as I've put 'em in my shop notes:

For a predetermined width (AB):

• Create a square ABCD of (width) dimensions.
• Bisect the square. (xy) There's dozens of methods for doing this, even without a ruler
• Place a compass at point x and scribe an arc from corner D to intersect with the line congruent to BC thus determining point E
• BE is the desired length, giving the Golden Rectangle of ABEF

For a predetermined length (AB):

• Create a 2:1 rectangle ABCD of (length)x(0.5 x length) dimensions.
• Mark the diagonal AC
• Place a compass at point C and scribe an arc from point B to intersect with AC, thus determining point x
• Place the compass at point A and scribe an arc from point x to intersect with the line congruent to AD, thus determining point F
• AF is the desired width, giving the Golden Rectangle of ABEF

Proofs included, just in case...

Cheers Skew. Your method is very clever. Whoever came up with it must be very clever. Mine is the dumb way but I just like to have a go.

Your way'd work, I think, but has a bit of trial'n'error in it. Without sitting down and doing the math, I can't really say... and I'm inherently lazy. I prefer to just ask the question and have you lot answer. "My" way is generally attributed to the Greek Pythagoreans and I guess you can't get much cleverer men than them... but by the same token I've seen references where the same principle was used by far older civilisations in their art'n'architecture.

So attribution doesn't equate to discoverer. As usual.

Good one Skew - Just printed my copy for the shed!

Fascinating thread... thanks , Skew. I'm always using the Golden Mean to help in designing my paintings, but hadn't delved too deeply into it (near enough being good enough in this case.)



The following rough and ready (but accurate way of doing it ) comes from this site :

http://community.middlebury.edu/~harris/Humanities/TheGoldenMean.html



The 1 : 2 rectangle and the Egyptian Rope Measurers

It has always been a problem to undertand how the Greek architect and his consruction workers managed to incorporate into the design of large-scale temples like the Parthenon the "irrational" measurements which the Golden Mean requires. The Greeks had no system for handling irrational numbers in a theoretical manner, let alone applying irrational measurements to an actual conctruction project. Extending the numbers of the GM proportion from one place to another on a building in the process of construction would seem to have been impossible.
But the proportions are clearly there in fact. So at this point I want to introduce a method, which I take to be an independent discovery on my part, and the key to the use of the GM ratio in large scale applications in architecture, for example in Iktinos' GM based designs for the Parthenon..
a) I construct a 1 : 2 rectangle of any size, depending on what scale I am working with.
b) I fix a non-elastic string or tape to the lower left hand corner of this rectangle, and run it around a point (a nail) at the upper right hand corner then draw it down to the lower right corner. This adds the short side of the rectangle (1) to the diagonal (sqrt5).
c) I then take my string, hold the ends together, and stretching it out double, I halve its length. This is now (sqrt 5+1)/2 or numerically 1.618....., the number have been seeking for comparison to one (1).
d) I can take this string/number and use it as short side of a new larger rectangle, and construct a new larger rectangular figure with the same proportions preserved.
e) But I may want to get smaller, that is find the inverse (1/x) of 1.618 (which is. 618), I can do this by the string method too. I draw my line from left lower to right upper corner, bring down the line to lower right corner, and folding that back along the diagonal, I mark that point, which represents the subtraction of one (1) from sqrt5. If I take that remaining length of my line from the start to the mark, and fold it double, I get. 618 or the inverse of 1.618 (1/x).

Now, if I could only remove the bold type from the top of this post, life would be perfect!