Box - golden ratio

[lurker]

I've ask a similar question in the past & did not get an answer I could understand.

I understand how to apply the ratio in two dimensions but not three

Say I was building a box and knew one dimension:

For arguements sake lets say my stock is 100 mm and thus my box will be 100 mm high. What would length & depth be??

And why.

 

[jpb]

Hi

The golden ratio, represented with the Greek letter phi has the value 1.618. Which means for a rectangle the long dimension is 1.618 times greater than the shorter dimension.

A box shape incorporates multiple golden rectangles that are proportionate to one another.

Using your example for a box hight of 100 mm, call (A = 100mm)

The width if calculated by A * phi which is 100 * 1.618 = 161.8 mm

The depth is calculated by A/phi which is 100/ 1.618 = 61.8 mm


There is a good article on the fine woodworking web site. It was also in the designing furniture FW extra that was in WH Smith a couple of weeks ago.

James

 

[woodbloke]

I've ask a similar question in the past & did not get an answer I could understand.

I understand how to apply the ratio in two dimensions but not three

Say I was building a box and knew one dimension:

For arguements sake lets say my stock is 100 mm and thus my box will be 100 mm high. What would length & depth be??

And why.

Jim - the GR is 1:1.618 and in a three dimensional object you can't have all three dimensions adhering to the GR...only two. If the front of the box is 100mm high then the length would be 161mm and the short side would be 62mm (100 divided by 1.618) The top would then be 161x62mm which doesn't conform to the GR whichever way you slice it.
What's done normally is a compromise bit of juggling with the dimensions so that it looks OK but doesn't necessarily sit exactly with the GR proportions
A space occupied by a frame can also adopt GR proportions - Rob

 

[StevieB]

Well the golden ratio works out to about 1.6 to 1. So if you have 100mm as the smaller of the two dimensions, the other should be approx 160mm.

A ratio is the product of two dimensions, not 3. So its impossible to get it to work for both the side and the front of a box unless height equals depth and both are in the ratio 1.6:1 to the length. Where it is traditionally used is in graduated drawers where each is larger by a ratio or 1.6:1 as you go down the series. It can also be calculated for a rectangle (as in the length of a box in relation to its height).

As to why - it is simply what looks visually appealing. It also occurs in nature and as such was thought (and possibly still is in some instances) to be a design principle introduced by God. Typically quoted examples are spirals such as nautilus shells, certain distances on the human body such as arm length and numerous other examples. In these instances its likely to be a tipping point for maximum efficiency against minimum effort such that any deviation requires either more effort or produces less efficiency. So more a product of environment than design.

HTH,

Steve.

 

[bodgermatic]

Hmmm, this one could get contentious I think I see what you're getting at. If the height is x, the length along the side which has that x as it's vertical is 1.618x. At this point, either one end or the top will have to be square. I think it's impossible to have all 6 sides of a rectangular box with golden ratios.

 

[woodbloke]

Hmmm, this one could get contentious I think I see what you're getting at. If the height is x, the length along the side which has that x as it's vertical is 1.618x. At this point, either one end or the top will have to be square. I think it's impossible to have all 6 sides of a rectangular box with golden ratios.

...as I said. Only four can adhere directly strictly to the GR - Rob

 

[oddsocks]

I don't think you can have a three dimensional object that conforms on each face. The ratio is 1.618 (..to be baffled : http://en.wikipedia.org/wiki/Golden_ratio) or (for golden rectangle)...http://en.wikipedia.org/wiki/Golden_rectangle...extract from the latter....A golden rectangle is one whose side lengths are in the golden ratio,.. approximately 1:1.618.
A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle; that is, with the same proportions as the first. Square removal can be repeated infinitely..

If your box is 100 high then the width is either 162 or 62 (100x1.62 or 100/1.62). Assuming you go with that face being 100h x 162w, the top face then needs to be 162w x 100d or 162w x 262d to maintain the ratio. Whichever of these you chose, the remaining side face is not in the ratio (it will be either square 100hx100d or rectangle at 100hx262d).

Hope that makes sense = sketch it and see. I've rounded to whole numbers!

Dave




EDIT - seems others did the same examples but beat me to it
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[lurker]

Thanks chaps,

I think I was expecting too much from it.

I thought some classic buildings were of the golden ratio & buildings are 3 dimensional.

JPB - I'll look in on FW website - assume Mags are now gone??

 

[woodbloke]

Thanks chaps,

I think I was expecting too much from it.

I thought some classic buildings were of the golden ratio & buildings are 3 dimensional.

JPB - I'll look in on FW website - assume Mags are now gone??

The classic case is the Parthenon in Athens which conforms to the GR, but probably not in all three planes - Rob

 

[MikeG.]

The classic case is the Parthenon in Athens which conforms to the GR, but probably not in all three planes - Rob

To be really pedantic, it doesn't conform to the ratio in anything except plan, because all its vertical elements taper or curve slightly.

To add to the pedantry, 1.618 is only an approximation. It is a geometrically derived ratio, which is quite easy to draw. One of its beauties, for the mathmatically minded, is that the inverse of the golden ratio is the same as the golden ratio minus one (ie 1/1.618..... is 0.618.......). As Einstein repeatedly said...."beautiful!"

Mike

 

[JoinerySolutions]

It can get even more complex due to subdivisions derived from the basic geometry. These are used for moulding sizes, heights they are place at etc.
However, for your box to be correct at 100mm high to conform to the Golden Ratio it would be height times 1.618 (=161.8mm) and length would be height times 1.618+height (=261.8mm).
This is not the only method that can be used, the sq root is a common one.
Draw a square and extend the bottom line of it out to the right, then set your compasses pointy bit in bottom left corner and pencil to top right corner, now scribe an arc down to meet bottom line of the square. This will increase the square by a factor of 1.414.
This is often used to find the height of a top drawer in a cabinet, though when a variety of widths are to be used a mean size is used.

Incidently if you did the same square excersise but set the compass point to the middle of the square base line to top right the extension would be a factor of 1.618.

 

[MikeG.]

..........and just because I am really bored.........a box in which 4 sides are in golden ratio proportions could also have the other two sides in the proportions 1.618..... :1/1.618......, which of course is a ratio of 1:2.618.... Rather pretty, if not fully beautiful!

Mike

edit PS which is another way of saying what Rob just said only I didn't know he was going to say it.....

 

[woodbloke]

The classic case is the Parthenon in Athens which conforms to the GR, but probably not in all three planes - Rob

To be really pedantic, it doesn't conform to the ratio in anything except plan, because all its vertical elements taper or curve slightly.


Mike

I was going to add that the exterior view from the ground is also misleading because as you say, columns and 'vertical' surfaces are not really vertical...damn clever these Chinese - Rob

 

[JoinerySolutions]

I am pretty sure the term for the curvature of columns is "Entarsis". However I cannot find the details I had when making some moulds for fake ones.
The reason for it was that as the column gained height the diameter would increase then at a set point it would decrease again, thus deceiving the eye and making the column appear to be straight or parallel.

Maybe Mike can elaborate, it was many years ago that I needed to understand it, I'm sure I have a cad drawing saved somewhere that illustrates it.

 

[jimi43]

What if I work all this out for my chisel box only to find that they don't fit?



Sorry guys...could not resist that one!!

Great thread...real eye-opener for me...thanks!

Jim

 

[chipchaser]

Entasis

more than one way of setting out, a couple here:

http://chestofbooks.com/architecture/Cy ... olumn.html

not everyone believes in the golden section etc:

http://www.maa.org/devlin/devlin_05_07.html

Graham

 

[MikeG.]

Yep, its entasis, and the theory is that it was to make the buildings appear grander by having them tend to a "fake" vanishing point vertically. In other words, they distort perspective to make a thing appear taller. Entasis escaped from grand buildings into all sorts of other forms of contruction and decoration.

I didn't use the term in my original post for fear of driving the few remaining readers of the thread into a near coma-like state out of mind-sapping boredom!

Mike

 

[Ironballs]

Or just get Box Making Basics by Freedman (Taunton). Great for beginners and advanced and covers this topic at length. Think I built a 3:2:1 box based on this

 

[devonwoody]

If GR is pursued continuously things can get boring. The word pedantic come to mind again.

 

[lurker]

Or just get Box Making Basics by Freedman (Taunton). Great for beginners and advanced and covers this topic at length. Think I built a 3:2:1 box based on this

Actually I'm going to build an Oak bench seat. :lol:
I'm using a lovely design I saw in Brit wood but could not decide on the proportions and I'm hopeless at aesthetics.

I have decided on a comfortable height so with guidance from this thread I wanted to sort depth & length - mentally, Joinery Solutions proportions sound about right - will do a mock up at the weekend.

Thanks again chaps :lol: