# Smoothly increasing/decreasing graduations



## marcus (18 Oct 2012)

Just wondering what methods there are for working out increasing or decreasing drawer sizes etc that look harmonious. I tend to either eyeball it, or add on the same amount to each drawer, or use fibonacci for big size differences, but I'm sure there must be more elegant ways that I should know. Anyone got any cunning formulas?


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## Jacob (18 Oct 2012)

My cunning formula is to look at an example which looks alright and to copy that. You could be surprised at the arbitrary way things are laid out without seeming at all odd. So called 'harmonious' systems are really just ways of randomising your decision on the basis of semi mystical reasoning.


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## marcus (22 Oct 2012)

Not sure I'm with you on this. Almost all the great buildings ever built have used various mathematical/geometrical formulas of one sort or another in their design. There are reasons for this, mostly based on the not particularly mystical observation that certain proportions and relationships reliably look good and harmonious, making them a sensible starting point (not finishing point!) in thinking about a particular design. The human brain is very adept at tuning in to, and responding to mathematical relationships, whether in music, art, design, whatever. Like Bach fugues which are deeply mathematical, and some of the best and most satisfying music ever written.

Needless to say maths isn't everything, and on it's own can't produce good design, but I'm not willing to concede that the designers of Chatres Cathedral or, say, the Nereid Monument (British Museum, it's amazing) didn't know exactly what they were doing. That sort of architectural power doesn't happen by accident. Systems of proportion were the bedrock of training for architects (and furniture makers) for centuries — oddly enough right up until the time that people started making ugly buildings as a matter of course for the first time in human history. This may not be a complete coincidence....

Usually if you find something that looks good —even if it has been made totally by eye and intuition— and you analyse it thoroughly, you will find that it has an underlying geometrical order. Have you considered that the reason why the things you are looking seem arbitrary may sometimes simply be that you don't understand the principles on which they were laid out, so you can't see them when you look?


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## Jacob (22 Oct 2012)

It's a popular myth which doesn't stand up to examination. Yes some classy productions have had an order imposed on them by the maker, but many don't. People have always been searching for the right 'order' and magical/mystical solutions.
A certain level of 'harmony' may be unavoidable - things have to be structurally sound, and simple proportions 1/2, 1/3, 2/3...... come naturally. And of course in the natural world mathematics rules.
If you search for the golden ratio on any building facade of enough complexity, you may well find it, mainly because there are so many options of where and which lines to choose.



> Usually if you find something that looks good —even if it has been made totally by eye and intuition— and you analyse it thoroughly, you will find that it has an underlying geometrical order.


You may, you may not


> Have you considered that the reason why the things you are looking seem arbitrary may sometimes simply be that you don't understand the principles on which they were laid out, so you can't see them when you look?


I've considered this and don't think it's the case. It works both ways - something can comply exactly with a set of 'aesthetic' rules but still be garbage. 
If you spend long enough working over anything you can often find a delusory mathematical relationship - if that's what you are looking for!


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## marcus (22 Oct 2012)

It's not really a question of looking for complexity where there is none — the people involved have generally been quite open about their use of geometry and proportion, it's incontrovertible historical record that this is how things were done in top end work. There's an enormous body of evidence, starting with the testimony of the builders and cabinet makers themselves. These were busy and practical men working in a difficult profession for demanding and discerning clients in often very scary times; they were not in a position to waste time on things that didn't help them. 

Obviously just blindly following proportional rules is not going to produce great work, but that is not how such systems are intended to be used. They are flexible principles, not laws.



> It's a popular myth which doesn't stand up to examination.



Whose examination? Anyone can make something small and simple that looks nice by winging it, but are you really in a position to make such a judgment — ie have you made something seriously complex which is astoundingly beautiful and widely acknowledged to be so? If not I'm going to go with the judgment of those who have — on the basis that the quality of their work indicates that they're probably not such fools as you seem to take them for.


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## Jacob (22 Oct 2012)

marcus":xoiq057b said:


> It's not really a question of looking for complexity where there is none — the people involved have generally been quite open about their use of geometry and proportion, it's incontrovertible historical record that this is how things were done in top end work. There's an enormous body of evidence, starting with the testimony of the builders and cabinet makers themselves.


Can you point us to, or show us, to a bit of this evidence?

Here's the classic example of the theory being made to fit the facts http://www.mathsisfun.com/numbers/golden-ratio.html







It's obvious that lines could be drawn equally convincingly in many other positions, to prove almost anything you like!

My personal favourite bit of nonsense is a drawing by Prof Thom showing how a line drawn through a particular row of stones is supposed to align with probably sunrise, or Arcturus rising 2000 years ago etc.
Doubtful on one count - any line drawn anywhere will align with something or other sooner or later.
But also wrong on another count - if you lifted the page up and looked down the line of stones, Thom's line clearly wanders off from the best guess at the median line. If he'd noticed this he would no doubt have found an alignment with another astronomical event.

It's all harmless fun though!

PS the most obvious rectangle of the front of the Parthenon is (IMHO) the area between the floor and the cornice, which is close to 1:2. 
I bet that's what they started with - other ratios being accidental.


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## marcus (22 Oct 2012)

*Thomas Chippendale good enough for you?*

The Gentleman's and Cabinet Makers Director — http://digicoll.library.wisc.edu/cg...&id=DLDecArts.ChippGentCab&isize=M&pview=hide\\ 

*Andrea Palladio?* (ie the man who is widely agreed to be the most influential architect in the Western Tradtion through his four books, which contain chapters and chapters on mathematical proportion). http://www.aboutscotland.com/harmony/prop3.html

*Leonardo de Vinci OK?*

http://en.wikipedia.org/wiki/Vitruvian_Man

*Vitruvius?* - a roman architect in the 1st century BC whose books have been hugely influential and are a very good source on design practice at that time.

http://www.amazon.co.uk/On-Architec...=UTF8&qid=1350913006&sr=8-1#reader_0141441682

Really I could go on for pages, but I can't be bothered. There's heaps of further stuff on this, and on new proportioning systems used by the better modern architects, and on the gothic etc. etc. 

There's a huge difference between justifiably dismissing someone's ill-informed attempt to find the golden ration in all things, or new age star alignments in pyramids, or whatever, and being ignorant of basic and undisputed architectural history.

You know, by the by I can't help noticing that your 'seat of the pants' approach to design is not a million miles away from a certain ... Jim Krenov :wink:


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## Jacob (22 Oct 2012)

Yes we all know that there was a huge effort to find these relationships but taking just one example from your links, Palladio's room size theory recommends as follows:

_1. Circular 
2. Square 1:1 
3. The diagonal of the square 1:1.414....etc. 
4. A square plus a third 3:4 
5. A square plus a half 2:3 
6. A square plus two-thirds 3:5 
7. Double square 1:2 1. Circular 
2. Square 1:1 
3. The diagonal of the square 1:1.414....etc. 
4. A square plus a third 3:4 
5. A square plus a half 2:3 
6. A square plus two-thirds 3:5 
7. Double square 1:2_

It's pretty obvious that this covers just about every possibility (not including curves and other heathen variations). Anything bigger than a double square is going to look like a corridor ( but there are plenty of examples where that works perfectly well as a room - think of 'long galleries' - how horribly incorrect :roll: ). Anything smaller, down to a square, is going to be close enough to one of these options as to make no difference. It follows that any room you care to look at it is also going to be close - whether or not that was the intention.

But then there are a whole other set of rules with other variations - in fact _everything_ is probably conforming to one or another of them!

In the end it's down to taste - personally I prefer wild, primitive, over decorated St Marks to harmonious, sterile and tidy St Georges




> new proportioning systems used by the better modern architects,


Some do some don't. Le Corbusier's modular thing was more a stylistic device than a practical reality. Other builder's modular systems are more related to material sizes and are just practical. Le Corbusier's 'Modulor' was based on 2.2 metres. Building regs ceiling height is 2.3 metres


> and on the gothic etc. etc.


What was their system then? Obviously very different from the classical. I thought they were mainly preoccupied by going ever upwards and structural issues were the main concern.


Getting back to your harmonious drawer proportions - which system would you prefer? Classical, neo classical, Gothic, ditto revival, muslim, "Le modulor" etc etc or just take a punt on it - make them each an inch bigger than the preceding one.
Or - to be utterly harmonious, make them all the same size!


PS These are interesting. You could just scale one off with dividers but I reckon equal increments bigger than say 1/2" would probably look good. Any finer tuning might not be apparent.


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## marcus (22 Oct 2012)

> It's pretty obvious that this covers just about every possibility



Yes, if you just take one shape in isolation, but they're not supposed to be taken in isolation, the proportions relate to other proportions in a system. Also that's a very small part of what he is proposing. It is much less likely, for example, that the harmonic proportions of the Greek orders would just happen if worked out in any detail.



> think of 'long galleries' - how horribly incorrect



That's a bit of a straw man. The idea is not that anything that does not fit perfectly within a system is wrong. The system is a loose structure within which variation and departures are not only acceptable but necessary. But it's only the fact that there _is_ a system that makes the departure mean something... There's a dialogue between system and freedom, control and intuition. Rather like life in fact. It's a cliche that in art true freedom requires limits. Some of these limits are provided by physical reality (the height of a desk etc., the mechanics of an arch, etc) and some can be provided by man made systems (greek orders, building regs) which have different emotional effects, and contain various coded values and social messages which can be upheld or subverted. It's a rich inheritance to play with.



> What was their system then? [ie gothic]



I think it wasn't so much a system as a process. It was circle based, growing organically out of the decision to use pointed arches formed from the arcs of circles, worked out within the constraints of how a church needed to be arranged, a sort of naturally flowing geometry, but still impossible without the basic geometry of the arch that underlay it, and the system of church layout....



> In the end it's down to taste - personally I prefer wild, primitive, over decorated St Marks to harmonious, sterile and tidy St Georges



I like both, for different reasons....



> Getting back to your harmonious drawer proportions - which system would you prefer?



I wasn't looking to subscribe to a particular system — for what it's worth I draw on many different approaches, mixed up with a good dose of intuition. 

I did want more choices and to save some time. For example one thing I've found quite tricky is to get a smoothly accelerating transition between narrow and wide where the difference between the narrowest and widest is quite small. This is a subtle effect I like a lot, but it takes me ages of fiddling around —ie winging it— whenever I do it. An equation would be practical and timesaving which is another advantage of mathematical approaches — used wisely they're another option to help you get down the road — every little helps.


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## Jacob (22 Oct 2012)

marcus":1busiwep said:


> ..... An equation would be practical and timesaving which is another advantage of mathematical approaches — used wisely they're another option to help you get down the road — every little helps.


I think you just make up your own. That's what they all did. 
Perhaps choose smallest and largest first, then size the others in equal steps between?


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## Jacob (27 Oct 2012)

So Marcus did you find a cunning formula?


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## marcus (27 Oct 2012)

> That's what they all did.



Whatever.



> Perhaps choose smallest and largest first, then size the others in equal steps between



If that was all I wanted to do I wouldn't have needed to ask the question.

I'm interested in ways that go beyond the basic maths I already have ie functions which when plotted do not lead to straight line, but perhaps something like this: 

*
*
*
-*
-*
---*
-----*
---------*
------------------*
---------------------------------------*
----------------------------------------------------------------------------------------*

Or perhaps a different kind of curve. But not a straight line, which is what you get when you increase size in equal steps.

I want a variety of approaches to the problem which I can adapt and quickly and reliably draw on when I need them, rather than relying only on trial and error, which is time consuming.

Any one good at maths out there?


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## Jacob (27 Oct 2012)

h = height of drawer, a + starting height (top drawer?), n = number of drawer (0,1,2 etc), x = any number you choose except 0
then h = a + nx gives you graduated spaces in a straight line
or h = a + n^x gives a curve (n^x means n to the power x).

Or you could make each drawer height be a multiple of the one before e.g. times the golden ratio (1.61803398875).

There are no end of possible formulas!


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## marcus (27 Oct 2012)

Great, that's the sort of thing.

This morning I've come across another that looks nice - the harmonic series ie

1 1/2 1/3 1/4 1/5 1/6 1/7

So that's:

Fibonacci
Playing around with powers
Multiplying each product by x (where x is greater than 1)
Harmonic series

Any more to play with?


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## Phil Pascoe (27 Oct 2012)

A catenary curve is a fibonacci series - maybe that's why it looks natural. They occur in nature, e.g. free hanging honeycomb.


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## AndyT (1 Nov 2012)

I've had a look at some old books and some old furniture.

This design, from Practical Home Woodworking (1951) shows a tallboy with graduated drawers. It looks quite carefully thought out. The system seems to be that the increment increases as you go down the stack. 






Sizes are:
6 1/8"; 6 5/8"; 7 1/4"; 8"; 9 1/8"

so the increments are:

1/2"; 5/8" 3/4"; 1 1/8"

The increase is a bit clearer if we reduce all those sizes to 1/8ths:

4; 5; 6; 9


(It's also interesting to see that the drawing suggests gradated positions for the drawer pulls too.)

I also looked at some furniture: a dressing table - probably Edwardian:






This has drawers

5 7/8"; 7 1/8"; 8 5/8"

which gives increasing increments of

1 1/4"; 1 1/2"

- somewhat like the design in the book.

However, the drawers in this wardrobe - of similar vintage - which would be hidden by doors any way






go like this:

6 3/4"; 7 3/4"; 8 3/4"

which suggests the possibility that the maker just used standard boards of nominal 7", 8" and 9" breadth and planed them to finished sizes. 

So I expect you don't need a complicated formula for five drawers or fewer, and economy of material will often matter more.

(Sorry for the picture quality - it's oddly sunny and the light is all wrong!)


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## Jacob (1 Nov 2012)

The main reason for graduating drawer sizes has nothing to do with aesthetics - it's about the contents. Small things in the top little drawers - socks, hankies etc, to big bulky things in the bottom - Jumpers etc.
So you could get all drawers the same size in an office piece, but nobody would comment on the aesthetics I'm certain.

One problem with mags and books is that the writer often wants to appear to know something which the readers don't. Hence elaborate layout as with these drawers. Or all those confident assertions about specific angles for so many tool edges, dovetails etc. All nonsense! You can make them up anyhow and somebody somewhere will believe you.


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## AndyT (1 Nov 2012)

Fair points Jacob.

I guess sometimes, especially in instructional books for beginners, what the author has to do is helpfully make a series of choices on behalf of the reader. So instead of saying 'make the top drawer suitable for socks' he says to make it six inches deep; instead of saying 'mark the dovetails at a sensible angle to suit the wood' he says to cut them at exactly 1:6. Without some of that removal of choice, the decisions can be bewildering.

The mistake would then be - as you say - to regard those beginner's prescriptions as always right, and anything else wrong.

A bit like always ordering the safe set menu, never eating a la carte, and never devising a meal of your own.


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## bugbear (1 Nov 2012)

Jacob":alotaisz said:


> The main reason for graduating drawer sizes has nothing to do with aesthetics - it's about the contents.



The real magic of good design is to fulfil practical requirments and aesthetics simultaneously. Doing one or the other is a good deal easier.

BugBra


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## Jacob (1 Nov 2012)

bugbear":hmik57d1 said:


> Jacob":hmik57d1 said:
> 
> 
> > The main reason for graduating drawer sizes has nothing to do with aesthetics - it's about the contents.
> ...


BugBra? :shock: How does it compare with Maidenform?
Perhaps nothing to do with aesthetics - it's about the contents?


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## bugbear (2 Nov 2012)

Jacob":hmp8wi0k said:


> bugbear":hmp8wi0k said:
> 
> 
> > Jacob":hmp8wi0k said:
> ...



It's true what the Tabloids say; there is filth on the internet, and Jacob Butler knows where to find it!

:lol: :lol: :lol: 

BugBear

(you might want to clear your browser history before Mrs Butler uses that computer again)


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