A Practical Handbook of Geometrical Composition and Design, Matila Ghyka, 1952c 1964

I ,

!I

A Practical Handbook oI

GEOMETRICAL COMPOSITION AND DESIGN

I !I I !

by

, MATILA

GHYKA

l

I l,

.

Il,l I

. ,1 ,I

I:

.11

'Il

ALEC TIRANTI LTD. . 72 CHARLOTTE STlU:ET LONDON,

W.I

., ..... :....;.

,

L·

OTIIER

Esthétiqu«

des Proportions

PUBLICATIONS

(Gallimard,

BY MATILA

(;IIYKA

Paris)

[.e Nombre d'Or, 2 volurncs, with prctace by Paul Valcry (Gallunard, Essai sur I. Rhythme (Gallimard, Paris) Tour d'Horizon Philosophiqu« «(;allilllan.l, Pluu d'Etoiles (Novcl) (Gallimard,

Paris:

Paris)

Sorli/lges du Verbe. Pr ••Iace hy Leon-Paul ..!.I!ail/ Dn« Day (Novelj

(Ml'thu"~l,

Paris)

London

Farguc «;allilllanl,

Paris)

i

(;tolllelry oJ Ari al/d Lif« (Shccd & Ward, N"", Ynrk ) Articlrs

in QJlarltr/y Recicu: Niuctccnth

Rau« de Paris, Reoue Hebdomadioe,

(;C7/I/I~~,

llnrizon,

Lif«

ctc., etc.

LECTURE/ES~A ON :\RT

YS

Numhcr

Published

1~52

reprinted

1956

reprinted

1964-

Mml,' '/IId ')till/d

;11

/11, UII;/~

K/II.~dom

ali"

Lctters 'l oday,

I Proportion oJ a Rcclallglc Static and Dynamic Rectangles . HE concept or proportion is in composition thc moxt important une, whether it is used consciously. or unconsciously. It is itself derived frorn the concept of ratio which has to be defined first. Ralio.-Ratio is the quantitative comparison betwccn two things,'aggregates or magnitudes, belonging to the same kind -or species. This cornparison, or which ratio is the syrnbol ami " thc result, is a particular case or judgll/C1/1 in gencral, or the ' r most important operation perforrncd by intelligencc. This (judgment) consists of: l. Perceiving a functionai relationship or a hierarchy or valucs ; " 2. Discerning the relationship, making a compansnn 01" values, qualitative or quantitative. ~"'hcn this comparison produces a definite rncasuring, a quantitative " weighing ", thc result is a ratio. Il" we are dealing with segmcnts or a straight linc, thc ratio bctween two segments AB and Re will bc syrnbolizcd by -AB a Be' or b' il' a and b ar,c the lengths or thcse scgmcnts mcasurcd Proportion in Generai -

T

with the same unit. a

"

"The ratio b has not ònly the appearancc

but al! the propcr-

ties of a fraction,

and can be cxprcsscd by a nurnbcr, result 8 of the division of a by b. For example the ratio ""is equivalcnt

5

lo the number 1.6. Proporlion.-::::'I'he definition of proportion follows irnmediatcly that of ratio. Proportion is the equality of two ratios.

-

.

3

If we have ,.established

two

" magnitudes " (comparable objects or quantities) A and B on one side, and two other magnitudes, C and D, A

I I

II

l ·1

I

B l

to D) expresses that A,

B, C, D, are connected

ADe CI

t

RATiO

lo

•

a

tìon b

l

T

the kind generally in

proportion,

generaI

identical,

These

b -~.

continuous

proportions,

or .continuous,

dismay

have any number of terrns, as in:

b

= - = c

c dc

d

5_

••

1 1

"I

,

UCTANCLf

1ft':" .,S

,

p

,.'"

J~'"

.> , .••••. --; 1 IIfC TAHCiLf FIC.

ft I

STATI e AND DYNAMIt: RECTANCLES

b-d--t-h a

~

RA T10·PROPORTION.

a_c_e_g

or b

"

case

as in

UCTANCLf

(.../T,\ ~" ,~\

It is- called

when a, b, c, d, are different, and continuous if two of those four are

+-Jf

used in com-

the

1'''7

m

these IUCTANCLl

posi tion and designo

numbers a b

o

E PAOPOATION

d

discontinuous

l'

IIfCTANCLE

the propor-

=-.c'

This is thegeometrical

•

by a pro-

between these measurements, theequaliry,

A

C

CI

by a, b, c, d, we have

numbers,

twò

A ..Io c 0----~-----ee_~~~e

portion. Here too, if A, B, C, D, are . segments of , straight . lines measured

betwccn

B'n'

on the other, the equality C. . = D (A compared to B is as C o

compared Fig.

A C

ratios

";,

)

1\\

I~ ,

, We have In both cases thc pennauency

rif..a charactcristic ratio;

this cxpIains why the notions of ratio and proportion

~__1

~ Il''',

I I

:

JS I

I

~.I A

I ' '

I I

11

+

~ .AI AI 8C

a.....

I

I

O

o

•

1...

.•..

'

"="v.

\ \

e>'

,

"

I AC

............. \'\'~

A

to thc other;

~c

.rs., .•,

I

é .... >.0 •

arc oftcn

confused. We see that this concept of proportì'on' introduces, besides the simpie comparison or measurement, this idea of a permanent quality transmitted from one ratio

-

1

C

it is this analogical

inoariant which .br;ngs out an order-

l

ing principle. The geometrica I proportion is -thc mathematical

I

mctaphor

IS

thc

EFCO AGHO (. AGIE , RECTANGLE S UfE

,1

"eH

RECTANGLE

AI.IC.CO

ne

Fl(;S .• -:; C.ONSTRUCTION OI' GOI.DEN SECTION. 'l'HI; GOLDEN REC'TANGU, IN THE SQUARE. THE ~ l'ROGRESSIOl'\

Fig

From the point or view of proportion, the most important piane figure in composition is the rcctangle; the most important charactcristie of a reetangle is indeed its proportion or characteristic ratio, thc ratio between its longer and its shorter side. AlI the rectangles having the sa me proportions are, of course, simiIar, but may differ in sizc. .I'he pro portio n or characteristie ratio of a rectangle ABCD having sides measured by a and' b is ~, and speaking about it we shall b' , cali it: thc rcctangle

a 4 8 3 " can be a rational numbcr Iikc -, ", . l) 351

Il

literary

aspcet).

c

I

aspect of the very generai and important concept of analogy (of : which

H

I

~. b

= 3, or an irrationai 5

2

li i

I

I

onc likc ~~

=

I

'/3", ~~,

2

V3,

rectangle 3, the rectangle rectanglcs

~,~~,g. I

I

-having

as

etc. Wc shall say in practice : thc the rcctangle ~, meaning thc

proportions

(charaetcristic

ratios)

2

Static and Dynamic Reclangles.--Thc

Fig.

1

rcetangles such as ~, ~, ~, 245 3, etc., of which the proportions show only rational numbers, are called static rectangles.

- - .- '5

such as V2, V3, V5, '\I-, cf>

Therectangles

2

'5 + I =. '\1--2

(the Golden Rectangle which wc shall meet farther on), showing irrational numbers in their proportions, are callcd dynamic rectangles. As we shall see, these latter are the ones most used in geometrica l composition, specially in the techniquc rediscovered by Jay Hambidge and called by him Dynamic ~ymmetry. . The reason is that they allow much more flexibility and a much greater variety of choice than the static rectanglcs, spccially when used in order to establish the comrnodulation by proportion of the elements and the whole of an architec~ turai, pictorial or decorative composition. The

rectangles

(!.

I

I

(~ = ~~)

this

2

or

itself relatcd

The

diagonal

V4 rcctangle to ~5"'±_!, 2

6

the

squarc,'

and

2,

.I

the doublc squarc, arc at' ~he same time static

and clynamic.

V5;

= ~~),

of the double

square

is thus relatccl to thc rcctanglc the Goldcn Section proportion.

2

is

VS,

II Thc Golden Section Thc Goldcn Scction is the simplest asymmetrical proportion obtaincd whcn in the continucd geometrica l proportion a b b = ~ wc try (applying " Ockham's Razor") to reduce to two the three elements a, b, c; the simplest way is to suppose a b c = a + b, and the pro portio n becomes -b = '---'--b- or

a+

b = a-T + b ; ab ~

~

('m w h'lC h IJ lS is bi19ger t h an a )'

lS

t h e c h arac-

teristic ratio or .proportion we arc trying to calculatc, and wc see that its logical dcfinition is: "'l'his proportion exists, between two measurable quantities or any kind, when thc ratio bctwccn the biggcr and the smaller onc is equal to thc ratio betwecn the sum or the two and the biggcr one." The numerical

~=

a

a

b value of - is easy to obtain. a

+ 1:> gives b

(~) a

b in which - is the unknown. a " fi es t h'lS equation "b wh'lC h satis

2 ~

~

a

_

1

=

o

The value (the positive root) of-

b a

v- ..+_..

1 = /5. = I. 6 18 .,. a 2 This was called by "the Greeks" The Section ", by Luca Pacioli (1509) "The Divine Proportion", by Leonardo and

after him the' Gol~en

lS -

Section.

This number

~

+! 2

1.618 . , . generally symbolized the most remarkable arithmetical, properties.

by the Greek letter' é, has algebraical and geometrica l

7

A

c

V

T H

-, ,

,

A,.,.,--,/ l'

\

I

\

\

I

Q

\ \

,

\ I

I

.'\,\, 1\ l'

" ~ \,

\

J

I

I

,

, ~--- .- " , ... ...• I

I I /

/

\0

C (

le

U

..••.....

AI H llCTANCL( GHHN VTWU UCO

H

,1

PQ,U AI HC

$

.,1 ATWO l'IG. (;

ATK( THE

Wc have ~

=

1.618

- =

0.618

~2=

2.618

l

-~

From the expression

"H MoIO THf.

(~)

2

J5

91

Jf

IH'

~ SQUARE

'\(CTANGL'

IIMG

,

RECTA:>IGI.ES

~

l

=

O

f, y5

+, ~,

or

= ~ -\-1

~2

wc obtain (rnultiplying both tcrms by ~ any nurnbcr of timcs) ~"= ~"- I + ~n - 2, that is: in any gl'owing progression or scrics of terms having ~ as ratio betwcen the successive terms, each term is equal to the sum or the two preccding ones . (in a descending Fig. 5

Figs.

!l";)

Fig.

1

1

~ = ~"+ l

progression 1

+ 4>" +2)'

This

having allows

1

~ as ratio, an

easy

we have

geometrical

manipulation or the scrics; with two givcn successive terrns, wc' can construct ali the other terms by simple moves or the compasso Figs. 2-.5 show the two most important constructions connected with the Golden Section. I. Given a sègment or line AB, find the third point C such AB AC that BC = AB = "'.

,~

Givcn a scgmcnt or line AC find thc intermcdiary point B giving the sarnc prcportion. 2.

The two constructions

Fi,. 3

can bc verificd

by ealculating

on cach figure AB by thc BC thcorem or Pythagoras, and verifying that t hiIS

rauo.

equa ls

VS ..__ + ._-.l

A

2 .

Fi, .•

t hird (fu

eonstruetion shows how to divide thc vertieal side or a square aecording to thc

~

l'

"I I

Il

Golden

Seetion;

the reetangle

then: a

Golden

Reetangle

and

thc rectanglc

(~: =

EFCD

DC ( ED -

ABFE a 4>2 reetanglc

Fig. 6 shows the relationship angles

the double-square,

VS, 4>, "';4>

Passing

.J. )

'l'

4>2).

the square, A

is

and

betwcen

Fi,.6

thc rect-

4>2.

now to the geometrieal

pro-

perties or the Golden Seetion, we shall see that this proportion the star-pentagon

eontrols the pentagon, (or pentagram), thc

deeagon and the star-de~agon, Figs. 7 to 14. In a regular pentagon

..

~.~.~ Me; A"

N"

the diagonal Golden

Fi,I. 7.14

as shown in

the ratio between

and the side is equal to the

Seetion also

ratio, .

diagonal

lS

pentagon

or pentagram.

the

AD AB side

=

.J..

'l'

or the

This star-

9

Fil. 7

FIGS.13-14 THE TWO CONSTRUCTI0NS OF THE PENTAGON

, Fi,.

IO

Fi,. 12

lf we draw the five diagonals of the pentagon, we obtain thc star-pentagon or pentagram, in which the rp ratio or proportion is very much in evidence. lf we also draw the diagonals of the inner pentagon, and so on, we obtain an indefinite recurrence, a " nest" of Golden Section progressions. The regular decagon and star-decagon are also intimately related to the Golden Section in the following way: The ratio between the radius of a circle and the side of the R inscribed decagon is dr = rp (the Golden Section). The ratio between the side of the star-decagon and thc radius of the circurnscribed circle is also We might hcrc remember that the radius of any circle is equal to the side of the inscribed hexagon. Figs. 13-14 show two important constructions, second only in importance to the construction of the Golden Section as shown in Figs. They are: I. Given a circle, to inscribe a regular pentagon i '. that circle, that is, construct the side AC of the inscribed pentagon; thc side BO of the regular decagon is incidentally obtained on é.

Fi,S.13-14

2-5.

Fig, 13

.

R

the same diagram (because of d,

r

= rp).

2. Givcn a. certain Iength (a segmcnt of a straight line) MN, to construct the regular pentagon having its sidcs equal IU

II

'I

I

..~' ............

:::

o"

:::

'~'"''

>....

'É--\-~Otr+-~ -,

'~

..•

..'

'- •

o

.0

~~

.' :

.

; \"

-" .- .•.....

l

II

..................

FIO.

I.)

VARIATIONS ON THE P~NTAGON

I

I

, V~ and 4>2 (that is rectangles such that the ratio~ between their longer and their shorter sides are equal to these numbers). We have already seen that the rectangles I and 2 can be considered either as static -',, ' ;' ';.-:: \ or dynamic. ", ~ rrc,

19

DYNAMIC

RECTANGLES

/(f>"

»:

~~

V·

"'''~'' ~, /~,.

\

~

\

\

\ \

\

\

\ \

\

.. .. "

\ \

\

.: ","',,:.A.. " ...." \\ / ,.,.f'

;"'"

Most of those dynamic rectangles are shown again in Figs. 18 and ,19; the rectangles 4>, 4>2,V~' and Vs are ali

Fill,11I-19

related, but they are not related to V; and V3. This notion of relationship between rectangles, that is,' between their proportions, derives its importance from a law of composition already mentioned by L. B. Alberti and redìscovered by Hambidge, the "Iaw of the non-mixing of proportions or lhemes in a pIane composition "; in such a composition, only •• related " themes must be used, " antagonistic " themes must not be mixed (the analogy with music is correct, the modulations of proportions is similar to the passing from one related

,I

v~

scale or key to another), The proportion = 1.273 ... has a certain importance because it happens that many vertical frames of paintings have this proportion; this applies as well to paintings in museums as to modern commerciaI frames bought in the trade. We can, according to the abovementioned law, "attack" a composition placed in ~uch a frame either by its own proportion (V~), or by any related one, like 4>. Fig. 20 shows this rectangle V~ with what l:lambid.$~\:é ...•. :

l··.

~

"

.

-

•

.

•

FIG. 01 "1"\\"0 GOTHIC STANIJAIU) l'LANS (~\OESSEL)

Fig. 6.

FiR· 62

Fi~_ ~ 1 shows thc two standard plans composcd by Moessel to include most or thc plans (and sornctirnes thc clevat ion) thc major ity or Gothic cathedrals ancl churchcs, Fi~_ b2 shows the origina! Gothic diagrarn Ior thc cl cvatiou or Milan Cathedral, published in 1521 by Csesar Cicsariau. l, architcct and Master or thc Works or thc cathcdral. Thc

or

directing circle figures on it (it is part of the originai drawing), also thc words symmctry, proportion and curhythrny. HCIT thc plan is worked out " in the Ccrrnan mode", havi llg as leading ·H

figure and proportion

tlic cquilatcral

trianglc

and

V']_

fOtA GIO~CM. AJCHTTICTONlCAI: U ICHNOl.:UPHlÀ SVMnA· VT pn.AMVS~l,.,tAS f'OSStH'T' PI ~ OUHOORA.PHlAM AC 5CAJ:NO~PHJAM P[I\PVC[Rt OHNfS QV~SCVNQ..vA[ ~,,"'ON lOlVM AD CLR("1NI C['H'n.VM· SE.D Q'AE. A -n..JGONO ET QVADAATO AVT AllO QVOVlSMQOO P[lNT NIVNT PQHINI NVM HA8[.,E llS'ONSV'M • "rVM r(p .. rvnv't HMI/\ M pROPOR. ••• ,.IONATA..M ~ '('TlAM.J: SYMM E TUAL QVANT I TATE M O!'.OINARIAM AC PlA.. OPlPIS'DECOR,.AnoNtM OSTENO[R,E..V"TI JJIA.M H(C QVI\E. A GOtMANICO HOJU:J.uNE:t N'IVNT l.llnIUJlVl:NTV~ PENI. Q,V.EM."O'HOOVl'1 .sAettA CATHtOI'lAln AlDlS MlOlol...l\lil PAT(T .• [~'" j> .•. 1'1 •. C ••.C .• A.P .•.. V1 .•. QJ.C •• A.ç A.f .• D," y

1((;,",

I)()~IL I(:

A:'\11~u:n():'\ ,,,",)

01' ~!!I.A:'\, LI.L\i:\T[(),'\ r\I:~ .. \I(

(::\r.~Alu:\:'\U,

FIG. 6:1

Plat.XII

THE

r

RECTANGLE

rn.

WIENER)

Plate XII shows thc analysis or Raphacl's Crucifìxiun (National GaIIery) by Moessel's mcthod ; a regular pentagon and a regular dccagon produce alI the important points or thc composition, whose balanced geornetry seerns to bc most consciously thought out.

l'late

XIl I rcproduccs

gl'alTI, su pportcd

thc corresponding structural lJy thc " trianglc or the pentagon ".

dia-

PI.I, XllI

HamlJidgc's and Mocsscl's hypctheses, quite indcpcndcntly or their archaiological valuc, oflcr also to architects, paintcrs,

rlccorators, sculptors, two very useful techniques or composition, which are being widcly uscd by artists in England, Francc, the Unitcd Statcs, Brazil, Switzcrland, llclgium, etc. Fig. 63 represents OIlC

mctrc

" harrnonic

a Go!dcn

to 0.618) cornposition

Rectangle

(rea! size about

uscd by certa in painters

as help

to

Fig. 63

". (I). Platcs

x rv ss, XV

repre-

Plaus XIV·XV

sent a paintlng

com-

posed by D. Wiener

un

these lines. F ig . shows

64 the

section through

a

silv er cup, with its construetional diagram, executed by Puiforcat,

J. REGU/.A'nr:G

J)/i\Glti\J\1

FIG. 64 UF CUI' (J. l'U;FURCAT)

l'l In ordr-r lo divide t he vcrtical tr> ihe Goldcn Sccrion, une multiplics

or horizontaI sidc 01' a frame according thc Iength or this sidc by 0.6Ill.

Fig. 6.•

~

/ r-. ~~

\

IV IV W tJ / i \ \

v

V

v

~

\

/

.,I

I I

l

i! J

!

I! ,

1'1(;. (;:.

1)f"(;I(A~f

h.!.;. li,) shovvx in Paris planucd

t hc li)"

or

!":\(;AOE 01' "/"[I-"h\:\\·.\ lA. S()\'TII\\"I