A Practical Handbook of GEOMETRICAL COMPOSITION AND DESIGN by » MATILA GHYKA ALEC TIRANTI LTD. 7a Cuartorre STREET Lonpox, W.1 HEATTHIER PUBLICATIONS BY MATILA G1I¥KA Esthbigne des Proportions (Gallimard, Pacis) Lz Nombre d'Or, 2 volumes, with preface by Paul Valery (Gallimard, Pars) Fssai sr le Rhythme (Gallinard, Paris) ‘Tour d'Horizon Phitesephique (Gallisnard, Paris) Pluie d’Ewiles (Novel) (Gallimard, Paris) Sorildes du Verbe. Preface by Leon-Paul Fargue (Gallimard, B Again One Day (Novel) (Me London) Geometry of Art and Life (Sheed & Ward, New York) Articles in Quarterly Reciew, Nineteenth Century, Horizon, Life and Letters Vodey, Recue de Paris, Reoue Hebdomadive, etc, ete. URE/ESSAYS ON ART Number 2 Published 1952 reprinted 1956 reprinted 1964 Marte wd printed tv the United KinedowFigs If we have established two ratios “ magnitudes ” (comparable objects or quantities) A and B on one side, and two other magnitudes, C and D, on the other, the equality A B compared to D) expresses that A, B, C, D, are connected by a pro- portion. Here too, if A, B, C, D, are segments of straight lines measured by a, b, c, d, we have between these measurements, these numbers, the equality, the propor- aire Hon = 5, Es a = 5 (A compared to B is as G ‘This is the geometrical proportion, the kind generally used in com- position and design, It is"ealled discontinuous in the general case when a, b, ¢, d, are different, and continuous if two of those four numbers are identical, as in a Ee b ‘These proportions, dis- c » between two B’D’ 4 . accrancue fF continuous or continuous, may have any number of terms, as in: FIG. ieee Ne RATIO-PROPORTION. 88 STALG AND Sega ieee DYNAMIC RECTANGLES bod. a b ye = beI Proportion in General — Proportion of a Rectangle Static and Dynamic Rectangles . HE concept of proportion is in composition the most important one, whether it is used consciously, or uncon sciously, It is itself derived from the concept of ratia which has to be defined first. Ratio—Ratio is the quantitative comparison between two things, aggregates or magnitudes, belonging to the same kind or species, ‘This comparison, of which ratio is the symbol and the result, is a particular case of judgment in general, of the * most important operation performed by intelligence, ‘This (judgment) consists of 1, Perceiving a functional relationship or a hicrarchy of values; 2, Discerning the relationship, making a comparison of values, qualitative or quantitative, When this comparison produces a definite measuring, a quantitative “ weighing”, the result is a ratio, If we are dealing with segments of a straight line, the ratio between two segments AB and BC will be symbolized by AB oa Bee with the same unit. ‘a and 6 are the lengths of these segments measured a ‘The ratio. has not only the appearance but all the proper- ties of a fraction, and can be expressed by a number, result 8 of the division of «by 6. For example the ratio is equivalent to the number 1.6. Proportion.—The definition of proportion follows immediately that of ratio, Proportion is the equality of two ratios.Figs If we have established two ratios “ magnitudes ” (comparable objects or quantities) A and B on one side, and two other magnitudes, C and D, on the other, the equality A B compared to D) expresses that A, B, C, D, are connected by a pro- portion. Here too, if A, B, C, D, are segments of straight lines measured by a, b, c, d, we have between these measurements, these numbers, the equality, the propor- aire Hon = 5, Es a = 5 (A compared to B is as G ‘This is the geometrical proportion, the kind generally used in com- position and design, It is"ealled discontinuous in the general case when a, b, ¢, d, are different, and continuous if two of those four numbers are identical, as in a Ee b ‘These proportions, dis- c » between two B’D’ 4 . accrancue fF continuous or continuous, may have any number of terms, as in: FIG. ieee Ne RATIO-PROPORTION. 88 STALG AND Sega ieee DYNAMIC RECTANGLES bod. a b ye = beEFCO AGHD & AGIE IFCH > RECTANGLES Abte $7 RECTANGLE O— 49099 -- eS AbeBCeCD ETC FIGS, 23 CONSTRUCTION OF | GOLDEN SECTION. THE GOLDEN’ RECTA INTHE SQUARE. | THE ¢ PROGRESSION We have in both cases the fermanency of a characteristic ratio; this explains why the notions of ratio and proportion are often confused. We sec that this concept of proportion ‘introduces, besides the simple comparison or measure- ment, this idea of a permanent quality transmitted from one ratio to the other; it is this analogical invariant which brings out an order- ing principle. ‘The geometrical proportion is the mathematical aspect of the very general and portant concept of analogy (of which metaphor is the literary aspect). From the point of view of pro- portion, the most important plane figure in composition is the rect- + angle; the most important charac- teristic of a rectangle is indecd its proportion or characteristic ratio, the ratio between its longer and its shorter side. All the rectangles having the same proportions are, of course, similar, but may differ in size. ‘The proportion or charac- teristic ratio of a rectangle ABCD having sides measured by a and b is P and speaking about it we shall * call it: the rectangle - 83 j, can be a rational number like Z : § = 5, or an irrational 5 FeaFis ane ike n/S rectangle 3, the rectangle V/3, the rectangle 3, meaning the wf 5, ete. We shall say in practice: the 2 rectangles having as proportions (characteristic ratios) 3/3 3. pata’ ve 3 B15 Static and Dynamic Rectangles.—'he rectangles such as 3, 4, <5 3, etc., of which the proportions show only rational numbers, are called static rectangles. ‘The rectangles such as V2, V3, V5, v5, $= Js! (the Golden Rectangle which we shall meet farther on), showing irrational numbers in their proportions, are called dmamic rectangles. ‘As we shall sce, these latter are the ones most used in geometrical composition, specially in the technique re- discovered by Jay Hambidge and called by him Dynamic Symmetry. ‘The reason is that they allow much more flexibility and a much greater variety of choice than the static rectangles, specially when used in order to establish the commodulation by proportion of the elements and the whole of an architec- tural, pictorial or decorative composition. The rectangles 1 @ -v ‘), the square, and 2, = v4) the double square, are at the same time static 1 and dynamic. ‘The diagonal of the double square 2 is V5; this 2 or V/4 rectangle is thus related to the rectangle V'5, itaineeiaead wn "| the Golden Section proportion. 2 6Il The Golden Section on is the simplest asymmetrical proportion in the continued geometrical proportion The Golden Sect obtained when a ob we try (applying “ Ockham’s Razor”) to reduce to bre two the three elements a, b, 0; the simplest way is to suppose . ab €=a-+b, and the proportion becomes j= Wi, oF b bb :7 *y 3 (in which b is bigger than a) is the charac teristic ratio or proportion we are trying to calculate, and we see that its logical definition is: “‘Lhis proportion exists, between two measurable quantities of any kind, when the ratio between the bigger and the smaller one is equal to the ratio between the sum of the two and the bigger onc.” b The numerical value of: is easy to obtain. bo atb. /b\? b pop ae (2)' 2-1 a0 b in which cds the unknown. The value (the positive root) off which satisfies this equation is ® — nf SES 618)... a a This was called by ‘the Grecks “he Scction”, by Luca Pacioli (1509) “ The Divine Proportion”, by Leonardo and after him the Golden Section. This number fats - 2 1,618 . . . generally symbolized by the Greek letter ¢, has the most remarkable arithmetical, algebraical and geometrical properties.Fes Fign a3 Fig ABFE RECTANGLE aired rans ncrancit JE, vrww. ro Abn treo atwo arec” p utr raus une é HIG.6 ‘THE SQUARE AND THE RECEAN( We have ¢ = 1.618 « 0618... $= 2.618... _ (b\? b From the expresion (7) —2—1=0 or g§ = +1 we obtain (multiplying both terms by ¢ any number of times) ° = 6°" "4 $4, thats: in any growing progression or series of terms having ¢ as ratio between the successive terms, cach term is equal to the sum of the two preceding ones™ ' (in a descending progression having 4 as ratio, we have This allows an easy geometrical manipulation of the series; with two given successive terms, we can construct all the other terms by simple moves of the compass. Figs. 2-5 show the two most important constructions connected the Golden Section, 1. Given a ségment of line AB, find the third point C such AB AG that 5G = ap =?FIGS. 7-48 2, Given a segment of line AG find the intermediary point B giving the same preportion. “The two constructions can be verified by calculating on each’ figure 2 by the theorem of Pythagoras, and verifying that this ratio equals ee ‘A third construction shows how to divide the vertical side of a square according to the Golden Section; the rectangle EFCD is then’ a Golden Rectangle (3 me $) and the rectangle ABFE a ¢* rectangle AB _ ’) (ae =P Fig. 6 shows the relationship between the square, the double-square, the rect- angles 1/5, $, /¢ and *. Passing now to the geometrical pro- perties of the Golden Section, we shall sce that this proportion controls the pentagon, the star-pentagon (or pentagram), the decagon and the star-decagon, as shown in Figs. 7 to 14. Ina regular pentagon the ratio between the diagonal and the side is equal to the Golden Section ratio, 2 =$. This diagonal is also the side of the star- pentagon or pentagram. Fes Fina Fig 6 Fiv 74 FerFigo Fig Figs 9-14 Fie.tg ‘AC «Pp SIDE PENTAGON OBadp SIDE DECAGON BQ SIDE STAR DECACON ‘iB Ag If we draw the five diagonals of the pentagon, we obtain the star-pentagon or pentagram, in which the ¢ ratio or proportion is very much in evidence. If 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 7 = # (the Golden Section). The ratio tee "the side of the star-decagon and the radius of the circumscribed circle is also ¢. We might here 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. 2-5, They are: 1. Given a circle, to inscribe a regular pentagon id that cle, that is, construct the side AC of the inscribed pentagon; the side BO of the regular decagon is incidentally obtained on the same diagram (because oe = 4). 2. Given a certain length (a segment of a straight line) MN, to construct the regular pentagon having its sides equal 10LRN : | ' SSCA ol 4FIG. 17 YHE-TRIANGLE OF THE PENTAGON, HARMONIC COMPOSITION to AB, The connexion between the Golden Section VW & t ig the pentagon and the decagon; that is, the affinity of the Golden Section and the pentagonal symmetry in general (which includes decagonal symmetry), derives from the +1 2 the two constructions can be controlled by the formulas: presence of V5 in the expression .. The accuracy of Rye Ps => Vio + 2V5 (for the side of the star-pentagon) and Ry - Pr=> Vi0— 2V5 (for the side of the regular pentagon), Ps with also (to explain Fig. 14) the formula 5 = 4. Figs. 15, 16 and 17 show the role of the “ triangle of the pentagon” (having as vertices the extremities of a side of the regular pentagon and the opposite vertex) in different “‘ harmonic ” compositions derived from the pentagon. 3 Figg Fig azey Fig ts07mW Dynamic Symmetry Jay Hambidge found this expression in Plato; symmetry for Greek and Roman, and also Gothic architects, meant the commedulation or linking of all the elements of the planned RECTANGLE FIG. #8 DYNAMIC REGTANGLES yar Wh YS 4 whole through a certain pro- portion or a set of related proportions, As Vitruvius states in what is the key- sentence of his treatise on architecture: when this sym- metry or commodulation be- tween the elements and be- tween the elements and the whole is achieved in the right way, we obtain curhythmy. Dynamic Symmetry, or symmetry “in the second power" means that although the linear elements (segments of straight lines) used in the composition are irrational (not commen- surable “in the first power ”), the surfaces built on them may be commensurable, linked through a rational proportion, For example: The surfaces of squares built on lengths proportional to \/2 and /3 will be to each other as 2 is to 3 (commensurable in the square or second power). So that in order to obtain dynamic symmetry or com-modulation, we must use the rectangles already introduced as dynamic, that is the rectangles V2, V3, V5, 4, V¢ and 6% (that is rectangles such that the ratios between their longer and their shorter sides are equal to these numbers). We have already scen that the rectangles 1 and 2 can. be considered cither as static or dynamic, FIG. 19 DYNAMIC RECTANGLES Most of those dynamic rectangles are shown again in Figs. 18 and 19; the rectangles 4, $2, V# and V5 are all Fig. sei9 related, but they are not related to V2 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 rediscovered by Hambidge, the “law of the non-mixing of proportions or themes in a plane composition”; in such a composition, only “ related ” themes must be used, " antagonistic " themes must not be mixed (the analogy with music is correct, the modula- tions of proportions is similar to the passing from one related scale or key to another). The proportion Wd = 1.273... « has a certain importance because it happens that many vertical frames of paintings have this proportion; this applics as well to paintings in museums as to modern commercial frames bought in the trade, We can, according to the above- mentioned law, “attack” a composition placed in such a frame either by its own Proportion (V4), or by any related one, like ¢. wei 20 shows this rectangle V4 with what Hambid TRE: “harmonic subdivisions”; it is here attacked bothby Viti i vi and the proportions. ‘The painter Seurat invhis very _ geometrical compositions generally used the Gol len Secti and as his frames often were in the V'¢ proportion, th are very satisfactoiy, as in the famous “ Circus”. \@FIG.22 ‘THE 7 RECTANGLE, HARMONIC DECOMPOSITIONS of gmmetry or harmonious commodulation, and that the subtler questions of symmetry “‘ have to be solved by the use of irrational proportions”, that is by dynamic symmetry, Hambidge introduced as the method probably used by the Greek architects his technique cf harmonic sub-divisions of the square and of the dynamic rectangles by the tracing of | | |FIG. 23 THE Ys RECTANGLE, HARMONIC DECOMPOSIFIONS diagonals and perpendiculars to diagonals. The diagrams on Figs. 21, 22, 23, 24, show how this method can be applied in Fie. a2 order to obtain the harmonic sub-divisions of the square and of the dynamic rectangles V2, V3, V5 and ¢. Hambidge’s method takes into consideration the fact that the overall frame of a two-dimensional plan or composition, whether archi-FIG. 44 THE # RECTANGLE, HARMONIC DECOMPOSITIONS tectural, pictorial or decorative, is generally a rectangle or a complex of rectangles; in his method of harmonic sub-division or analysis, these rectangles arc ‘* treated” by the diagonal, That is, we draw a diagonal, and from one of the opposite summits a perpendicular to this diagonal; this can be repeated with the of] agonal, and from. the points of intersection between diagonals, perpendiculars and. sides, are drawnRECTANGLES ABCD & BCFE spac «SIMILAR ‘AD EB RECTANGLE @ FIG, 25 THE DIAGONAL IN DYNAMIC RECTANGLES, parallels to the sides. ‘This process can he con- tinued with many varia- tions, aind every diagram thus obtained is what Hambidge calls a “ har- monic sub-division”, that is, produces a perfect commodulation. ° of | sur- faces, and obeys the “ law of non-mixing themes”. An important property of this method is that the perpendicular to the diagonal produces inside the original rectangle a smaller rectangle similar to it, obeying thus the general “Principle of Analogy " formulated by ‘Vhiersch and which we shall meet below (Chapter IX), Fig. 25 shows this property of the perpen- dicular to the diagonal in several rectangles, ine cluding the Golden Rect- angle; in this case the original ' Golden Rect- angle is ‘divided into a smaller one and a square. If one continues this con- Fig. 23D coo ABCD EBCF @ RECTANGLES AEPEF EFCO GHD ENG FH) NIKL LKIM JB RECTANGLES FIG. a6 RECTANGLE §, OF THEE WHIRLING SQUARES. REGIANGLE J struction, one sees why Hambidge called the ¢ rectangle the 4.28 “rectangle of the whirling squares ” (Fig. 26). One sces als that on each rectangle the successive points obtained by this construction are on a certain spiral, having as leading proportion between cach pair of perpendicular radii the proportion of the rectangle,Fig 28 ° F ¢ op co c ABCD EBCF G RECTANGLES AgeEF EFCO GHCD ENG IFHJ NIKL UKIM JB RECTANGLES FIG.s6 RECTANGLE f OF THE WHIRLING SQUARES, REGTANG a struction, one sees why Hambidge called the ¢ rectangle the “ rectangle of the whirling squares ” (Fig. 26).. One sees also that on each rectangle the successive points obtained by this construction are on a certain spiral, having as leading proportion between cach pair of perpendicular radii the proportion of the rectangle.v Regular Polygons, Partitions and Equi-partitions of Space ‘The number of regular polygons, that is polygons with equal sides and angles which can be inscribed in a circle, is illimited; we can have regular polygons with 3, 4, 5, 6, 7, 8, . . ., n sides, u being as big as one wishes. Not all of these polygons can be rigorously constructed with rule and com- pass; for example, the regular polygon with seven sides cannot be construeted. We have seen how to consiruct the regular pentagon. ‘The decagon is obtained by halving the segments of the arcs situated between the summits of the pentagon. FIGS. a7-30 HEXAGON, HEXAGRAM, OCTAGON, STAR-OCTAGON, ‘The side of the regular hexagon is equal to the radius of the circumscribed circle, hence a simple construction for the hexagon and the equilateral triangle inscribed in a given circle, The square and the octagon derived from it present no difficulty. We have already met the star-pentagon or pentagram, There is no true “ hexagram ” or star-hexagon; Fig. 28 is really composed of two equilateral triangles whose lines do not fiow into each other; this figure is sometimes called the “ Seal of Solomon” or “ Shield of David” and plays an important role in Hebraié Symbolism (the number 6 was the number of Justice and Balance in the Kabbala). ‘The penta- gram, or five-pointed star on the other side was the secret symbol of the Pythagoreans; in their number mystic five was the number of health, harmony and love. ay Figs 13914 Fig 27-99 Fig 8Among the regular polygons, only the triangle, the square ‘ Fes: and the hexagon allow a complete equipartition or filling of the plane; they are the only polygons whose angles at the summit (60°, go®, 120°) are sub-multiples of 360°, SEAL) WAVAVAVAVAN AAAAY MAY FIG. 31 REGULAR THONS FIG. 92 WO REGULAR. OF THE PLANE PARTITIONS OF THE PLANE. g i THREE REGULAR HG. 94 “tWO REGULAR HONS ‘OF THE PLANE PAICTETHONS Ok Yat POANE 24It is easier to fill the plane when several kinds of regular polygons are used; these partitions produce a number of Fie. s240 patterns which can be used for pavements or mosaics. 6. SEMEREGULAR HIG. 35 REGULAK HG, PARTITION OF THE PLANE. PARTITION OF THE PLANE. FAG. 99) SEMME PARTITION OF FIG. 98 SEMI-REGULAR PARTITION OF THE PLANEvi Regular and Semi-Regutar Solidi or Palyhedra in Space Whereas, as we have seen, the number of kinds of regular polygons on the plane is illimited, in three-dimensinnal space the situation is quite different: there are only five regular solids, also called the platonic solids (they are solids with equal sides or edges, Equal regular faces, equal solid a inscribable in a sphere) Number of swnmits of sides of faces oie 6 4 triangles Cube we 8 12 G squares Octahedron 6 12 B triangles Dodecahedron .. a) 4012 pentagons Icosahedron 12 yo 20 triangles ee These five solids are shown in Fig. 41. We can immediately state that the dodecahedron and the icosahedron, which are FIG. 4) THE IVE REGULAR SOLIDS the developments of the pentagon in three dimensions, are dominated by the Golden Section proportion (incidentally the 2 summits of the icosahedron are also the summits of three Golden Rectangles perpendicular w each other). ‘Those two hodies are also reciprocal: by taking the centres of all the faces of a dodecahedron, we obtain the 12 summits of an icosahedron, and inversely. ‘The same property éf reciprocity belongs to the cube and the octahedron. ‘The tetrahedron is auto-reciprocal, In the plane, no regular construction allows us to pass from the square to the equilateral triangle or to the oH Figs 4sFig 43 Fig as Wet pentagon; these three syx s are antagonistic, hence the “Jaw of the non-mixing of themes”. This is not true in three-dimensional space; we can pass from the dodecahedron to the cube by taking eight of its summits. ‘The 20 vertices or summits of a dodecahedron arc thus the vertices of five cubes, or of five tetrahedra. ‘The 12 vertices of the icosahedron are on the surface of a cube; the eight vertices of a cube are the vertices of two intersecting tetrahedra, By taking the middles of the sides of a tetrahedron, wwe obtain the six vertices of an octahedron, So that the law of the non-mixing of themes does not apply in space. A building (like many Byzantine churches) may have its horizontal plane and its volumes derived from the square, the cube and the sphere’ (leading proportion y/2), but its elevations, wall decorations, etc., modulated-in the Golden Section. ‘There arc also 13 semi-regular “ archimedian” polyhedra, which can equally be inscribed in a sphere but have as faecs two or three kinds of regular polygons, We shall mention only two: * phy sae? FIG. 43, FROM TEPRAUEDRON 1) OCTAHEDRON FROM CUBE ETRALEDRON 1. ‘The cubsetahedron with 14 faces (eight equilateral triangles and six squares of equal sides), obtained by taking the middles of the sides ofa cube or of an octahedron; it plays an important role in crystallography, as its vertices and centres correspond to the close-packing of equal spheres in space. It corresponds in space to the hexagon (its side is equal to the radius of the circumscribed sphere). au———— 2. The polyhedron of Lord Kelvin with 14 faces (eight hexagons and six squares of equal sides). Tt can fill space by repetition, like the cube, and is the only one of the 13 archimedian solids which has this property. It is produced by dividing in three equal parts the sides of an octahedron, Itis also a development of the hexagon in three-dimensional space, as is the regular hexagonal prism, which can equally fill space by repetition, ‘The Star-Polyhedra.-—Il' we lengthen out all the sides (or the planes of the laces—the result is the same) of a dodeca- hedron until they meet, we obtain a star-dodecahedron, the vertices of which are also the vertices of an enveloping icosahedron; if we do the same with the sides of an icosa- hedron, we abtain another star-polyhedron (equally called star-dodeeahedron because its intersceting faces are also 12 pentagrams) whose vertices are also the vertices of an enveloping dodceahedron, ‘The orthogonal projections of the two star-docecahedra show the role of the Golden Section in their structure (sec the smaller figures in Plates II and III), To WE NARS POLTHEDRON amxagaSall usa ‘These projections appear also in the structural diagrams of the human body and of the human face. ‘The Plates 1V and V show the same star-dodccahedra in solid perspective, If the above-mentioned process (lengthening of the sides or faces) is repeated, we obtain an indefinite succession of alter- nate icosahedra and dodecahedra with a growth-pulsation dominated by the Golden Section. Fie 4s Fig as Plates WALL Plates VVFigs. 7 Few vu Proportions of the Human Body and of the Human Face ‘The Greek sculptors Polyclitus and Lysippus are stated to have established definite canons of proportion for their statues. At the beginning of the First Renaissance, the monk Lau Pacioli in his book on the Golden Section called “ Di Proportione ” (Venice, 1509), and illustrated by Leonardo da Vinci, underlined the role of this proportion in the human body and living shapes in general. About 1850 Zeysing in Germany showed that as an average for a certain number of human bodies, the navel divides the human body in the proportion of the Golden Section. Jay Hamnbidge, the rediscoverer of Dynamic Symmetry, agreed with Zeysing, but preferred to use skeletons for his measurements. Every human body for Hambidge was like i symphony or melody on a certain theme or proportion, this proportion being either the Golden Section, 4, or a related proportion like V5. In fact many human bodies had the rectangle wi 3 as over-all frame (with arms extended hori- 2 zontally); others could be inscribed in a square (like in the famous Leonardo diagram). Figs. 47 and 48 show two diagrams of proportions of the human body having respec- tively as over-all frames the rectangle a/' 2 and the square. 2 49 shows the abstract geometrical structure of the body of Fig. 48. ‘The nay 1 does indeed, as an average and in well-propor bodies, divide the vertical height according to ection (the same proportion, but with the major the Golden a0term on top, is produced by the finger tip of the middle finger of the naturally downwards stretched hand). But the key to the whole diagram is given by the fact that the height of the head from chin to top of skull is equal to the vertical vel and the fork, distance between the sx =sA st-00-50 Kaye FIG.47- MALE BODY IN 1, WET DIAGRAM.FIG. 48 ATHLETE'S BODY, HARMONIC ANALYSISPlate VE Plate Vit FIG. 50, HEAD STRUCTURE (A. DARVILLE) If one examines the diagram corresponding to a mathema- tically perfect face, one realizes ‘hat the structural diagram of the human face, in this frontal projection, reflects and reproduces the diagram of the whole body. It also coincides with the projection of the 12-pointed star-dodecahedron as shown on Plate III, Plate VI shows a perfect profile with the same proportion (Golden Section dominating), the profile of Beatrice d’Este by Leonardo. After comparing the measurements of a certain nuraber of human heads belonging to Greek statues, to Renaissance paintings and to living people, it appears that the idcal 4FIG. 51 HEAD STRUCTURE (A. DARVILLE) average Western faces fall, as regards proportions, in two main types, both framed frontally by a Golden Rectangle, divided into two equal parts by the horizontal line joining the centres of the eyes. In the first type (Plate VI) the forehead is higher (in proportion to the whole head) than in the second type (Fig. 5), where the line of the eyebrows divides the whole height of the face according to the Golden Section, The Figs sos: Figs. 50 and 51 correspond to measurements taken by the ished’ Belgian sculptor Alphonse Darville and set by ye diagrams,Play Vin1x vill The Golden Section in the Morphology of Living Organisms — Pentagonal Symmetry ‘The Golden Section plays a capital role not only in the proportions of the human body but alo in the curves and diagrams connected with the growth and shapes of plants and flowers, also of many marine organisms (like starfishes and sea-urchins) where the pentagonal symmetry, related to the Golden Section, appears very often. Plates VIII and IX show examples of pentagonal symmetry in flowers and marine animals. ‘The presence of the Golden Section in botany is shown not only in the prevailing pen- tagonal symmetry but also in the diagrams expressing the growth and disposition of branches, leaves and seeds; the presence there of the Golden Scetion or ¢ progression is easily: noticed, specially in the form of the Fibonaci Series 1,2, 3, 5, 8, 13, 21, 34, 55, 89, 144... where (as in the ¢ series) each term is equal to the sum of the two preceding ones, and the ratio between two successive terms tends very rapidly to 618 (Ex. * $ = 1.618 (Ex. Be tye? ene Bills 5 3: 55 ‘The mathematical curve most intimately related to living growth and to the pulsations of the rigorous ¢ series and of its approximation, the Fibonaci Series, is the equiangular or logarithmic spiral in which the angles between the radii grow in arithmetical progression but the radii themselves in an exponential progression (p — a®). ‘This curve has the property of gnomonic growth, that is: two of its ares are always “similar” to each other, varying in dimension but not in shape (in the same spiral), and the same applies to the surfaces determined by the vector radii and even the volumes contolled by logarithmic spirals as in marine shellsFIGS. sans Each logarithmic spiral is associated to a characteristic ectangle and proportion, and the spirals whose quandrantal oportions are and \ q are the most frequently met with in organic growth and especially shells. Figs, 52, 53 and Plate X show examples of logarithmic spirals with quadrantal proportion or pulsation Vf and 4, and examples of shells and their controlling spirals. 37 Fig 9233 Plate X.Plate Xt PiG.s4 PENTAGON SYMMETRY — FIG. 93 PSELDOSPIRAL OF SUIRAR OF s FIUANACCIAN GROWL ‘Yo return to the pentagonal symmetry found in many living organisms (and most obvious in flowers where the number of petals is 5, 10 or a multiple of 5, like the genus rosa and the flowers bf all edible fruit or berry-bearing trees and plants), we have to point out here that this pentagonal symmetry, that is, shapes or lattices based on the pentagon, the decagon and their representatives in three-dimensional space, dodecahedron, icosahedron, star-dedecahedra, do not appear, and cannot appear, in crystallography or in any homogenous equipartitions of space, and that for the very simple reason that the vertex angle of the pentagon (108°) is not a sub-multiple of 360° (whereas the vertex angles of equilaicral triangle, square and hexagon, 60°, go® and 120°, are such sub-miltiples). A case in point is the shape of the crystals which constitute snow-flakes; 9,000 different snow- crystals photographed by Mr. Bentley show different varia- tions on the hexagon or triangle, never a pentagon. Plate XI shows an example of’snow-flake hexagonal symmetry; Figs. 54 8and 59 show abstract diagrams of pentagonal symmetry associated (0 fi them. Whereas ion are never found , configurations (like cl cubic pentagonal symmetry in the geometry of inanimate, inorgani crystals), the reciprocal is not true; the hexagonal symmetries, the proportions V3 and V2, characteristic of crystalline la cs are also found in the geometry of i flowers with six petals like the lily, the tulip, the poppy; these may be said ty be more crystalline in morphology than the rose, the rhododendron, the columbine, the azalea, the orchid, the passion-flower, all ruled by pentagonal symmetry. A more general law regulating the morphology of inanimate closed sysicms (including crystals) is that their evolution is always regulated by the Principle of Least Action, whereas living, arganic systems constantly violate this principle (which appears in ‘Thermodynamics as the Second Law, or Principle of the Degradation of Energy, or of the Growth of Entropy). 1X The Golden Section and Architecture — Practical Apptications ‘The study of Greck and Gothic planning and canons of proportion during the last 4o years has produced two main archeological and asthetical theories concerning the regu- lating diagram supposed to have been used by Greek and Gothic architects, Both these theories found their clue in the treatise on architecture written by the Roman architect Vitruvius and in Plato's expression “ Dynamic Syrumetry ” (Wheactetus). We have already seen Jay Hambidge’s hypothesis, based on the supposed use of “dynamic ™ rectangles and proportions (this too is confirmed by Vitruvius), specially of the rectangles V5 and (the Golden Rectangle), and his technique of monic analysis” by diagonal and perpendicular.HG ay Hoa oF AAIESES 1, HARMON ANALYSIS frewtal views Fig, yy shows the analysis of the tomb af fess Ramses TV and its triple sarcophagus after a contemporary papyrus (the inner rectangle is a double square, the middle one ai doreetinele, the outer one two @ rectangles equal to the muddle one mM o SOUARE AND @ RECTANGLE FIG. sf DIAGRAMS OF GY FIAN PECTORALS, zn often used as plan for Egyptian Fi 56 xvi Fig, 58 shows a desi ss. 59 and Plate XVI show analyses by Larnbidg pectorals. Fig and D'Caskey of Greek vases.Fig. 06 FIG. 48 THE PARTHENON, HARMONIC ANALYSIS (HAMBIDGE) ‘The harmonic analysis by Hambidge of the plans of many Greek temples, vases, ritual implements, scem to confirm his interpretation of Plato and Vitruvius; the latter states that commadulation {* symmetria ”) between the elements of a plan and between the elements and the whole is produced by proportion (“* analogia ”) so as to obtain “ eurhythmy ”, and also that“ the more subtle questions of commodulation have to be treated by irrational proportions” (this being “ dyna- mic” symmetry). We have seen Hambidge’s treatment of dynamic rectangles in Figs. 21, 22, 23, 24. Fig. 56 shows Hambidge’s analysis of the Parthenon's 40HG ay Hoa oF AAIESES 1, HARMON ANALYSIS frewtal views Fig, yy shows the analysis of the tomb af fess Ramses TV and its triple sarcophagus after a contemporary papyrus (the inner rectangle is a double square, the middle one ai doreetinele, the outer one two @ rectangles equal to the muddle one mM o SOUARE AND @ RECTANGLE FIG. sf DIAGRAMS OF GY FIAN PECTORALS, zn often used as plan for Egyptian Fi 56 xvi Fig, 58 shows a desi ss. 59 and Plate XVI show analyses by Larnbidg pectorals. Fig and D'Caskey of Greek vases.Fu. 60 ‘These diagrams and the controlling diagrams of many Renaissance buildings show the conscious or unconscious use of dynamic rectangles and of the Golden Section, which produces, because of its characteristic properties, “the re-assuring in the diversity of evolution” (Timmerding), or of what is “a particular case of a more general rule, the recurrence of the same proportion in the clements of a whole ” (Id,). FIG. 59, GREEK VASE (STAMNG (EROM Gromtry ofthe Grea Var, BY DR. CASK Another theory deals with Gothic plans; itis Ernst Moessel’s theory of the sub-division of the circle of orientation of temples or churches. ‘This supposes that the circle of orienta- tion, traced on the ground itself, becomes on the plan ths frecting circle divided in 5, 10 or 20 equal parts; this, because of the relationship between the Golden Section and the pentagon and decagon, introduces also the Golden Section as dominating proportion. So that, although Hambidge’s and Moessel’s techniques proceed from different geometrical foundations, the resulting diagrams are often the same, Fig. 60 shows, after Moessel, the plan of an Egyptian ‘Temple and the elevation of the Rock Tomb at Mira (the use of the pentagon is very striking). #ROCK TOMI AT MILA (MOESSEL) 43e-6: to include most of the plans (and sometimes the Fa. 6 HG.61 Two uoitic STANDAID BLAM 61 shows the two standard plans composed by Moesel tien) of the majority of Gothic cathedrals and churches, . 2 shows the original Gothic diagram for the elevation of Milan Cathedral, published in 1521 by Casar CG architect and Master of the Works of the cathedral, ‘The directing circle figures on it (it is part of the original drawing), also the words symmetry, proportion and curhythmy. Here the plan is worked out “in the German mode”, hi Yas leading figure and proportion the equilateral triangle and v's 4Heche DOME, OF AULA MANATION, AND SECTION SAKCATSARIANO, 180)SCN, nid Away SANZ Ud fl A VNVA TIX AAW FIG. 6) THE f REGIANGLE (D. WHENEN Plate XII shows the analysis of Raphael's Crucifixion rutexit (Natio y) by Moessel’s method; a regular pentagon and a regular deeagon produce all the important points of the composition, whose balanced geometry seems to be most consciously thought out. <6Plate XI reproduces the corresponding structural dia- gram, supported by the “ triangle of the pentagon ”. Hambidge’s and Moessel’s hypctheses, quite independently of their archivological value, offer also to architects, painters, decorators, sculptors, two very useful techniques of composi tion, which are being widely used by artists in England, rance, the United States, Brazil, Switzerland, Belgium, etc. Fig. 63 represents a Golden Rectangle (real size about one metre to 0.618) used by certain painters as help to “ harmonic composition ”. (!) ais Plates XIV and XV repre- sent a paint ing com- posed by D, Wiener on these lines. Fig. 64 shows the section through a silver cup, with its con- structional diagram, executed by J. Puiforcat. FIG. 64 REGULATING DIAGRAM OF CUP (1. PUIFORCAT) 14) In order to le the vertical or horizontal side of a frame accordi te the Golden Secti = lcs the length of this side by 0,610, a7 rate XU Fie. 63 Plate, ‘Xiv-xv Fir. 6FIG. G3 IAGHAM OF EAGADL OF LIVFANY § GO, PARIS, ‘A. SOU THC ‘ade of Tiffany & Co. fi5 shows the diagram of the in Paris planned by A. Southwick. Both techniques, Hambidge's and Moessel’s, produce automatically sets of proportions which obey the “ rule of hon-mixing of themes”, and reflect the laws of * harmonious tance with the “ Principle of Analogy ” formulated by Uhierseh in the roth century. “We have found, in considering the most remarkable productions of all periods, that in cach of these one fundamental shop ix repeated, so that the parts by their adjustment and growth ", in_acee Uisposition reproduce similar figures. Harmony results only from the repetition of the principal figure throughout the subdivisions of the whole". 48Plate 1 Seurat. The Circus.Plate II] Star-Dodecahedron with twelve verticesPlate IV Model of Star-Dodecahedron with twenty vertices.Plate V Model of Star-Dodecahedron with twelve vertices.Plate VI Miss Helen Wills. Harmonic Analysis,Plate VII Isabella d'Este. Harmonic Analysis (Photo: Alinari)Plate VIII Pentagonal Symmetry in Flowers (Photo: Wasmuth Berlin)Plate IX Pentagonal Symmetry in Marine Animals (Photo: Bibliographisches Institut, Leipzig)Plate X Logarithmic Spiral and Shell-Growth.Reproduced by courtesy of the Trustees, National Gallery, London Plate XII Raphael Crucifixion, Decagon and Pentagon SY, eS TY - ePlate XIII Structural diagram of Raphael's CrucifixionPlate XIV Lilies (D, Wiener)Plate XV Diagram for Lilies (D. Wiener)premsdianldasssl Plate XVI Greek Vase (Kylix). Harmonic Analysis. (from The Diagonal, Yale University Press)