Pach ac totes , ora i Sata btals Bietaey sate elste 14 i Carag tr [tieia ais hy etaleidl ° FS Het As Mah 4 ct . Britten ticanrs hart as ; ry - ‘ te ‘ ‘ te tet He wea ety tens ote te ae a ele s ais ae te pepe ey Wise gale fe ABS IM A: ot ihe Bee ot eos eee btw eet ee) eitit e145 berets eid! srl ser tteags : olapeTel ys aise eae ] t. CECE Ich $ Cewrerk tr tt Wes vie = oS +4 a Th (att —antecvadeosdl eam c * ome ai pee or oa d v dons os ein * ot ~~ a = + Ae ‘deeaty a a : Rha Did & REZ $ Z ‘ c! . = q 2 ; . ard if a f % a4 & Gab cout 7 ¢ “ + 5 @ ar A sa ce . a eee sor > ep * Le * “ Ps . Leer LTT) ROEM +e * r ss oD NRE 1, RS hE Oa CRETE a Cie & it ie a ( bison t | s ~ © ie. Re Gea Oe i Ee ” & sete a CEP. e roo bal Br Pal ecole. ee a. ie 2 4% iH ‘ oe if ie + ee ie OURS Se a: > tome 8 a * ea. i hme * + A » ae w a bg * ® i rR Ge : lle Rt Gas ce . of ad * a il s Bee & i ee fx 2 aeege ws 7 » ee -o eS fi Hes i ie ri ‘e : S- | OY n ‘ Lt a? h 4 = yn 2 u a) wy A . » Yi : \ Dae ” ; ' f Ds ji 4 , S ’ 3 4 ‘ * u 1 t hs ) * ye 1% eT ae pr. BY THE SAME A The Golden Person in the Episodes from an Unw A Primer of Higher Spac Four-Dimensional Vistas. as A 4 ~ > “oe Pee dl a } * f 7 > i ¢ ‘3 \ Ne oe i. ox * r an a P ai oUF > “ORNAMENT Claude on EW_YOR ALFRED A’ KNOPF 1927 , Copyright 1915 by Claude Bragd , All rights reserved—no part of this boo tes 0 3 Males ve Ms ? aoe ou Gat wa) DEDICATED TO E. B. CONTENTS THE NEED OF A NEW FORM LANGUAGE ORNAMENT AND PSYCHOLOGY THE KEY TO PROJECTIVE ORNAMENT THREE REGULAR POLYHEDROIDS FOLDING DOWN - MAGIC LINES IN MAGIC SQUARES A PHILOSOPHY OF ORNAMENT THE USES OF PROJECTIVE ORNAMENT FOREWORD ANY sincere workers in the field of art have realized the aesthetic poverty into which the modern world has fallen. Designers are reduced either to dig in the boneyard of dead civilizations, or to develop a purely personal style and method. The latter is rarely successful: city dwellers that we are for the most part, and self-divorced from Nature, she witholds her intimate secrets from us. Our ignorance and superficiality stand pitifully revealed. Is there not some source, some secret spring of fresh beauty undiscovered, to satisfy our thirsty souls? Having all his life asked himself this ques- tion, the author at last undertook its quest. Such results as have up to the present rewarded his search are here set forth. Their value and import- ance will be determined, as all things are determined, by use and time, but this much must be admitted— they are drawn from a deep well. The author desires to acknowledge his indebted- ness to the following sources for material contained in this volume: The Fourth Dimension, by C. Howard Hinton, M. A.; Geometry of Four Dimen- sions, by Henry Parker Manning, Ph. D.; Obser- vational Geometry, by William T. Campbell, A. M.; Mathematical Essays and Recreations, by Hermann PROJECTIVE ORNZAARE Schubert; also to an essay entitled Regular Figures in n-dimensional Space, by W. I. Stringham, in the third volume of the American Journal of Mathematics, and an article on Magic Squares in the Eleventh Edition of the Encyclopaedia Britannica. The chapter entitled 4 Philosophy of Ornament is enriched by certain ideas first suggested in a lec- ture by Mr. Irving K. Pond. With no desire to wear borrowed plumes, the author yet found it im- possible in this instance to avoid doing so, they are so woven into the very texture of his thought. In the circumstances he can only make grateful acknow- - ledgement to Mr. Pond. The author desires to express his gratitude to Mr. Frederick L. Trautmann for his admirable inter- pretations of Projective Ornament in color, of which the frontispiece gives an idea—and only an idea. “A “ee ie “at feces \ A Wi 7 2 : Sat ) a , ‘ ’ ‘ ‘ § ’ ‘ 4 ‘ ' 1 { ‘ i ' 1 1 s ‘ i ' 1 ' ’ ’ ‘ ‘ 42s —_ ~OAN. te \i2— = Weng / <_s.~ oi ING | Ad) \) eee oe” le —; THE NEED OF A NEW FORM LANGUAGE We are without a form language suitable to the needs of today. Archi- tecture and ornament constitute such a language. Structural necessity may be depended upon to evolve fit and expressive architectural forms, but the same thing is not true of ornament. This necessary element might be supplied by an individual genius, it might be derived from the conventionalization of natural forms, or lastly it might be devel- oped from geometry. The geometric source is richest in promise. ARCHITECTURE AND ORNAMENT N contemplating the surviving relics of any period in which the soul of a people achieved aesthetic utterance through the arts of space, it is clear that in their architecture and in their ornament they had a form language as distinctive and adequate as any spoken language. Today we have no such language. This is equivalent to saying that we have not at- tained to aesthetic utterance through the arts of space. That we shall attain to it, that we shall develop a new form language, it is impossible to doubt; but not until after we realize our need, and set about supplying it. PROJECTIVE ORNAZARe Consider the present status of architecture, which is preéminently the art of space. Modern architecture, except on its en- gineering side, has not yet found itself: the style of a building is determined, not by necessity, but by the whim of the designer; it is made up of borrowings and survivals. So urgent is the need of more appropriate and indigenous architectural forms with which to clothe the steel framework for which some sort of protective covering is of first importance, that some architects have ceased search- ing in the cemetery of a too Pontshatenids sacredly cherished past. They are seeking to solve their problems rather by a process of elimination, using the most elementary forms and the materials readiest to hand. In thus facing their difficulty they are re- creating their chosen art, and not abrogating it. / (am. © 4 J \) The development of new architectural forms appropriate to the new structural methods is already under way, and its successful issue may safely be left to necessity and to time; but the no less urgent need of fresh motifs in ornament has not: yet even begun to be met. So far as architecture is con- cerned, the need is acute only for those who are determined to be modern. Having perforce abandon- ed the structural methods of the past, and the forms 2 PROJECTIVE ORNAMENT associated with these methods, they nevertheless continue to use the ornament associated with what they have abandoned: the clothes are new, but not the collar and necktie. The reason for this failure of invention is that while common sense, and a feeling for fitness and proportion, serve to produce the clothing of a building, the faculty for originating appropriate and beautiful ornament is one of the rarest in the whole range of art. Those arts of space which involve the element of decoration suffer from the same lack, and for a similar reason. Three possible sources of supply suggest them- selves for this needed element in a new form language. Ornament might be the single-handed creation of an original genius in this partic- ular field; it might be de- rived from the conventional- ization of native flora, as it was in the past; or it might be developed from geometry. Let us examine each of these possibilities in turn. The first we must reject. ch Even supposing that this art Pegt saviour should appear as some rarely gifted and resourceful creator of ornament, it would be calamitous to impose the idiosyncratic space rhythm of a single individual upon an entire architecture. Fortu- Tesseracts: Cubes nately such a thing is impossible. In Mr. Louis Sullivan, for example, we have an ornamentalist 3 PROJECTIVE ORNAMEW® of the highest distinction (quite aside from his sterling qualities as an architect), but from the work of his imitators it is clear that his secret is in- communicable. It would be better for his disciples to de- velop an individual manner of their own, and this a few of them are doing. Mr. Sullivan will leave his little legacy of beauty for the en- richment of those who come after, but our hope for an ornament less personal, more universal and generic, will be as far from realization as before. NATURE Tetrahedrons: Tesseracts: : ‘ Icositetrahedroid Such a saviour being by the very necessities of the case ‘denied, us, may, we not go directly to Nature and choose whatever patterns suit our fancy from the rich garment which she weaves and wears? There is no lack of precedent for such a procedure. The Egyptian lotus, the Greek honeysuckle, the acanthus, the Indian palmette, achieved, in this way, their apotheosis in art. The Japanese use their chrysanthemum, their wisteria and bamboo, in similar fashion; so why may not we do likewise? The thing has already been attempted, but never consistently nor successfully. While far from solving the problem of a new language of ornament, for reasons presently to 4 PROJECTIVE ORNAMENT appear, the conventionalization of our native grains, fruits and flowers, would undoubtedly introduce a note of fresh beauty and ap- propriateness into our archi- tecture. Teachers of design might put the problem of such conventionalizations before their pupils to their advantage, and to the advancement of art. There is, however, one diffi- culty that presents itself. By reason of scientific agriculture, intensive cultivation under glass, and because of the ease and freedom of present-day transportation, vegetation in civilized countries has lost much of its local character and significance. Corn, buck- .wheat, cotton, tobacco, though native to America, are less distinctively American than they once were. Moreover, dwellers in cities, where for the most part the giant flora of architecture lifts its skyscraping heads, know nothing of buckwheat except in pan- cakes, of cotton except as cloth or in the bale. Corn in the can is more familiar to them than corn on the cob, and not one smoker in ten would recognize tobacco as it grows in the fields. Our divorce from nature has become so complete that we no longer dwell in the old-time intimate communion with her visible forms. Pentahedroids: Tesseracts PROJECTIVE ORNAMERT GEOMETRY There remains at least one other possibility, and it is that upon which we shall now concentrate all our attention, for it seems indeed an open door. Geometry and number are at the root of every kind of formal beauty. That the tapestry of nature is woven on a mathematical framework is known to every sincere student. As Emerson says, “Nature geometrizes . . . moon, plant, gas, crystal, are con- crete geometry and number.” Art is nature selected, ar- ranged, sublimated, triply re- fined, but still nature, how- ever refracted in and by con- sciousness. If art is a higher power of nature, the former must needs submit itself to mathematical analysis too. The larger aspect of this whole matter—the various vistas that the application of geom- etry to design opens up—has been treated by the author in a previous volume*. Narrow- ed down to the subject of ornament, our question is, what promise does geometry hold of a new ornamental mode? \ ya Ae ’ se. & a C/ ¥. —— ¢ 0. en7 Dx | f., ‘s . . . Tesseract In the past, geometry has given birth to many characteristic and consistent systems of ornamenta- tion, and from its very nature is capable of giving *The Beautiful Necessity. mmeOoyeCclIVE ORNAMENT birth to many more. Much of Hindu, Chinese, and Japanese ornament was derived from geometry, yet these all differ from one another, and from Moorish ornament, which owes its origin to the same source. Gothic tracery, from Perpendicular to Flamboyant, is nothing but a system of straight lines, circles, and the intersecting arcs of circles, variously ar- ranged and combined. ‘The interesting development of ornament in Germany which has taken place of late years, contains few elements other than the square and the circle, the parallelogram and the ellipse. It is a remarkable fact that ornamentation, in its primitive manifestations, is geometrical rather than naturalistic, though the geometrical source is the more abstract and purely intellectual of the two. Is not this a point in its favor? The great war undoubtedly ends an era: “the old order changeth.” Our task is to create the art of the future: let us ney draw our inspiration from the deepest, purest well. Geometry is an inexhaustible well of formal beauty from which to fill our bucket; but before the draught is fit for use it should be examined, analyzed, and filtered through the consciousness of the artist. If with the zeal of the convert we set at once to work with IT square and compass to devise a new system of ornament from geometry, we shall proba- bly end where we began. Let us, therefore, by a purely intellectual process of analysis and selection, try to discover some system of geometrical forms and configurations which shall yield that new orna- mental mode of which we are in search. y Ka> Cena = = esl Erap , ——— ~~ | fa 4 j : r= : a Gy sx | ae, (# << | _ We a beset Seed AS a \8 II ORNAMENT AND PSYCHOLOGY Ornament is the outgrowth of no practical necessity, but of a striving toward beauty. Our zeal for efficiency has resulted in a corresponding aesthetic infertility. Signs are not lacking that consciousness is NOW looking in a new direction—away from the contemplation of the facts of materiality towards the mysteries of the supersensuous life. This transfer of attention should give birth to a new aesthetic, expres- sive of the changing psychological mood. The new direction of con- sciousness is well suggested in the phrase, The Fourth Dimension of Space, and the decorative motifs of the new aesthetic may appropri- ately be sought in four-dimensional geometry. THE ORNAMENTAL MODE AND THE PSYCHOLOGICAL MOOD RCHITECTURAL forms and features, such as the column, the lintel, the arch, the vault, are the outgrowth of structural necessity, but this is not true of ornament. Ornament develops not from the need and the power to build, but from the need and the power to beautify. Arising from a psychological impulse rather than from a physical necessity, it re- flects the national and racial consciousness. ‘To such a degree is this true that any mutilated and time- worn fragment out of the great past when art was a language can without difficulty be assigned its place and period. Granted a dependence of the ornamental mode upon the psychological mood, our first business is to discover what that mood may be. A great change has come over the collective consciousness: we are turning from the accumula- 9 PROJECTIVE ORNAMENT tion of facts to the contemplation of mysteries. Science is discovering infirmities in the very founda- tions of knowledge. Mathematics, through the questioning of certain postu- lates accepted as axiomatic for thousands of years, is concerning itself with prob- lems not alone of one-, two-, and three-, but of n-dimen- sional spaces. Psychology, no longer content with super- ficial manifestations, is plung- ing deeper and deeper into the examination of the sub- conscious mind. Philosophy, despairing of translating life by the rational method, in terms of inertia, is attempting to apprehend the universal flux by the aid of intuition. Religion is abandoning its man-made moralities of a superior prudence in favor of a quest for that mystical experience which fore- goes all to gain all. In brief, there is a renascence of wonder; and art must attune itself to this new key-note of the modern world. Icositetrahedroid THE FOURTH DIMENSION To express our sense of all this Newness many phrases have been invented. Of these the Fourth Dimension has obtained a currency quite outside the domain of mathematics, where it originated, and is frequently used as a synonym for what is new and 10 PeeyenG ll VE ORNAMENT strange. But a sure intuition lies behind this loose use of a loose phrase—the perception, namely, that consciousness is moving in a new direction; that it is glimpsing vistas which it must needs explore. Here, then, is the hint we have been seeking: consciousness is moving towards the conquest of a new space; ornament must indicate this movement of consciousness; geometry is the field in which we have staked out our particular claim. It follows, therefore, that in the soil of the geometry of four dimensions we should plant our metaphysical spade. The fourth dimension may be roughly defined as a direc- tion at right angles to every known direction. It is a hyperspace related to our space of three dimensions as the surface of a solid is re- lated to its volume; it is the withinness of the within, the outside of externality. “But this thou must not think to find With eyes of body but of mind.” We cannot point to it, we cannot picture it, though every point is the beginning of a pathway out of and into It. Double Prisms FOUR-DIMENSIONAL GEOMETRY However little the mathematician may be prepared to grant the physical reality of hyperspace—or, more 1] PROJECTIVE ORNAM Ee properly, the hyperdimensionality of matter—its mathematical reality he would never call in question. Our plane and solid geometries are but the beginnings of this science. Four-dimen- sional geometry is far more extensive than three-dimen- sional. The numberof figures, and their variety, increases more and more rapidly as we mount to higher and higher spaces, each space extending in a direction not existing in the next lower space. More- over, these figures of hyper- space, though they are un- known to the senses, are known to the mind in great minuteness of detail. To the artist the richness of the field is not of great im- portance. He need concern himself with only a few of the more elementary figures of four-dimensional geometry, and only the most cursory acquaintance with the mathematical concepts involved in this geometry will give him all the material he seeks. Base of Icosahedroid: Cubes In the ensuing exposition, the willfulness and im- patience of the artistic temperament towards every- thing it cannot turn to practical account will be indulged to the extent of omitting all explanations and speculations not strictly germane to the purely aesthetic aspect of the matter. To such readers as are disposed to dig deeper, however, the author’s 12 PROJECTIVE ORNAMENT A Primer of Higher Space may be found useful, and there is besides a literature upon the subject. If after reviewing this literature the reader is disposed to regard the fourth dimension as a mere mathematical convention, it matters not in the least, so long as he is able to make practical use of it. He may likewise, with equal justice, question the existence of minus quantities, for example, but they produce practical results. With this brief explanation the author now turns up his shovelful, leaving it to the discerning to determine whether it contains any gold. >. OES Pe SS SEE eae CERI 2K D, cece Ws 7 ats VEO SES SES RR Ue es PE ae <éK2) EAR PQ OD Ww KS | a) ASD ar re < Va Y—— VW [J M “ : fre eet g py > eer ot 3 - ie seed RCo fo All WS Ke > f) Lg KO Oy ll i =e =o << y Ly < De IIT THE KEY TO PROJECTIVE ORNAMENT The idea of a fourth dimension is in conformity with reason, however foreign to experience. By means of projective geometry it is possible to represent a polyhedron (a three-dimensional figure) in the two dimensions of a plane. By an extension of the same method it is no less possible to represent a polyhedroid (a four-dimensional figure). Such representations in plane projection of solids and hypersolids constitute the raw material of Projective Ornament. THE DEVELOPMENT OF THE EQUILATERAL TRIANGLE IN HIGHER SPACES ‘THE concept of a fourth dimension is so simple that almost anyone can understand it if only he will not limit his thought of that which is possible by his opinion of that which is practicable. It is not reason, but experience, that balks at the idea of four mutually perpendicular directions. Grant, therefore, if only for the sake of intellectual adven- ture, that there is a direction towards which we cannot point, at right angles to every one of the so- called three dimensions of space, and then see where we are able to come out. It is possible to locate in a plane (a two-dimen- sional space) three points, and only three, whose mutual distances are equal. This mathematical fact finds graphic expression in the equilateral triangle. (A, Figure 1). In three-dimensional, or solid space, it is possible to add a point, and the mutual equal distances, six 15 PROJECTIVE ORNAMENT PLANE, PROJECTIONS OF CORRESPONDING FIGURES OF THREE: AND OF FOUR:DIMENSIONAL SPACE<—— LX & AIAN TETRAHEDRAL CELLS Of PENTAHEDROD'D” TETRAHEDRAL CELLS OF PENTAHEDROID'E” in number, between the four points, will be expressed by the edges of a regular tetrahedron whose vertices are the four points. But in order to represent this solid in a plane, we must have recourse to projective geometry. The most simple and obvious way to do this is to locate the fourth point in the center of the equilateral triangle and draw lines from this central point to the three vertices. Then we have a rep- resentation of a regular tetrahedron as seen di- rectly from above, the central point representing the apex opposite the base (B, Figure 1). But suppose we imagine the tetrahedron to be tilted far enough over for this upper apex to fall (in plane projection) outside of the equilateral triangle representing the base. In such a position the latter would foreshorten to an isosceles triangle, and at a certain stage of this motion the plane projection of the tetrahedron appears as a square, its every apex representing an apex of the tetrahedron,whose edges are repre- sented by the sides and diagonals of the square (C, 16 PROJECTIVE ORNAMENT Figure 1). In this representation, though the points are equidistant on a plane, as they are equidistant in solid space, the six lines are not of the same length, and the four triangles are no longer truly equilateral. But this is owing to the exigencies of representation on a surface. If we imagine that we are not looking az a plane figure, but zn#o a solid, the necessary corrections are made automatically by the mind, and we have no difficulty in identifying the figure as a tetrahedron. Now if we concede to space another independent direction, in that fourth dimension we can add another point equidistant from all four vertices of the tetrahedron. ‘The mutual distances between these five points will be ten in number and all equal. The hypersolid formed—a pentahedroid—will be bounded by five equal tetrahedrons in the same way that a tetrahedron is bounded by four equal equilateral triangles, and each of these by three equal lines. We cannot construct this figure, for to do so would require a space of four dimensions, but we can rep- resent it in plane projection, just as we are able to represent a tetrahedron. We have only to add another point and connect it by lines with every point representing an apex of the original tetrahe- dron (D, Figure 1); or according to our second method we can arrange five points in such fashion as to coincide with the vertices of a regular pentagon and connect every one with every other one by means of straight lines (E, Figure 1). In either case by convention we have a plane representation of a hypertetrahedron or pentahedroid. 17 PROJECTIVE ORNAMENT If we have really achieved the plane representa- tion of a pentahedroid, it should be easy to identify the projections of the five tetrahedral cells or bound- ing tetrahedrons, just as we are able to identify the four equilateral sides of the tetrahedron in plane pro- jection. We find that it is possible to do this. For convenience of identification, these are separately shown. By dint of continued gazing at this pentagon circumscribing a five-pointed star, and by trying to recognize all its intricate inter-relations, we may come finally to the feeling that it is not merely a figure on a plane, but that it represents a hypersolid of hyper- space, related to the tetrahedron as that is related to the triangle. THE CORRESPONDING HIGHER DEVELOPMENTS OF THE SQUARE Let us next consider the series beginning with the square. The cube may be conceived of as developed by the movement of a TESSERACT GENERATION AND | square in a direction at fs PROJECTION right angles to its two dimensions, a distance equal to the length of La one of its sides. The direction of this move- TESSERACT@HvERcux| Ment can be represented CUBE on a plane anywhere we wish. Suppose we “ establish it as diagonal- ly downward andtotheright. The resultant figure is a cube in isometric perspective, for each of the four 18 PROJECTIVE ORNAMENT points has traced out a line, and each line has devel- oped a (foreshortened) square (Figure 2). The mind easily identifies the figure as a cube, notwithstanding the fact that the sides are not all squares, that the angles are not all equal, and that the edges are not all mutually perpendicular. Next let us, in thought, develop a hypercube, or tes- seract. To do this it will be necessary to conceive of a cube as moving into the fourth dimension a distance equal to the length of one of its sides. For plane representation we can, as before, assume this direction to be anywhere we like. Let it be diagonally downward, to the left. In this position we draw a second cube, to represent the first at the end of its motion into the fourth dimension. And : because each point has traced out a line, each line a square, and each square a cube, we must connect by lines all the vertices of the first cube with the corresponding vertices of the second. The resultant figure will be a perspective of a tesseract, or rather the perspective of a perspec- tive, for it is a two-dimensional representation of a three-dimensional representation of a four-dimen- sional form (Figure 2). THE EIGHT CUBES OF A TESSERACT SNK ACG 19 PROJECTIVE ORNAMENT If we have achieved the plane projection of : a tesseract we should be able to identify the LNINT NESTS S eight cubes by whee it TRIANGLES is bounds an pa at es ASIAN Sie tt Balt the beginning and end o LNDNINIAAAGAACA the iaoteel ar the six TETRAHEDRON developed by the move- Fed SAMs WE OS ment of the six faces of LBP LEAT PLBR OY the cube into four- PN NY NY |sdimensional space. We bGAA ey find that we can do this. For convenience of iden- tification the eight cubes are separately shown in Figure 3. SQUARES TE TRUTH TO THE MIND IS BEAUTY TO THE EYE. Ornament is largely a matter of the arrange- ment and repetition ofa few well chosen motifs. The basis of ornament is geometry. If we arrange these various geometrical figures in sequence and in groups we have the rudiments of ornament (Figure 4). Al- though all these are plane figures, there is this im- portant difference between them: the triangle and the square speak to the mind only in terms of two dimen- 20 WX PeeorerarlVyE ORNAMENT sions; the plane representations of the tetrahedron and the cube portray certain relations in solid space, while those of the pentahedroid and the tesseract portray relations peculiar to four-dimensional space. It will be observed that the decorative value of the figures increases as they proceed from space to space: the higher- dimensional developments are more beautiful and carry a greater weight of meaning. This accords well with the dictum, “Beauty is Truth; Truth, Beauty.” The above exercises consti- tute the only clue needed to understand the system of orna- ment here illustrated. Every symmetrical plane figure has its three-dimensional correla- tive, to which it is relatedas Terie poems a boundary or a cross-section. These solids may in turn be conceived of as boundaries or cross-sections of corresponding figures in four- dimensional space. The plane projections of these hypersolids are the motifs mainly used in Projective Ornament. | NX = \ 2G “SS oe ¢ = ( ADIGPAMAMAABDABIAIABIGAIG SSSSESSESESSSSSSSSSS << ( CILA IAI GIGI GIGIGs*GIGi BiG GsBiGes 7 G4 “ eZ iL ts Z 7) 7) *? =P, Zz % aM 7 eZ. 7) v7) a 21 ran ZT; SS A A NET iS Fr LIL oa 7) —iMEaCD a, Laeo LEP TBE KDE a aD ZT. S ELEC CLE LE CLOCKS = oy di. Vlas ty 7 _)7} ae SP = v 7 “a SS —N 5 — IV THREE REGULAR POLYHEDROIDS The paradoxes of four-dimensional geometry are best understood by referring them to the corresponding truisms of plane and of solid geometry. This may profitably be done in the case of the pentahe- droid, the tesseract, and the 16-hedroid, the four-fold figures of most use in Projective Ornament. In the plane representation of four-fold figures for decorative purposes certain conventions should be observed, conventions which, though they serve aesthetic ends, find justification in optical and physical laws. TWO-, THREE-, AND FOUR-FOLD FIGURES oe most effective method for a novice to approach an understanding of any four-dimensional figure can becompared tothe athletic exercise called the hop, skip and jump. In this the cumulative impetus given by the hop and the skip 1s concentrated and expended in the supreme effort of the jump. The jump into the fourth dimension is best prepared for, in any given case, by a preliminary hop in plane space, and a skip in solid space. In the following cursory consideration of the three simplest regular polyhedroids of four-dimensional space let us apply this method. Even at the risk of wearisome reiteration let us resolve the paradoxes of hyperspace by referring them to the truisms of lower spaces. A regular polygon—a two-fold figure—consists of equal straight lines so joined as to enclose symmetri- cally a portion of plane space. A regular polyhedron a three-fold figure—consists of a number of equal 23 PROJECTIVE ORNAM ESS regular polygons, together with their interiors, the polygons being joined by their edges so as to enclose symmetrically a portion of solid space. A regular polyhedroid consists of anum- ber of equal regular poly- hedrons, together with their interiors, the polyhedrons be- ing joined by their faces so as to enclose symmetrically a portion of hyperspace. In the foregoing chapter we have considered the two simplest regular polyhedroids: the regular pentahedroid, or hypertetrahedron, and the tesseract, or hypercube. To these let us now add the hexadekahedroid, or 16-hed- roid, bounded by 16-tetrahed- rons. These regular hyper- Octahedrons: Tetrahedrons solids are of such importance in Projective Ornament that their elements should be familiar, and their construction understood. THE PENTAHEDROID A regular pentahedroid is a regular figure of four- dimensional space bounded by five regular tetrahe- drons: it has five vertices, ten edges, ten faces, and five cells. | If we take an equilateral triangle and draw a line through its center perpendicular to its plane, every point of this line will be equidistant from the three 24 BeeOyeClTIVE ORNAMENT vertices of the triangle, and if we take for a fourth vertex that point on this line whose distance from the three vertices is equal to one of the sides of the triangle, we have then atetra- hedron in which the edges are all equal. If through the center of this regular tetrahedron we could draw a line perpendicu- lar to its hyperplane every point of this line would be similarly, as above, equidis- tant from the four vertices of the tetrahedron, and we could take for a fifth vertex a point at a distance from the four vertices equal to one of the edges of the tetrahedron. We would have then a penta- hedroid in which the ten edges would all be equal. All the parts of any one kind—face angles, dihedral angles, faces, etc.—would be equal; for the penta- hedroid is congruent to itself in sixty different ways and can be made to coincide with itself, any part coinciding with any other part of the same kind. Tetrahedrons: Icosahedrons As every regular polyhedroid can be inscribed in a hypersphere in the same way that a regular polygon can be inscribed in a circle, and every re- gular polyhedron in a sphere, the pentahedroid is most truly represented in plane projection as in- scribed within a circle representing this hypersphere. Radii perpendicular to the cells of the pentahedroid 25 PROJECTIVE ORNAMENT meet the hypersphere in five points which are the vertices of a second regular pentahedroid symmetric- ally situated to the first with respect to the center, and therefore equal to the first. Representing these vertices by equidistant in- termediate points on the circle circumscribing the pentahedroid and complet- ing the figure, we have a graphic representation of this fact (Figure 5). These PLANE PROJECTION OF TWO SYMMETRICALLY PLACED PEN'TAHEDROIDS' IN A HYPER SPHERE intersecting pentahedroids inscribed within a hyper- sphere have their analogue in plane space in two symmetrically intersecting equilateral triangles in- scribed within a circle, and in solid space in two symmetrical intersecting tetrahedrons inscribed with- in a sphere (Figure 6). THE TESSERACT The tesseract, or hypercube, lee ut is a regular figure of four- TERE ACIRCLE dimensional space having RON | eight cubical cells, twenty- four square faces, (each a common face of two cubes), Lip NV, thirty-two equal edges, and sixteen vertices. It con- tains four axes lying in lines which also form a rectangu- 6 lar system. 26 PROJECTIVE ORNAM EW | CORRESPONDING PRO- | JECTIONS OF CUBE, AND TESSERACT bee A b In order to comprehend the tesseract in plane repre- resentation, let us first con- sider the corresponding plane representation of the cube. In parallel perspec- tive a cube appears as a square inside of another square, with oblique lines connecting the four vertices (A, Figure 7). By reason of our tactile and visual ex- perience, the inner and smaller square is thought of as the same size as the outer and larger, and the four intermediate quadri- lateral figures are thought of as squares also. If the cube is shown not in parallel, but oblique perspective, the mind easily identifies the two figures (B, Figure 7). These two ways of representing a cube in plane space may be followed in the case of the tesseract also (A’ and B’, Figure 7). We can think of the first as representing the ap- pearance of the tesse- ract as we look down into it, and the second as we stand a little to one side. In each case it is possible to identify the eight cubes whose in- teriors form the cells of GENERATION OF TESSERACT 28 Meee ye Cl IVE ORNAMENT the tesseract. The fact that they are not cubes except by ieee ard se iy convention is owing to the Eeecnvics of representation: ag LZ in four-dimensional space the , es cells are perfect cubes, and EQS Lip are correlated into a figure whose four dimensions are WS all equal. EWS SOS In order to familiarize vip ye, ourselves with this, for our Ze purposes the most impor- tant of all four-fold figures, ry let us again consider the Kd —YYj manner of its generation, be- \S TO pe ginning with the point. Let SES the point A, Figure 8, move to the right, terminating with \ We the point B. Next let the QR / \ ‘Un line AB move downward a distance equal to its length, tracing out the square AD. NY) (ZS THE TESSERACT IN THREE DIFFERENT ASPECTS PROJECTIVE ORNGM Es This square shall now move backward the same distance, generating the (stretched out) cube A H. And now, having exhausted the three mutually perpendicular directions of solid space, and undaunted by the physical impracticability of the thing, let this cube move off in a direction perpendicular to its every dimension (the fourth dimension) represented by the arrow. This will generate the tesseract Al. It will be found to contain eight cubical cells. For convenience of identification these are shown in Figure 9. Other aspects of the tesseract are shown in Figure 10; and in GENERATION OF TESSERACT| Figure 11 it is shown with an intermediate or cross-sectional square in each of the cubes, which square in the tesseract becomes an intermediate cube. Whenever, in the figure, we have three squares in the same straight line, we know that we have a cube. There are eight of these groups of three, the cubi- cal cells of the tesseract. If instead of represent- ing the fourth direction outside the generating cube we choose to con- ceive of it as inward, the resultant figure is that shown at the bot- 30 PROJECTIVE ORNAMENT tom of Figure 11, the innermost of these cubes cor- responding with the furthermost of the upper figure. THE 16-HEDROID After the pentahedroid or hypertetrahedron, and the tesseract or hypercube, already considered, we have as the next regular polyhedroid the hexadeka- hedroid, or, more briefly, the 16-hedroid. If we lay off a given distance in both directions on each of four mutually perpendicular lines inter- secting at a point, the eight points so obtained are the vertices of a regular polyhedroid which has four diagonals along the four |THE HEXADEKAHEDROID given lines. This is the 16- hedroid. It has, as the name implies, sixteen cells, (each a tetrahedron), thirty- two triangular faces, (each face common to two tetra- hedrons), twenty-four edges, and eight vertices. Figure 12 represents its projection upon a plane. The sixteen cells are ABCD, A’B’C’D’, AB’C’D’, Mpc. ABCD, A’BC’D, ABC’D, A‘B’CD’, pee b O'D),. ABC’D’, A’B/CD, A’BC’D, meow ABCD’, AB/C’D. The accented letters are the antipodes of the unaccented ones. Figure 13 represents another plane projection of this poly- hedroid. 31 PROJECTIVE ORNGH THE DECORATIVE VALUE OF THESE FIGURES As this is a handbook for artists and not a geome- trical treatise, the description of regular polyhedroids need not be carried further than this. The reader who is ambitious to continue, from the 24-hedroid even unto the 600-hedroid, is referred to the geo- metry of four dimensions; upon this he can exer- cise his mind and experience for himself the stern joy of the conquest of new spaces. But the designer has already, in the penta- hedroid, the hypercube, and the hexadekahedroid, ample material on which to exer- cise his skill. It should be remembered that just as in plane geometry a regular polygon can always be in- scribed in a circle, and in geometry of three dimen- 13 sions a regular polyhedron can always be inscribed in a sphere, so in four-dimensional geometry every regular polyhedroid can be inscribed in a hyper- sphere. In plane projection this hypersphere would be represented by a circle circumscribing the plane figure representing the polyhedroid. Almost any random arrangement on the page of these three hypersolids¥in plane projection will serve to indicate what largess of beauty is here— they are like cut jewels, like flowers, and like frost. Combined symmetrically they form patterns of endless variety. THE HEXADEKAHE DROID 32 MerewinGl tl VE ORNAMENT THE CONVENTIONS EMPLOYED IN THEIR REPRESENTATION There is a reason why the plane projections of hypersolids are shown as transparent. Our senses operate two-dimensionally—that is, we see and contact only surfaces. Were our sense mechanism truly three-dimensional, we should have X-ray vision, and the surfaces of solids would offer no re- sistance to the touch. In dealing with four-dimen- sional space we are at liberty to imagine ourselves in full possession of this augmented power of sight and touch. The mind having ascended into the fourth dimension, there would follow a _ corresponding augmentation on the part of the senses, by reason of which the interiors of solids would be as open as are the interiors of plane figures. There is justification also for the attenuation of all lines towards their center. It is in obedience to the optical law that when the light is behind an object it so impinges upon the intercepting object as to produce the effect of a OPTICAL EFFECTS | thinning towards the center. The actual form of the bars of a leaded glass window, for example, is as shown in A, ‘A Figure 14, but their optical LIGHT effect when seen against the our S light is as in B. Because in X-ray vision some substances are Opaque, and some trans- lucent, we are at liberty to attribute opacity to any part 33 OPAQUE CENTER, PROJECTIVE ORNAMENT we please, and thus to add a new factor of variation as in C. We are also at liberty to stretch, twist or shear the figures in any manner we like. By the use of tones, of color, or by mitigating the crystalline rigidity of the figures through their combination with floral forms, we can create a new ornamental mode well adapted to the needs of today. eB) | at &, ra bx ora ] J KA BINDING ATESSERACT OF STRETCHED 3-CUBES 34 7 | : on Wh WW, en if: | Oe NAAR a Pe Ed bo AVAD : xl aN LIA HY BF Ll ee NG 5 NN Od Ks ASO” 7K ~ we Si N Uh SN be WE AN: va : NAN ZING , VIX Xi ae ay ay ie WA yi V. FOLDING DOWN Regular polyhedroids of four-dimensional space may be unfolded in three-dimensional space, and these again unfolded in a space of two- dimensions; or, contrariwise, they may be built up by assembling the regular polyhedrons which compose them. In this way new and valu- able decorative material is obtained. ANOTHER METHOD OF REPRESENTING THE HIGHER IN THE LOWER ‘THE perspective method is not the only one whereby four-fold figures may be represented in three-dimensional and in two-dimensional space. Polyhedroids may be conceived of as cut apart along certain planes, and folded down into three-dimen- sional space in a manner analogous to that by which a cardboard box may be cut along certain of its edges and folded down into a plane. As the bounda- ries of a polyhedroid are polyhedrons, an unfolded polyhedroid will consist of a number of related polyhedrons. ‘These can in turn be unfolded, and the aggregation of polygons—each a plane boundary of the solid boundary of a hypersolid—will represent a four-fold figure unfolded in a space of two dimen- sions. An unfolded cube becomes a cruciform plane figure, made up of six squares, each one a boundary of the cube (A, Figure 15). Similarly, if we imagine a tesseract to be unfolded, its eight cubical cells will occupy three-dimensional space in the shape of a double-armed cross (B, Figure 15). In four-dimen- 37 PROJECTIVE ORNAMENT sional space these cubes can be turned in upon one another to form a symmetrical figure just as in three-dimensional space the six squares can be re- united to form a cube. A regular tetrahedron unfolded yields an equilat- eral triangle enclosed by three other equilateral triangles (C, Figure 15). Similarly, an un- folded pentahedroid, or hyper tetrahedron, would consist of a cen- tral tetrahedron with four others resting on its four faces (D, Figure 15). The pentahedroid could be re-formed by turning these towards one another in four- dimensional space, until they came com- pletely together again. A regular triangular prism unfolded yields three parallelograms, 15 its sides; and two equilateral triangles, its ends (E, Figure 15). Similarly, a regular hyper- prism would unfold into four equal and similar triangular prisms and two tetrahedrons (F, Figure 15). In four-dimensional space we could turn these prisms around the faces of the tetrahedron upon which they rest and the other tetrahedron around the face by which it is attached to one of the prisms, 38 FOLDED-DOWN FIGURES OF HIGHER SPACE mer ol lV E ORNAMENT and bring them all together, each prism with a lateral face resting upon a lateral face of each of the others, and each of the four faces of the second tetrahedron resting upon one of the prisms. This could be done without separating any of the figures, or distorting them in any way, and the figure thus folded up would then enclose completely a portion of four-dimensional space. THE POLYHEDRAL BOUNDARIES OF FOUR-DIMENSIONAL REGULAR ANGLES A regular angle for any dimensional space is one all of whose boundaries are the same inform and magni- tude. The summits of all regular figures in any space form regular angles since the distribution of their boundaries is sym- metrical and equal. G and H, Figure 16, repre- | BOUNDARIES OF REGULAR, sent respectively the en OFS AND CF'4 SPACE summits, one in each figure, of the tetrahedron | OO and the cube, with the two-dimensional bound- - . BOUNDARIES OF THE SUMMITS OF #& aries of the summit | tr et eae vei ea spread out symmetri- cally in a plane. The boundaries of the sum- mits of a four-dimension- G H’ al figure being solids, G’ | pounpartes oF THE SUMMITS OFA BeimerepresentTespec= ee eC tively the summits, one | DIMENSIONAL, $pack in each figure, of the higher correlatives of the 16 39 OUT F OMIM TICALLY IN PLANE SPACE PROJECTIVE ORNAMENT tetrahedron and the cube—the pentahedroid and the tesseract—spread out in three-dimensional space. That is, they represent, in three-dimensional per- spective, the symmetrical arrangement of the four boundaries of regular four-dimensional angles. In four-dimensional space the faces of those figures which lie adjacent to the common vertex are brought into coincidence, just as in three-dimensional space the edges of the triangles and squares adjacent to the common vertex are brought into coincidence, orn the summits of the tetrahedron and the cube. THE CONSTRUCTION OF THE 24-HEDROID It is possible to build up any regular polyhedroid by putting together a set of polyhedrons. We take them in succession in such order that each is joined to those already taken by a set of polygons like the incomplete polyhedron. Take the case of the four-fold icositetrahedroid or 24-hedroid. I, Figure 17, shows a summit with six octahedral boundaries arranged about it symmetric- ally in three-dimensional space. Conceive I to be transported into four-dimensional space, and the interstices between the adjacent triangular faces to be closed up by joining those faces two and two; the figure assumes a form whose projection is represented in J with dotted lines omitted. Adjust to this figure twelve other octahedrons in a symmetrical manner; three of these octahedrons are represented by the dotted lines of J. Again, close up the interstices between the adjacent faces; the outline of the figure assumes a form whose projection is represented in K. 40 eH =; 3 Tamar ANAL Na EVN ia By 14 ) @ |}: e ——— 2 ~—_" = JU) Rg p's (0; 1 4p. PROJECTIVE ORNAME Now conceive this figure to be turned inside out. There will be left in the middle of the figure a vacant space of exactly the form of J with the dotted lines omitted (L, Figure 17): such a group of six octahedronsis therefore required to complete the four-fold figure. By counting it is found that all the constituent octa- hedral summits of the four-fold figure are filled to saturation, and that the figure is in other respects complete and regular. The number of octahedral boundaries or cells is twenty-four; of summits, twenty-four; of triangular faces, ninety- six; of edges, ninety-six. CONSTRUCTION OF A 24 -HEDROID ae, : A ] Nose nd K <) 17 TESSERACT SECTIONS In the same way that it is easy to conceive all regular polygons as two-dimensional boundaries or cross-sections of regular polyhedrons, it is possible, though not so easy, to conceive of these same polygons as boundaries or cross-sections of corresponding polyhedroids. The various figures are represented in perspective projection, but they may be unfolded, after the manner of the cardboard box. If this be done the bounding polygons will be free from the distortions incident to perspective representation, but the result 42 Peo CLIVE ORNAMENT REGULAR ICOSAHEDRON UNFOLDED IN A PLANE Ee OF ICOS. | OF ICOsAHEDEGN AHEDRON;S; ONE! INSIDE THE OTHER. 18 in most cases is the monotonous and uninteresting repetition of units (Figure 18). What we require for amanicnt is Bicates contrast and variety o saint form, and this may be DOWN AND ae IN A PLANE obtained without going farther than the won- der-box of the tesseract itself. There are certain interesting polyhe- droids embedded, as it were, in the tesseract. Such are the tetratesse- ract, and the octatesse- act. Lhis: lastis: ob-= tained by cutting off every corner of the tesseract Just as an Oc- tahedron 1s left if every corner of a cube is cut off. Three such poly- PROJECTIVE ORNAMENT hedral sections of a tesseract, unfolded, repeated, and arranged symmetrically with relation to one another, roduce the highly decorative pattern shown in igure: 19. jie Z y__\ \__/ Tt dg N Bm FN ua SS See ee ee eee eee aaa BINDING: FOUR TESSERACTS AND FOUR CUBES 44 ! h | a i We D [E | / i} E | Jes] SS VI MAGIC LINES IN MAGIC SQUARES The numerical harmony inherent in magic squares finds graphic expression in the magic lines which may be traced in them. These lines, trans- lated into ornament, yield patterns often of amazing richness and variety, beyond the power of the unaided aesthetic sense to compass. Magic lines have relations to spaces higher than a plane—they, too, are Projective Ornament. THE HISTORY OF MAGIC SQUARES AiMost everyone knows what a magic square Is. Briefly, it is a numerical acrostic, an arrangement of numbers in the form of a square, which, when added in vertical and horizontal rows and along the diago- nals, yield the same sum. Magic squares are of Eastern and ancient origin. There is a magic square of 4 carved in Sanskrit characters on the gate of the fort at Gwalior, in India (Figure 20). Engraved on stone and metal, magic squares are worn at the present day in the East as talismans or amulets. They are known to have occupied the attention of Mediaeval philosophers, astrologers, and mystics. Albrecht Direr introduced what is perhaps the most remarkable of all magic squares into his etching Melancholia (Figure 21). ‘Today they find place in the puzzle departments of magazines. Their laws and formulas have engaged the serious attention of eminent mathematicians, and the discovery of so- called magical relations between numbers, not alone 47 , PROJECTIVE ORNAMEae in squares, but in cubes and hyper-cubes, is one of the recreations of the science of mathematics.* The artist, impatient of concept, but questing the AHINDU-SQUARE as[io]se haul] 27 noe beautiful, will care little about the mathematical aspect of the matter, but it should interest him to know that the magic lines of magic squares are rich in decora- tive possibilities. A magic line is that endless line formed by following the numbers of a magic square in their natural sequence from cell to cell and returning to the point of departure. Because most magic squares are developed by arranging the numbers in their natural order in the form of a square and then subjecting them to certain rotations, the whole thing may be compared to the formation of string figures—the cat’s cradle of one’s childhood— in which a loop of string is made to assume various intricate and often amazing patterns—magic lines in space. *See Philip Henry Wynne’s Magic Tesseract in the author’s Primer of Higher Space. 48 memoyeCcTIVE ORNAMENT THEIR FORMATION Without going at all deeply into the arcana of the subject it will not be amiss to suggest one of the methods of magic square forma- tion by the simplest possible example, the magic square of 3. Arrange the digits in sequence in three hori- zontal lines, and re- late them to the cells of a square as shown in Figure 22. This will leave four cells empty and four numbers outside the perimeter. Dispose these numbers, not in the empty cells which they adjoin, but in the ones opposite; in other words, rotate the outside numbers in a direction at right angles to the plane of the paper, about the lines which FORMATION OF THE MAGIC SQUARE OF THREE 22 49 PROJECTIVE ORNAA ESS MAGIC LINES IN MAGIC SQUARES 47] 10 | 25| | 49] 2 | 9] 6 | 2 63] 45| 2 | 62] S50] 3 | raf sfer ae] 1/2] 7 | exla[a ale [oe | re [i [25 |0[ 8 [4*] 8 [> afslals[a[a]4] [as] s9] a] a] 6 |x| | EIEYED COENEN CAEN EN a Ea Ea MAGIC SQUARE! OF 4.» MAGIC SQUARE OF 7 CHESS-BOARD 2ATH OF KNIGHT THE MAGIC LINE INA 1C SQUARE’ Is DISCOV ERED By Tec N¢ THe NUMERAL IN se ORDER F CELL, TO CELL, AND BACK TO THE BEGINNING NUMBER ewe ‘@ MAGIC LINE CF3 Ri MAGIC LINE OF 4 MAGIC LINE OF 7 severally bound the central cell. By this operation each outside number will fall in its proper place. These rotations are indicated by dotted lines. The result is the magic square of 3. Each line, in each of the two dimensions of the square, adds to 15, and the two diagonals yield the same sum. Now with a pencil, using a free-hand curve, follow the numbers in their order from 1 to 9 and back again to 1. The result is the magic line of the 50 PROJECTIVE ORNAMENT magic square of 3 (Figure 22). We have here a configuration of great beauty and interest, readily translatable into orna- ment. As the number of magic squares is practically infinite, and as each containsa magic line, here is a rich field for the designer, even though not all magic lines lend them selves to decorative treat- ment. Figures 23 and 24, show some of them which do so lend them- selves, and Figures 25, 26 and 27 show the translation of a few of these into ornament. eae ah © SQUARES MAGIC aii 5 MAGIC LINFON4 MAGIC LINE O'S 24 THE KNIGHT’S TOUR It is a common feat of chess players to make the tour of the board by the knight’s move (two squares forward and one to right or left), starting at any “a a MAGIC LINES 25 square, touching at each square once, and returning to the point of departure. Keller, the magician, intro- duced this trick into his per- formance, permitting any member of the audience to designate the initial square. 51 PROJECTIVE ORNAYE ee It is a simple feat of mnemonics. ‘The per- former must remember 64 numbers in their or- der, the sequence which yields the magic line in the magic square of 8. The plotting of this line is shown in Figure 23; its decorative applica- tion in the binding of The Beautiful Necessity. Euler, the great mathe- matician, constructed knight’s move squares of 5 and of 6, having peculiar properties. In 26 one diagram of Figure 28 the natural numbers show the path of a knight moving in such a manner that the sum of the pairs of numbers opposite to and equidistant from the middle figure is its double. In the other diagram the knight returns to its starting cell in such a manner that the difference between the pairs of numbers opposite to and equidis- tant from the middle point is 18. PATTERN FROM MAGIC SQUAEES CLINE OF 3 aes PATH TRACED BY THE KNIGHT IN MAKING WHAT Is KNOWN AS THE KNIGHT'S TOURS INTERLACES Figure 28 shows interlaces derived from these two magic squares. They so resemble the braided bands found on Celtic crosses that one 52 Nave 04 CA IIPAS an 2 0.25 0 Ow oO” .“@.. Tere ek eee? Breasts, q oe oct tos! ~Or1 etbeaeenseee ‘i; ©” 4 ~~ e yO ye OR Q ‘ PISO PS, PROJECTIVE ORNAMENT FULERS KNIGHT S-TOUR SQUARES | pola Ts [oes] [7 | 6 | 29 [20] 5 [| fs [3s| [27 ref 92 far roofs] [a MAGIC LINE POM EULER su. Albrecht Durer, whose ac- quaintance with magic squares is a matter of record, is known to have expended a part of his inventive genius in designing interlacing knots. Leonardo da Vinci also amused himself in this way. The element of the mystic and mysterious entered into the genius of both these masters of the Renais- sance. One wonders if this may not have been due to some secret afhliation with an occult fraternity of adepts, whose existence and claims to the possession of extraordinary knowledge and power have 54 28 naturally wonders if their unknown and ad- mirable artists may not have possessed the secret of deriving orna- ment from magic nu- merical arrangements, for these arrangements are not limited to the square, but embrace polygons of every des- cription. Here is an- other curious fact in this connection: PATTERNS FROM EULER'S KNIGHTS MOVE SQUARES << = CREED Peon Cli VE ORNAMENT been the subject of much debate. Were these knots of theirs not only ornaments, but symbols—password and counter-sign pointing to knowledge not possess- ed by the generality of men? These patterns show forth in graphic form the symphonic harmony which abides in mathematics, a fact of sweeping significance, inasmuch as it involves the philosophical problem of the world- order. [he same order that prevails in these figures permeates the universe; through them one may sense ae cosmic har- mony of the spheres, just as it is possible to fy [is [is [| hear the ocean in a 5 | shell. MAGIC SQUARE AND CUBE OF 4. MAGIC THE PROJECTED MAGIC LINE In answer to any question which may arise in the mind of the reader as to the relevancy of magic squares to the subject of Projective Orna- ment, it may be stated that magic lines are Projective Ornament in a very strict sense. These lines, though figures on a_ plane, represent an extension PROJECTIVE ORNAMENT tion at right angles to the plane, and they have rela- tions to the third and higher dimensions. As this is a fact of considerable interest and importance, the attempt will be made to carry its demonstration at least far enough to assure the reader of its sub- stantial truth. Let us examine the three-dimensional aspect of the magic line in a magic square of 4. Figure 30 represents one of the most remarkable magic squares. Each horizontal, each vertical and each diagonal column adds 34. ‘The four corner cells add 34, and the four central cells add 34. The two middle cells of the top row add 34 with the two middle cells of the bottom row. The middle cells of the right and left columns similarly add 34. Go round the square clock-wise; the first cell beyond the first corner, plus the first beyond the second corner, plus the third, plus the fourth, equals 34. Take any number at random, find the three other numbers corresponding to it in any manner that respects symmetrically two dimensions, and the sum of the numbers is 34. In Figure 30 is also represented the magic cube of 4. It is made up of 64 cubical cells, each con- taining one of the numbers from 1 to 64, inclusive. This cube can be sliced into four vertical sections from left to right, or it can be separated into four other vertical sections by cutting planes perpendi- cular to the edge A B, proceeding from front to back, or the four sections may be horizontal, made by planes perpendicular to AD. Now each of these twelve sections presents a magic square in which each row and each column adds 130. The diagonals of these squares do not 56 Pewee oll) E ORNAMENT S*. OFA ry Cen BINDING: THE KNIGHTS TOUR (MAGIC LINE CF 8;SQUARE) add 130, but the four diagonals of the cube do add 130. The essential correspondence of the magic square of 4 to the magic cube of 4 is clearly apparent. Now if we plot that portion of the magic line of the magic cube of 4 embraced by the numbers from 1 to 16 and compare it with the magic line of the magic square of 4, it is seen that the latter is a plane projection of the former. In other words, shut the four sections of the cube up so that the front section, A C, in 1-16 fits over the back section, 49-64; then using only the numbers 57 PROJECTIVE ORNAMENT 1 to 16, they will be found to fall magically into the same places they occupy in the magic square of 4. Because all magic lines in magic squares have, in their corresponding cubes, this three-dimensional ex- tension, the patterns derived from magic squares come properly under the head of Projective Orna- ment. 7A | [A i LS ah ASTISAISED EME ME SA A MANMAE ISLS \*/ —<— KIA LAS p24 VN Zn : Sy. TS CTAB ARE LAM SNZH SZ VY < WAN 58 oe elt ate i WSEAS ie a al igs ‘ose 2 NE Sih sENeS sa: N24 |)3( — ’ i iY 4 = | > i | oo ma Vill THE USES OF PROJECTIVE ORNAMENT Projective Ornament, being directly derived from geometry, is universal in its nature. It is not a compendium of patterns, but a system for the creation of patterns. Its principles are simple and comprehensive and their application to particular problems stimulates and develops the aesthetic sense, the mind, and the imagination. THE FIELD AND FUNCTION OF PROJECTIVE ORNAMENT PROJECTIVE Ornament is that rhythmic sub- division of space expressed through the figures of Projective Geometry. As rhythmic space sub- division is of the very essence of ornament, Pro- jective Ornament possesses the element of univer- sality, though it lends itself to some uses more readily than to others. To those crafts which employ linear design, such as lace-work, lead-work, book-tooling, and the art of the jeweler, it is particu- larly well suited; with color it lends itself admirably to stained glass, textiles, and ceramics. On the other hand, it must be considerably modified to give to wrought iron an appropriate expression: its application to cast iron and wood-inlaying pre- sents fewer difficulties. Its three-dimensional, as well as its two-dimensional aspects, come into play in architecture, and from its many admirable geo- metrical forms there might be developed architectural detail pleasing alike to the mind and to the eye. A crying need of the time would thus be met. The drab 71 PROJECTIVE OKNAG aN BR RL Ae e monotony of broad cement ww, Catt “os surfaces could be relieved by a 2 oN Vx \ A means of incrusted ornament a: ow, in colored tiles arranged in yob,° ° Xi, <4! patterns developed by the i AS 7 a Nig " methods described. 67,9 Various applications of Projective Ornament to prac- tical problems are suggested in the page illustrations dis- persed throughout this vol- ume, but a careful study of Y WSS the text will be more profit- A “> able to the designer than any ‘3 as SA copying of the designs. If evar pacenias the rationale of the system See ran is thoroughly grasped, a de- signer will no longer need to copy patterns, since he will have gained the power to create new ones for himself. To copy is the death of art. No worse fate could befall this book, or the person who would profit by it, than to use it merely as a book of patterns ‘These should be looked upon only as illustrative of certain fundamental principles susceptible of endless application. Mr. Sullivan, from sad experience, predicted that the zeal of any converts that the book might make would be expended in sedulous imitation rather than in original creation. The author, however, takes a more hopeful view. HOW TO AWAKEN THE SLEEPING BEAUTY The principles here set forth are eminently com- municable and understandable. They present no 72 Peo ymeo lI YE ORNAMENT difficulty, even to an intelligent child. Indeed, the fashioning and folding up of elementary geometrical solids is a kindergarten exercise. The great impedi- ment to success in this field is a proud and sophisti- ‘cated mind. Let the learner “become as a little child,’ therefore: let him at all times exercise him- self in Observational Geometry—that is, look for the simple geometrical forms and relations of the objects that come under his every-day notice. He should come to recognize that the myriad forms in the animal, vegetable, and : mineral kingdoms furnish an GING 7 } unending variety of symmetri- = fs Zar cal and complex geometric ; forms which may be discovered and applied to his own prob- lems. This should create an appetite for the study of Formal Geometry. From that study a fresh apprehension of the beauty of arithmetical relations is sure to follow. Enamored of this beauty, the disciple will seek out the basic geometrical ground rhythms latent in nature and in human life. The development of Hexadekahedroids faculty will follow on the awakening of perception: the elements and relations grasped by the mind will externalize themselves in the work of the hand. Not content with the known and familiar space re- lationships, the student will essay to explore the 73 PROJECTIVE ORNAWEW field of hyperspace. But let him not seek to achieve results too easily and too quickly. In all his work he should follow an orderly sequence, quarrying his gold before refining it, and fashioning it to his uses only after it is refined: that is, he should endeavor to understand the figures before he draws them, and he should draw them as geometrical diagrams before he attempts to alter and combine them for decorative use. It is the author’s experience that they will require very little alteration; that they are in themselves decorative. The filling in of certain spaces for the purpose of achieving notan (contrast) is all that is usually required. This done, the application of color is the next step in the process: first comes line, then light and dark, and lastly color values. Such is the method of the Japanese, those masters of decorative design. THE ILLUSTRATIONS AND DIAGRAMS The black-and-white designs interspersed throughout the text represent Projective Orna- ment removed only one degree from geometrical diagrams, — yet they are seen to be highly decorative even in this form. At the pleasure of the designer they may be elongated, con- tracted, sheared, twisted, translated from straight lines 74 AG BY; We Si V7 f AG <7 O== Zk Cres ey YAyp Sos EN Ze (CW Wa (GS h wa A\ PROJECTIVE ORNGA Ye into curves; and by subjecting them to these modi- fications their beauty is often augmented. Yet if their geometrical truth and integrity be too much tampered with, they will be found to have lost a certain precious quality. It would seem as though they were beautiful to the eye in proportion as they -|}- W771 Neo, AL (4 \Y he 7 “No ( 2. Fd 1 Vi ee 1 ee ed 1 : Av? Hl mars ae As bal kc Ly q ee NS VW N oe", ( (xi »@ Iw @ = A SN74S2 72 NA] VAS 76 feel ly EF ORNAMENT are interesting to the mind. For the sake of variety the figures are presented in three different ways; that is, in the form of mons, borders, and fields— corresponding to the point, the line, and the plane. It is clear that all-over patterns quite as interesting as those shown may be formed by repeating some of the unit figures. With this scant alphabet it is possible to spell more words than one or two. Projective Ornament, derived as it is from Pro- jective Geometry, is a new utterance of the trans- cendental truth of things. Whatever of beauty the figures in this book show forth has its source, not in any aesthetic idiosyncracy of the illustrator, but in that world order which number and geometry represent. ‘hese figures illustrate anew the idea, old as philosophy itself, that all forms are projections on the lighted screen of a material universe of archetypal ideas: that all of animate creation is one vast moving picture of the play of the Cosmic Mind. With the falling away of all our sophistries, this great truth will again startle and console man- kind—that creation is beautiful and that it is necessitous, that the secret of beauty is necessity. *‘Let us build altars to the Beautiful Necessity.”’ CONCLUSION Emerson says, ‘‘ Perception makes. Perception has a destiny.” How can new beauty be born into the world except by the awakening of new percep- tion? Evolution is the master-key of modern science, but that very science ignores the evolution of consciousness—of perception. ‘This it treats as 77 PROJECTIVE ORN AWE f = é rr © Nb ae ii 4 =. gyntqe | - Hl i “ly f < oo — —_/:> ve 198 Fame ii 8H waar ii fixed, static. On the contrary, it is fluent, dynamic. Were it not so, there would be little hope of a new art. The modern mind has adventured far and fear- lessly in the new realms of thought opened up by research and discovery, but it has left no trail of beauty. ‘That it has not done so is the fault of the 78 PeeeCtiV E ORNAMENT artist, who has failed to interpret and portray the movement of the modern mind. Enamored of an outworn beauty, he has looked back, and like Lot’s - wife, he has become a pillar of salt. The outworn beauty is the beauty of mere appearances. The new beauty, which corresponds to the new knowledge, is the beauty of principles: not the world aspect, but the world order. The world order is most perfectly embodied in mathematics. This fact is recognized in a practical way by the scientist, who increasingly invokes the aid of mathematics. 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