BEYOND THE HORIZON:COHERENT GEOMETRIC SYSTEMS IN RODCHENKO'S PAINTING

PATRICIA RAILING and CAROLINE WALLISPATRICIA RAILING is an art historian who has publishedwidely on early tv /̂entieth-century art and on the Russianavant-garde in particular. She is Director of ArtistS'Bookworksin Sussex, England, remarkable for its facsimile reprints ofartists' books and v /̂ritings,

CAROLINE WALLIS is a biologist and cosmologistspecializing in bio-meteorology. She has developed newtechniques for long-range weather forecasting and is aconsultant to a number of international businesses.

An exhibition "Aleksandr Rodchenko" was held atthe Museum of Modern Art in New York in 1998 andin the catalogue the artist's grandson, AleksandrLavrentiev, describes his grandfather's interest inscience. In his article "On Priorities and Patents,"Lavrentiev writes that

Rodchenko. in his most intensive period ofexperiment with abstract geometric forms, wastrying to determine the iaws of constructiongoverning the physicai worid. The categoriesof space and time interested him iess asphiiosophicai concepts than as attributes of variousastronomic, geometric, and psychoiogicai modeisof the wor!d. !t is no coincidence that his iibrary of1917-20 included such works as C. H. Hinton'sThe Fourth Dimension and A New Era of Thought,Moritz Wiikomm's Die Wunder des Mikroskops(The Miracle of the Microscope), CamiiieFiammarion's L'Atmosphère (The Atmosphere),C. A. Young's The Sun, and the speeches andpapers of the German physioiogy professor MaxVerworn. Compieteiy in the spirit of Rodchenko'sconcept of anaiytic research was the titie ofa book by the German physicist Arthur Zart:Bausteine des Weltalls (The Building Blocksof World Creation). Aii of these books (whichremain in Rodchenko's iibrary today) containinformation on the physica!, bioiogicai, andconceptuai building biocks of the worid, theprima materia of its construction.'

The most fundamental components of this primamateria of the construction of the world, the buildingblocks, are the atom, a physical body, and geometry,that which organizes this physical body anddetermines how forces shape its creation andgovern its motion in space. What Rodchenkolearned in these scientific studies about the cell(where the microscope had been used), the sun and

the stars (where the telescope had been used), andthe atmosphere (where both instruments had beenused as well as the photographic camera) was thatthere are certain constants in geometry accordingto which all physical phenomena are organized. Itwas these constants that seemed to fascinateRodchenko and this is what he was exploring in avariety of ways in his paintings between 1917 and1920, "the most intensive period of experiment withabstract geometric forms," as Lavrentiev says.

A GEOMETRICAL CONSTANT—THE HEXAGON

One of the most fundamental geometrical constantsfound in Nature is the hexagon. It is what organizesthe honeycomb and the snowflake, among a myriadof other phenomena. The hexagon is perfectlydemonstrated in Eig. 1-Section of a hailstone,reproduced in the 1874 The Atmosphere by CamilleFlammarion, a French meteorologist working at theParis Observatory in the mid-nineteenth century.

Flammarion also reproduced a plate showing ninety-six snowflakes, a smail number compared to thoserecorded by several nineteenth-century scientists(including the American farmer Wilson Bently, who in1885 was the first to photograph a snow crystal andrecorded over 5,000 before his death in 1931, all ofwhich are unique).

As Flammarion writes:

The construction of snow-fiakes has iong aftractedthe attention of observers ... but it is oniy sincethe iaws of crystaiiization in gênerai have beenascertained that it has been possibie to throwany iight upon this subject.

in a cirde. of aii the poiygons which can beinscribed, there is but one whose sides areequai to its radius; that is, the reguiar hexagon,or figure with six sides,,..

The examination of the figures of snow leads toimpressions not iess marked as to the existenceof geometry. Number and Beauty, in the works ofnature, it is not mereiy a few ice-fiowers, such asthe above, which have been remarked anddesigned in the siender snow-fiakes, but thereare many hundred different kinds, aii constructedupon the same fundamentai angie of 60°. ̂

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Fig. 1. C. FLAMMARION, THE ATMOSPHERE.

SECTION OF A HAILSTONE. ENLARGED.

The same is true of the hexagon-order of thehailstone, i l lustrated above, each hexagoninscribed in the matrix being made up of sixequilateral triangles "constructed on the samefundamental angle of 60"."

TWO PAINTINGS BY RODCHENKO

Several paintings by Rodchenko have beenanalyzed in order to understand the geometries hewas using and how he was using them to organize apictorial construction. What has oome to light is thatthe hexagon has been found in a number of workswhere it is the basic structural element and maydetermine relationships among the parts. Inscribedinto a oircle, the circle is then used to create a matrixwhich extends throughout the surfaoe of theoomposition, its purpose being to unify pictorialspace and create points of structural reference.

Both modular hexagon and matrix of circles are thebasic constructive devices found in Rodchenko's1918 Composition No. 68 (Still Life) (Regional ArtMuseum in Perm, Russia) (Fig. 3).

Although a hexagon can be inscribed in any circle,that it is an authentic module in this painting canbe seen in the diagram (Fig. 4).

The yellow circle contains an equilateral triangleof 60°, sharing one of its sides with one of theequilateral triangles of the hexagon inscribed in thelarge black circle.

(This diagram and the subsequent geometricaldiagrams of Rodchenko's paintings [Figs. 5, 6, 9, 14,and 15] are by Caroline Wallis. It was she whoanalyzed Rodchenko's paintings with the aim ofdiscovering the internal geometrical orders that theartist was using to organize his paintings. Relying onclues that the artist himself gives in the paintings,some of which are discussed below, and knowing thatRodchenko's aims were "to determine the laws ofconstruction governing the physical world," as

Fig. 2, C, FLAMMARION, THE ATMOSPHERE. SNOW CRYSTALS.

Lavrentiev writes, Caroline Wallis found that themodule, the matrix, and geometries of progressionare what determine the regular structural orders inthe paintings. She then verified these orders bycomparison to Rodchenko's drawings and partioularlythe series of linocuts of 1921. Here she foundconfirmation that the artist's own structural systemsdid indeed turn on the module, the matrix, and certaingeometries of progression. The purpose of this articleis to reveal Rodchenko's systems and the ideas withinthem as they are found in the paintings discussed.)

Now, when two lines are drawn (Fig. 5) from thecenter of the large black circle extending on eitherside of the yellow circle, an equilateral trianglewould be formed if two congruent circles wereadded in. The equilateral triangle of 60°, one-sixthof a hexagon, is thus implied and is a fundamentalstructural principle in this painting.

Again taking the clue from the yellow circle,Rodchenko has constructed a matrix from it. Itoriginates from the center of the large black circle(Fig. 6).

When, from the yellow circle, a circle of the samesize is drawn whose circumference passes throughits center, it also passes through the center of theblack circle. Subsequent circles can be drawn, thecircumference of eaoh passing through the oenterof the previous circle, creating a succession of

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Fig. 3. A. RODCHENKO. COMPOSITION NO. 68 Fig. 4. DIAGRAM OF COMPOSITION NO.68 (STILL LIFE). 1918. Fig. 5. DIAGRAM OF COMPOSITION NO. 68

(STILL LIFE), 1918, OIL ON CANVAS. SHOWING HEXAGON. (STILL LIFE). 1918, SHOWING BEGINNING

REGIONAL ART MUSEUM, PERM, RUSSIA. OF ANOTHER HEXAGON AND ITS MATRIX.

interlooking circles. Another row of interlockingcircles can likewise be drawn in, the circumferencespassing through the centers of the previous circles.Rodohenko has constructed a matrix, one that canbe extended canvas-wide.

Now the red circle at the bottom of the composition,which is congruent to the yellow circle, is found tobe displaoed in this matrix. Suoh a deviation fromthe regular matrix is a common device for setting upa sensation of "differenoe," henoe of dynamism. Bymaking them the same size, Rodohenko has shownthat they are related. By giving them different colorsand by shifting the position of one of them within thematrix, Rodchenko has set up contrast.

Such matrices have been found to be typioal ofRodchenko's paintings of 1918. Indeed, he did alinocut in 1921 (Fig. 7) (when he was teaching atthe Moscow Vkhutemas, the Higher State Artisticand Technical Workshops, so it was perhaps usedas a teaching tool) which demonstrates thiselementary structural principle. From it, an infinitevariety of unique constructions can emerge.

What is important about the matrix is that it is first ofall a stable structure on which to "hang" thearrangement of geometrical elements, as it were. Thematrix provides a unified and regular field from whicha coherence in the relationships of the parts to each

other and to the whole can be established. Thestructure of the hailstone (Fig. 1) is characterized bya matrix determined by the hexagon, and theillustration shows that the prinoiple of the matrixis both structural, holding a body together, anddynamic, due to its potential for infinite and variableextension in space. These two ideas are inherent toRodohenko's paintings.

A matrix like that of Composition No. 68 (Still Life)is also the basic structural principle used in a verybeautiful painting of c. 1920, Untitled {F\Q, 8),

The key to discovering this matrix is found in thepart-circle on the far right that is bisected by a redbar. These two elements (part-circle and red bar)are placed just outside of the main composition—the diagonally oriented arrangement of circles andrectangles tipped to the left—and they set up acounterpoint dynamic.

In doing geometrical analyses of some of Rodchenko'spaintings, a oonsistent feature has been discovered:whenever there is an element or a group of elementssituated outside of, or in oontrast or counterpoint to, themain composition, Rodohenko has introduoed a newphenomenon, a dynamic, or an idea. It may also bethe key to unraveling the mystery of the entire painting.

In the case of Untitled, Rodchenko made the blackand white part-circle different from the blue, white.

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Fig 6 COMPOSITION NO 68 (STILL UFE).

1918, SHOWING MATRIX OF CIRCLES.

Fig. 7 A, RODCHENKO, CONSTRUCTION, 1921. LINOCUT.

PRIVATE COLLECTION.Fig. 8. A. RODCHENKO, UNTITLED. c. 1920,

OIL ON CANVAS. PRIVATE COLLECTION.

and black circle-spheres by modeling it in twocolors, and by giving it a sectional aspect and anopen structure. This part-circle provides a startingpoint for a geometrical analysis (Fig. 9).

When the part-circle is inscribed in a fully drawncircle and a second circle is drawn whose positionis determined by the inner section of the part-circle,a series of interlocking circles, the circumferencespassing through the centers of adjacent circles,appears along a tipped horizontal. This constructionof circles is extended upward to the top of thecomposition and downward to the bottom of thecomposition, creating a matrix. Within this matrixRodchenko has determined some of the extents ofcircles and bars. This is why the composition appearsstable, even though the overall orientation is on atipped vertical axis with all the elements depicted asif floating in space above the large black circle atthe bottom of the canvas.

Now it can be seen that the part-circle is displaced inrelation to the regular matrix, and so has the samefunction as the displaced circle in Composition No. 68(Still Life): to introduce a shift which suggests changeof direction or kind of movement. Indeed, the red barbisecting the part-circle is set on a tipped vertical axisin another direction to the main body of thecomposition. This produces a contrasting movement,hence a dynamic sensation in the painting.

Again the hexagon is the basic element inscribedwithin this part-circle, and is what determines theposition of several elements. This is revealed whena red and a blue line are drawn in. The red line isestablished by the vertically tipped red bar. Theblue line is established by connecting the center ofthe part-circle with the centers of the matrix ofcircles tipped along the horizontal. The red andblue lines cross at the center of the part-circle andform a 60° segment of an equilateral triangle, one-sixth of the hexagon which is found within.

Reading this painting with the awareness of its regularmatrix, animated by a displaced element, makes iteasier to understand why this—and all of Rodchenko'spaintings in this group—produces such an experienceof expansion into pictorial space accompanied by asense of dynamism. The nature of the matrix is infiniteextension in the four planar directions on the canvas^above and below, to the right and to the left. To thiscan be added extension forward into the spectator'sspace and backward into the depth of the pictorialplane. Now we have six directions in space, this sixfoldspatial order coincident with the sixfold hexagonalmodular arrangement of the parts. To extension in theplane is integrated extension in the perpendiculardirections of space.

Such spatial structure is a coherent order. It is alsointegral to the idea of Rodchenko's paintings where

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Fig. 9. DIAGRAM UNTITLED c. 1920

SHOWING MATRIX OF CIRCLES.

Fig. 10. A. RODCHENKO, COMPOSITION NO. 60

(CONCENTRATION OF COLOURS). 1918. OIL ON CANVAS.

RODCHENKO FAMILY ARCHIVE. MOSOOW.

Fig. 13. A. RODCHENKO, UNTITLED

(COMPOSITION). 1918. OIL ON PLYWOOD.

PRIVATE COLLECTION.

Fig. 11. C. YOUNG. THE SUN. SPECTRO-

HELIOGRAPH PHOTOGRAPH OF THE

ENTIRE CHROMOSPHERE.

Fig. 12. C. YOUNG, THE SUN. VENUS TOUCHING THE LIMB

OF THE SUN IN THE 1874 TRANSIT OF VENUS.

perspective along Farth's surface from the eye of anobserver has been abandoned in favor of extensionin the six directions of spaoe simultaneously. Thesubjeot matter of this Unfitted painting reveals whythis spatial organization is necessary. For Rodchenkohas constructed a composition in outer space wherethere are no fixed direotions until elements areintroduced to suggest them. The elements in thepainting—white sphere, blue sphere, blue conicalsection, green-black rectangle—appear to be risingup over the large black circle-sphere at the bottom ofthe oanvas, weightless in a spaoe without gravity.

There is also movement, a kind of counterpoint rhythmin the contrasts of direction in the way the red bardirects the eye toward the right, to be pulled baok tothe left due to the orientation of the arrangement of theprominent blue and green elements over a redluminious flare.

Being able to decipher the geometrical organizationmakes it possible to understand and so explainwhat is seen, and this adds to the way the paintingis experienoed by the observer. Without knowledgeof the geometry, the reasons for the experienoe ofdifferent sensations would remain a mystery.

COSMIC LIGHT PHENOMENA

This Untitled painting transmits a magnificentdisplay of light. The background glows with reds

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Fig, 14, GEOMETRICAL DIAGRAM

ON A, RODCHENKO. UNTITLED

(COMPOSITION). 1918,

Fig, 15, DIAGRAM WiTHIN A, RODCHENKO,

UNTITLED (COMPOSITION). 1918,Fig, 16, A, RODCHENKO, CONSTRUCTION

NO, 60. 1921. LINOCUT PRIVATE COLLECTION,

and yellow-oranges, and the ring of luminous redaround the large black circle at the bottom of thecanvas, of which only a part can be seen, indicatesthat this painting is inspired by solar phenomena ofeclipses, Rodchenko did a number of paintingsbetween 1918 and 1920 which are inspired bysuch phenomena, as well as by the flight of cometsthrough space. Here, the Untitied painting revealsthe fantastic luminosity of the sun, our viewobscured by the moon eclipsing it. Although in asolar eclipse the sky itself becomes dark whenseen from Earth, that Rodchenko should depict aluminous sky and a darkened sun now places us inouter space from where we observe magnificentsolar lights.

Rodchenko's knowledge of the sun, its corona, andsolar and lunar eclipses would have come, at leastin part, from Charles Augustus Young's The Sun.First published in 1881 by the Princeton UniversityProfessor of Astronomy, The Sun saw a sixth andcompletely revised American edition in 1895, whileseveral editions had also appeared, one after theother, in Russian translation, also from the 1890s. Atthe time it was the most thorough and authoritativetreatise on the sun worldwide.

Several paintings of 1918, 1919, and 1920 depict thevisual effects of solar lights and of eclipses. Thesubject, even, was topical since a total solar eclipse

had occurred in 1918 and received great coverage inthe press. The next year an equally famous eclipsewas much talked about. It took place on 29 May 1919and was viewed by teams of scientists around theworld, including in Russia, for the aim was to verifyAlbert Einstein's theory of relativity. The eclipse wasvisible and Einstein's theory was vindicated.

Rodchenko even makes direct reference to hisinterest in light in his entries to the 1919 catalogue ofthe "Xth State Exhibition of IZO Narkompros, Non-Objective Creation and Suprematism," held inMoscow. There he describes several of his paintingsof the "second half of 1918" as "Luminescence ofcolour" and "Light-colours [Svet-tsveta]." He laterwrote about these works, saying that in them he had"achieved extreme innovations (the light of colour,the tone of light) in painting." So they are all aboutlight and, indeed, about cosmic light.-'

Composition No, 60 (Concentration of Colours) of1918 (Fig. 10) is a fine example of pure light radiancewhich Rodchenko captures in the perfect relationshipof two luminous bodies (here in an eightfoldarrangement). The corona of the eclipsed sun (thesmall circle) is similar to photographs taken over theUnited States of the total eclipse of the sun of 8 June1918, which would have been published innewspapers around the world.

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Very similar to these photographs is one reproducedin Charles Young's The Sun, showing a total solareclipse (Fig. 11).

In Composition No. 60, the aura of shimmeringyellow light emitted around the outside of thesmaller body, the sun blackened by the eclipsingmoon, is contrasted to a ring of radiant blue light.Rodchenko has used a pair of complementarycolors, orange-yellow and blue, turning colors intolight effeots in the eye of the observer.

The oomposition itself also may have been inspired bya particularly significant celestial event, the transit ofVenus over the sun, to which Young devotes an entirechapter in The Sun. These transits occur in pairsabout every 130 years and the then most recent oneshad been in 1874 and 1882, so the information theyhad provided was timely for Young's book. Openingthe chapter. Young explains their significance;

The probiem of finding the distance of thesun is one of the most important and difficuitpresented by astronomy. Its importance lies inthis, that this distance—fhe radius of the earth'sorbit—is the base-iine by means of which wemeasure every other ceiestiai distance, exceptingonly that of the moon: so that error in this basepropagates itself in ail directions through allspace, affecting with a corresponding proportionof faisehood every measured line—the distanceof every star, the radius of every orbit, fhediameter of every pianet '

And that was only the beginning of the story forthe astronomer.

There is a black and white illustration (Fig. 12) in TheSun ("Venus touohing the limb of the sun in the 1874transit of Venus") whioh depicts the moment whenthe edge of the planet Venus meets the edge (these"edges" oalled the "limb") of the sun during the 1874transit of Venus. At this moment is observed whatlooks like a feathery aotivity which is due

partiy upon fhe essentiai nature of light, leadingto what is known as diffraction, and partiy uponthe action of the pianet's atmosphere. The twofirst-named causes [opticai instruments andthe human eye, and diffraction] produce whatis caiied irradiation, and operate to make theapparent diameter of the pianet, as seen on thesolar disk, smaller than it really is—smaller, too,by an amount which varies with the size of thetelescope, the perfection of its ienses, and thetint and brightness of the sun's image. The edgeof the pianet's image is also rendered slightlyhazy and indisfincf.

The pianet's atmosphere also causes its diskto be surrounded by a narrow ring of light,which becomes visibie long before the planettouches the sun, and at the moment of internalcontact produces an appearance of which theaccompanying figure is intended to give anidea, though on an exaggerated scale.''

Young's illustration could be related not only toComposition No. 60, suggesting that the painting mayhave been inspired by such a drawing and such anevent, but also to another Composition by Rodchenko,whose number in his house list is not known (Fig. 13).

This painting is clearly inspired by a planetary transit,Rodohenko depicting the transit itself to which he hasadded further stages of planetary movement in theupper circles on a curved trajectory. Significantly, theartist gave order to the transiting planet (the ochrecircle) over the sun (the large white sphere) by usingthe fundamental hexagon (Fig. 14).

Rodchenko established the hexagon at the pointswhere the trajeotory of the ochre circle touches thecircumference of the sun at the bottom (in thediagram) and at the uppermost point where itmeets the circumference of the red torus.

Now the white circle in the center of the torus iscongruent with the ochre oircle, indicating thepassage of the transiting planet to a subsequentstage of progression.

The geometry of this progression is demonstratedin Fig. 15, and it is the same geometry ofprogression that Rodchenko described in anotherlinocut of 1921, Construction No. 60 (Fig. 16).Although this geometry is not visible in thefinished painting, we sense an internal order andcoherent relationship among the parts, as well ashave a sensation of movement.

All of Rodchenko's geometrioally struoturedpaintings are complex and only some of thearrangements are discussed here. However muchhe may have introduced shifts into the relationships,the basic structural principles do not change. Theyremain the module and the matrix.

MODULE AND MATRIX

Rodchenko had been noticed by Vladimir Tatlin forhis very sophisticated geometrical line drawings,Tatlin then inviting him to exhibit for the first time asa member of the Russian Avant-garde in the 1916Moscow show "The Store."

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Geometrioian by nature, Rodchenko movedmethodically through the exploration of theserational systems in order to discover and thendescribe how forms exist in space, forms that arecreated by the laws of force, of dynamism. Thiswas necessarily a rational, coherent space, onemade visible by rational, coherent geometries.

To reveal space and the laws of form creation,Rodohenko aligned his method with the laws of naturalcreation which he discovered in his scientific reading.There he found "the laws of construction governing thephysical world," as Lavrentiev writes in his Museum ofModern Art catalogue article. Rodchenko selectedNature's elementary forms, the modules or buildingblooks, and he discovered the laws of generation inspace, the laws of orderly expansion in spaceaccording to the module of the matrix. In this way,Rodchenko also created a unified spatial field.

Knowledge of Rodchenko's methods not onlycontributes to ease of entering into his paintings,but also makes one realize that this "ease" isdependent upon being able to open up, open out,the mind. We have to move around in the spaceshe creates, and to do this we have to forget the oldhabit of one-point perspective.

Now we see many perspectives, different points ofview, numerous positions in space, a space that iswithout dimension and oertainly not bound to Earth.We are in a multidimensional space, an all-overspace, even a "pure" space. Rodchenko's paintingsteaoh us to expand our experience of space.

They also reveal order to our mind's eye.

And they inspire awe and wonder. That, we believe,was the artist's true intention in all his paintings,those discussed in this essay being but the gatewayto Rodohenko's realm beyond the horizon. •

NOTES

1. Aleksandr Lavrentiev, "On Priorities and Patents," in exh.cat. for "Aleksandr Rodchenko," Museum of Modern Art,New York, 1998,55.

2. Camille Flammarion, The Atmosphere (New York: Harperand Brothers, 1874), 230.

3. Aleksandr Rodchenko, diary entry of 2 January 1919 inExperiments for the Future—Diaries, Essays. Letters, andOther Writings, ed. Alexander N. Lavrentiev, trans. JameyGambrell, introduction by John E. Bowit (New York: TheMuseum of Modern Art, 2005), 89.

4. Charles Augustus Young, The Sun (Akron, OH: The WernerCompany, 1895), 10.

5. Ibid., 23-4.

DDD

A TRIBUTE

For fifty years. THE STRUCTURIST has been committedto publishing on art trends of the twentieth centurywhose inspirations arise out of the recognition of,and deepest respect for, the ways in whioh thehuman being. Nature, and the cosmos make up anintegrated dynamio whole.

To be aware of the world and of the life it generatesas an integrated whole demands a consciousnessthat is itself holistio.

Artists of the Russian Avant-garde were amongthose at the origins of this worldview, together withearly twentieth-century scientists epitomized in thepersonality and beliefs of the genius of AlbertEinstein. Indeed, Russian Avant-garde art wasimagined with an awareness of the holistio as itwas being revealed by this new science, and itwas creating with its laws.

In this context, the Russian artists were able tomake claims for a new oonsoiousness, for aholistic consciousness, and their art revealed theirbeliefs. That is why they oould assert that art canchange the world, make it a better place by raisingthe consciousness of all those who take the time tooontemplate it.

When we live in the awareness that "all is one," amotto that reverberates in every value and attitude,that appeals to respeot for all that lives and where itlives—on the Earth that is revolving in our cosmicuniverse—we are living in harmony with the laws thatgovern these great systems. We are the microcosmsof the macrocosms, we are part of these systems,and that is what Russian Avant-garde artistsreoognized. Painting, architecture, the graphic arts,the theater, are the means of revealing these lawsand reminding us of them.

May we just be able to remember this andaot aocordingly.

Finally, it is with thanks to THE STRUCTURIST and toits editor, Eli Bornstein, that this worldview hasbeen perpetuated among so many over theseyears. Let it reverberate in ever greater pulsationsthrough all of us, and throughout society.

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