The Circle by Bruno Munari, 1964-2012

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~I THE CIRCLE

... the circle is related to the divine: a simple circle has since ancient times represented eternity, since it has no beginning and no end. BRUNO MUNARI

ISBN 978-88-7570-040-9

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BRUNO M UNA RI

THE CIRCLE

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TT-IE CJH CLE

Whil e th e squ a re is closely linked to ma n a nd hi s constru ctio ns, to architec ture , ha rmo nio us stru ctures, writing, and so o n , th e circle is related to the di ,·ine: a simple circle ha s sin ce an cie nt times re prese nted ete rnity, since it has no beginning and no e nd. An ancie nt text says th at God is a circle w hose centre is everywhere but wh ose circumfe re nce is nowhere . T he circle is esse nti ally unstable a nd d ynamic: a ll rotary moveme nts and impossible sea rches fo r pe rpetu al mo tio n de rive fro m the circle. Despite be ing the simplest of the curves, it is co nside re d by math e m aticia ns as a p o lygon w ith an infinite number o f sides. If yo u re move a n in visible po int fro m the circumfe rence of a circle then it is no lo nger a circle but a pathocircle, w hich presents compli cated pro blems. A poin t ma rked o n its circumfe re nce eliminates the idea o f ete rnity, indica ting a beginning a nd th e refore an end to the circumfe rence itself. If thi s ci rcle rotates o n the fl at, the po int m arked

o n its circumfe re nce describes a cyclo id. The circle is easy to find in nature, a ll you have to do is th row a stone into still wate r. Instead , the sphe re appea rs spo nta neously in soa p bubbles. Trees g row fo llow ing a concentric circul ar patte rn: a sectio n shows its rings. A circle drawn by hand showed th e s ki ll o f Giotto. The first thing a child draws looks li ke a circle. People spo ntaneously a rrange th e mselves in a circle w he n they need to obse rve something close up, and this led to the orig in o f the a re na, the circus and the stock excha nge trading posts. One o f the o ldest sy mbo ls is a d isk made up o f two dy na mic equa l a nd o pposing pa rts: Ya ngYin , which re prese nt the balance o f o pposing forces in all li ving things. Famo us pa inte rs have pa inte d o n a c irc ul a r surface, eac h o f the m findin g co mpositio na l solutio ns closely ti ed to the circul a r shape. In ce rta in cases, such as in Botticelli 's Virg in w ith Child , the fin al effect of the work appea rs spherica l to th e eye. A di sk lying o n a flat surface ca nn ot be placed wrongly, w hich is why plates are a lmost a lways round ; and it is easier to arrange the m o n the tabl e. If they we re hexagona l o r squ are o r ova l it wo uld require g rea te r ca re to lay th e m o ut w itho ut crea ting a se nse of di so rde r. A circle instead is a lways tidy. This is eve n true r o f the sphe re, w hi ch ca nnot be ove rturn e d in a ny way. A sphe re is a lways the rig ht way up, so to spea k, in any position.

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AGRIPPA The magic circle of Agrippa.

AMATERAS Popular Japa nese divinity dressed in reel and standing o n a rock w ith th e solar disk o f the sun in her right hand . According to legend Amateras was born fro m the left eye o f the god I za nagui; from th e mo m ent she was born her resplendent beauty lit up the w ho le wo rld and l za nagui gave her the empire of th e sun .

ACONA' B!CONB I'

Three d imensiona l constru cti o n obta ined by repeating and jo ining equal elements in the shape of a ci rcular crow n . The overa ll shape changes depending o n the number of elements used. 7

TI-IE ARCHANGEL M l CI-IAEL

The magic c ircle o f the Archange l Michael.

TI-IE RING

The ring is sa id to o rig inate fro m Asia. Bo th H ebrews and Egyptians wore rings. Initial ly th e Ro mans o nl y wo re iro n rings w ith a sea l. Gold rings were the mark o f people of high birth. Each yea r, during th e reign of Pope Alexander Ill , the Venetian Doges would throw a ring into the sea on Ascensio n Day to symbo li se a marri age w ith the sea.

GROWTH RINGS

A cross-secti o n of a tree trunk. 8

HALO

MUSLIM AHCH

Structu ra l outline of an Arab ian-Moorish arch.

Portrait of St Francis by Simo ne Martini, Assisi. 10

NEWTON 'S RI NGS

TO HAVE FIN ISHED

If you put a slig htl y co nvex len s lit by a w hite

After a sacrifi ce the ancients would make a circle in the altar usi ng the b lood o f the v ictim s collected in a jar and th en th ey wou ld prono unce a ho ly Greek wo rd mea ning to have

light o n a fl at piece o f g lass a se ries o f conce ntric iridescen t rings app ea r at the spot where the two pi eces of g lass meet. If you use a red light in stea d o f a w hite o n e a large num ber o f conce ntrated, reg ular rings, altern ately red and dark , form aro und the point of contact; as you gradu all y m ove away from the black spot in th e ce ntre , th e distance b etween th e rings diminishes. New to n di scovered that the radii o f the dark rings are the sa me as the sq uare roots of consecuti ve even numbers.

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EQUAL AREA S

AN NUAL - BI ENN IAL

0e Bo tanica l sig n s for annual o r biennial plants.

The surface outside the b lack ring , conta ined within a wider circumference, is equa l to the surface inside the bl ack ring. The radius is div ided into fi ve equal parts. II

JAPAN ESE FLAG

O LYMPI C FLAG

DAVIDE BO RJAN J M agneti c surface. Kin eti c o bject, shown at th e Exhibition o f Prog rammed Art, O li vett i , May 1962, Milan. Th e object m easures 80 cm in diameter and co ntain s iro n powder kept in co nsta nt move ment by a number o f magnets th at move in different ways undern eath th e surfa ce, making an infinite number o f p atterns. 12

BAPTI STERY

SOAP BUBB LES

A natural sphe re.

MAX BILL

The Baptistery at Pisa, one of the most bea uti ful o ld bu ild ings o n a ci rcular base.

Design made fro m a se ri es o f circ les. 1942 . 14

THE STOCK EXCHANG E

BOWLS

A ga me o f b owls al Mo nte O limpino .

A circular trading p ost at th e Stock Exchange.

GOO D SPIR IT S

BOTTICE LLI

Th e Virg in and Child . Uffi zi Ga llery, Florence. Th e p arti cul ar co mpos iti o n and p ainting techni que g ive the round surface o f th e painting th e impressio n o f being a sphere.

A m agic circle to attract Good Spirits. 16

LANFRANCO BOMBELLI

Drawing, 1947. 18

A MATAKAM HOUSE At Mo ko lo in th e Cameroon are the houses of the Matakam . Each room is cylindrica l and made o f beaten ea rth crow ned by a conical thatched roof. The rooms form a large enclosure. Each room has a specific function; the number of rooms is de te rmin ed by the number of fami ly me mbe rs. There are n o openings for the light to e nte r the rooms and o ne circul ates as if in a dark circular maze .

ROU ND HUT The two o ldest types of dwe lling have e ithe r a sq uare or a round ground plan. The domed hut is fo und in Australia a nd amo ng many Africa n and Am erica n peoples.

Enclosure for a fam ily o f nineteen membe rs with the room (or ho use) of the head of th e family; ho use for the bull , ho use for the main wife, ho uses for the o ther wives and children , house for th e o ldest married son, house for an adult son, ho use for th e water tank, the kitchen, houses fo r the goa ts , larde rs, th e tank for the ashes with w hich sa lt is made, the outer wall. The Matakam keep the bu ll wa lled up in its ho use and it can o nl y communicate w ith the o utside through a small , very low o pening thro ugh which it ca nnot pass. There is anothe r opening for scraping o ut the manure. The bull is ke pt like this for three yea rs, during which time it is feel and looked afte r. It is let o ut on the feast of the ancestors and killed in a sole mn ce re mony performed unde r the direction of the Bu ll Master.

BALL BEARING 19

CARDIOID

CYCLOID

The cycl o id is th e path traced b y a fixed p o int on th e circumference of a circle that rolls along a given straight line. An interesting property of the cycloid wa s discovered by Galil eo: w ith th e help of th e cyclo id we ca n constru ct an area that is exactly the sa me as that of the given circl e. First o f all th e length of th e cyc lo id fro m cusp to cusp is equal to four tim es th e leng th o f the diameter o f th e generating circl e. On th e ba sis of thi s it ca n be demonstrated that the area delimited by th e p o rtion o f th e cyclo id b etween th e two cu sps and th e straight line that unites them is eq ual to three times th e area o f the circl e. Th erefore th e sp ace delimited b y each p art o f the circl e is exactly th e sa me as the area o f th e circle itse lf.

A curve described by a point situated on a ci rcle which rolls, without slipping, around th e circumference of another circl e.

Astrologica l circles to ca lcul ate co nfi gurations.

COMPASSES 20

CU RTATE CYCLOID

A point traced out on th e in side of a circle ro lling along a straight line generates a curtate cycloid.

PRO LATE CYCLOID CLEOPATRA

A point on the o utside o f a circle ro lling alo ng a straight line describes a prol ate cyc lo id.

Cleop atra's magic circl e.

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CYCLE

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A co nce pt introd uced by Laguerre: the cycle is a circle with an arrow m arked o n its circu mference. An eq ual circle w ith the arrow facing in the opp osite directio n is ano th er, different cycle .

CLUST ERS OF SPJ-IERES

The thickest cluster o f spheres is obtained when th e ce ntres of th e spheres form a rh ombohedric network. 22

TH E POLYGONAL ClRCLE

©000 0000 Circles w ith inscribed po lygons. The sa me is true for circumscribed polygons. The me th od of increasing or decreas ing polygons was kn ow n to Archimedes w ho, using 96-sided polygons, de monstrated that rr is less than 3 1/7 and mo re than 3 10/7 1. The area of the circle is to be fo und between th ese two fi gures.

TH E MAG IC CJRCLE OF TH E COVENANT

O PPOSITION

Two circles th at to uch , like two w heels th at move in th e o pposite directio n whe n they make contact, symbolise o ppositio n.

CONE SPHE RE

Mode l of ex perim ental geometry made by the School o f Ulm. 24

HORSE POWER

CIRC LE

A whee l in which a horse produces powe r by wa lking a long its inte rnal circumfe re nce. This was used in the past to move the paddl es o n the rive r boats. In China clogs were used to move th e w hee ls o f small mills and priso ne rs were used to bring water up to irrigate th e fi e lds.

The circle is o ne of the o ldest fi g ures in mathematics. The straig ht line is the simplest of lines but the circle is the simplest curve.

CU RVES INS IDE AND OUT

Draw a circle with any radius a nd choose six equidistant po ints o n the circumfe re nce. Ta ke three alternate arcs and turn the m inwa rd s. The pe rimete r re mains the same. The n trisect each inte rn a l o r ex ternal arc and inve rt th e ce ntral sectio n. By continuing this operation we obta in a fin a l curve whose perimete r is equal to the o riginal circle and an area equal to the inscri bed hexagon. 26

INSCRIBED CIRCLES

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Insc ribe a circle in a mi xtilinea r isosceles triangle.

Inscribe a circle in a curvilinear equilateral triangle w ith concave sides.

Inscribe a circle in a curvilinea r eq uilate ral tria ngle of w hi ch three sides are convex and o ne is concave.

Inscribe a circle in a curvilinear eq uilate ral triangle w ith convex sides.

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DIFFRACTI ON

Insc ribe a circle in a curvil inea r trian gle having as sid es a semicircle and two arcs w hose radii are eq ual to th e diameter o f th e semi circle itself

DECO RATI O

Diffra ctio n o f electro ns thro ugh a ve ry thin laye r o f silver. Thi s is proves th e wave nature o f th e electro n, and th erefore o f matter.

DANCE

Dan cing in a circle, bea tin g rh ythmi ca ll y, no o ne is first, no o ne last, all are th e sa m e, all b ea t in the sam e w ay. Th e start is slow th en th e rh ythm takes ove r, a se nse o f infinity arises fro m this human ring that turns and bea ts rh ythmi call y. Ph o to Michel Huet. 28

VILLARD DE H ONNECOURT

THE SUN GOD

The reli gio n of ancient Egypt w as based on the ador ation of the sun. Th e fo rm of the Sun Goel Amon -Ra was a h awk o r a man w ith th e appearance of a hawk w ith a solar disk, travelling throu g h th e sky. A n ancient c hant of Th ebes says : Amon- Ra , divine hawk with shining p l umage , tra ces with th e spread of his wi ngs a circle on th e va ult of th e sk ies. Amenophi s JV, accord in g to hieroglyph ic inLerpretation , started a new cul t w ith the adoration of th e rea l sun in place of th e Goel Ammon-Rf1. Since th en th e su n god is simpl y represen ted by a radia nt d isk.

GOD

"Goel is a circle w hose ce ntre is everywhere but w hose circumference is nowhere". O ld sayin g. O ne of the first drawings o f a perpetual mo tio n m achine.

MAXWELL'S DISK CHROMATIC D ISK

A turquoise and red disk in different adju stable parts. By rota ting this disk you obta in a neutral grey colour. The neutral shade of grey depends o n th e two colo urs being exactly complementary. If the amount of reel is greater, you get a redd ish grey and if the turqu o ise is grea ter the resu lt wi ll be a bluish green.

Diagram of complementary colours on a chro m atic di sk . The num be rs marked by a sma ll square indica te the relative positions o f the colours o n the normal spectrum and the numbers marked by a cross indicate the wave lengths in ten -millio nth s of a millimetre. 30

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