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 UNARI THE CIRCLE

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TT-IE CJHCLE

While the squa re is closely linked to ma n a nd hi s constructio ns, to architecture , ha rmo nious structures, writing, and so o n, the circle is re-lated to the di ,ine: a simple circle has since ancie nt times re presented ete rnity, since it has no beginning and no e nd. An ancie nt text says that God is a circle whose centre is everywhere but whose circumfe re nce is nowhere . The circle is essentially unstable a nd d ynamic: a ll rotary moveme nts and impossible sea rches fo r pe rpetual motio n de rive fro m the circle. Despite be ing the simplest of the curves, it is conside red by mathe maticia ns as a po lygon w ith an infinite number o f sides. If you re move a n invisible po int from the circumfe rence of a circle then it is no lo nger a circle but a patho-circle, w hich presents complicated pro blems. A point ma rked o n its circumfe re nce eliminates the idea o f ete rnity, indicating a beginning a nd the refore an end to the circumfe rence itself. If this ci rcle rotates o n the flat, the po int marked

on its circumfe rence 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 sphere appea rs spontaneously in soa p bubbles. Trees g row fo llowing a concentric circular pat-te rn: a section shows its rings. A circle drawn by hand showed the ski ll o f Giotto. The first thing a child draws looks li ke a circle. People spontaneously a rrange them-selves in a circle when they need to observe something close up, and this led to the orig in of the a rena, the circus and the stock exchange trading posts. One of the o ldest symbols is a d isk made up of two dynamic equa l and opposing parts: Ya ng-Yin , which represent the balance of opposing forces in all living things. Famous painte rs have pa inted o n a c ircula r surface, each o f them finding compositio nal solutions closely tied to the circula r shape. In certa in cases, such as in Botticelli 's Virg in with Child , the fin al effect of the work appea rs sphe-rica l to the eye. A disk lying on a flat surface ca nnot be placed wrongly, which is why plates are a lmost a lways round ; and it is easier to arrange them on the table. If they were hexagonal o r squ are o r oval it would require g reate r care to lay them out without creating a sense of disorde r. A circle instead is a lways tidy. This is even true r of the sphe re, w hich ca nnot be ove rturned in a ny way. A sphere is a lways the right way up, so to spea k, in any position.

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ACONA' B!CONBI'

Three d imensiona l construction obta ined by re-peating and jo ining equal elements in the shape of a ci rcular crown. The overa ll shape changes depending on the number of elements used.

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AGRIPPA

The magic circle of Agrippa.

AMATERAS

Popular Japanese divinity dressed in reel and standing on a rock w ith the solar disk o f the sun in her right hand . According to legend Amateras was born from the left eye o f the god I zanagui; from the moment she was born her resplendent beauty lit up the w ho le world and l za nagui gave her the empire of the sun.

TI-IE RING

The ring is sa id to o rig inate from Asia. Both H ebrews and Egyptians wore rings. Ini tial ly the Ro mans only wore iron rings w ith a sea l. Gold rings were the mark o f people of high birth. Each yea r, during the reign of Pope Alexander Ill , the Venetian Doges would throw a ring into the sea on Ascension Day to symbo lise a marria-ge w ith the sea.

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TI-IE ARCHANGEL Ml CI-IAEL

The magic c ircle o f the Archangel Michael.

GROWTH RINGS

A cross-section of a tree trunk.

HALO

Portrait of St Francis by Simone Martini , Assisi.

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MUSLIM AHCH

Structu ra l outline of an Arabian-Moorish arch.

NEWTON 'S RI NGS

If you put a slightly convex lens lit by a w hite light on a fl at piece o f g lass a series o f concen-tric ir idescent rings appea r at the spot where the two pieces of g lass meet. If you use a red light instead o f a w hite one a large number o f concentrated, regular rings, alternately red and dark , form around the point of contact; as you gradually move away from the black spot in the ce ntre , the distance between the rings diminishes. Newton discovered that the radii o f the dark rings are the sa me as the square roots of consecutive even numbers.

AN NUAL - BIENNIAL

0e Botanica l signs for annual o r biennial plants.

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TO HAVE FIN ISHED

After a sacrifi ce the ancients would make a cir-cle in the altar usi ng the blood o f the v ictims collected in a jar and then they wou ld pro-nounce a ho ly Greek word meaning to have .finished.

EQUAL AREAS

The surface outside the black ring, contained within a wider circumference, is equa l to the surface inside the black ring. The radius is di-v ided into five equal parts.

O LYMPIC FLAG

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JAPAN ESE FLAG

DAVIDE BO RJAN J

Magnetic surface. Kinetic object, shown at the Exhibition o f Programmed Art, O li vett i , May 1962, Milan. The object measures 80 cm in diameter and contains iron powder kept in constant movement by a number o f magnets th at move in different ways underneath the surface, making an infinite number o f patterns.

BAPTI STERY

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

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SOAP BUBBLES

A natural sphere.

MAX BILL

Design made from a series of circ les. 1942 .

THE STOCK EXCHANGE

A circular trading post at the Stock Exchange.

GOOD SPIR ITS

A m agic circle to attract Good Spirits. 16

BOWLS

A ga me o f bowls al Monte O limpino .

BOTTICE LLI

The Virg in and Child . Uffi zi Ga llery, Florence. The parti cular compositi on and painting techni-que g ive the round surface o f the painting the impression o f being a sphere.

LANFRANCO BOMBELLI

Drawing, 1947. 18

ROU ND HUT

The two o ldest types of dwelling have e ithe r a sq uare or a round ground plan. The domed hut is found in Australia and among many Africa n and American peoples.

BALL BEARING 19

A MATAKAM HOUSE

At Mokolo in the Cameroon are the houses of the Matakam. Each room is cylindrica l and made of beaten ea rth crowned by a conical thatched roof. The rooms form a large enclosure. Each room has a specific function; the number of rooms is de te rmined by the number of fami ly members. There are no openings for the light to ente r the rooms and one circulates as if in a dark circular maze .

Enclosure for a fam ily o f nineteen members with the room (or house) of the head of the family; house for the bull , house for the main wife, hou-ses for the o ther wives and children , house for the o ldest married son, house for an adult son, house for the water tank, the kitchen, houses fo r the goats , larde rs, the 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 house and it can only communicate with the outside through a small , very low o pening through which it cannot pass. There is anothe r opening for scraping out the manure. The bull is kept like this for three yea rs, during which time it is feel and looked af-te r. It is let o ut on the feast of the ancestors and killed in a solemn ceremony performed unde r the direction of the Bu ll Master.

CARDIOID

A curve described by a point situated on a ci rcle which rolls, without slipping, around the circu-mference of another circl e.

Astrologica l circles to ca lculate confi gurations.

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CYCLOID

The cyclo id is the path traced by a fixed po int on the circumference of a circle that rolls along a given straight line. An interesting property of the cycloid was discovered by Galileo: w ith the help of the cyclo id we ca n construct an area that is exactly the sa me as that of the given circl e. First o f all the length of the cyc lo id from cusp to cusp is equal to four times th e length of the diameter o f the generating circle. On the basis of this it ca n be demonstrated that the area delimited by the portion o f the cyclo id between the two cu-sps and the straight line that unites them is equal to three times the area o f the circle. Therefore the space delimited by each part o f the circl e is exactly the sa me as the area o f the circle itse lf.

COMPASSES

CU RTATE CYCLOID

A point traced out on the inside of a circle ro lling along a straight line generates a curtate cycloid.

PRO LATE CYCLOID

A point on the outside of a circle ro lling along a straight line describes a prolate cyc lo id.

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CLEOPATRA

Cleopatra's magic circl e.

CYCLE

A concept introduced by Laguerre: the cycle is a circle with an arrow marked o n its circumferen-ce. An equal circle w ith the arrow facing in the opposite directio n is ano ther, different cycle .

CLUSTERS OF SPJ-IERES

The thickest cluster o f spheres is obtained when the centres of the spheres form a rhombohedric network.

THE POLYGONAL ClRCLE

000 0000 Circles with inscribed po lygons. The sa me is true for circumscribed polygons. The me thod of increasing or decreas ing polygons was known to Archimedes w ho, using 96-sided polygons, demonstrated that rr is less than 3 1/7 and more than 3 10/71. The area of the circle is to be found between these two figures.

O PPOSITION

Two circles that touch, like two w heels that move in the opposite direction when they make contact, symbolise opposition.

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TH E MAG IC CJRCLE OF TH E COVENANT

CONE SPHERE

Mode l of experimental geometry made by the School o f Ulm.

HORSE POWER

A wheel in which a horse produces power by wa lking a long its inte rnal circumference. This was used in the past to move the paddles on the rive r boats. In China clogs were used to move the whee ls o f small mills and prisone rs were used to bring water up to irrigate the fi e lds.

CURVES INS IDE AND OUT

Draw a circle with any radius and choose six equidistant points on the circumference. Take three alternate arcs and turn them inwards. The perimete r remains the same. Then trisect each inte rnal o r external arc and invert the central section. By continuing this operation we obta in a fina l curve whose perimete r is equal to the o riginal circle and an area equal to the inscri -bed hexagon.

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CIRC LE

The circle is one of the o ldest figures in mathe-matics. The straight line is the simplest of lines but the circle is the simplest curve.

INSCRIBED CIRCLES

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Inscribe a circle in a mixtilinea r isosceles trian-gle.

Inscribe a circle in a curvilinea r equilate ral tria n-gle of w hich three sides are convex and o ne is concave.

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Inscribe a circle in a curvilinear equilateral trian-gle w ith concave sides.

Inscribe a circle in a curvilinear equilate ral trian-gle w ith convex sides.

Inscribe a circle in a curvi l inea r triangle having as sides a semicircle and two arcs w hose radii are equal to the diameter o f the semicircle it-self

DECO RATI O

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DIFFRACTION

Diffractio n o f electro ns through a very thin laye r o f silver. This is proves the wave nature o f th e electron, and therefore o f matter.

DANCE

Dancing in a circle, bea ting rhythmica ll y, no one is first, no one last, all are th e sa me, all bea t in the same w ay. The start is slow then the rhythm takes over, a sense o f infinity arises from this human ring that turns and bea ts rhythmicall y. Photo Michel Huet.

VILLARD DE H ONNECOURT

O ne of the first drawings of a perpetual motion machine.

MAXWELL'S DISK

A turquoise and red disk in different adjustable parts. By rota ting this disk you obta in a neutral grey colour. The neutral shade of grey depends on the two colours being exactly complemen-tary. If the amount of reel is greater, you get a redd ish grey and if the turquo ise is greater the resu lt wi ll be a bluish green.

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THE SUN GOD

The religion of ancient Egypt w as based on the adorat ion of the sun. The fo rm of the Sun Goel Amon-Ra was a hawk o r a man w ith the ap-pearance of a hawk w ith a solar disk, travelling through the sky. An ancient chant of Thebes says : Amon-Ra , divine hawk with shining p lu-mage, traces with the spread of his wings a cir-cle on the vault of the sk ies. Amenophis JV, accord ing to hieroglyph ic inLer-pretation , started a new cul t w ith the adoration of the rea l sun in place of the Goel Ammon-Rf1. Since then the su n god is simply represented by a radiant d isk.

GOD

"Goel is a circle w hose centre is everywhere but w hose circumference is nowhere". O ld saying.

CHROMATIC D ISK

Diagram of complementary colours on a chro-matic disk . The numbers marked by a sma ll square indica te the relative positions of the co-lours o n the normal spectrum and the numbers marked by a cross indicate the wavelengths in ten-millionths of a millimetre.

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MUSLIM DECORATION

Muslim decoration at Constantinople. 32

MARCEL DUCHAMP

Disks rotating at a constant speed w ith special optica l effects, devised by Marcel Duchamp in 1936

NEWTON'S DISK

A w heel made up o f sectio ns of different sizes in all the colours o f the spectrum, repeated four ti -mes. By ro tating the disk at speed all the colou rs blend into a luminous w hite.

COMPOSITIONS

Some examples o f figures obtained with circles. disks and parts o f them.

ETERNITY

Eastern eternity symbol.

The snake biting its tail is a symbol o f eternity.

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BENHAM'S DISK

Take a white disk and cover ha lf o f it in deep b lack, then paint concentric bands o n the o ther half. When it ro tates at a certain speed you w ill notice how instead o f seeing concentric grey rin-gs, there are coloured bands, dark and of low sa turation. This leads us to the disconce rting conclusion made by S. I. Vavilov that we cannot base ourselves simply on visual perception to define ligh t.

ACUSTJC EXPERIMENTS

In the US experiments have been made on the sound effects inside a sphere large enough to contain a person . The sound transmitted by the wa lls makes the w ho le body resonate: sku ll , thorax, abdominal cavity and bo nes all vibrate at once. To the man , it seems that the sound derives from his body. The musica l experience is activa ted to the fu ll , although the man remains passive. His body resounds and a read iness to assimilate and individual wi ll are of little impor-tance.

OPTICAL EFFECT

BUCKMI NSTER FULLER

One o f the many geodesic domes des igned by Fuller. In 1957 it took 22 hours to bui ld this eno r-mous dome in Hono lulu for a series o f concerts. Al l geodesic domes are built using prefabricated parts and are easily assembled . In ce rtain cases, some domes w ere hung from a metal cahle and transported by helicopter. Fuller created geode-sic domes in a wide range o f different materials, p lastic, wood and at the Milan Triennal o f 1954 he made one out o f cardboard with a radius o f approximately ten metres.

SPHERI CAL BUILDINGS

In aro und 1770 Claude-Nicolas Ledo ux designed a giant sphere as a symbo l o f eternit y for the cemetery o f Caux. He also designed a spheri ca l home for the caretakers ::is a symbo l of human solitude. In 1800 Lequeu designed a spherica l "Temple o f the Earth ". Later, the Russian compo-ser and theosophist Aleksa ndr Skrjabin designed a semi-spherical bui lding w hich would stand over the w ater so th at its refl ectio n wou ld com-plete the other half o f the sphere. The building w as to be constructed in India and w as to be the site o f a ho ly p lay about universa l redempt io n . Russian architects, w ho tended towa rds futuri -sm , designed a suspended spheri ca l huilding Lo be used as a planetarium and reading room fo r the Lenin Institute. In March 1958 Jo hann Lu do-vici submitted a spherical ho use made o f metal to the Londo n Exhibition .

BALANCE

The three stages o f balance o f a w heel : wi th

0 0 CD the weight at the to p it is unstable. With the w eight at the botto m it is stable. With the weight in the centre, it is neutral. Spo ntaneous mo tio n can only arise in the first case. This principle is always put into practice fo r models o f machines ----'"""'"'""'~-----===--------'"""""""''---- for perpetual motion . 36

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CYLINDRICAL FOUNTAIN

The outer cylinder is 2 metres high with a diame-te r o f 4 metres. It is made of iron with panels o f coloured plastic materia l curved like the blades

of a turbine. The other two inner cylinders are built on the sa me principle . The outer cylinder is activated by a slow motor and its colours a re neutral, from black to white. The medium cylin-der turns with the wind and its colours are warm and transparent, from yellow to purple and red . The small cylinder is prope lled by a jet of wate r and its colours are cold , from green to blue. The random blending of these colours as the cylin-ders rotate creates unexpected effects. Built for Montecatini at the Milan Fair 1961.

SPHERICAL GEOMETHY

When establishing his non-Euclidian principles of geometry, Lobacevskij bega n with the de fi -nition of the distance be tween two points as an invariant of movement; then on the bas is of the definiti on of the distance between two points he was able to acid the definiti on of the surface of the sphere as a locus of the points in space equi-distant from a given point. The definition of the circle o r, to be exact, the circumference , fo llows on from that of the sphe rica l surface in that the circle is defined as a locus of the po ints commo n to two sphe rical surfaces . It fo llows then that the plane can be defined as a locus of the circles of inte rsection of spheres having the sa me radius described around two fixed points which are also poles of the plane. In this way Lobacevskij defined a limited portion of the plane, the one inside the generating circle. The infinity of the plane exists not as a fa ct but as the possibility of extending the area in all directions. Lastly, he defined the stra ight line as the inte rsection of two planes; o r rathe r the segment of a stra ight line as the diamete r common to two grea t circles of a sphere .

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YELLOW AND DEEP BLUE

In his book "Concerning the Spiritual in Art" Kandinsky describes the optica l effect of co-lours. He says that if you fill two circles of equal size, o ne ye llow and o ne dark blue, you obser-ve , afte r concentrating briefl y on one and then the other, that the ye ll ow radiates outwards and almost visibly comes closer to us. The blue on the other hand develo ps centripe ta lly and moves away from us. The eye is stru ck by the ye llow circle while it is drawn into the blue one.

TOYS AND GAM ES

Many toys and ga mes a re based on the circle: ring-a- ring-a- roses, the merry-go- ro und , ro ule t-te, the hoop, the top , the ball , the pin-whee l and others.

PEOPLE IN A CIRCLE

People automatica lly arrange themselves in a circle when they want to look at something to-gethe r. This probably gave ri se to the form of the arena, the circus and the trading posts at the stock exchange. Photo by Lori Sammartino.

HINDU

Cosmic Hindu diagram.

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IGLOO

The ho 111e of the Eskimo is a ho llow half-sphere 111ade w ith b locks of snow. A long open corridor leading outside provides ventilation while pro-tecting the entrance fro 111 co ld winds. It is sa id that 'a new hut is warmer than an o ld one because it is 111ade o f snow, w hile an o ld hut is made o f ice'. Altho ugh it takes longer ro bui ld an ig loo than a tropica l hut, it is only a temporary ho me for hunters and is abandoned in spring w hen the snow o n the roof starts to 111elt and when the pudd les on the floor 111ake it inhabitable. Photo by Mario De Biasi.

JOSEPH

Joseph's problem is certa inly one of the most fa mous and one of the o ldest. Depending on the e ra in which the sto ry is set, the people are e ither Christians and Jews, Christians and Turks, blacks and whites, good and bad . The o rig inal sto ry was about Jose ph , who found himself in a cave with 40 other Jews firml y decided on killing themselves rather than fa ce an even worse fate at the hands of the Romans. Joseph , who had decided to save his own skin , arranged them in a ci rcle and established to general agreement that counting in threes aro und the circle , the pe rson who was third , sixth etc would be killed . In the

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circle of 41 people he kept 16th and 31st place fo r himself and fo r another fa rsighted person. Although he and his companio n remained to the encl they were able to avo id martyrdom. Another version of this problem has 15 Turks and 15 Christians on board a ship tossed by a storm; the ship would sink unless half the pas-sengers were thrown into the sea. Afte r a rran-ging everyone in a circle, the Christians sugge-sted that the every ninth , e ighteenth etc pe rson would be killed . Naturally enough, the infide ls were arranged so as to spare a ll the Christians.

KANDINSKY

Circles w ithin a circle, painting 1923. Louise and Wa lter Arensberg Collection . Philadelphia Mu-seum of A rt.

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LEONARDO DA VINCI

Variations on Leonardo 's principle of the per-petual motion machine based on free-moving spheres.

PAUL KLEE

Movement and counter-mo vement.

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Sensitisation f sitio T 0 layers a I . n. he raisin s tie centre mvolves the ... g of the point f changes po-1a1smg o f h o v1s1on I t e hori zo 1 a ways n me.

Patterns of r . energ f c iatton from I y o rmation o f h . growth of th t i_e centre and b t I e circle by ra-tside. e 1 adius from they tie progressive ms1de to the ou-

Synthesis o f static and dynamic re lationships.

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FRAN

MANDALA

A symbol o f tota li ty. There are many versions o f it: from the so-ca lled prehisto ric "circle o f the sun ", to the circle that surrounds and protects , to the alchemist's microcosm and , lastly, as the mode rn symbol that conta ins psych ic entirety.

RI NG MACHI NE

An iron ring is he ld by two cylinders attached to a curved arm . The ring is free and according to the invento r it should turn and the refore make the cylinders turn , because the fo rce of gravity w ill d raw it downwa rds. However the centre o f gravity o f the ring a lways stays in the sa me place so that the machine acquires perfect , immobile equilibrium.

MICHELANGE LO

The Ho ly Fa mi ly. Uffi zi Ga lle ry, Flo rence.

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BYZANTINE MONOGRAM

HYDROSTATIC MACHINE

Diagram fo r pe rpe tual motion with liquid wei-ght.

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MERCURY MACHINE

Two cylinde rs in pe rfect contact with each othe r and rotating in opposite directions inside a sea-led box. The mercury has been placed in the uppe r, shaded pa rt and this should put pressure o n the cylinde rs, making them turn. All liquids put pressure on the bottom of the recipient in w hich they are contained but in this case the re-cipient has no bottom but only wa lls. Therefore the pressure is put on the wa lls and, because it is d irected towa rds the centre o f the cylinders , these remain still .

MACHINE WITH SPHERES

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' I \ ' '

Perpetual motion machine.

l Being greater than the weight o f the other sphe-res, the w eight o f the sphere around the curve ought to bring about the motion .

Variation o f the same principle.

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ENZO MARI

Composition no. 383. Aluminium and lamina-ted plastic, 67x23x20 cm , b lack , w hite and blue. 1959

Pheno l ic resin sphere containing cubic meshing. Structure no . 969, diameter 12 cm .

TRADEMARKS

Symbols and trademarks e ither round in shape o r made up of e lements de riving from the cir-cle.

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WATCH

The turning of the hands dete rmines the most logical shape of this object.

NURAGHI

Ancient Sardinian constructions from the Bronze Age o r perhaps ea rlier. They have a round base and are bui lt w ith la rge stones. Some are as high as 16 metres ta ll wi th a base diameter of about 30 metres w hich then tapers towa rds the top.

EIGHT CIRCLES

In a given circle inscribe e ight circles of different sizes, placed symmetrica lly and tange nt to each othe r and to the given circle.

GIOTTO'S 0

At Pisa Giotto made such a marvellous painting that.. "the work brought him in that city and e lsewhere such fame that pope Benedict IX, ha-ving the desire to ca rry out some pa intings in St. Peter's , sent one of his courtiers from Trevisi to Tuscany to see w hat kind of man this Giotto was and what his works were like. The courtier on his way to see Giotto lea rned that the re were in Flo rence other excellent maste rs in pa inting and mosa ic, and spoke in Siena to many ma-sters. Then, afte r obtai ning drawings from them he went to Flo rence. Going one mo rning into the studio of Giotto w ho was at work, he descri -bed to him w hat the Pope had in mind and how he wished to make use of his work , and fina lly

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he asked him fo r a few drawings to send to His Holiness. Giotto, w ho was most well-manne red, took some paper and using a paintbrush clip-ped in red, held his arm still at the side like a compass and , turning his hand , drew a circle so perfect in curve and profile that it was a marve l to beho ld . After this, grinning at the courtie r, he sa id : - Here is your drawing -. Thinking he was be ing mocked he sa id : - Will I have any other drawing besides this' - This is more than enough; - rep lied Giotto - send it along with the othe rs and you w ill see that it will be well received .

"The envoy, seeing that he would be g iven no more, left most clissatisfiecl , and fea ring that he had been tricked . Nevertheless, he sent the Pope the othe r drawings and the names of those w ho had clone them. He also sent Giotto's drawing, explaining how he had drawn his circle without moving his arm and without using a compass. This is how the Pope and his many expert cour-tiers lea rned that Giotto's excellence far surpas-sed that of the o ther painte rs of his time. Once the news became known it gave o rigin to the saying still in use today about dim-witted men: "You are rounde r than Giotto 's O". This is a fine proverb not only due to the event which gave rise to it , but all the more so because its meaning is ambiguous, since in the Tuscan language the word rondo does not only mea n 'pe rfect circle ' but a lso one who is none too sharp.

''The Pope invited him to Rome w here, prai-sing him grea tly and recognising his high me-rit, he engaged him to pa int fi ve sto ries of the li fe of Christ in the tribune of St Peter's , and to execute the main painting in the sacristy; he executed these works with such great diligence that no better work ever came from his hands. To rewa rd his merit and as a sign of gratitude the Pope paid him six hundred gold ducats and bestowed upon him many favours, so that his name beca me known throughout Italy".

DEMOSTRATIVE OBJECT SKF

The large glass disk has a metal rim with a groove conta ining a number of steel balls. The disk is slightly inclined and it rotates very slowly so that every so often a ball separates from the top part and ro lls half way round the circle until it jo ins up with the first ba ll at the other end . Immedia-tely, ano ther ball separates from the others and so on ...

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INITIAL PO INT

If we put an initial point on the circumference of a circle we obtain a new fi gure which is used in the theory of polygene o r complex fun ctions.

GOLD

Magic circle used fo r making gold.

UMBRELLA

Japanese ba mboo umbrella with waterproof va r-nished pa pe r. The spokes of the umbre lla come from a single bamboo ca ne . When the umbre lla is closed the spokes fit back into the shape of the ca ne they were cut from and the umbrella is enclosed inside it.

GESTALT PSYCHOLOGY

After staring for a whi le at the centre of this fi -gure, most people suddenly see a different sche-me. The radii that in the first drawing served as edges to the narrow sections become the ed-ges of the wider sections. Clearly, the layout of the scheme has changed and it tends to change again. The rhythm at which the two figures alter-nate will gradually increase as the pe rson stares at the centre of the figure .

ASTROLOGICAL PLANISPHERE

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1T

In the Book of Kings and in the Chronicles, rr is given as 3. Egyptian mathematicians gave it the va lue of 3. 16; the decimal 3.1416 was known at the time of Pto lemy in the yea r 150 BC.

PATHOCIRCLE

Another variation on the circle was introduced by the eminent American mathematician , C. ]. Geyser. This consisted in taking a ci rcle and re-moving a po int from it. Although the fi gure can-not be drawn beca use a point has no extension, it was an important change in concept. Keyser ca lled the new figure a pathocircle, meaning a pathological circle, and he used it to discuss the axioms of logic.

THE PROBLEM OF APOLLONIUS

Given three circles, find the ci rcle tangent to them. The problem can be solved simply using rule r and compass and there are e ight possible solutions.

CONCENTRJC WAVES

One of the natural forms of the circle.

POLAROID

Graphic symbol o f Po laro id filters.

PSEUDOSPHERE

Geometric form determined by the rotation o f a tractrix around the longitudinal axis. The tractrix is a curve perpendicular to a family of equal circles, p laced at equal distances, whose centres are on a straight line.

56

PAFFARD KEATINGE CLAY

Plan o f a building to be used for experimental performances o f dance, music and light effects designed for the Ca rl Cherry Foundation in Ca-li fornia .

X HOUR

Objects o f kinetic art, created by Munari in 1945. Fifty numbered pieces were made by Danese o f Milan. The half-disks in the centre of the object are transparent and turn by clockwork, thus composing continuously changing geometrica l fi gures.

GIOVANNI PI NTORJ

D rawings.

58

INDIAN BALL

A flexib le wi re structure made o f circular arcs. A series o f geometric figures ca n be obtained by chang ing its shape in different w ays.

LUCA PACIO LI

Letter of the "dignissimo antico" alphabet by Luca Pacio li.

60

Variations on the shape of the Ind ian ba ll obta i-ned by bend ing the object in va rious ways.

RAPHAEL

The Madonna of the Chair. Pitti Ga lle ry, Flo re n-ce.

ROMANESQUE

Cross of Trinity Church at Caen . 61

CIRCULAR PENTAGRAM

Musical annotatio n of a sound-making object, with no beginning or end.

BERNARD REDER

Sphe rica l theatre. The audience is seated along a continuously moving inte rnal spira l. The action takes place in the centre of the sphere. Very light mate rials need to be used in its construction.

62

THE WHEEL

The ea rliest archaeologica l evidence of the wheel dates back to the urba n cultures of Mesopota-mia . The first wheels were simply solid wooden disks firml y jo ined to an axle that turned a long with the wheels. Late r o n, hubs were invented and holes were made in the solid disk. These holes gradually increased in size until the idea of spokes was introduced. Wheels with spokes existed in Asia Minor as ea rly as 2700 BC.

THE WHEEL AS A SYMBOL

The wheel is also a symbo l of the sun , of d ivi-nity and good luck and it has always been used as an o rnament. To celebrate the solstice in the past blazing wheels were ro lled clown slopes and narrow wooden hoops were th rown into the a ir.

GOTHIC ROSE-WINDOW

Rose-window of the Sa nta Chia ra Bas ilica at As-sisi. Photo by Paolo Mo nti .

RADAR

Radar at the Vandenberg auxiliary station follows the path of the artificial satellites through space. The network of the radar stations extends from the US coast o f the Pac ific to Hawa ii and no rth to Alaska.

CELTIC SEAL

64

DECREASING RADI US

Draw a circle. Inside this, d raw an equilate ral triangle . Draw anothe r circle inside the triangle. Draw a square inside the second circle. Draw another circle inside this sq uare and fo llow this with a pentagon. Continue, adding an extra side to each po lygon. The radius o f the circles may appear to get shorter and sho rte r, tending to zero, but it is not so. What rea lly happens is that the process o f contraction itse lf tends to a limit that appears when the po lygon and circle have become a lmost equal. The decreasing radius ap-proaches a limit of about 1/ 12 of the radius o f the o rig ina l circle.

INCREASING RADI US

Closely connected to the problem of the decrea-sing radius is the proble m o f circumscribing ra-the r than inscribing po lygons and circles.

In this case it may appea r that the radius, in-creasing beyond every limi t, becomes infinite. Instead , it comes close to a limit of about twelve times that of the rad ius of the o rigina l circle. It is in te resting to note how at the limit the in-creasing rad ius has become reciproca l to the de-creasing o ne.

65

POTTER'S WHEEL

The po tte r's wheel was known in Egypt before the beginning of the third millennium BC. Cre-tan craftsmen used it in the earliest stages o f the Bronze Age and it was a lso known in many parts of India. It appea red it Europe, in France and southern Germany, in a round 500 BC, while it was unknown on the America n continent.

WRITI NG

The sign of the circle lies at the base of almost all alphabets o r ideograms. These signs are com-mon to Azilian, Mega litic, Cretan, American , Ca-nary Island writing and cave painting.

CONNECTED SPHERES

A search for pe rpetual motion using connected spheres. The greate r weight of the spheres on the longer arm should ca use the mechanism to turn.

Variation of the idea with connected sphe res on inclined surfaces.

66

COSTS

Little disks engraved on obe lisks o r pyramids would appea r to refe r to the money spent o n these monuments. If the disks are arranged in a circle it mea ns that public money was spent , if instead they are in a line it was financed by priva te funds.

DREAM

In his book Flying Sa ucers: A Modern Myth of Things Seen in the Skies, Ca rl Gustav Jung de-scribes the fo llowing dream to ld to him by an acquaintance: .. . I must stress how bea utiful the sphere was with its soft grey o r opaq ue whi te aga inst the night sky. When we rea lised that a te rrible collisio n was about to take place with the ea rth , we were, of course, afra id but it was a fear in which reverence preva iled . It was a cosmic event that caused great awe . While we were a ll absorbed in that vision a second and then a third sphere appea red, and then others approached at great speed . Each sphere exploded o n the ea rth like a bomb but evidently at so great a d istance that I was unable to define the nature o f the explosio n, o r detonation, o r whatever it was. In one case at least I had the impression I had seen a fl ash of lightning. The spheres fe ll a ll around us at inte rva ls, but all so fa r away that it was impossible to pe rceive the ir destructive action. There seemed to be some sort of a danger of shra pnel effect o r something s imil ar. Then - I must have gone ho me - I found myself talking to a g irl sitting on a wicke r chair ...

SHINTO

Shinto symbol: the revolving of the universe.

EMPIRE STYLE

Decorative pattern on French cloth. 67

GOLDEN SECTION

Construction of the golden section on the ra-dium of a given circle .

STONEHENGE

Site of the fa mous ruins of a circular stone ring erected in around 1400 BC on Salisbury Plain in England. The 80 roughly hewn pillars and architraves that make up the main circle, each weighing about 50 tons, seem no t to have been found on the site but to have come from the Prescelly Hills in Wales. The stones were phy-sica lly transported overland for many miles and

then were carried o n ra fts made of tree trunks to coastland nea re r Stonehenge, w here they were aga in moved by human force. Around the circu-lar group of stones are other circles of holes at equal distances. Excavation work has brought to light Irish tools and weapons, gold Mycenaea n jewelle ry and amber objects from centra l Euro-pe which point to a fl ourishing trade with other peoples. The central ring, still standing today, consists of a more recent circle of stones which were cut and worked nea rer Stonehenge. In the centre is a large block of sa ndstone measuring a lmost 5 metres (16 ft) in length . An adventurous Greek architect might have come to Stonehenge and used the techniques lea rned from his people to build this stone circle , o r henge.

A COLUMN OF SPHERES

A mobile object, Programmed Art. The spheres are held by three sheets of glass. The first sphere at the bottom sits on the pulley of a very slow motor. All the spheres turn by friction , thereby constantly changing the combination of the cur-ved white strips painted on them. Munari 1962, Olivetti collection. Photo by Mulas.

KARLHEINZ STOCKHAUSEN

Karlheinz Stockhausen imagines an idea l au-ditorium fo r e lectronic music, in the form of a sphere . In the centre the public are seated on a platform bui lt to let the sound pass th rough so that they ca n hea r the music from all directions. Loudspeakers are placed all a round the inte rio r of the sphere.

68

69

METERO LOGJCAL SIGNS

From left to right: clea r sky, cloudy sky, solar halo , lunar halo, rainbow, mirage, solar corona , lunar corona.

LAYER OF CIRCLES

If at a ll the knots of a network made up of equi-late ral triangles we draw some circumferences whose radii are equal to half the length of the sides of the triangles, we obtain a system of cir-cles which touch but never overlap. This system of circles is called a layer of circles.

SAME AREA

The two shaded parts have the same area as the circle that generated them.

SOLOMON

The great magic circle of Solomon. 70

SYMBOLS OF MARTYRDOM

CHINESE SPHERE

A ca rved ivo ry ball containing e leven o ther balls, one inside the other, all de rived from the same block using a conica l drill bit and other tools. All the balls move freely inside each o ther, from the smallest in the centre to the largest on the outside.

TARGET SHOOTING

TRINITY

One of the symbols of the Trinity. 72

TURBINES

A turbine consists of an infinite number of con-tinuous elements arranged so that the external part is placed over a circumference and the in-te rnal parts must all incline equ~lly with respect to the circumference. There is a whole geometry of turbines that dea ls with continuous groups of transformations connected to differentia l eq ua-tions and diffe rential geometry, introduced by Edward Kasner.

MUSLIM TALISMAN

THE ROUND TABLE

According to Breton trad ition , the legendary kni -ghts of King Arthur used to sit at a round table to symbolise the fac t that a ll were equally bound to do the ir duty.

73

TRULLI

Ancient stone dwellings, almost all of which are one storey high. They have a circular base and a conica l roof. Houses with more than one room are made up of cyli ndrica l constructions, one for each room, connected by passageways. The most famous trulli are at Albe ro be llo and in the va lley around Martina Franca.

TANGENTS

Draw several circles tangent to each other and to the outside of a given circle.

'' ' ' ' ' ' ' \ ',

', ' .... __ , ' ' '

'

'

'

Inscribe a circle in a curvilinea r isosceles trian-gle.

74

Draw four circles tangent to three straight lines which are symmetrica ll y crossed.

I I

'

' I I

I \

\ I I

I I

'

' ' '

'

'

I

I I ,

\ ' ','

, I

Given a straight line and two points outside it, draw a circle passing through these two points and a point on the stra ight line.

'' \ '',,'',,,',,,'

,,,,

Draw several circles tangent to each other and to the inside of a given circle .

'

" : \

" ' ' ' '

-.::::: _______ i_ - -

Circles tangent to each other and to the sides of a given triangle.

76

Draw two circles tangent to a stra ight line and to a g iven circle .

-----'=--r-..~=----- ........ ~ ... j/ ,...I...._ v

Draw a circle tangent to ano ther o ne at a gi-ven po int and to a stra ight line at another given point.

Draw a circle with a g iven radius, tangent to two othe r given circles.

Draw a circle tangent to another one at a g iven point and passing through another one situated inside o r outside the ci rcle itself.

77

Draw three circles tangent to each other and in-side a given equilate ral triangle.

Draw three circles tangent to each other and in-side a given curved equilate ral triangle.

Draw two tangents common to two given ci rcles with different radii .

Solution A

,-

... -... - - - - -;- - - - - -......... i_-----

Solution B

78

ALL IS WELL

Sign o f understanding among beggars .

MARY VIEIRA

Circle + movement= forms. Object made of ano-dised aluminium, diameter 33 cm Basie 1953-1958

Mobile circular surfaces in the ir sphe rical space. Anodised aluminium, Mary Vieira , Basie 1953-1958.

80

YIN - YANG

In around 1000 BC, some ano nymo us Chinese sages noticed that e ve rything in nature results from the unio n of two o pposing fo rces . They ca lled these Yin and Yang and re presented the m w ith a disk made up of two equal shapes in comple mentary colo urs, such as black a nd w hi-te. Each fo rce has its own ma ny qualities and nuances: Ya ng, the positive force, re presents masculini ty, actio n , wa rmth , hardness, dryness, brilliancy and firmness; it is the essence o f fire and light , the foo t o f a hill , the rive r source, and so o n. Yin o n the other hand is the negative principl e: feminine, coldness, humidity, softn ess, da rkness, myste ry, secrecy, evanescence, nebu-losity, turbidity and inaction . It is the essence of shadow a nd wate r, the no rth side of a hill , the mouth of a rive r .... De pe nding o n w hethe r Yang o r Yin preva ils, things appea r diffe re ntly: Ya ng is predominant in the sky, Yin o n the ea rth. O ne of the principles may preva il in a give n thing at a certain time and the n the othe r may do minate.

8 1

LU DO LPH VAN CEULEN

A fa mous German mathe matician. In 1596 he ca lculated the value of rr to 35 decimal places . The result he o bta ined was 3.1 4159 26535 89793 23846. According to his wishes the number was e ngraved o n his headsto ne as his e pitaph .

MARQUIS O F WO RCESTER

Model of a perpetual motio n machine . In 1663 the Marquis published a collectio n o f his very bizarre inventions . He became famous howeve r for his sa uce.

FAN

A se ries of equal sticks determines the shape of this type of fo lding fan .

82

MONOCYCLE

American mode l.

INDEX

As all the subjects in this book have been ar-ranged in a lphabe tica l o rde r (as far as make-up allows) no index is necessa ry.

Books and publications consulted:

N. I. Lobacevskij: I NUOV! PRINCIPI DELLA GEOMETR IA. Einaud i, Torino 1955. S. I. Vav ilov: L'OCCHIO E IL SOLE. Feltrinelli , Milano 1955. Hugo Steinhaus: MATHEMATICAL SNAPSH OTS. Oxford University Press , New York 1960. ENCICLOPEDIA DELLA CIVILTA ATOMICA. JI Saggiatore, Milano 1959. David Diringer: L'ALFABETO NELLA STORIA DELLA CI VILTA. Barbera , Firenze 1953. Rudolf Koch: THE BOOK OF SIGNS. Dover publicat ions inc. , New York 1955. Ernst Lehner: THE PICTURE BOOK OF SYMBOLS. Penn, New York 1956. ARTE PROGRAMMATA. O livetti , Milano. Ce leste Malavasi: SETTECENTOCJNQUANTA MECCANISMI. Hoepli , Milano 1937. Camillo Bruno: IL PROBLEMA DEL MOTO PERPETUO. Lavagno lo, Torino. Pau l K lee: TEORIA DELLA FORMA E DELLA FIGURAZ!ONE. Feltrinel li , Milano 1956. Helmut Th. Bossert: ENCYCLOPEDIE DE L'ORNEMENT. Albert Morance, Paris 1955. Ca rl G. Jung: SU COSE CHE SJ VEDONO NEL CIELO. Bompiani, Mi lano 1960. Edward Kasner / James Newman: MATEMATICA E lMM AGINAZlONE. Bompiani, Milano 1948. Alexander Speltz: LES STYLES DE L'ORNEMENT. Hoepli , Milano 1949. Julius E. Lips: L'ORl GlNE DELLE COSE. Sansoni, Firenze 1947. Pierre Schaeffer: A LA RECHERCHE D 'UNE MUSIQUE CONCRETE. Ed itions du Seui l , Paris 1952. Gyorgy Kepes: THE NEW LANDSCAPE. Paul T heobald , Chicago 1956. Michel Huet et K . Fodeba: LES HOMMES DE LA DANSE. Editio ns Clairefonta ine, Lausanne 1954. Umberto Eco e G. B. Zorzoli: STORIA lLLUSTRATA DELLE INVENZIONI. Bompiani , Milano 1960. IL MONDO DELLA NATURA. Mondadori , Milano 1962. Robert W. Marks: THE DYMAXJON WORLD OF BUCKM I NSTER FULLER. Reinhold , New York 1960. Edouard Fer: SOLFEGE DE LA COULEUR. Dunod , Paris. ARCHITECTURES FANTASTIQUES. L'architecture d 'aujourd 'hui , Paris. SCIENTIFIC AMERICAN. Scient i fic American inc. , New York . Matila C. Ghyka: ESTETIQUE DES PROPORTIONS. Ga llimard, Paris 1946. Sigfried Giedion: BREVIARIO DI ARCHITETTURA. Garzant i , Milano 1961. David Hilbert e S. Cohn-Vossen: GEOMETRIA INTU ITIVA. Boringhieri, Torino 1952. Wassily Kandinsk y: DELLA SPIRITUALITA NELL'ARTE. Religio, Roma 1946.

THE CIRCLE DISCOVERY OF THE CIRCLE Bruno Munari

1964 Bruno Munari All rights reserved by Maurizio Corraini s.r.I. No part of this book may be reproduced or transmitted in any form or by any means (electronic or mechanical , including photocopying, recording or any information retrieval system) without permission in writing from the publisher.

Origina l first ed ition published in 1964 Scheiwil ler, M ilano Mau rizio Corraini first edition May 2006 Maurizio Corra ini second rep rinr June 2012 The publisher will be at complete disposal to whom might be related to the unidentified sources printed in this book.

Translatio n co rrainiScudio Printed in Ita ly by Sti lgraf, Viadana 2nd reprint June 20 12

Maurizio Corraini s.r. I. Via Ippolito Nievo, 7/A 46100 Manrova Tel. 0039 0376 322753 Fax 0039 0376 365566 e-mail : sito@corraini .com www.co rraini .com