Geometrical Properties of Cusa's Infinite Circle

8/12/2019 Geometrical Properties of Cusa's Infinite Circle

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THE NEW FEDERALIST November 27, 1987 Page 8

American Almanac

The Geometrical Properties of Cardinal

Nicolaus of Cusa's 'Infinite Circle'

by Robert Gallagher

Renaissance architects, students of geometer Nicholas of Cusa, often used

circular designs, particularly in churches, to evoe the conception of God as

the unifying principle of universal perfection! Above, a detail of Raphael's

Marriage of the Virgin.

In the Aug. 28 issue of New Federalist's American Amanac,Robert

Gallagher challenged our readers to wrestle with a problem in geometry

that he asserted was by its nature impossible to sole. Benoit Chalifoux

!"ontreal# andAnders Best !$an Francisco# proposed interesting solutions%

now reported in this follow&up.

In !"e 1#!" cen!$r%, &ar'ina Nicoa$( o) &$(a *1+1-.+/ )o$n'e' !"e

mo'ern (cien!i)ic o$!oo0 T"ro$g" ($c" ri!ing( a( "i( 1++#e docta

ignorantia *34n Learne' Ignorance3/, &$(a in!ervene' in !"e or0 o)

(cience in "i( 'a% A! !"a! !ime, a( no, mo(! (cien!i(!( beieve' !"e

)o$n'a!ion( o) (cien!i)ic in5$ir% re(!e' 6rimari% $6on care)$ mea($remen!


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o) "a! e e6erience i!" o$r (en(e(, an' genera% reec!e' !"e i'ea !"a!

$niver(a 6rinci6e( ("o$' g$i'e !"e )orm$a!ion o) "%6o!"e(e(, !"e

carr%ing-o$! o) e6erimen!(, or !"e ma0ing o) ob(erva!ion(

&$(a em6"a(ie' !"a! (cien!i(!( m$(! oo0 be%on' "a! !"e% can mea($re

i!" !"eir (en(e( an' in(!r$men!(, !o 'i(cover !"e a( $n'er%ing !"e$niver(e, !"a! !"e% m$(! oo0 be%on' !"e )ini!e !oar' !"e in)ini!e Ine

docta ignorantia% &$(a ro!e:

Sen(e 0noe'ge i( a imi!e' 0in' o) 0noe'ge T"e (en(e( 0no

on% !"e in'ivi'$a obec! &om6are' !o (en(e-0noe'ge, in!eec!$a

0noe'ge o) !"e $niver(a, i( ab(o$!e; i! i( ab(!rac!e' )rom !"e

imi!a!ion( o) !"e 6ar!ic$ar T"e more e ab(!rac! )rom (en(e

con'i!ion(, !"e more cer!ain an' (oi' o$r 0noe'ge i(

To comm$nica!e !"i( i'ea !o men immer(e' in (en(e e6erience, &$(aem6"a(ie' !"e nee' !o 'eveo6 a 6e'agog% !"ro$g" "ic" !o conve% ($c"

conce6!( via "a! can be e6erience' b% !"e (en(e( T"e &ar'ina

'eveo6e' !"e $(e o) (%n!"e!ic geome!rica con(!r$c!ion( )or !"i( roe

For !"e rea'er e m$(! $(e 'raing( a( i$(!ra!ion(, b$! "e m$(! ri(e

above !"e(e, an' eave a(i'e "a! i( (en($o$( in !"em, in or'er !o

arrive $nim6e'e' a! "a! i( 6$re% incor6orea

In !"e (ec!ion o)e docta ignorantia 'evo!e' !o (%n!"e!ic geome!r%, &$(a

goe( rig"! !o !"e 6oin!:


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mo(! cear% 6erceive !"a! in !"e in)ini!e !"ere i( !"e ab(o$!e mai-

m$m o) (!raig"!ne(( i!" !"e ab(o$!e minim$m o) c$rva!$re

"igure #!

This figure from Cardinal Nicolaus of Cusa's $%n &earned Ignorance,$illustrates the nonlinear transformation from finite circles G, (", and C),

to the $infinite circle$ A*, +hose circumference is a straight line!

In !"i( a%, &$(a 'eveo6e' (%n!"e!ic geome!r% a( !"e (!$'% o) !ran()orma-

!ion(, ra!"er !"an o) in'ivi'$a geome!rica )ig$re( He em6"a(ie' !"a!

!ran()orma!ion( are on!oogica% 6rior !o a (6eci)ic con(!r$c!ion(

He "o o$' gra(6 !"e =aim$m in a (im6e in!$i!ion, m$(! ri(e

above !"e 'i))erence( an' 'iver(i!ie( o) !"ing( an' above a ma!"e-

ma!ica )ig$re(

Proective Properties of Circles

&$(a>( me!"o' a( a'o6!e' b% !"e grea! geome!er( o) "i(!or%?Leonar'o 'a

@inci, o"anne( Be6er, Co!!)rie' Leibni, Bar Ca$((, acob S!einer,


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In re(6on(e !o !"e c"aenge I ma'e !oNew Federalistrea'er( in !"e A$g 28

i(($e, !o !r% an' )in' a a% !o ma6 !"e 6oin!( a! !"e area o) 'i(con!in$i!%,

( 3in)ini!e circe3


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S!ar! b% 'raing a ra'i$( )rom !"e cen!er o) circe = !"ro$g" 6oin! P *(ee

Fig2b/ T"i( ra'i$( in!er(ec!( !"e circ$m)erence o) !"e circe = a! a

'e)ini!e 6oin! corre(6on'( !o ra'i$( = in Fig 2a gro( in (ie in'e)ini!e% "ie !"e 6oin! A>

*"ere i!( circ$m)erence in!er(ec!( !"e ai( connec!ing !"e cen!er( o) circe(

=> an' =/ remain( )ie' A( circe => gro(, i!( ra'i$( eng!"en( an' i!(

cen!er i( 6$("e' )ar!"er an' )ar!"er aa% )rom !"e (maer circe !ima!e-

%, !"e cen!er i( in)ini!e% 'i(!an!, !"e ra'ii are in)ini!e% ong, an' !"e cir-

c$m)erence o) !"e circe, "ic" i( i!(e) in)ini!e% ong, i( 6er6en'ic$ar !o

!"e ai( connec!ing !"e cen!er( a! 6oin! A>, a( ("on in Fig aHo i( !"e 6o(i!ion o) !"e in!erna 6oin! o) (imiari!% a))ec!e' b% !"i(

!ran()orma!ion A( circe => become( arger an' arger, !"e ange be!een

!"e in!erna !angen!( become( arger an' arger, a( !"e !angen!( o6en $6 !o

accommo'a!e !"e groing circe => T"e )ini!e circe = )i!( more an' more

in!o !"e a6e o) !"e ange ma'e b% !"e !angen!(, a( !"a! ange i'en( T"e

in!erna 6oin! o) (imiari!% con(e5$en!% move( co(er an' co(er !o !"e

circ$m)erence o) circe =

Fina%, "en circe => become( 3in)ini!e% arge,3 !"e ange be!een !"e

!o !angen!( o6en( $6 !o 18 'egree( *a (!raig"! ine, or no ange/; !"e%

coa6(e in!o a (inge !angen!, a (!raig"! ine 6er6en'ic$ar !o !"e ai( =>=

"ere !"a! ai( in!er(ec!( !"e (ma circe a! I *(ee Fig a/ I! i !o$c" !"e

in)ini!e circe a! on% one 6oin!?a 6oin! a! in)ini!% Poin! I a! !"e in!er(ec-

!ion o) !"e circ$m)erence o) circe = an' !"e ai( connec!ing !"e cen!er( o)

bo!" circe(, i( !"e in!erna 6oin! o) (imiari!% Simiar%, a( 'i(c$((e' in !"e


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A$g 28 i(($e, !"e !o e!erna !angen!( a(o coa6(e in!o one, 'e!ermining

!"e e!erna 6oin! o) (imiari!% A !o be a! !"e o!"er en' o) circe = )rom

6oin! I

"igure 0a! "igure 0b!

Construction of e.ternal point /apping points from a finiteof similarity A and internal point of circle / to an $infinite

similarity, I, bet+een a finite and circle$ via corresponding

an $infinite$ circle! radii!

We ma6 corre(6on'ing 6oin!( )rom a )ini!e circe !o an in)ini!e circe in !"e

(ame a% a( e 'i' above be!een !o )ini!e circe(, b% )in'ing

corre(6on'ing ra'ii T"e 6ec$iari!% o) !"e in)ini!e circe i( !"a! a ra'ii are

6arae !o eac" o!"er, (ince !"e% emana!e )rom an in)ini!e% 'i(!an! 6oin!;

!"e% are a 6er6en'ic$ar !o !"e (!raig"! ine 'eno!ing !"e vi(ibe 6or!ion o)

i!( circ$m)erence

To ma6 6oin! P )rom !"e )ini!e circe = !o !"e in)ini!e circe =>, re6ea! !"e

6roce'$re ("on in Fig 2b Dra ra'i$( =< !"ro$g" P, an' 6roec! 6oin! !"a! corre(6on'( !o 6oin! P o) !"e )ini!e circe, 6roec!

6oin! P !"ro$g" !"e in!erna 6oin! o) (imiari!% $n!i !"e ine o) 6roec!ionin!er(ec!( !"e ra'i$( => o) !"e 3in)ini!e circe3 T"i( 6oin! o) in!er(ec!ion i(

6oin! P>

"igure 1a! "igure 1b!

/apping points along the a.is /apping points alongconnecting t+o finite circles the a.is via corresponding

via corresponding secants! concentric circular arcs!


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Points of (.ception

T"e 6roce'$re o) )in'ing corre(6on'ing ra'ii, 6ermi!( $( !o ma6 !"e 6oin!(

o) one )ini!e circe in!o ano!"er, or !"e 6oin!( o) a )ini!e circe in!o an

3in)ini!e circe,3 ece6! )or !"o(e 6oin!( !"a! ie aong !"e ai( ==>

connec!ing !"e cen!er( o) !"e circe( Since !"e ra'ii on "ic" !"e(e 6oin!(ie, ie aong !"e ai(, !"ere i( no a% !o )in' a $ni5$e 3ine o) (ig"!3 !o

6roec! a 6oin! aong !"e ai( !"e a% e can )or 6oin!( no! aong !"e ai(, a(

in Fig 2b For !"e ca(e o) )ini!e circe( e can $(e !"e 6ro6er!% o) !"e

(imiari!% o) a )ini!e circe( !o eac" o!"er, !o con(!r$c! corre(6on'ing

(ecan!(, or corre(6on'ing arc(, !o 6roec! an% 6oin! o) eac" circe on!o i!(

corre(6on'ing 6oin! in !"e o!"er

Secan!( can be $(e' !o ma6 corre(6on'ing 6oin!( !"a! ie aong !"e ai(

connec!ing !o )ini!e circe(, b$! "o co$' !"e% be $(e' in !"e in)ini!e

ca(e A (ecan! i( a ine connec!ing an% !o 6oin!( on a circ$m)erence In

!"e in)ini!e ca(e, i) !"a! ine ere (!raig"!, i! o$' coinci'e i!" !"e

circ$m)erence I) a (ecan! in an 3in)ini!e circe3 i( c$rve', "o co$' e

'e!ermine !"e co$r(e !"a! !"a! c$rve o$' !a0e be!een !"e circ$m)erence

an' 3in)ini!%3

An'er( o)

circe =>, beca$(e e can )in' a $ni5$e ine o) (ig"! )or !"em


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We !"en 'ra !"e ra'ii =>S> an' =>T> !"a! corre(6on' !o ra'ii =S an' =T o)

circe =>, an' 6roec! 6oin!( G an' R !"ro$g" !"e in!erna 6oin! o) (imiari!%

!o )in' !"eir corre(6on'ing 6oin!( G> an' R>, "ic" ie on ra'ii =>S> an'

=>T> Wi!" o$r com6a(( e 'ra !"e circ$ar arc 'e)ine' b% !"e(e 6oin!(

abo$! => T"e 6oin! a! "ic" !"a! arc R>G> in!er(ec!( !"e ai(, i( P>, !"e 6oin!

corre(6on'ing !o 6oin! P o) circe =

"igure 1c!

Concentric arcs in a finite circle map to straight lines in the $infinite circle!$

We no "ave ano!"er me!"o' )or ma66ing corre(6on'ing 6oin!( !"a! ie

aong !"e ai( connec!ing !o )ini!e circe(, b$! !"i( me!"o' a(o canno! be

$(e' !o ma6 !"e(e 6oin!( in !"e in)ini!e ca(e &oncen!ric circ$ar arc( i!"in!"e )ini!e circe, ma6 !o (!raig"! ine( in !"e in)ini!e circe "ic" in!er(ec! a!

6oin! A> on !"e circ$m)erence o) !"e in)ini!e circe *(ee Fig+c/

In !"e !ran(i!ion )rom !"e )ini!e !o !"e in)ini!e e 6a(( !"ro$g" a rea

'i(con!in$i!%, an' (o in !"e A$g 28 i(($e I o))ere' 3!"e 6rie o) a )ree co6%

o) !"e Engi(" ang$age !ran(a!ion o) S!einer>( )onstructions *ith a


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$traight +dge Gien a Fi,ed )ircle and its )enter% !o an%one "o can inven!

a (%n!"e!ic con(!r$c!ion )or )in'ing 6oin!( !"a! corre(6on' be!een !"e )ini!e

circe an' !"e in)ini!e circe aong !"eir ai(, i!" re(6ec! !o ei!"er 6oin! o)

(imiari!%3

( (o$!ion ("o(, on% one-"a) o) !"e ai( o) !"e

)ini!e circe can be ma66e' in!o !"e in)ini!e one; !"e o!"er "a) canno! be

ma66e', an' (o !"e c"aenge (!an'( vin'ica!e' T"e 6robem "a( no

(o$!ion beca$(e in na!$re, !"e )ini!e an' !"e in)ini!e are incommen($rabe

In !"i( a% e (ee !"a! geome!rica !ran()orma!ion( can in(!r$c! $( !"a! !"ere

are )$n'amen!a c"arac!eri(!ic( o) !"e $niver(e !"a! canno! be (en(e'

'irec!%; !"e% are incor6orea

An (lliptical "unction2

=r &"ai)o$ a66ie' !"e 6roce'$re re6re(en!e' in Fig b, !o ma6 a (!raig"!

ine 6arae !o !"e circ$m)erence o) !"e 3in)ini!e circe3 in!o !"e )ini!e circe

*(ee Fig#/ T"e re($!ing image i( an ei6(e, "ic" "a( one o) i!( !o )oci

a! !"e cen!er o) !"e )ini!e circe 4n !"e o!"er "an', i) e con(!r$c! an ei6(e

in(i'e !"e )ini!e circe "o(e cen!er (erve( a( one )oc$( o) !"e ei6(e "ic"

e!en'( !o !"e in!erna 6oin! o) (imiari!% I, !"e ei6(e i 6roec! in!o !"e

in)ini!e circe a( a (!raig"! ine T"e 6oin! P a! "ic" !"e ei6(e in!er(ec!(

!"e ai( ==> connec!ing !"e circe(, corre(6on'( !o 6oin! P> a! "ic" !"e(!raig"! ine 6arae !o !"e circ$m)erence o) !"e 3in)ini!e circe,3 in!er(ec!(

!"e (ame ai(

We 0no !"a! !"e image o) !"e (!raig"! ine i( ac!$a% an ei6(e )or !"e

)ooing rea(on(

1/


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"igure 3!

alf the a.is of a finite circle can be mapped into the $infinite circle$ via an

ellipse!

/ I) e can 6rove !"a! !"e co(e' c$rve i( a conic?!"a! i(, a c$rve genera!e'

b% a c$! !"ro$g" a cone?!"en, (ince i! i( a co(e' c$rve, i! m$(! be ei!"er a

circe or an ei6(e ( T"eorem, !"e image i( a conic Since !"e

image i( a(%mme!ric in !"e 'irec!ion 6er6en'ic$ar !o !"e ai( connec!ing !"e

circe(, i! i( no! a circe, b$! an ei6(e

T"ere are (evera in!ere(!ing !"ing( !o no!e abo$! =r &"ai)o$>(

con(!r$c!ion "ic" i$(!ra!e !"e 'i(con!in$i!% be!een !"e )ini!e an' !"e

in)ini!e

1/ W"en ma66ing )ini!e circe( !o )ini!e circe(, !"e en!ire!% o) one circe i(

ma66e' on!o !"e en!ire% o) !"e o!"er


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Ever% 6o((ibe ine in !"e in)ini!e circe, 6arae !o i!( circ$m)erence,

corre(6on'( !o an ei6(e in !"e )ini!e circe T"e )$r!"er !"i( ine i( )rom !"e

circ$m)erence o) !"e in)ini!e circe, !"e more eccen!ric *or e(( i0e a circe/

i i!( ei6!ica image be For a ine 3in)ini!e% 'i(!an!,3 !"e ei6!ica image

coa6(e( in!o !"e ine (egmen! =I

2/ T"$( e (!i canno! ma6 !"e "oe ai( Since !"e ei6(e in !"e )ini!e

circe m$(! "ave !"e cen!er a( one )oc$(, !"e 6oin!( be!een = an' I canno!

be ma66e' via !"e in!erna 6oin! o) (imiari!% !o !"e 3in)ini!e circe3 a! a A

(ecan! 'ran !"ro$g" !"e $66er "a) o) !"e )ini!e circe, 6er6en'ic$ar !o !"e

ai( I=, i 6roec! in!o !"e in)ini!e circe a( a "%6erboa T"e 6oin! "ere

!"e (ecan! in!er(ec!( !"e ai( I=, o$' "ave !o corre(6on' !o a 6oin! o)

in!er(ec!ion o) !"e !o branc"e( o) !"e "%6erboa

Geometrical Properties of Cusa's Infinite Circle

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