sixty degree spaces

8/10/2019 Sixty Degree Spaces

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a(d B. otate ) /ro$ +, such that the (e* l!(e se"$e(t ) e(do!(ts + a(d B a(d a"a!(, rotate ) /ro$ B such thata(other l!(e se"$e(t ) has e(do!(ts B a(d +. here + B co!(c!de, a( e3u!lateral, e3u!a("ular tr!a("le !s /or$ed*!th s!des ), ), a(d ). et ) - 7, a(d let ) - . 9ro$ th

o!(t P o( ), ete(d a l!(e to 7 arallel to , a(d a l!(e y/ro$ P to arallel to 7. 'hese t*o l!(es a(d y /or$ t*o

!(ter(al e3u!lateral tr!a("les 7) a(d y) *!th!( the 7tr!a("le, each hav!(" a s!de *h!ch !s a se"$e(t o/ ), t*ocor(ers o/ *h!ch $eet at P. +lso, 7 !s a se"$e(t o/ 7, a(d !s a se"$e(t o/ . +(y l!(e ete(d!(" /ro$ o(e s!de o/ a(

e3u!lateral tr!a("le to the oos!te s!de a(d arallel to the th!rd s!de /or$s a(other e3u!lateraltr!a("le. 'he l!(e se"$e(t o( ) !s e3ual !( le("th to the !(ter(al l!(e , a(d the l!(e se"$e(o( ) !s e3ual !( le("th to the !(ter(al l!(e y. hether *e are talk!(" a%out o( ) or %et*ee) a(d 7, a(d *hether *e are talk!(" a%out y o( ) or y %et*ee( ) a(d , a(d y are thecoord!(ates o/ the o!(t P o( ).

:e/!(!t!o(; a(d y o( ) are o(e d!$e(s!o(al coord!(ates, *h!le a(d y *!th!( the 7tr!a("le are t*o d!$e(s!o(al coord!(ates.


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'he coord!(ates o/ P o( ) (o* %eco$e =2?, reco"(!F!(" that tak!(" the tr!a("ular rooo/ a (u$%er rerese(t!(" a( area o/ a( e3u!lateral tr!a("le %eco$es a l!(e se"$e(t.

A Generalization of z = x + y

t has %ee( sho*( that a(y l!(e se"$e(t F has a le("th such that F - y. t ca( %eclearly see( that F - =0 ? =0 y?. 'h!s rerese(ts a(y l!(e, such as , *h!ch l!(e has %

rotated *!th !ts e(do!(t attached to the !(tersect!o( o/ a(d 7 a(d !s a s!de o/ the e3u!lateratr!a("le 7). +s !s rotated, !t cuts across the l!(e ) *h!ch !s oos!te the 7 !(tersect!o(.a$ us!(" a co(ve(t!o( o/ e(u$erat!(" the s!des o/ the tr!a("le a(d (ot the aees to (a$e thtr!a("le.?

'h!s ca( %e !llustrated *!th the !(tersect!o( o/ t*o l!(es l1a(d l2 as they !(tersect at a 6a("le. et there %e a l!(e se"$e(t o( l1des!"(ated as G+ a(d the o!(t o/ !(tersect!o( *!th l+. et G+ rotate a%out G u(t!l a(other 60oa("le occurs *!th l2. Bet*ee( these t*o!(tersect!o(s o( l2!s a(other l!(e se"$e(t des!"(ated +B. herever G+ !(tersects *!th +B a

o!(t P, there are t*o other l!(e se"$e(ts created. 'hey are +P a(d PB. Because the a("le G+as *ell as the a("le +BG are 60oa("les, the( so !s the a("le BG+, /or$!(" a( e3u!a("ular,e3u!lateral tr!a("le G+B. 'h!s $ea(s that G+ - +B. 'here/ore, G+ - +P PB %ecause +B +P PB.

et G+ - a(d +B - ), a(d *he(ever !s rotated, let - r. et +P - y a(d PB - .'he( *!th a(y rotat!o( o/ r %et*ee( + a(d B, ) !s art!t!o(ed !(to t*o l!(e se"$e(ts a(d yet F %e the le("th o/ ), there/ore, F - y. But s!(ce - 7, r - y. S!(ce r rotates a%out a!s ere(d!cular to the la(e, the le("th o/ r - =0 ? =0 y?. +(d !/ F - r the(

F - =0 ? =0 y?.

t ca( %e "e(eral!Fed that a(y l!(e se"$e(t r hav!(" a(y rotat!o( *!th!( &60 de"rees !se3ual to the su$ o/ t*o coord!(ates. (stead o/ hav!(" r !(tersect o(ly o(e l!(e, !t !s su//!c!e(

rove that r !s e3ual to t*o sets o/ coord!(ates as r !(tersects t*o arallel l!(es. 'h!s ca( %eacco$l!shed %y hav!(" r rotate a%out the ce(ter o/ a hea"o( *h!le !(tersect!(" a(y t*ooos!(" s!des o/ the hea"o(.

et l1!(tersect t*o arallel l!(es l2a(d l&such that %oth a("les o/ !(tersect!o( are 60o.'here !s a $!do!(t G o( l1. otate l1a%out G u(t!l the (et a("les o/ !(tersect!o( reach 60o. the o!(ts o/ !(tersect!o( %e des!"(ated as P o( l2a(d P o( l&. t has %ee( sho*( that *here l

has sl!ced throu"h l2there are t*o l!(e se"$e(ts a a(d % created such that *he( added to"ethare the le("th o/ GP. o* the sa$e s!tuat!o( e!sts /or l1s!$ulta(eously sl!c!(" throu"h l& . 'l!(e se"$e(ts d a(d c are created as l1sl!ces throu"h l&. +dd!(" d a(d c, *e have the le("th oGP. So, add!(", GP GP - l1 - =a %? =d e?. 'here/ore, a(y l!(e

l - =u v? =u v?*here u a(d v are a(y co$%!(at!o( o/ H I , H I y or H I F to de(ote a(y t*o oos!(" s!do/ the hea"o(.

+(y l!(e l has a le("th such that l - $ (. 'he Pytha"orea( 'heore$ !s (ot (eeded to/!(d the le("th o/ a l!(e. u$%ers ca( %e le/t rat!o(al, %ut !t takes a coord!(ate syste$co$osed o/ the aes o/ the hea"o(.


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Chapter ne

!he "ia#onal of a $%uare

!he Isotropic &ector Matrix

+ll the *orld !s a cu%e, or so *e thou"ht. t !ssta(dard to use the >0ocoord!(ate syste$ !( all$athdo$. Jolu$e !s $easured !( cu%!c ce(t!$eters.+rea !s $easured !( s3uare $eters, etc. he(eversc!e(t!sts or $athe$at!c!a(s sou"ht /or sace %eyo(dthe cu%e, they !(ve(ted the hyercu%e *h!ch !s(Cd!$e(s!o(al.

hat !/ they *ere look!(" !( the *ro(" d!rect!o(Khat !/ there !s a *hole d!//ere(t "eo$etry inside thecu%eK Lust as a tr!a("le !s the $ost %as!c oly"o( that

e(closes area !(stead o/ as3uare, that !s, !t has thes$allest (u$%er o/ s!des *h!ch ca( de/!(e a la(e, the tetrahedro( !the s$allest olyhedro( !(stead o/ a cu%e *h!ch e(closes volu$e. +tetrahedro( has o(ly 4 /aces, *hereas a cu%e has 6. 7ou ca( /!t lesssace !(to a tetrahedro(. ( /act, !t re3u!res 5 tetrahedro(s to $ake 1

cu%e.

/ you take three co((ect!(" d!a"o(also( three /aces o/ the cu%e, you see a(

e3u!lateral tr!a("ular la(e !(s!de the cu%e. Each /ac!al d!a"o(al !s 45o

/ro$ the ed"e o/ the cu%e, a(d a(y t*o co((ect!(" d!a"o(als are 60o

a*ay /ro$ each other o( that tr!a("ular la(e !(s!de the cu%e. So !(s!dethe cu%e !s a(other "eo$etry alto"ether 45oa*ay /ro$ the (or$al >0o

coord!(ate syste$.

'ake the a%ove cu%e a(d cut !t !(to 8 s$aller cu%es. Each /ace o/ the cu%e !s cut !(to /os$aller /aces. :ra* the d!a"o(als o/ the outs!de /aces o/ the s$aller cu%es so they co((ect atthe $!ddle o/ each ed"e o/ the lar"er cu%e. 'hese d!a"o(als are auto$at!cally co((ected at thvertees o/ /our hea"o(s or a sol!d %y the (a$e o/ cu%octahedro(. t !s also l!ke tak!(" e!"htu(!t tetrahedro(s /ro$ the cor(ers o/ the lar"er cu%e. 'h!s cu%octahedro( !s the $a(!/old o/ a(e* "eo$etry a(d a (e* *ay o/ $easur!(" volu$e a(d area. t could also rove to %e the %ao/ a (e* al"e%ra a(d calculus.


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9ro$ th!s !((er "eo$etry *e ca( etraolate a /e* th!("s. / each ed"e o/ thecu%octahedro( !s co(s!dered to %e t*o vectors =the d!a"o(als o/ the s$aller cu%es? o!(t!(" aeach other, o(e co$!(" /ro$ o(e verte a(d the other /ro$ the oos!te verte, a(d %oth e3uthe( *e call !t a( !sotro!c vector $atr! or vector e3u!l!%r!u$ or shorte(ed to JE .

'he r!$ary d!//ere(ce %et*ee( th!s !sotro!c vector $atr! a(d theold cu%!c vector $atr! !s that the d!a"o(als *!th!( a u(!t s3uare o( the/aces o/ a cu%e are o/ u(!t value $ak!(" the s!des o/ each s3uare 1I2 or2 I 2 *h!ch !s the value o/ %oth s!( a(d cos a(d are !rrat!o(al. 'h!sleaves the cu%e *!th a( !rrat!o(al $easure a(d the $easure o/ th!s !((er"eo$etry rat!o(al.

et $e co$e at th!s /ro$ a d!//ere(t a("le.

'he Isotropic &ector Matrix co$es /ro$ the closest ack!(" o/ u(!t rad!us sheres.

Each shere *!th!( the !sotro!c vector $atr! has 12 surrou(d!(" sheres. No((ect!(" thece(ters o/ each o/ the 12 sheres to the ce(ter o/ the (uclear shere are 12 dou%le rad!!rad!at!(" /ro$ the (uclear shere. =G(e rad!us /ro$ each shere co((ected to o(e rad!us /ro$the (uclear shere.? Each a!s !s searated %y 60o /ro$ a( adace(t a!s. 'h!s a("le o/ 60o !s

roerty o/ the adace(cy o/ the sheres.

Con'ertin# fro( Irrational to )ational

9our*,u(!t sheres *!th!( the

!sotro!c vector $atr! co$e to"ether*!th!( the cu%!c vector $atr! such that/our o/ the cor(ers correso(d to thece(ters o/ the sheres, a(d the cu%es/ace d!a"o(als co!(c!de *!th thesheres rad!!, $ak!(" the d!a"o(als o/the s3uare s!des t*o u(!ts each, that !s,t*o oos!(" vectors. 'here/ore, eachs3uare s!de o( the cu%e has a d!a"o(al o/ 2, a(d each ed"e o/ the

cu%e has a le("th o/ 2 . 'he area o/ each o(e o/ these s3uares !s there/ore =2 ?2- 2 trad!t!o


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u(!t s3uares. But /or our uroses, a u(!t s3uare has a u(!t d!a"o(al *!th s!des o/ 2 .

'he trad!t!o(al area o/ the cu%e us!(" the !rrat!o(al s!des o/ 2 !s =2 ?& - 2.828428.'he u(!t octahedro( !s $ade u o/ /our u(!t tetrahedro(s. 'here/ore, the volu$e o/ theoctahedro( !s 4 tetrahedro(s. Nut t*o o/ these tetrahedro(s !( hal/ to $ake /our -Htetrahedro(s, each hav!(" a volu$e o/ -. he( /our -Htetrahedro(s are added to each /aceo(e u(!t tetrahedro(, the s$allest cu%e !s created %ecause 4=*? 1 - &, a( eas!er *ay o/calculat!(" the s$allest cu%e tha( us!(" the s!de o/ 2.

No$ar!(" th!s rat!o(al volu$e o/ & to the calculated volu$e us!(" the 2, *e "et thesy(er"et!cs co(vers!o( /actor o/ /.,,0,- 1.06066 . Us!(" th!s co(vers!o( /actor o(co(ve(t!o(al areas a(d volu$es, they are co(verted to rat!o(al areas a(d volu$es. 'h!sco(vers!o( /actor o/ 1.06066 - =>I8? .

9or areas, 2 d!$e(s!o(s,=>I8? !s tr!a("led to %eco$e ==>I8? ?2- 1, .9or volu$es, & d!$e(s!o(s, =>I8? !s tratrahedro(edO to %eco$e ==>I8? ?&- 1.1>&24&&24& - 5 $!ll!o( tetrahedro(s.

u$%ers %eco$e rat!o(al *!th!( th!s !((er "eo$etry. 'h!s !s due to $ak!(" the d!a"o(


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*!th!( each s3uare the u(!t !(stead o/ the $!stake o/ $ak!(" the s!de o/ the s3uare the u(!t.

O='etrahedro(!(" *!ll %e ela!(ed later, %e!(" e3u!vale(t *!th cu%!(".?

&olu(e as the !etrahedral 2art

!th!( the cu%octahedro( !s ala(e, a hea"o(, $ade u o/ s!e3u!lateral, e3u!a("ular tr!a("les,hav!(" three aes /ro$ each cor(er tothe oos!te s!de d!v!d!(" the tr!a("le!(to 6 r!"ht tr!a("les. ( a u(!ttetrahedro(, these aes o( each o/ the/aces o/ the tetrahedro( %eco$e

la(es *!th!( the tetrahedro(, a(dthey d!v!de the tetrahedro( !(to 4 6 - 24 $odules, each o(e %e!("called a( +HQua(ta Module. =4 /aces 6 aes?

+ *hole octahedro( has a volu$e o/ 4. 1I8tho/ that !s a tetrahedro( hav!(" a volu$e o-. !ak!(" a*ay1I4 o/ that tetrahedro( "!ves you a 3uarter volu$e. :!v!d!(" that 3uartervolu$( %y 6 "!ves you a 1I24thvolu$e. Nall that the B Qua(ta Module. 'he + a(d B Qua(taModules are e3ual !( volu$e.

e!ther the tetrahedro( (or the octahedro( are all sace /!llers. t takes %oth to /!ll all o

sace. 'hat !s *hy to descr!%e a(y art o/ sace, you (eed a collect!o( o/ + a(d B 3ua(ta


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$odules, +s co$!(" /ro$ tetrahedro(s, a(d Bs co$!(" /ro$ octahedro(s.

&olu(e of a $phere

9or a u(!t shere, the rad!us %e!(" o(e, the co(ve(t!o(alvolu$e !s 4I& pr/- 4.188D>0 cu%es. Mult!ly!(" %y thesy(er"et!cs co(sta(t /or & d!$e(s!o(s, R=>I8?/, that !s,4.188D>0 1.1>&24&, *e "et 4.>>8425, that !s, 5 tetrahedro(s.t has there/ore %eco$e rat!o(al a(d the (e* /or$ula /orsher!cal volu$e should %e & = 5r/ tetrahedro(s.

+( !cosahedro( has 20 e3u!lateral tr!a("les o( !ts sur/ace.Proect!(" the !cosahedro( o(to the sur/ace o/ a surrou(d!("shere a(d tak!(" each o/ the 20 (o* sher!cal e3u!lateraltr!a("les a(d d!v!d!(" the$ !(to 6 r!"ht tr!a("les, you "et 20 6- 120 tr!a("les. Ete(d!(" the$ to the ce(ter to /or$ 120tetrahedro(s, each has the volu$e o/ a( + or B Qua(ta Module, or a volu$e o/1I24tho/ a u(!

tetrahedro(. 'here/ore, 120 *0- 5, the volu$e o/ the u(!t shere. 'h!s !s eas!er tha( us!("the co(ve(t!o(al /or$ula /or sher!cal volu$e.

'he volu$e o/ the u(!t cu%e !s &. 'ak!(" a*ay the 8 cor(ers, such that each !s**3th o/u(!t tetrahedro(, roduces the Jector E3u!l!%r!u$. 8 **3- -, so the su$ o/ the cor(ers taka*ay !s -o/ a u(!t tetrahedro(, sho*!(" that the volu$e o/ a u(!t or %as!c Jector E3u!l!%r!u$!s & H -- 2 -. 'here/ore, the volu$e o/ a u(!t shere, %e!(" 5, !s the sa$e as the volu$e o/t*o Jector E3u!l!%r!u$s a(d has the sa$e volu$e as 120 + a(d B Qua(ta Modules.

9ro$ the ea$les thus "!ve(, !t !s see( that us!(" "eo$etry, that !s, the !((er "eo$etr/ou(d !(s!de the cu%e a(d 45oa*ay /ro$ !t, to /!(d area a(d volu$e !s s!$ler tha( us!("

co(ve(t!o(al $ea(s. 'h!s (e* $ethod "!ves rat!o(al solut!o(s.+ll sy$$etr!c /or$s ca( %e $easured s!$ly us!(" the + a(d B Qua(ta Modules as thu(!t o/ $easure. 'h!s !s *!thout the use o/ p.

'h!s /ollo*!(" chart sho*s so$e ea$les o/ the volu$es o/ so$e sol!ds %ased o( thevolu$e o/ the tetrahedro( as u(!ty.

SYMMETRICAL FORM

=%ased uo( the closestack!(" o/ u(!t rad!us sheres?

TETRA VOLUMES

=the u(!t o/ volu$e %e!(" o(eu(!t tetrahedro(?

A and B QUANTA

MODULES

=$ult!les o/ 12 sheres

surrou(d!(" a (uclear sher'etrahedro( 1 24 - 2 12

Jector E3u!l!%r!u$ 2 T 60 - 5 12

Nu%e & D2 - 6 12

Gctahedro( 4 >6 - 8 12

uclear Shere 5 120 - 10 12

ho$%!c :odecahedro( 6 144 - 12 12


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+ll o/ the a%ove sy$$etr!cal /or$s are o/ the /or$ 4r/

*here !s the (u$%er o/tetrahedro(s !( the volu$e o/ the %as!c /or$ a(d r !s the /re3ue(cy o/ that /or$. 9or ea$leus!(" r as e3ual to 2(d /re3ue(cy, r

&- 2

&- 8 =/re3ue(cy as ea(s!o( o/ t!$e throu"h sace?

SYMMETRICAL FORM

r&TETRA VOLUMES

r&=r - 2?

A and B

QUANTA MODULES

tetrahedro(8

08 24 - 1>2 - 8 24

vector e3u!l!%r!u$ 20 20 24 - 480 - 8 60

cu%e 24 24 24 - 5D6 - 8 D2

octahedro( &2 &2 24 - D68 - 8 >6

(uclear shere 40 40 24 - >60 - 8 120

rho$%!c dodecahedro( 48 48 24 - 1152 - 8 144

e$e$%er that a( + or B 3ua(ta $odule !s 1I24thvolu$e.

'he (u$%er o/ + a(d B 3ua(ta $odules are sho*( here as $ult!les o/ tetrahedro(s o//re3ue(cy 2 t!$es the (u$%er o/ + a(d B 3ua(ta $odules !( the!r r!$ary /or$s.

'he (u$%er o/ + a(d B 3ua(ta $odules are also the sa$e (u$%er as !$orta(t a("les*!th a syste$ o/ a("les, l!(es a(d la(es $ak!(" u three d!$e(s!o(al /or$s.

!he 5exa#onal 2lane

'he Jector E3u!l!%r!u$ !s a 4Hd!$e(s!o(al$a(!/old. t e!sts 45o/ro$ a( e(clos!(" cu%e. t ca( %esl!ced /our *ays to roduce 4 d!//ere(t la(es a(d has 4aes, each a!s %e!(" cola(ar *!th each o/ the 4 la(es

Each o(e o/ these la(es !s a hea"o(. 'he hea"o( hasthree aes, each o(e dra*( /ro$ cor(er to oos!te cor(all three $eet!(" !( the ce(ter a(d /or$!(" s! e3u!a("ue3u!lateral tr!a("les. Each o(e o/ these hea"o(s !( theJector E3u!l!%r!u$ !s a &Hd!$e(s!o(al $a(!/old roecteo(to a 2Hd!$e(s!o(al la(e.


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!s called ( tr!a("led, %ecause !t !s the $ult!l!cat!o( o/ t*o s!des o/ a( e3u!lateral tr!a("le!(stead o/ the t*o s!des o/ a s3uare. :!v!d!(" all s!des o/ a s3uare %y ( a(d co((ect!(" each

o!(t to !ts oos!te o!(t *!th a l!(e, the s3uare !s d!v!ded !(to (=s3uared? s3uares. / the s!

o/ a( e3u!lateral tr!a("le are d!v!ded %y (, a(d each o!(t !s co((ected to !ts t*o oos!te o!=at 60

oa("les? *!th a l!(e, the tr!a("le !s d!v!ded !(to (

=tr!a("led? tr!a("les. 'he (u$%er ( h

%ee( tr!a("led.

!he Area of a !rian#le

'he s3uare (, o/ a (u$%er (, has a o(eHtoHo(ecorreso(de(ce *!th the tr!a("le o/ the (u$%er (. :!v!d!(" as3uare !(to (s!$!lar s3uares, !s the sa$e (u$%er *he( youd!v!de a( e3u!lateral tr!a("le !(to (s!$!lar tr!a("les. ( the/!"ure, 4- 16 tr!a("les. 'he tr!a("le o/ a (u$%er !s the area o/a( e3u!a("ular e3u!lateral tr!a("le. 'h!s ca( %e "e(eral!Fed !(toa(y tr!a("le. +lso, !/ ($ !s the area o/ a(y recta("le, the( T ($ !sthe area o/ a(y tr!a("le *here ( !s the %ase a(d $ !s the he!"ht o/the tr!a("le. But !/ *e su%st!tute the area o/ a( !rre"ular tr!a("le*!th !ts e3u!vale(t area !( a( e3u!lateral tr!a("le, tak!(" ( as thed!v!sor o/ a(y s!de, the( (!s the (u$%er o/ s!$!lar tr!a("les *!th!( the lar"er tr!a("le. So thearea o/ a(y tr!a("le ca( %e eressed as (.

6indin# the !rian#ular )oot of a 4u(ber

et a( e3u!lateral tr!a("le %e d!v!ded !(to (s!$!lar tr!a("les. By the de/!(!t!o( otr!a("l!(" a (u$%er (, each s!de o/ the tr!a("le !s d!v!ded !(to ( arts. 'here/ore, the tr!a("ulroot =(? - (. o* !/ t - (, the( 7t - (. 'he tr!a("ular root o/ t !s e3ual to (, *here t !s the(u$%er o/ s!$!lar tr!a("les *!th!( a( e3u!lateral tr!a("le *!th s!des $easur!(" ( u(!ts. 'hetr!a("ular root %eco$es the scale o/ a(y tr!a("le. ( /act, the tr!a("ular root o/ a(y (u$%er !sthe le("th o/ a l!(e, *here as the tr!a("le o/ a (u$%er !s a( area.

!he trian#ular root of an area beco(es a line

'he tr!a("ular root o/ a( area %eco$es a l!(e. 'h!s !s true *hether the area !s = y?2o

=y?2. So !/ a %!(o$!al !s a( area, the tr!a("ular root o/ !t !s a l!(e/ tak!(" a( ord!(ary seco(d de"ree e3uat!o( rerese(t!("so$eth!(" !( t*o d!$e(s!o(s, tak!(" the tr!a("ular root o/ !tcha("es !t to o(e d!$e(s!o(. t *ould see$ that a s!$!lar oerat!o( three d!$e(s!o(s such as a cu%e *ould /latte( the threed!$e(s!o(s !(to t*oCd!$e(s!o(al sace such as a hea"o( *h!chhas s! e3u!lateral tr!a("les *!th aes , y, a(d F. 'he three aes!(s!de a hea"o( rerese(t the three sac!al d!$e(s!o(s o/ the cu


12/25

!he Bino(ial as a &olu(e

Nha("!(" the s!"(s *!th!( the %!(o$!al *!ll "!ve you the d!//ere(t seta(ts o/ thehea"o(. Based uo( F - y, you ca( access the

=, y, F? seta(t !( the = y?2%!(o$!al, the=y, F, ? seta(t !( the =y F?2%!(o$!al, the=F, H , y? seta(t !( the =F H ?2%!(o$!al, the=H , H y, H F? seta(t !( the =H H y?2%!(o$!al, the=H y, H F, H ? seta(t !( the =H y H F?2%!(o$!al, the=H F, , H y? seta(t !( the = H F?2%!(o$!al, the

'h!s ca( %e do(e %y tak!(" the tr!a("ular root o/ a %!(o$!al. t *!ll "!ve you o(e o/ theaes o/ the hea"o( *h!ch rerese(ts a /latte(ed >0ocoord!(ate syste$.

'here/ore,

F - R= y?2

- R=y F?2

y - R=F H ?2

H F - R=H H y?2

H - R=H y H F?2

H y - R= H F?2

Lust as tak!(" the tr!a("le o/ o(e o/ the aes o/ a hea"o( "!ves you a %!(o$!al, such ac2- =a %?2, tak!(" the tr!a("ular root o/ a %!(o$!al !s the e3u!vale(t o/ cha("!(" t*od!$e(s!o(s !(to o(e d!$e(s!o(.

8sin# the Bino(ial to 6ind the 9en#th of an Arbitrary 9ine

s there a arallel to the Pytha"orea( 'heore$ !( the 60oNoord!(ate Syste$K :ra*!("a("led l!(e *!th!( the e3u!lateral e3u!a("ular tr!a("le, *a(ted to k(o* !ts le("th. (ot!ced d!//ere(t tr!a("les a(d the!r relat!o(sh!s *!th the

arallelo"ra$ !( clos!(" the l!(e.

have already co$e u *!th the !dea that a(y l!(ese"$e(t F - y, *here a(d y are the coord!(ates o/ a(y

o!(t o( F. 'he( /or a(y rotated l!(e *!th!( the e3u!lateraltr!a("le, r - 1I2 =0Icos y0Is!( ?. Measur!(" the l!(e dre*, ca$e u *!th & 1>I&2!(ches. et, $easured thet*o s!des o/ the arallelo"ra$ to "et & !(ches a(d 1 !(ch.Us!(" the Pytha"orea( 'heore$, =&2 12? - =10?. 'hatdoes(t *ork. But *hat do k(o* !s that

F - y, a(d

F2

- = y?2

, %ut


13/25

= y?2 - 2 y y2.G(e o/ the $a!( !deas o/ the 60oNoord!(ate Syste$ !s that the s3uare o/ the s!de o/ the

tr!a("le !s the area o/ the tr!a("le. So !/ k(o* the area o/ the e3u!lateral tr!a("le *hose s!dethe l!(e dre*, ca( take the tr!a("ular root o/ the area a(d "et the le("th o/ the l!(e. S!(ce Fthe le("th o/ the l!(e, F2 !s the area o/ the e3u!lateral tr!a("le F %elo("s to, a(d %y the a%ovee3uat!o(s, F2- 2 y y2. 'here/ore, the le("th o/ the l!(e F - =2 y y2 ?. lu""ed !(the values o/ a(d y a(d ca$e u *!th F - =&2 &x1 12? - =1&? - & 1>I&2!(ches. Jo!la t

*orks accord!(" to $y ruler $easure$e(ts.

Gettin# )id of :

+ c!rcle trad!t!o(ally rerese(ts a cycle. 'he $easure o/ a cycle !s !( V u(!ts *h!ch !s thrat!o o/ the rad!us to c!rcu$/ere(ce o/ the c!rcle. Lust as 2V rerese(ts o(e cycle, so the s!des a hea"o( hav!(" s! u(!t s!des, rerese(ts o(e cycle. So 6 u(!ts rerese(ts o(e cycle. G(e s!o/ the er!$eter o/ the hea"o( !s thus the u(!t o/ cycl!c $easure !(stead o/ V.

/ *e do co(s!der the arcs o/ a c!rcle, surely, there are the s! arcs cut /ro$ thec!rcu$/ere(ce o/ a c!rcle %y the s! chords o/ the !(scr!%ed hea"o(. 'hese ca( %e used /or$easur!(" de"rees, *hereas the chords ca( %e used /or rad!a( $easure.

!he 4atural ;ay of Measurin# a Circle

/ th!s hea"o( Icoord!(ate syste$ !s c!rcu$scr!%ed *here each coro/ the hea"o( touches the er!$eter o/ a c!rcle, *e ca( "et r!d o/ V also. 'hea"o( rov!des 6 chords that d!v!des the c!rcle !(to 6 arcs. (stead o/try!(" to lay the rad!us o/ the c!rcle out across !ts c!rcu$/ere(ce *here !tcovers a( !(co$lete (u$%er o/ t!$es, !t !s $ore lo"!cal to ust d!v!de thec!rcle !(to 6 to descr!%e a cycle or a art o/ a cycle. 'here/ore, thetr!"o(o$etr!c /u(ct!o(s do (ot dee(d uo( V, %ut uo( a rat!o(al (u$%er

+( ea$le has %ee( "!ve( o/ us!(" o(e "eo$etr!c shae to calculate the volu$e o/ a(other"eo$etr!c shae. t $ay %e oss!%le to do that *!th other shaes such as the c!rcle a(d shere'he secret !s to use the u(!t tr!a("le or the u(!t tetrahedro(.

'he trad!t!o(al *ay o/ $easur!(" the area o/ a c!rcle !s to use V . 9or a u(!t c!rcle, r - 1

the co(ve(t!o(al area !s Vr2

or V - &.1415>. Mult!ly!(" th!s %y the sy(er"et!cs co(sta(t /or 2d!$e(s!o(s, &.1415> >I8 - & T s3uares. 'hats cha("!(" !t /ro$ !rrat!o(al to rat!o(al. 'he ao/ a u(!t e3u!lateral tr!a("le =each s!de !s e3ual to o(e? !s T h% - T =&?I2 1 - =&?I4 - .4&&01& s3uaresItr!a("le. 'o cha("e the & T s3uares to tr!a("ular u(!ts, take the rec!rocal o/.4&&01& s3uaresItr!a("le, *h!ch !s 2.&0>401 tr!a("lesIs3uare.

Mult!ly!(" the u(!t c!rcle o/ &.1415> s3uares %y 2.&0>401 tr!a("lesIs3uare, *e "etD.2551>D or D W e3u!lateral tr!a("les. =Mult!ly .2551>D %y 16 or &2 a(d you "et W.? So eachu(!t c!rcle !s D W tr!a("les. 'he sace %et*ee( the 6 chords o/ the hea"o( e(cased %y the c!r

a(d the 6 arcs o/ the c!rcle over the hea"o( !s D W H 6 - 1 W e3u!lateral tr!a("les.


14/25

+rea - D W r2tr!a("les/or a(y c!rcle. 'here !s(o: !(volved

'he c!rcu$/ere(ce o/ a u(!t c!rcle !s de/!(ed as 6 arc le("ths, %e!(" %ased uo( thechords o/ the !(scr!%ed hea"o( !(stead o/ o( p .'he /or$ula /or th!s c!rcu$/ere(ce !s 2pr. Garc le("th !s e3ual to 2 pr6. he( r - 1, 2 pr6 - 1.04D1>D. Mult!ly!(" that %y theco(vers!o( /actor o/ 1.06066, *e "et a( arc le("th o/ 1.110D20 or ust 1. So let p = 3, a(d,

"e(erally seak!(", o(e arc *!ll %e 1r so the c!rcu$/ere(ce *!ll %e 6r arcs.

t has %ee( d!scovered that each c!rcle has D W e3u!lateral tr!a("les (o $atter *hat therad!us !s e3ual to. + c!rcles area o/ the (et h!"her !(te"ral rad!us or /re3ue(cy !s ust r2D W I4 &2

- > D W - 65 W

- 4 4 42 - 16 D W - 116

#

Osur/ace area o/ u(!t shere =4 "reat u(!t c!rcles are used to /!(d the sur/ace o/ the shere?

#sur/ace area o/ shere *!th r - 2 !s the sur/ace area o/ a u(!t shere !( e3u!lateral tr!a("les. 'he( the sur/area o/ a(y shere !s 2>r.


15/25

!he )elationship Bet>een 2eri(eter and Area

'he tr!a("le co$es !( $a(y shaes a(d s!Fes, yet each tr!a("le has a( area, a er!$etera(d a he!"ht. 9or a(y "!ve( area, a tr!a("le ca( %e stretched hor!Fo(tally, vert!cally, ord!a"o(ally a(d kee the sa$e area. G(ly the shae !s cha("ed. 'here/ore, a(y tr!a("le *!th a"!ve( area ca( %e rerese(ted %y a( e3u!lateral tr!a("le hav!(" the sa$e area. :!v!d!(" eachs!de o/ that e3u!lateral tr!a("le !(to e3ual se"$e(ts, 2!s e3ual to the area s o/ the tr!a("le athe tr!a("ular root s o/ the area s !s e3ual to the (u$%er . he( each o!(t o/ d!v!s!o(

%et*ee( each se"$e(t o( o(e s!de o/ the tr!a("le !s co((ected to !ts correso(d!(" o!(t !(stra!"ht l!(es to each oos!te s!de =a(d th!s !s do(e /or each s!de?, the area s o/ the tr!a("le !sd!v!ded !(to 2s!$!lar e3u!lateral tr!a("les. 'here/ore, the er!$eter o/ the e3u!lateraltr!a("le !s e3ual to three t!$es the tr!a("ular root o/ the area s o/ the tr!a("le, !.e. - & .'here %e!(" a o(eHtoHo(e correso(de(ce %et*ee( the o!(ts o( the er!$eter o/ the e3u!lattr!a("le to a(y other tr!a("le, the er!$eter o/ anytr!a("le !s e3ual to three t!$es the tr!a("ulroot o/ the area o/ that tr!a("le.

'h!s relat!o(sh! %et*ee( the area o/ the tr!a("le a(d !ts er!$eter ca( %e ete(ded to oly"o(s re"ular a(d !rre"ular. Each oly"o( ca( %e d!v!ded !(to tr!a("les. e"ular oly"o(s!(to s!$!lar tr!a("les. So /ro$ a s!("le tr!a("le *!th a er!$eter o/ - & s, "o!(" out*ard /rthe ce(ter o/ a(y oly"o( to the er!$eter,

/or a s3uare, - 4s,/or a e(ta"o(, - 5s,/or a hea"o(, - 6s,a(d so o( /or a(y re"ular oly"o( o/ ( s!des, - (s.

'h!s !s talk!(" a%out the (u$%er o/ d!v!s!o(s - s o( each s!de o/ the oly"o( a(d (othe le("th o/ each d!v!s!o(, so - (.


16/25

Chapter !hree

!(ear Measure$e(t

!o>ards a 6or(ula for "istance

!ve( a(y l!(e F d!v!ded !(to t*o se"$e(ts a(d y, de/!(e the le("th o/ the l!(e to %e y. 'here/ore, the le("th o/ a( !(terval $ust %e F - y. 'h!s !s the /or$ula /or d!sta(ce !ts s!$lest /or$.

'o rove !t, let us use the e3u!lateral tr!a("le. a%el the s!des , y, a(d F. 'ake a l!(e 0/ro$ y to F arallel to a(d take a l!(e y0/ro$ the !(tersect!o( o/ 0a(d F to arallel to y.

Because 0a(d y0/or$ t*o other e3u!lateral tr!a("les *!th!( thee(co$ass!(" e3u!lateral tr!a("le, !( *h!ch o(e tr!a("le has all s!des e3uto 0a(d the other e3u!lateral tr!a("le has all s!des e3ual to y0, F !s there/

d!v!ded !(to t*o se"$e(ts 0a(d y0o( each s!de o/ a o!(t P=0, y0?. 'ht*o l!(e se"$e(ts 0a(d y0add u to /or$ F, so that 0 y0- F. 0a(d yare called the coord!(ates o/ P o( F.

e $ust there/ore co(clude that /or a(y l!(e se"$e(t F /ro$ a to %,hav!(" a o!(t P, the d!sta(ce /ro$ a to % !s F - y *here !s the d!sta(ce /ro$ P to a, a(d !s the d!sta(ce /ro$ P to %.

:e/!(!t!o(; a o!(t o( a(y l!(e F !( a la(e !s des!"(ated as P= 0, y0?.

'heore$; the coord!(ates o/ a(y o!(t P o( a l!(e F stretch!(" /ro$ a to % are the t*o lse"$e(ts 0a(d y0, the d!v!s!o(s o/ that l!(e a%ove a(d %elo* o!(t P= 0, y0?.

Norollary; the t*o d!$e(s!o(al coord!(ates o/ a(y o!(t P o( a l!(e F *he( F !s a s!de oa( e3u!lateral tr!a("le !s the l!(e 0arallel to /ro$ y to P a(d the l!(e y0arallel to y /ro$ Pto .

!he 9en#th of a 9ine or the End of the 2ytha#orean !heore(

'ake a(y l!(e r /ro$ the or!"!( o/ the hea"o(al la(e arallelto a(y o/ the 6 aes such that !t crosses a(other l!(e s 60oa*ay /ro$

r. ( rotat!(" r to a(y a("le /ro$ !ts or"!(al os!t!o(, r crosses s !( ara("e o/ 0

oto 60

o. 'he arc so !(scr!%ed !s o(e o/ the s! arcs $ak!("

u a c!rcle. 'h!s ca( %e rerese(ted !( a( e3u!lateral tr!a("le *!ths!des , y, a(d F. 'he le("th o/ r !s the le("th o/ a(y o/ the s!des o/the e3u!lateral tr!a("le. 'he e(d o/ r !s !(dee(de(t o/ a(y o!(talo(" the arc. 'he o!(t a(d coord!(ates *e are !(terested !( !s the

o!(t *here r crosses F %ecause !/ the coord!(ates o/ that o!(t are

added u, they e3ual the le("th o/ r as *ell as a(y o/ the s!des o/ the e3u!lateral tr!a("le. So a


17/25

l!(e r, at a(y a("le /ro$ 0oto &60

o, *!th!( the hea"o(al la(e has a le("th o/ y. 'he le("

o/ the l!(e r !s (ot dee(de(t o( !ts e(do!(ts /or $easur!(" !ts le("th, o(ly the coord!(ates*here !t crosses the cord u(der the arc o/ the c!rcle !(scr!%!(" the hea"o(. 'hus, *e have doa*ay *!th the Pytha"orea( 'heore$ as a $aor *ay o/ $easur!(" le("ths o/ l!(es. 'h!s takecare o/ a(y l!(e se"$e(t *!th!(" the hea"o(al la(e %y the co(struct!o( o/ a( e3u!lateraltr!a("le a(y*here *!th!( that la(e.

;hat About the 2ytha#orean !heore(?


18/25

had a( e!ha(y. s there a arallel to the Pytha"orea( 'heore$ !( the 60oNoord!(ateSyste$K :ra*!(" a( a("led l!(e *!th!( the e3u!lateral e3u!a("ular tr!a("le, *a(ted to k(o*!ts le("th. (ot!ced the d!//ere(t tr!a("les a(d the!r relat!o(sh!s *!th the arallelo"ra$ !(clos!(" the l!(e.

have already co$e u *!th the !dea that a(y l!(e se"$e(t F - y, *here a(d y arethe coord!(ates o/ a(y o!(t o( F. Measur!(" the l!(e dre*, ca$e u *!th & 1>I&2!(ches.

et, $easured the t*o s!des o/ the arallelo"ra$ to "et & !(ches a(d 1 !(ch. Us!(" thePytha"orea( 'heore$, =&2 12? - =10?. 'hat does(t *ork. But *hat do k(o* !s that

F - y, a(dF2- = y?2 , %ut

= y?2 - 2 y y2.G(e o/ the $a!( !deas o/ the 60oNoord!(ate Syste$ !s that the tr!a("le o/ the s!de o/ th

tr!a("le !s the area o/ the tr!a("le. So !/ k(o* the area o/ the e3u!lateral tr!a("le *hose s!dethe l!(e dre*, ca( take the tr!a("ular root o/ the area a(d "et the le("th o/ the l!(e. S!(ce Fthe le("th o/ the l!(e, F2 !s the area o/ the e3u!lateral tr!a("le *hose s!de !s F, a(d %y the a%o

e3uat!o(s, F2- 2 y y2. 'here/ore, the le("th o/ the l!(e r - =2 y y2 ?. lu""ed !(the values o/ a(d y a(d ca$e u *!th r - =&2 &x1 12? - =1&? - & 1>I&2!(ches. Jo!la t*orks accord!(" to $y ruler $easure$e(ts.

!he len#th of the lines = %@

o*, /ro$ the or!"!( G =*here a(d y co$e to"ether?, dra* a l!(eto the o!(t P o( F. +s the a("le %et*ee( r a(d aroaches 0

o, r a(d

aroach a(d as aroaches 60o, r a(d y0aroach y. Nall 0a(d y0th

co$o(e(ts o/ r such that the d!sta(ce r /ro$ G to P !s

r - 1I2 =0Icos y0Is!( ?*h!ch !s the avera"e o/ t*o tr!"o(o$etr!c /u(ct!o(s.

et r s*!(" do*( to creat!(" a( arc s. 9ro$ the !(tersect!o( o/ s

a(d , dra* a l!(e l arallel to F. 'he result!(" tr!a("le thus has s!des thele("th o/ r. +s P sl!des do*( s, the coord!(ate 0!(creases to the s!Fe o/ r,a(d the coord!(ate y0decreases to Fero a(d r - 0 . +s P sl!des u a(other arc

s to the to o/ l, the coord!(ate y0!(creases to the s!Fe o/ r, a(d thecoord!(ate 0decreases to Fero a(d r - y0 . 'hese act!o(s ca( %e !(terretedas 0%e!(" co$ared to cos a(d y0%e!(" co$ared to the s!( . 9ro$

tr!"o(o$etry *e have - r cos a(d y - r s!( , a(d s!(ce r - l, l- Icos or l- yIs!( . B

these t*o 3ua(t!t!es are (ot eact, so *e take the avera"e o/ the t*o to /!(d the le("th o/ l. '"!ves us the e3uat!o( o/

r - 1I2 =0Icos y0Is!( ?.

et r %e the %ase o/ a(other e3u!lateral tr!a("le that has s!des , y, a(d F. otat!(" r


19/25

crosses F. here r a(d F cross, there e!sts a o!(t P=, y? *!th coord!(ates a(d y d!v!d!("

!(to t*o se"$e(ts a(d y. S!(ce F !s co("rue(t *!th la(d !s !( /act e3ual to l, a(d F - ythe( r - y. 'here/ore, a(y l!(e F or a(y rotated l!(e r - y. But the o!(t P=, y? !s al*

o( la(d (ot o( the curve s.

S!(ce r - l a(d !/ l- y - F, /or a rotat!(" r, *here a(d y are the se"$e(ts o/ la

cuts a(d d!v!des l, I0- yIy0.

ote; the ere(d!cular d!sta(ce %et*ee( F a(d F !s F H = cos s!( ? I 2.

So. X(o*!(" the a(d y coord!(ates o/ the o!(ts o/ the l!(e se"$e(t *e ca( k(o* thle("th o/ the l!(e !( three d!//ere(t *ays.

1. F - y2. F - =2 y y2 ?

&. r - 1I2 =0Icos y0Is!( ?

!he "ifference Bet>een !>o 2oints

S!(ce F - y, *he( F !s e!ther $ore or less tha( F, a(there !s a o!(t P=, y? o( F a(d a o!(t P=, y? o( F, the( F - = y? H = y? !s a l!(e %et*ee( P a(d P.

et r - F H F, the( r - = H ? =y H y?.+"a!(, *he( c - a %, a a(d % %e!(" the coord!(ates, th

le("th o/ r !s (ot the le("th %et*ee( P=a, %? a(d P=a, %?, %ut le("th c, the su$ o/ the t*o coord!(ates.

'here/ore, !/ = H ? - $, a(d =y H y? - (,the( r - $ (

a(d, as r rotates, crosses the chord u(der the arc o/ thee(clos!(" c!rcle.


20/25

Chapter 6our

+ 60oNoord!(ate Syste$

ote; ut !( the *ork o( co$le (u$%ers here.

A &ector $pace


21/25

+ l!st o/ oss!%le aes are *h!ch /or$ a $atr!.v11v12v1&v21v22v2&v&1v&2v&&v41v42v4&

Each o/ the 4 la(es *!th!( the JE have 6 %as!s vectors %ut & aes. t !s a( error to th!(the( there are 4 6 - 24 %as!s vectors. 6 aes are shared *!th other la(es as the la(es!(tersect. / 2 la(es !(tersect, there are (ot 2 & - 6 aes, %ut 2 & - 5. G(e !s ecluded. & la(es !(tersect, 1 2 - & aes are ecluded a(d the aes add u to =& & - >? H & - 6. So*he( 4 la(es !(tersect, 1 2 & - 6 aes are ecluded, leav!(" =4 & - 12? H 6 - 6.

'he e3uat!o( that deals *!th co$%!(ator!cs !sN(- =(?IR=(C?,

so the co$%!(at!o( o/ the !(d!ces o/ the a!al $atr! are 4 la(es take( 2 %as!s vectors at a t!$N24- =4?IR2=4 H 2? - 6 co$%!(at!o(s or 6 aes.Each aes has t*o %as!s vectors. 'hat !s, there are 6 os!t!ve a(d 6 (e"at!ve %as!s vecto

$ak!(" u 12.

Each o!(t !( a JE !s 12Cd!$e(s!o(al.


22/25

ote; e have so$e really !$orta(t d!scover!es erta!(!(" to the al"e%ra o/ a 60ocoord!(atesyste$.

1. 'he d!a"o(al o/ a s3uare %e!(" u(!ty.2. 'he death o/ V . 'he $easure o/ c!rcular le("th %e!(" %ased o( & a(d 6.&. y - F as the d!sta(ce e3uat!o( !(stead o/ F- =2 y2?,4. - =2?, sho*!(" that the tr!a("ular root o/ a( area !s a l!(e,5. = y?2 - 2 y y2 =(ot 2 2y y2?, a(d

a. + arallelo"ra$ !s treated as a recta("le y *he( !ts !(ter(al a("les are 60o a(120o,

6. 2- J2 -[ - J =!/ 2!s a /actor o/ J2?.

ote; yn

2- y2I=2( 1?, yn

2R=( 1?I=2( 1? (I=2( 1? - y2- \2 ] let 9 %e called a 9our!e

so 9 - \2H yn

2 =(I=2( 1??

ote; all the tr!a("les *!th!(" the $aor tr!a("le are cou(ted us!(" tr!a("ular (u$%ers. 'h!sleads to %e!(" a( a(alo" /or har$o(!cs o/ a str!(". 'h!s leads to the ortho"o(al!ty o/ s!( a(d

cos!(e /u(ct!o(s. 'h!s leads to the %as!s vectors o/ the 60o

coord!(ate syste$.


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!he !hree $paces

1. l!(ear sace2. var!a%le l!(ear sace&. (o(l!(ear sace

!he 6ractal 4ature of the 4u(ber 9ine

+ lot has %ee( sa!d o/ the co(t!(u!ty o/ the (atural (u$%ers or the (u$%er l!(e,so$et!$es called E1. t has %ee( sa!d that *!th!( a(y !(terval =a, %? o( E1that there !s a( !(/!(


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ser!es S o/ d!v!s!o(s 1I(, such that a ^ 1I( ^ %, as ( aroaches !(/!(!ty. But *hat !s %et*ee(each d!v!s!o(K 9or ea$le, *!th!( 1Iaaaa $ ^ 1Iaaaa 3 ^ 1Iaaaa (, there !s al*ays such that $ ^ 3 ^ (, $ea(!(" that all the d!"ets aaaa are the sa$e a(d o(ly $, 3, a(d ( ared!//ere(t a(d 3 !s %et*ee( $ a(d ( o( E1.

Appendix

'here !s a rat!o !(volved !( the volu$e o/shere. t !s 15I&. t co$es /ro$ =120I8?I=24I8?, 8

%ecause o/ the sher!cal octahedro( *here there are/aces. 'he sher!cal r!"ht tr!a("le *!th!( the sher!ce3u!lateral tr!a("le !s 1I120tho/ the sur/ace area. (

%oth the sher!cal !cosahedro( a(d the sher!cal

octahedro(, there are 15 + a(d B 3ua(ta $odules !(o(e o/ the sher!cal tr!a("ular /aces.

+(other rat!o !s 20I4 related to the sher!cu%octahedro( $ade u o/ 60 + a(d B Qua(taModules. 'h!s also has to do *!th the volu$e o/ ashere. =15I&? 4 - 60I12 - =20I4? &. 'here are 4

la(es !( the Jector E3u!l!%r!u$ a(d three aes !(each o/ the 4. 'he Jector E3u!l!%r!u$ !s the key to t

reaso( *hy the u(!t shere !s 5. 'he cu%octahedro( has 60 + a(d B Qua(ta Modules. 60I12 -

'here !s a relat!o(sh! %et*ee( the !cosahedro(, the cu%octahedro(, a(d thedodecahedro(. 'he cu%octahedro( a(d the !cosahedro( have the sa$e (u$%er o/ vertees,*here the closest ack!(" o/ sheres have the!r ce(ters. 'ak!(" out the ce(tral shere /ro$ thcu%octahedro(, !t co(tracts to a $ore sy$$etr!cal co(/!"urat!o(, the !cosahedro(, %ut the(u$%er o/ vertees, *h!ch !s 12, re$a!( co(sta(t. 'he dodecahedro( has 20 vertees, %ut !t h12 /aces. 'he sher!cal dodecahedro( thus has 12 e3ually saced ce(ters o( !ts sur/ace *herethe vertees o/ the e(cased !cosahedro( touches the sur/ace, as does the cu%octahedro(. So threlat!o(sh! o/ each o/ these sheres !s th!s 12 e3ually saced o!(ts. 'h!s o%v!ously co$es

/ro$ the *ay 12 sheres ack closely arou(d a ce(tral shere. 'hus co$$o( de(o$!(ator o/12.

6or(ation of the "odecahedron

'ake 6 c!rcles surrou(d!(" o(e c!rcle a(d ush the$ o(to the ce(ter c!rcle *!th e3ual/orce. 'he ce(ter c!rcle %eco$es a hea"o(. Beeh!ves are the result o/ the $ost eco(o$!cal uo/ c!rcular sace. 'he dodecahedro( rerese(ts the $ost eco(o$!cal use o/ three d!$e(s!o(asace, a(d l!ke the cu%e !s a( allHsace /!ller. / you take 12 so/t sheres surrou(d!(" a ce(tra

so/t shere a(d each outer shere !s ushed *!th the sa$e /orce to*ards the ce(tral shere, a


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dodecahedro( results.

!he Bino(ial

'he bino(ial theore(states that

*here - N=(, ( H ? - =(I=( H ??,

or e3u!vale(tly, .

But accord!(" to the de/!(!t!o( = y?2 - 2 y y2, *here ( - 2 a(d - 1, the

%!(o$!al theore$ /or the 60ocoord!(ate syste$ %eco$es =a %?( - a( H %.

9or ( - 1 a(d - 0, =a %?( - a. 9or ( - 2 a(d - 1, =a %?( - a2 a% %2.

e talked a%out tak!(" a l!(e a(d tr!a("l!(" !t to $ake !t !(to a( area, lets (o* talk a%the %!(o$!al = y?2 a(d de"rad!(" !t !(to a l!(e.

sixty degree spaces

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