,,! ,,molecul> ,,tactics ,,(    ,a      
,,cry/al      ,,"l ,,kelv9      ,london 
,,h5ry ,,fr[de    ,,ox=d ,,univ]sity    
,,press ,,w>eh\se    ,,am5 ,,corn]1     
,e4,c4    ,illu/ration      ,new ,york  
,,macmillan  ,,co41 #ff ,,fif? ,,av5ue  
,,!    ,,molecul> ,,tactics ,,(    ,a   
,,cry/al      ,,by      ,,"l ,,kelv91   
,p4,r4,s4      ,,pr(essor ,,( ,,natural 
,,philosophy ,,9 ,,! ,,univ]sity ,,(    
,,glasg[    ,,& ,,fell[ ,,( ,,pet]h\se1 
,,cambridge        ,be+ ! ,second       
,,rob]t ,,boyle ,,lecture1 deliv]$ be=e 
! ,ox=d ,univ]sity ,junior ,sci5tific   
,club    on ,tuesday1 ,may #af1 #ahic   
,,) ,,tw5ty ,,illu/rations        ,ox=d 
,,at ,,! ,,cl>5don ,,press    #ahid     
,ox=d    ,,pr9t$ ,,at ,,! ,,cl>5don     
,,press    ,,by ,,horace ,,h>t1 ,,pr9t] 
,,to ,,! ,,univ]sity          ,,on ,,!  
,,molecul> ,,tactics ,,( ,a ,,cry/al    
,by ,,"l ,,kelv91 ,p4,r4,s4       #a4   
,my subject ?is ev5+ is n ! physical    
prop]ties (  cry/als1 n ev5 _! dynamics2
 x is m]ely ! geometry ( !  /ructure--! 

arrangem5t ( ! molecules 9 ! con/itution
 ( a  cry/al4 ,e cry/al is a homog5e\s  
assemblage ( small bodies  or molecules4
 ,! conv]se proposition is sc>cely true1
 unless 9  a v ext5d$ s5se ( ! t]m      
cry/al 7 #bj bel[74 ,i c be/  expla9 a  
homog5e\s assemblage ( molecules by ask+
 y to ?9k  ( a homog5e\s assemblage ( p4
 ,to be homog5e\s e p]son  ( !          
assemblage mu/ be equal & simil> to e   
o!r3 !y mu/  be seat$ 9 r[s or /&+ 9 r[s
 9 a p]fectly simil> mann]4  ,ea* p]son1
 except ^? on ! bord]s ( ! assemblage1  
mu/ h  a neighb\r on "o side & an       
equi-di/ant neighb\r on ! o!r3 a        
neighb\r on ! left front & an           
equi-di/ant neighb\r beh9d on !  "r1 a  
neighb\r on ! "r front & an equi-di/ant 
neighb\r  beh9d on ! left4 ,8 two       
neighb\rs 9 front & 8 two neighb\rs     
beh9d are memb]s ( two r[s equal &      
simil> to ! r[s consi/+  ( himself & 8  
"r-h& & left-h& neighb\rs1 & _!         
neighb\rs' neighb\rs indef9itely to "r &
 left4 ,9 p>ticul>  cases ! ne>e/ ( !   

front & re> neighb\rs may be "r 9  front
 & "r 9 re>2 b we mu/ n conf9e our      
att5tion to !  rectangul>ly gr\p$       
assemblages ?us con/itut$4 ,n[ let !re  
be  equal & simil> assemblages on floors
 above & bel[ t :i*  we h be5 consid]+1 
& let !re be any indef9itely great      
numb] ( floors at equal di/ances f "o   
ano!r above & bel[4  ,?9k ( any "o p]son
 on any int]m$iate floor & ( 8 ne>e/    
neighb\rs on ! floors above & bel[4 ,^! 
?ree p]sons mu/ be  exactly 9 "o l9e2   
?is1 9 virtue ( ! homog5e\sness ( !     
assemblages on ! ?ree floors1 w secure t
 e p]son on !  int]m$iate floor is      
exactly 9 l9e ) 8 ne>e/ neighb\rs above 
& bel[4 ,! same condition ( alignm5t mu/
 be fulfill$ by e  ?ree consecutive     
floors1 & we ?us h a homog5e\s          
assemblage (  p 9 ?ree dim5sions (      
space4 ,9 p>ticul> cases e p]son's      
ne>e/ neighb\r 9 ! floor above may be   
v]tically ov] him1 b  we mu/ n conf9e   
our att5tion to assemblages ?us         
rectangul>ly  gr\p$ 9 v]tical l9es4     

#b4 ,consid] n[ any p>ticul> p]son ,c   
7,fig4 #a7 on any  int]m$iate floor1 ,d 
& ,d 8 ne>e/ neighb\rs1 ,e & ,e  8 next 
ne>e/ neighb\rs all on 8 own floor4 ,8  
next next ne>e/  neighb\rs on t floor w 
be 9 ! positions ,f & ,f 9 !  diagram4  
,?us we see t ea* p]son ,c is surr\nd$  
by six p]sons1  ,,dd1 ,,ee & ,,ff1 be+ 8
 ne>e/1 8 next ne>e/1 & 8  next next    
ne>e/ neighb\rs on 8 own floor4 ,exclud+
 = simplicity  ! special cases (        
rectangul> gr\p+1 we see t ! angles (  !
 six equal & simil> triangles ,,cde1    
,,cef1 c41 are all acute3  & because !  
six triangles are equal & simil> we see 
t !  ?ree pairs ( mutually remote sides 
( ! hexagon ,,defd,e,f are  equal &     
p>allel4    ,illu/ration3 ,,fig4 #a     
#c4 ,let n[ ,a1 ,a1 ,a1 c41 d5ote places
 ( p]sons ( !  homog5e\s assemblage on !
 floor imm$iately above1 & ,b1 ,b1  ,b1 
c4 on ! floor imm$iately bel[1 ! floor (
 ,c4 ,9 !  diagram let a1 a1 a be po9ts 
9 :i* ! floor ( ,,cde is  cut by        
p]p5dicul>s to x ?r\gh ,a1 ,a1 ,a ( !   

floor above1  & b1 b1 b by p]p5dicul>s f
 ,b1 ,b1 ,b ( ! floor  bel[4 ,( all !   
p]p5dicul>s f ! floors imm$iately above 
& bel[1 j two1 "o f ea*1 cut ! area ( ! 
p>allelogram  ,,cdef3 & !y cut x 9 po9ts
 simil>ly situat$ 9 respect to  !       
oppositely ori5t$ triangles into :i* x  
is divid$ by ei!r  ( its diagonals4     
,h5ce if a lies 9 ! triangle ,,cde1 !   
o!r  five triangles ( ! hexagon mu/ be  
cut 9 ! correspond+ po9ts1  z %[n 9 !   
diagram4 ,?us1 if we ?9k only ( ! floor 
( ,c  & ( ! floor imm$iately above x1 we
 h po9ts ,a1 ,a1 ,a  v]tically above a1 
a1 a4 ,imag9e n[ a triangul> pyramid1   
or tetrah$ron1 /&+ on ! base ,,cde &    
hav+ ,a = v]tex3  we see t ea* ( its    
sides ,,acd1 ,,ade1 ,,aec1 is an acute  
angl$  triangle1 because1 z we h already
 se51 ,,cde is an acute angl$  triangle1
 & because ! %orte/ ( ! ?ree di/ances1  
,,ca1 ,,da1  ,,ea1 is 7 #b7 great] ?an  
,,ce 7?\gh x may be ei!r great] ?an  or 
less ?an ,,de74 ,h5ce ! tetrah$ron      
,,cdea has all its angles  acute2 n only

 ! angles ( its triangul> faces1 b ! six
 angles  betwe5 ! planes ( its f\r      
faces4 ,?is important !orem reg>d+      
homog5e\s assemblages 0 giv5 by         
,bravais1 to :om we owe ! :ole  doctr9e 
( homog5e\s assemblages 9 its mo/ p]fect
 simplicity &  complete g5]ality4       
,simil>ly we see t we h equal & simil>  
tetrah$rons on ! bases ,d,,cf1 ,e,f,c2 &
 ?ree o!r tetrah$rons  bel[ ! floor (   
,c1 hav+ ! oppositely ori5t$ triangles  
,,cd,e1 c4 = _! bases & ,b1 ,b1 ,b = _! 
v]tices4  ,^! ?ree tetrah$rons are equal
 & het]o*irally#a simil> to !  fir/     
?ree4 ,! consid]ation ( ^! acute angl$  
tetrah$rons1 is  ( fundam5tal importance
 9 respect to ! eng9e]+ ( an ela/ic     
solid1 or cry/al1 accord+ to ,boscovi*4 
,s also is ! consid]ation  ( ! clu/] (  
?irte5 po9ts ,c & ! six neighb\rs       
,,defd,e,f 9 ! plane ( ! diagram1 & !   
?ree neighb\rs  ,,aa,a on ! floor above1
 & ,,bb,b on ! floor bel[4     #d4 ,!   
case 9 :i* ea* ( ! f\r faces ( ea* ( !  
tetrah$rons (  #c is an equilat]al      

triangle is p>ticul>ly  int]e/+4 ,an    
assemblage fulfill+ ?is condition may   
conv5i5tly  be call$ an #8equilat]al    
homog5e\s assemblage1' or1 = brevity1 an
  #8equilat]al assemblage4' ,9 an       
equilat]al assemblage ,c's twelve       
neighb\rs are all equi-di/ant f x4 ,i   
hold 9 my h& a clu/] (  ?irte5 little   
black balls1 made up by tak+ "o ( !m &  
plac+  ! twelve o!rs 9 contact ) x 7&   
!re=e pack$ 9 !  close/ possible ord]71 
& fix+ !m all toge!r by fi%-glue4  ,y   
see x looks1 9 size1 col\r1 & %ape1 q l 
a mulb]ry4  ,! accompany+ diagram %[s a 
/]eoscopic view ( a simil> clu/]  (     
balls pa9t$ :ite = ! photograph4        
,illu/ration3 ,,fig4 #b4     #e4 ,by    
add+ ball aft] ball to su* a clu/] (    
?irte51 &  always tak+ c>e to place ea* 
additional ball 9 "s position 9  :i* x  
is prop]ly 9 l9e ) o!rs1 s z to make !  
:ole  assemblage homog5e\s1 we c ex]cise
 ourselves 9 a v int]e/+  mann] 9 !     
build+ up ( any possible =m ( cry/al ( !
 class  call$ #8cubic' by "s writ]s &   

#8octah$ral' by o!rs4 ,y see  be=e y    
sev]al examples4 ,i advise any ( y :o   
wi% to /udy  cry/allography to contract 
) a wood-turn]1 or a mak] ( beads =     
furniture tassels or = ros>ies1 = a ?\s&
 wood5 balls ( ab\t  half an in* diamet]
 ea*4 ,holes ?r\gh !m w d no h>m & may  
ev5 be useful2 b make sure t ! balls are
 z ne>ly equal to "o  ano!r1 & ea* z    
ne>ly sph]ical1 z possible4             
,illu/ration3 ,,fig4 #c4       #f4 ,y   
see "h be=e y a l>ge model :i* ,i h made
 to  illu/rate a homog5e\s assemblage ( 
po9ts1 on a plan fir/ giv51  ,i believe1
 by ,mr4 ,william ,b>l[ 7,nature1       
,decemb] #bj & #bg1 #ahhc74  ,! ro( ( ! 
model is a lattice-frame 7,fig4 #c7     
consi/+ ( two  sets ( eight p>allel     
wood5 b>s cross+ "o ano!r1 & kept       
toge!r by p9s ?r\gh ! middles ( !       
cross+s4 ,z y see1 ,i  c alt] x to make 
p>allelograms ( all degrees ( obliquity 
till  ! b>s t\*1 & aga9 y see ,i c make 
!m all squ>es4       #g4 ,! jo9t pivots 
are 7= *eapness ( con/ruction7 ( copp]  

wire1 ea* b5t to make a hook bel[ !     
lattice frame4 ,on ^!  sixty-f\r hooks  
are hung sixty-f\r f9e cords1 firmly    
/ret*$ by  little lead weights4 ,ea* (  
^! cords 7,fig4 #d7 be>s eight %ort     
p]=at$ wood5 cyl9d]s1 :i* may be slipp$ 
up & d[n to any  desir$ position#b4 ,!y 
are at pres5t actually plac$ at di/ances
  consecutively ea* equal to ! di/ance f
 jo9t to jo9t ( !  lattice frame4       
,illu/ration3 ,,fig4 #d4       #h4 ,!   
ro( ( ! model is hung by f\r cords1     
ne>ly v]tical1 (  indep5d5tly v>iable   
l5g?s1 pass+ ov] hooks f fix$ po9ts     
above1 & kept /ret*$ by weights1 ea*    
equal to "o qu>t] (  ! weight ( ro( &   
p5dants4 ,y see n[ by alt]+ ! angles  ( 
! lattice "w & plac+ x horizontal or 9  
any incl9$  plane1 z ,i am all[$ to d   
readily by ! mann] 9 :i* x is  hung1 ,i 
h ?ree indep5d5t v>iables1 by v>y+ :i*  
,i c %[  y all v>ieties ( homog5e\s     
assemblages1 9 :i* ?ree ( !  neighb\rs (
 e po9t are at equal di/ances f x4 ,y   
see  "h1 = example1 we h ! equilat]al   

assemblage4 ,i h adju/$  ! lattice ro(  
to ! prop] angle1 & its plane to ! prop]
  incl9ation to ! v]tical1 to make a    
:olly equilat]al assemblage  ( ! little 
cyl9d]s ( wood on ! v]tical cords1 a    
case1 z  we h se51 ( special importance4
 ,if ,i v>y also ! di/ances  betwe5 !   
little pieces ( wood on ! cords2 & !    
di/ances  betwe5 ! jo9ts ( ! lattice "w 
7v>iations easily und]/ood1  ?\gh n     
conv5i5tly producible 9 "o model )\t m (
  me*anical con/ruction ?an w\ld be wor?
 mak+71 ,i h ?ree  o!r indep5d5t        
v>iables4 ,by prop]ly v>y+ ^! six       
indep5d5t  v>iables1 ?ree angles & ?ree 
l5g?s1 we may give any assign$  value to
 ea* edge ( "o ( ! fundam5tal           
tetrah$rons (  #c4       #i4 ,our       
assemblage ( p w\ld n be homog5e\s      
unless its  memb]s 7 all equal & simil> 
& 9 precisely simil> attitudes1  & 7 all
 look+ ! same way4 ,y und]/& :at a numb]
 (  p seat$ or /&+ on a floor or pla9 & 
look+ ! same way  means4 ,b ! expression
 #8look+' is n conv5i5tly applicable to 

?+s t h no eyes1 & we want a m          
compreh5sive mode (  expression4 ,we h x
 9 ! words #8ori5tation1' #8ori5t$1' &  
7v]b7 #8to ori5t1' sugge/$ by an        
ext5sion ( ! idea involv$ 9  ! ^w       
#8ori5tation1' fir/ us$ to signify      
positions relatively  to ea/ & we/ (    
anci5t ,greek & ,egyptian temples &     
,*ri/ian  *ur*es4 ,b = ! ori5tation ( a 
h\se or temple we h only  "o angle1 & t 
angle is call$ #8azimu?' 7! "n giv5 to  
an  angle 9 a horizontal plane74 ,=     
ori5tation 9 ?ree dim5sions (  space we 
mu/ ext5d our ideas & consid] position )
 ref]5ce to  ea/ & we/ & up & d[n4 ,a   
man ly+ on 8 side ) 8 head  to ! nor? & 
look+ ea/1 w\ld n be simil>ly ori5t$ to 
a  man /&+ upright & look+ ea/4 ,to     
provide = ! complete  specification ( h[
 a body is ori5t$ 9 space we mu/ h 9 !  
body a plane ( ref]5ce1 & a l9e (       
ref]5ce 9 ?is plane1  belong+ to ! body 
& mov+ ) x4 ,we mu/ also h a fix$  plane
 & a fix$ l9e ( ref]5ce 9 x1 relatively 
to :i* !  ori5tation ( ! moveable body  

is to be specifi$2 z1 = example1  a     
horizontal plane & ! ea/ & we/          
horizontal l9e 9 x4 ,!  position ( a    
body is completely specifi$ :5 ! angle  
betwe5 !  plane ( ref]5ce belong+ to x1 
& ! fix$ plane is giv52 &  :5 ! angles  
betwe5 ! l9e ( int]section ( ! two      
planes &  ! l9es ( ref]5ce 9 !m are also
 giv54 ,?us we see t ?ree  angles are   
necess>y & suffici5t to specify !       
ori5tation ( a  moveable body1 & we see 
h[ ! specification is conv5i5tly giv5  9
 t]ms ( ?ree angles4       #aj4 ,to     
illu/rate ?is take a book ly+ on ! table
 be=e y )  its side next ! title-page   
up1 & its back to ! nor?4 ,i n[ lift  ! 
ea/ edge 7! top ( ! book71 keep+ !      
bottom edge nor? &  s\? on ! table till 
! book is incl9$1 let u say1 #bj to !   
table4 ,next1 )\t alt]+ ?is angle ( #bj1
 betwe5 ! side (  ! book & ! table1 ,i  
turn ! book r\nd a v]tical axis1 ?r\gh  
#de till ! bottom edge lies nor?-ea/ &  
s\?-we/4 ,la/ly1  keep+ ! book 9 ! plane
 to :i* x has be5 ?us br\ght1 ,i turn  x

 r\nd 9 ?is plane ?r\gh #ce4 ,^! ?ree   
angles ( #bj1 #de1  & #ce1 specify1 )   
ref]5ce to ! horizontal plane ( ! table 
& ! ea/ & we/ l9e 9 x1 ! ori5tation ( ! 
book 9 !  position to :i* y h se5 me br+
 x1 & 9 :i* ,i hold x  be=e y4          
#aa4 ,9 ,figs4 #e & #f y see two        
assemblages1 ea* ( twelve equal  &      
simil> molecules 9 a plane4 ,fig4 #e1 9 
:i* ! molecules are  all same-ways      
ori5t$1 is "o homog5e\s assemblage (    
tw5ty-f\r  molecules4 ,fig4 #f1 9 :i* 9 
"o set ( r[s ! molecules are  alt]nately
 ori5t$ two diff]5t ways1 may ei!r be   
reg>d$ z  two homog5e\s assemblages1 ea*
 ( twelve s+le molecules2 or "o         
homog5e\s assemblage ( twelve pairs ( ^?
 s+le molecules4    ,illu/ration3 ,,fig4
 #e4       #ab4 ,i mu/ n[ call y\r      
att5tion to a purely geometrical  "q#c (
 vital int]e/ ) respect to homog5e\s    
assemblages  9 g5]al1 & p>ticul>ly !    
homog5e\s assemblage ( molecules        
con/itut+ a cry/al3--:at c we take z    
#8!' b\nd>y or #8a'  b\nd>y enclos+ ea* 

molecule ) :atev] portion ( space ar\nd 
x we are at lib]ty to *oose = x1 &      
sep>at+ x f  neighb\rs & _! portions (  
space giv5 to !m 9 homog5e\s  fairness8 
,illu/ration3 ,,fig4 #f4       #ac4 ,if 
we _h only ma!matical po9ts to consid]  
we %\ld be at  lib]ty to *oose ! simple 
obvi\s p>tition+ by ?ree sets (  p>allel
 planes4 ,ev5 ?is may be done 9 an      
inf9ite numb] ( ways1  ?us3--,beg9n+ )  
any po9t ,p ( ! assemblage1 *oose any   
o!r  ?ree po9ts ,a1 ,b1 ,c1 f> or ne>1  
provid$ only t !y are  n 9 "o plane )   
,p1 & t !re is no o!r po9t ( !          
assemblage 9 ! l9es ,,pa1 ,,pb1 ,,pc1 or
 )9 ! volume ( !  p>allelepip$ ( :i* ^! 
l9es are cont]m9\s edges1 or )9  ! areas
 ( any ( ! faces ( ?is p>allelepip$4    
,!re w be  po9ts ( ! assemblage at ea* (
 ! corn]s ( ?is p>allelepip$  & at all !
 corn]s ( ! p>allelepip$s equal & simil>
 to  x :i* we f9d by draw+ sets (       
equi-di/ant planes p>allel to  its ?ree 
pairs ( faces4 7,a diagram is           
unnecess>y47 ,e po9t (  ! assemblage is 

?us at ! int]section ( ?ree planes1 :i* 
is  also ! po9t ( meet+ ( eight         
neighb\r+ p>allelepip$s4 ,%ift  n[ any  
"o ( ! po9ts ( ! assemblage to a        
position )9 !  volume ( any "o ( ! eight
 p>allelepip$s1 & give equal p>allel    
motions to all ! o!r po9ts ( !          
assemblage4 ,?us we h e  po9t 9 a       
p>allelepip$al cell ( its own1 & all !  
po9ts ( !  assemblage are simil>ly plac$
 9 _! cells1 :i* are !mselves  equal &  
simil>4       #ad4 ,b n[ if1 in/ead ( a 
s+le po9t = ea* memb] ( !  assemblage1  
we h a gr\p ( po9ts1 or a globe or cube 
or o!r  geometrical figure1 or an       
individual ( a homog5e\s assemblage (   
equal1 simil>1 simil>ly dress$1 &       
simil>ly ori5t$ ladies1  sitt+ 9 r[s1 or
 a homog5e\s assemblage ( trees closely 
plant$  9 regul> geometrical ord] on a  
plane ) equal & simil>  di/ributions (  
molecules1 & p>allel planes above &     
bel[1  we may f9d t ! be/ condition$    
plane-fac$ p>allelepip$al  p>tition+ :i*
 we c *oose w\ld cut (f portions prop]ly

  belong+ to "o molecule ( ! assemblage 
& give !m to ! cells  ( neighb\rs4 ,to  
f9d a cell enclos+ all t belongs to ea* 
individual1 = example1 e "p ( ea* lady's
 dress1 h[ev]  complexly x may be fold$ 
among portions ( ! equal & simil>       
dresses ( neighb\rs2 or1 e twig1 leaf1 &
 rootlet ( ea* "o  ( ! homog5e\s        
assemblage ( trees2 we mu/ alt] ! b\nd>y
 by  give-&-take across ! plane faces ( 
! primitive p>allelepip$al  cells1 s t  
ea* cell %all enclose all t belongs to  
"o  molecule1 & !re=e 7because ( !      
homog5e\sness ( !  p>tition+7 no?+      
belong+ to any o!r molecule4 ,!         
geometrical  problem ?us pres5t$1       
wond]fully complex z x may be 9 cases   
su* z "s ( ^? :i* ,i h sugge/$1 is      
easily p]=m$ =  any possible case if we 
beg9 ) any p>ticul> p>allelepip$al      
p>tition+ det]m9$ = correspond+ po9ts ( 
! assemblage  z expla9$ 9  #ac1 = any   
homog5e\s assemblage ( s+le  po9ts4 ,we 
may prescribe to ourselves t ! corn]s   
are to rema9  un*ang$1 b if s !y mu/ to 

beg9 ) ei!r 9 int]faces (  contact      
betwe5 ! individual molecules1 or 9     
vacant space among  ! molecules4 ,if ?is
 condition is fulfill$ = "o corn] x is  
fulfill$ = all1 z ! corn]s are          
ess5tially correspond+ po9ts  relatively
 to ! assemblage4       #ae4 ,beg9 n[ ) 
any "o ( ! twelve /raight l9es betwe5   
corn]s :i* con/itute ! twelve edges ( ! 
p>allelepip$1 &  alt] x arbitr>ily to   
any curv$ or crook$ l9e betwe5 ! same   
pair ( corn]s1 subject only to !        
conditions 7#a7 t x does n  p5etrate !  
sub/ance ( any memb] ( ! assemblage1 &  
7#b7 t  x is n cut by equal & simil>    
p>allel curves#d betwe5 o!r  pairs (    
corn]s4    ,illu/ration3 ,,fig4 #g4     
,consid]+ n[ ! ?ree f\rs ( p>allel edges
 ( !  p>allelepip$1 let ! /raight l9es (
 "o set ( f\r be alt]$ to  equal &      
simil> p>allel curves 9 ! mann] :i* ,i h
 describ$2  & proce$ by ! same rule = ! 
o!r two sets ( f\r edges4 ,we  ?us h    
?ree f\rs ( p>allel curv$ edges in/ead (
 ! ?ree  f\rs ( p>allel /raight edges ( 

our primitive p>allelepip$ )  corn]s    
7ea* a po9t ( int]section ( ?ree edges7 
un*ang$4  ,take n[ ! quadrilat]al ( f\r 
curves sub/itut$ = ! f\r  /raight edges 
( "o face ( ! p>allelepip$4 ,we may call
 ?is  quadrilat]al a curvil9eal         
p>allelogram1 because x is a circuit    
compos$ ( two pairs ( equal p>allel     
curves4 ,draw n[ a curv$  surface 7an   
inf9itely ?9 %eet ( p]fectly ext5sible  
india-rubb]  if y please to ?9k ( x s7  
bord]$ by ! f\r edges ( our  curvil9eal 
p>allelogram1 & s %ap$ z n to cut any ( 
!  sub/ance ( any molecule ( !          
assemblage4 ,d ! same ?+ )  an exactly  
simil> & p>allel %eet relatively to !   
opposite face  ( ! p>allelepip$2 & aga9 
! same = ea* ( ! two o!r  pairs (       
p>allel faces4 ,we ?us h a curv$-fac$   
p>allelepip$  enclos+ ! :ole ( "o       
molecule & no "p ( any o!r2 & by  simil>
 proc$ure we f9d a simil> b\nd>y = e o!r
 molecule  ( ! assemblage4 ,ea* wall (  
ea* ( ^! cells is common to two         
neighb\r+ molecules1 & !re is no vacant 

space any:]e betwe5  !m or at corn]s4   
,fig4 #g illu/rates ?is k9d ( p>tition+ 
by  %[+ a plane section p>allel to "o   
pair ( plane faces ( !  primitive       
p>allelepip$1 = an ideal case4 ,! plane 
diagram is 9  fact a realization ( !    
two-dim5sional problem ( p>tition+ !    
p9e patt]n ( a ,p]sian c>pet by         
p>allelograms ab\t z ne>ly  rectil9e> z 
we c make !m4 ,9 ! diagram fa9t /raight 
l9es  are drawn to %[ ! primitive       
p>allelogrammatic p>tition+4  ,x w be   
se5 t ( all ! cross+s 7m>k$ ) dots 9 !  
diagram7 e "o is simil>ly situat$ to e  
o!r 9 respect to  ! homog5e\sly repeat$ 
patt]n figures3 ,a1 ,b1 ,c1 ,d are f\r  
( !m at ! corn]s ( "o cell4       #af4  
,conf9+ our att5tion = a %ort "t to !   
homog5e\s  division ( a plane1 rem>k t !
 division into p>allelograms  by two    
sets ( cross+ p>allels is s+ul> 9 ?is   
respect--ea*  cell is contigu\s ) ?ree  
neighb\rs at e corn]4 ,any  %ift+1 l>ge 
or small1 ( ! p>allelograms by relative 
slid+  9 "o direction or ano!r violates 

?is condition1 br+s u to  a             
configuration l t ( ! faces ( regul>ly  
hewn /ones 9  ord9>y bond$ masonry1 &   
gives a p>tition+ :i* fulfils !         
condition t at ea* corn] ea* cell has   
only two neighb\rs4 ,ea*  cell is n[    
virtually a hexagon1 z w be se5 by !    
lett]s ,a1  ,b1 ,c1 ,d1 ,e1 ,f 9 !      
diagram ,fig4 #h4 ,a & ,d are to be     
reckon$ z corn]s1 ea* ) an int]ior angle
 ( #ahj4 ,9 ?is  diagram ! cont9u\s     
heavy l9es & ! cont9u\s fa9t l9es       
cross+ !m %[ a primitive                
p>allelogrammatic p>tition by two sets  
( cont9u\s p>allel int]sect+ l9es4 ,!   
int]rupt$ cross+  l9es 7heavy7 %[1 = !  
same homog5e\s di/ribution ( s+le  po9ts
 or molecules1 ! virtually hexagonal    
p>tition+ :i* we get  by %ift+ ! b\nd>y 
f ea* portion ( "o ( ! light l9es to  ! 
heavy l9e next x betwe5 ! same cont9u\s 
p>allels4    ,illu/ration3 ,,fig4 #h4   
,fig4 #h bis repres5ts a fur!r          
modification ( ! b\nd>y by  :i* ! #ahj  
angles ,a1 ,d1 become angles ( less ?an 

#ahj4 ,!  cont9u\s p>allel l9es 7light7 
& ! %ort light portions ( !  cross+ l9es
 %[ ! configuration accord+ to ,fig4 #h1
 f :i*  ?is diagram is d]iv$4       #ag4
 ,9 ^! diagrams 7,figs4 #h & #h bis7 !  
object enclos$  is small en\gh to be    
enclosable by a primitive               
p>allelogrammatic  p>tition+ ( two sets 
( cont9u\s cross+ p>allel /raight  l9es1
 & by ! p>tition+ ( #8bond$'            
p>allelograms bo?  repres5t$ 9 ,fig4 #h1
 & by ! d]iv$ hexagonal p>tition+       
repres5t$ 9 ,fig4 #h bis1 ) fa9t l9es   
%[+ ! primitive &  ! second>y           
p>allelograms4 ,9 ,fig4 #g ! objects    
enclos$ 7  too l>ge to be enclosable by 
any rectil9e> p>allelogrammatic  or     
hexagonal p>tition+4 ,! two sets (      
p>allel fa9t l9es 9  ,fig4 #g %[ a      
primitive p>allelogrammatic p>tition+ & 
!  correspond+ pairs ( p>allel curves   
int]sect+ at ! corn]s (  ^!             
p>allelograms1 ( :i* ,a1,b1,c1,d is a   
specim51 %[ a  correspond+ p>tition+ by 
curvil9eal p>allelograms4 ,fig4 #i %[s  

= ! same homog5e\s di/ribution ( objects
 a bett] condition$  p>tition+1 by      
hexagons 9 ea* ( :i* "o pair ( p>allel  
edges is curv$4 ,! sets ( int]sect+     
p>allel /raight l9es 9  ,fig4 #i %[ !   
same primitive p>allelogrammatic        
p>tition+ z  9 ,fig4 #g1 & ! same       
slightly %ift$ to suit po9ts *os5 =     
well-condition$ness ( hexagonal         
p>tition+4    ,illu/ration3 ,,fig4 #h   
bis4    ,illu/ration3 ,,fig4 #i4        
#ah4 ,= ! division ( cont9u\s           
?ree-dim5sional space#e into  equal1    
simil>1 & simil>ly ori5t$ cells1 q a    
correspond+  trans=mation f p>tition+ by
 ?ree sets ( cont9u\s mutually          
int]sect+ p>allel planes to any possible
 mode ( homog5e\s  p>tition+1 may be    
inve/igat$ by work+ out !               
?ree-dim5sional  analogue (  #af-ag4    
,?us we f9d t ! mo/ g5]al possible      
homog5e\s p>tition+ ( space ) plane     
int]faces betwe5 !  cells gives u f\rte5
 walls to ea* cell1 ( :i* six are ?ree  
pairs ( equal & p>allel p>allelograms1 &

 ! o!r eight are  f\r pairs ( equal &   
p>allel hexagons1 ea* hexagon be+ b\nd$ 
by ?ree pairs ( equal & p>allel /raight 
l9es4 ,?is figure1 be+  b\nd$ by f\rte5 
plane faces1 is call$ a                 
tetrakaidekah$ron4 ,x  has ?irty-six    
edges ( int]section betwe5 faces2 &     
tw5ty-f\r  corn]s1 9 ea* ( :i* ?ree     
faces int]sect4 ,a p>ticul> case  ( x1  
:i* ,i call an or?ic tetrakaidekah$ron1 
be+ t 9  :i* ! six p>allelograms are    
equal squ>es1 ! eight hexagonal  faces  
are equal equilat]al & equiangul>       
hexagons1 & ! l9es  jo9+ correspond+    
po9ts 9 ! sev5 pairs ( p>allel faces    
are p]p5dicul> to ! planes ( ! faces1 is
 repres5t$ by  a /]eoscopic picture 9   
,fig4 #aj4 ,! ?irty-six edges & !       
tw5ty-f\r corn]s1 :i* are easily c\nt$ 9
 ?is diagram1 occur  9 ! same relative  
ord] 9 ! mo/ g5]al possible p>tition+1  
:e!r by plane-fac$ tetrakaidekah$rons or
 by ! g5]aliz$  tetrakaidekah$ron       
describ$ 9  #ai4       #ai4 ,! mo/ g5]al
 homog5e\s division ( space is n  limit$

 to plane-fac$ cells2 b x /ill consi/s  
ess5tially (  tetrakaidekah$ronal cells1
 ea* b\nd$ by ?ree pairs ( equal &      
p>allel quadrilat]al faces1 & f\r pairs 
( equal & p>allel  hexagonal faces1     
nei!r ! quadrilat]als nor ! hexagons be+
  necess>ily plane4 ,ea* ( ! ?irty-six  
edges may be /raight  or crook$ or      
curv$2 ! pairs ( opposite edges1 :e!r ( 
!  quadrilat]als or hexagons1 ne$ n be  
equal & p>allel2 nei!r  ! f\r corn]s (  
ea* quadrilat]al nor ! six corn]s ( ea* 
hexagon ne$ be 9 "o plane4 ,b e pair (  
correspond+ edges (  e pair ( p>allel   
correspond+ faces1 :e!r quadrilat]al or 
hexagonal1 mu/ be equal & p>allel4 ,i h 
describ$ an int]e/+  case ( p>tition+ by
 tetrakaidekah$rons ( curv$ faces )     
curv$  edges 9 a pap]#f publi%$ ab\t    
sev5 ye>s ago4 ,9 ?is case ea*  ( !     
quadrilat]al faces is plane4 ,ea*       
hexagonal face is a slightly  curv$     
surface hav+ ?ree rectil9eal diagonals  
?r\gh its c5tre 9  "o plane4            
,illu/ration3 ,,fig4 #aj4    ,! six     

sectors ( ! face betwe5 ^! diagonals lie
 alt]nately  on opposite sides ( _!     
plane1 & are bord]$ by six arcs  ( plane
 curves ly+ on ?ree pairs ( p>allel     
planes4 ,?is  tetrakaidekah$ronal       
p>tition+ fulfils ! condition t !       
angles betwe5 ?ree planes meet+ 9 an    
edge are ev]y:]e ea*  #abj2 a condition 
t _c be fulfill$ 9 any plane-fac$       
tetrakaidekah$ron4 ,ea* hexagonal wall  
is an anticla/ic surface (  equal       
opposite curvatures at e po9t1 be+ !    
surfaces ( m9imum  area bord]$ by six   
curv$ edges4 ,x is %[n easily &         
beautifully1  & ) a fair approa* to     
accuracy1 by *oos+ six little circul>   
arcs ( wire1 & sold]+ !m toge!r by _!   
ends 9 prop]  planes = ! six edges ( !  
hexagon2 & dipp+ x 9 soap  solution &   
tak+ x out4       #bj4 ,return+ n[ to ! 
tactics ( a homog5e\s assemblage1 rem>k 
t ! qualities ( ! assemblage z a :ole   
dep5d bo? ^u !  "* & ori5tation ( ea*   
molecule1 & on ! "* (  ! homog5e\s      
assemblage =m$ by correspond+ po9ts ( ! 

molecules4 ,aft] le>n+ ! simple         
ma!matics ( cry/allography1  ) its      
indicial sy/em#g = def9+ ! faces & edges
 ( a  cry/al accord+ to ! ,bravais r[s &
 nets & tetrah$rons (  molecules 9 :i*  
we ?9k only ( a homog5e\s assemblage (  
po9ts1  we are apt to =get t ! true     
cry/all9e molecule1 :atev] its  nature  
may be1 has sides1 & t g5]ally two      
opposite sides ( ea*  molecule may be   
expect$ to be v diff]5t 9 quality1 & we 
are  almo/ surpris$ :5 m9]alogi/s tell u
 t two p>allel faces on  two sides ( a  
cry/al h v diff]5t qualities 9 _m       
natural  cry/als4 ,we might almo/ z well
 be surpris$ to f9d t an army 9  battle 
array1 :i* is a k9d ( l>ge-gra9$ cry/al1
 pres5ts v  diff]5t appe>ance to any "o 
look+ at x f outside1 accord+  z e man 9
 ! ranks ) 8 rifle & bayonet faces to ! 
front  or to ! re> or to "o flank or to 
! o!r4       #ba4 ,consid]1 = example1 !
 ideal case ( a cry/al consi/+  ( h>d   
equal & simil> tetrah$ronal solids all  
same-ways ori5t$4  ,a ?9 plate ( cry/al 

cut p>allel to any "o set ( ! faces (  !
 con/itu5t tetrah$rons w\ld h v diff]5t 
prop]ties on  its two sides2 z !        
con/itu5t molecules w\ld all pres5t     
po9ts  outw>ds on "o side & flat        
surfaces on ! o!r4 ,we might expect  t !
 two sides ( su* a plate ( cry/al w\ld  
become oppositely  electrifi$ :5 rubb$  
by "o & ! same rubb]2 &1 rememb]+  t a  
piece ( glass ) "p ( its surface f9ely  
gr\nd b n  poli%$ & o!r p>ts poli%$     
becomes1 :5 rubb$ ) :ite  silk1         
positively electrifi$ ov] ! poli%$ p>ts 
& negatively  electrifi$ ov] !          
non-poli%$ p>ts1 we might almo/ expect t
  ! side ( our suppos$ cry/all9e plate  
t[>ds :i* flat faces  ( ! con/itu5t     
molecules are turn$ w\ld become         
positively  electrifi$1 & ! opposite    
side1 %[+ free molecul> corn]s1  w\ld   
become negatively electrifi$1 :5 bo? are
 rubb$ by a rubb]  ( int]m$iate electric
 quality4 ,we might also f elem5t>y  k (
 ! fact ( piezo-electricity1 t is to    
say1 !  developm5t ( opposite           

electricities on ! two sides ( a cry/al 
by pressure1 expect t our suppos$       
cry/all9e plate1 if press$  p]p5dicul>ly
 on its two sides1 w\ld become          
positively electrifi$  on "o ( !m &     
negatively on ! o!r4       #bb4         
,intimately connect$ ) ! subject (      
enclos+ cells =  molecules ( giv5 %ape1 
assembl$ homog5e\sly1 is ! homog5e\s    
pack+ toge!r ( equal & simil> molecules 
( any giv5 %ape4 ,9  e possible case (  
any inf9itely great numb] ( simil>      
bodies  ! solution is a homog5e\s       
assemblage4 ,b x may be a homog5e\s     
assemblage ( s+le solids all ori5t$ !   
same way1 or x may be  a homog5e\s      
assemblage ( clu/]s ( two or m ( !m     
plac$  toge!r 9 diff]5t ori5tations4 ,= 
example1 let ! giv5 bodies  be halves   
7oblique or n oblique7 ( any            
p>allelepip$ on ! two  sides ( a divid+ 
plane ?r\gh a pair ( p>allel edges4 ,!  
two  halves are homo*irally#h simil>2 &1
 be+ equal1 we may make a  homog5e\s    
assemblage ( !m by ori5t+ !m all ! same 

way  & plac+ !m prop]ly 9 r[s4 ,b !     
close/ pack+ ( ?is  assemblage w\ld     
necess>ily leave vacant spaces betwe5 ! 
bodies3  & we get 9 reality ! close/    
possible pack+ ( ! giv5 bodies  by tak+ 
!m 9 pairs oppositely ori5t$ & plac$    
toge!r to =m  p>allelepip$s4 ,^! clu/]s 
may be pack$ toge!r s z to leave  no    
unoccupi$ space4    ,:atev] ! numb] (   
pieces 9 a clu/] 9 ! close/ possible    
pack+ ( solids may be = any p>ticul>    
%ape1 we may consid] ea*  clu/] z itself
 a giv5 s+le body1 & ?us r$uce ! problem
 to  ! pack+ closely toge!r (           
assemblages ( individuals all sameways  
ori5t$2 & to ?is problem !re=e x is     
conv5i5t t we %\ld  n[ conf9e our       
att5tion4       #bc4 ,to avoid          
complexities su* z ^? :i* we f9d 9 !    
famili>  problem ( homog5e\s pack+ ( =ks
 or spoons or tea-cups or b[ls1  ( any  
ord9>y %ape1 we %all suppose ! giv5 body
 to be ( su*  %ape t no two ( !m        
simil>ly ori5t$ c t\* "o ano!r  9 m ?an 
"o po9t4 ,:olly convex bodies ess5tially

 fulfil ?is  condition2 b x may also be 
fulfill$ by bodies n :olly convex1 z  is
 illu/rat$ 9 ,fig4 #aa4    ,illu/ration3
 ,fig4 #aa4       #bd4 ,to f9d close &  
close/ pack+ ( any numb] ( our solids   
,s#a1 ,s#b1 ,s#c 444 ( %ape fulfill+ !  
condition (  #bc  proce$ ?us3--    7#a7 
,br+ ,s#b to t\* ,s#a at any *os5 po9t p
 ( its  surface 7,fig4 #ab74    7#b7    
,br+ ,s#c to t\* ,s#a & ,s#b1 at r & q  
respectively4    7#c7 ,br+ ,s#d 7n %[n 9
 ! diagram7 to t\* ,s#a1 ,s#b1  & ,s#c4 
7#d7 ,place1 any numb] ( ! bodies toge!r
 9 ?ree r[s cont9u+  ! l9es ( ,s#a,s#b1 
,s#a,s#c1 ,s#a,s#d1 & 9 ?ree  sets (    
equi-di/ant r[s p>allel to ^!4 ,?is     
makes a homog5e\s  assemblage4 ,9 !     
assemblage s =m$ ! molecules are        
necess>ily  f\nd to be 9 ?ree sets ( r[s
 p>allel respectively to ! ?ree  pairs  
,s#b,s#c1 ,s#c,s#d1 ,s#d,s#b4 ,! :ole   
space  occupi$ by an assemblage ( n (   
our solids ?us arrang$ has  cle>ly #fn  
times ! volume ( a tetrah$ron (         
correspond+ po9ts  ( ,s#a1 ,s#b1 ,s#c1  

,s#d4 ,h5ce ! close/ ( !  close pack+s  
obta9$ by ! op]ations 7#a7 444 7#d7 is  
f\nd if we  p]=m ! op]ations 7#a71 7#b71
 & 7#c7 z to make ! volume ( ?is        
tetrah$ron lea/ possible4               
,illu/ration3 ,,fig4 #ab       #be4 ,x  
is to be rem>k$ t op]ations 7#a7 & 7#b7 
leave = 7#c7  no lib]ty ( *oice = !     
place ( ,s#d1 except betwe5 two         
det]m9ate positions on opposite sides ( 
! gr\p ,s#a1 ,s#b1  ,s#c4 ,! volume ( ! 
tetrah$ron w g5]ally be diff]5t =  ^!   
two positions ( ,s#d1 &1 ev5 if ! volume
 *ance to be  equal 9 any case1 we h    
diff]5tly %ap$ assemblages accord+ z  we
 *oose "o or o!r ( ! two places = ,s#d4 
,?is w be und]/ood by look+ at ,fig4    
#ab1 %[+ ,s#a  & neighb\rs on ea* side (
 x 9 ! r[s ( ,s#a,s#b1  ,s#a,s#c1 & 9 a 
r[ p>allel to t ( ,s#b,s#c4 ,! plane  ( 
! diagram is p>allel to ! planes (      
correspond+ po9ts (  ^! sev5 bodies1 & !
 diagram is a projection ( ^! bodies  by
 l9es p>allel to ! int]sections ( !     
tang5t planes ?r\gh  p & r4 ,if ! ?ree  

tang5t planes ?r\gh p1 q1 & r1          
int]sect$ 9 p>allel l9es1 q w\ld be se5 
l p & r z a  po9t ( contact betwe5 !    
outl9es ( two ( ! bodies2 b ?is is  only
 a p>ticul> case1 & 9 g5]al q mu/1 z    
indicat$ 9 !  diagram1 be conceal$ by "o
 or o!r ( ! two bodies ( :i* x is  !    
po9t ( contact4 ,n[ imag9+1 to fix our  
ideas & facilitate  brevity (           
expression1 t ! planes ( correspond+    
po9ts ( !  sev5 bodies are horizontal1  
we see cle>ly t ,s#d may be br\ght  into
 prop] position to t\* ,s#a1 ,s#b1 &    
,s#c ei!r f  above or f bel[2 & t !re is
 "o det]m9ate place = x if  we br+ x    
into position f above1 & ano!r det]m9ate
 place =  x if we br+ x f bel[4         
#bf4 ,if we look f above at ! solids (  
:i* ,fig4 #ab %[s !  outl9e1 we see     
ess5tially a holl[ lead+ d[n to a       
p]=ation  betwe5 ,s#a1 ,s#b1 ,s#c1 & if 
we look f bel[ we see a  holl[ lead+    
upw>ds to ! same p]=ation3 ?is = brevity
 we  %all call ! p]=ation pqr4 ,!       
diagram %[s ar\nd ,s#a  six holl[s lead+

 d[n to p]=ations1 ( :i* two are simil> 
to pqr1 & ! o!r ?ree1 ( :i* pqr         
indicates "o1 are  simil> "o to ano!r b 
are dissimil> to pqr4 ,if we br+ ,s#d  f
 above into position to t\* ,s#a1 ,s#b1 
& ,s#c1 its  place ?us f\nd is 9 ! holl[
 pqr1 & ! places ( all !  solids 9 !    
lay] above t ( ! diagram are necess>ily 
9 !  holl[s simil> to pqr4 ,9 ?is case !
 solids 9 ! lay] bel[  t ( ! diagram mu/
 lie 9 ! holl[s bel[ ! p]=ations        
dissimil> to pqr1 9 ord] to make a s+le 
homog5e\s assemblage4  ,9 ! o!r case1   
,s#d br\ght up f bel[ f9ds its place on 
!  "u side ( ! holl[ pqr1 & all solids (
 ! l[] lay] f9d  simil> places3 :ile    
solids 9 ! lay] above t ( ! diagram  f9d
 _! places 9 ! holl[s simil> to pqr4 ,9 
! fir/  case !re are no bodies ( ! upp] 
lay] 9 ! holl[s above !  p]=ations      
simil> to pqr1 & no bodies ( ! l[] lay] 
9  ! holl[s bel[ ! p]=ations simil> to  
pqr4 ,9 ! second  case !re are no bodies
 ( ! upp] lay] 9 ! holl[s above !       
p]=ations simil> to pqr1 & none ( ! "u  

lay] 9 !  holl[s bel[ ! p]=ations simil>
 to pqr4       #bg4 ,go+ back n[ to     
op]ation 7#a7 (  #bc1 rem>k t :5 !  po9t
 ( contact p is arbitr>ily *os5 on "o ( 
! two bodies  ,s#a1 ! po9t ( contact on 
! o!r w be ! po9t on x  correspond+ to !
 po9t or "o ( ! po9ts ( ,s#a1 :]e its   
tang5t plane is p>allel to ! tang5t     
plane at p4 ,if ,s#a  is :olly convex x 
has only two po9ts at :i* ! tang5t      
planes  are p>allel to a giv5 plane1 &  
!re=e ! op]ation 7#a7 is  det]m9ate &   
unambigu\s4 ,b if !re is any concavity  
!re w  be f\r or "s great] ev5 numb] (  
tang5t planes p>allel to any  "o ( "s   
planes1 :ile !re w be o!r planes to ea* 
( :i*  only "o pair ( tang5t planes is  
p>allel4 ,h5ce1 op]ation 7#a71  ?\gh    
/ill det]m9ate1 w h a multiplicity (    
solutions1 or  only a s+le solution1    
accord+ to ! *oice made ( ! position (  
p4    ,h5ce=? h[ev]1 to avoid ne$less   
complications ( ideas1 we %all  suppose 
our solids to be :olly convex2 & ( "s   
su* unsymmetrical  %ape z ^? indicat$ 9 

,fig4 #ab (  #be1 & %[n by /]eoscopic   
photograph 9 ,fig4 #ac (  #cf4 ,) or )\t
 ?is conv5i5t  limitation1 op]ation 7#a7
 has two fre$oms1 z p may be *os5       
freely on ! surface ( ,s#a2 & op]ation  
7#b7 has cle>ly j  "o fre$om aft]       
op]ation 7#a7 has be5 p]=m$4 ,?us1 = a  
solid  ( any giv5 %ape1 we h ?ree       
disposables1 or1 z commonly call$  9    
ma!matics1 ?ree #8indep5d5t v>iables1'  
all free = mak+ a  homog5e\s assemblage 
accord+ to ! rule (  #bb4       #bh4 ,9 
! homog5e\s assemblage def9$ 9  #bd1 ea*
 solid1  ,s#a1 is t\*$ at twelve po9ts1 
be+ ! ?ree po9ts (  contact ) ,s#b1     
,s#c1 ,s#d1 & ! ?ree #c's ( po9ts  on   
,s#a correspond+ to ! po9ts on ,s#b1    
,s#c1 ,s#d1  at :i* ^! bodies are t\*$  
by ! o!rs ( ! qu>tet4 ,?is  /atem5t is  
some:at difficult to foll[1 & we see m  
cle>ly  ! twelve po9ts ( contact by n   
conf9+ our att5tion to !  qu>tet ,s#a1  
,s#b1 ,s#c1 ,s#d 7conv5i5t z ?is is  =  
"s purposes71 b complet+ ! assemblage & 
consid]+  six neighb\rs ar\nd ,s#a 9 "o 

plane lay] ( ! solids z  %[n 9 ,fig4    
#ab1 ) _! six po9ts prqprq ( contact )  
,s#a2 & ! ?ree neighb\rs ( ! two adjac5t
 p>allel lay]s  :i* t\* x above & bel[4 
,?is clu/] ( ?irte51 ,s#a  & twelve     
neighb\rs1 is %[n = ! case ( sph]ical   
bodies 9  ! /]eoscopic photograph (  #d 
above4 ,we might ( c\rse1 if we  pleas$1
 h begun ) ! plane lay] ( :i* ,s#a1     
,s#b1  ,s#d are memb]s1 or ) t ( :i*    
,s#a1 ,s#c1 ,s#d  are memb]s1 or ) !    
plane lay] p>allel to ! f\r? side  ,s#b 
,s#c ,s#d ( ! tetrah$ron3 & ?us we h f\r
  diff]5t ways ( gr\p+ ! twelve po9ts ( 
contact on ,s#a into  "o set ( six & two
 sets ( ?ree4       #bi4 ,9 ?is         
assemblage we h :at ,i call #8close     
ord]' or #8close  pack+4' ,= close/ (   
close pack+s ! volume ( ! tetrah$ron  7 
#bd7 ( correspond+ po9ts ( ,s#a1 ,s#b1  
,s#c1 &  ,s#d mu/ be a m9imum1 & ! lea/ 
( m9imums if1 z g5]ally  w be ! case1   
!re are two m diff]5t configurations =  
ea* ( :i* ! volume is a m9imum4 ,!re w 9
 g5]al also be  configurations ( m9imax 

volume & ( maximum volume1 subject to  !
 condition t ea* body is t\*$ by twelve 
simil>ly ori5t$  neighb\rs4       #cj4  
,pause = a mom5t to consid] ! int]e/+   
k9ematical  & dynamical problems pres5t$
 by a close homog5e\s assemblage  (     
smoo? solid bodies ( giv5 convex %ape1  
:e!r p]fectly  frictionless or ex]t+    
resi/ance aga9/ mutual slid+ accord+ to 
! ord9>ily /at$ law ( friction betwe5   
dry h>d solid bodies4  ,fir/ imag9e t !y
 are all simil>ly ori5t$ & ea* 9 contact
  ) twelve neighb\rs1 except outly+     
individuals 7:i* !re mu/  be at ! b\nd>y
 if ! assemblage is f9ite1 & ea* ( :i*  
is  t\*$ by "s numb] ( neighb\rs less   
?an twelve74 ,! coh]5t  assemblage ?us  
def9$ con/itutes a k9ematic frame or    
skeleton  = an ela/ic solid ( v peculi> 
prop]ties4 ,in/ead ( ! six  fre$oms1 or 
disposables1 ( /ra9 pres5t$ by a natural
 solid x has  only ?ree4 ,*ange ( %ape (
 ! :ole c only take place 9 virtue  (   
rotation ( ! con/itu5t p>ts relatively  
to any "o *os5 r[  ( !m1 & ! plane ?r\gh

 x & ano!r *os5 r[4       #ca4 ,suppose 
fir/ ! solids to be n only p]fectly     
smoo? b  p]fectly frictionless4 ,let !  
assemblage be subject$ to equal         
positive or negative pressure inw>ds all
 ar\nd its b\nd>y4 ,e  position (       
m9imum1 m9imax1 or maximum volume w be a
 position (  equilibrium4 ,if ! pressure
 is positive ! equilibrium w be /able   
if1 & un/able unless1 ! volume is a     
m9imum4 ,if ! pressure  is negative !   
equilibrium w be /able if1 & un/able    
unless1  ! volume is a maximum4         
,configurations ( m9imax volume w be    
ess5tially un/able4       #cb4 ,consid] 
n[ ! assemblage (  #ca 9 a position (   
/able  equilibrium "u ! influ5ce ( a    
giv5 con/ant uni=m pressure  inw>ds all 
r\nd its b\nd>y4 ,x w h rigidity 9      
simple  proportion to ! am\nt ( ?is     
pressure4 ,if n[ by ! sup]position  (   
non-uni=m pressure at ! b\nd>y1 =       
example equal & opposite  pressures on  
two sides ( ! assemblage1 a f9ite *ange 
( %ape is  produc$3 ! :ole assemblage   

ess5tially swells 9 bulk4 ,?is is !     
#8dilatancy' :i* ,osborne ,reynolds has 
describ$#i 9 an exce$+ly  int]e/+ mann] 
) ref]5ce to a sack ( :eat or s&1 or an 
india-rubb] bag tightly fill$ ) s& or   
ev5 small %ot4 ,consid]1  = example1 a  
sack ( :eat fill$ q full & /&+ up op54  
,x  is limp & flexible4 ,n[ %ake x d[n  
well1 fill x q full1 %ake  aga91 s z to 
get z mu* into x z possible1 & tie ! m\?
 v  tightly close4 ,! sack becomes almo/
 z /iff z a log ( wood (  ! same %ape4  
,op5 ! m\? p>tially1 & x becomes aga9   
limp1  especially 9 ! upp] p>ts ( ! bag4
 ,9 ,reynolds' obs]vations on           
india-rubb] bags ( small %ot 8          
#8dilatancy' dep5ds1 ess5tially  &      
:olly1 on brea*es ( "s ( ! contacts :i* 
exi/ betwe5 !  molecules 9 _!           
configuration ( m9imum volume3 & x is   
possible  t 9 all 8 cases ! dilatations 
:i* he obs]v$ are *iefly1  if n :olly1  
due to su* brea*es ( contact4    ,b x is
 possible1 x almo/ seems probable1 t 9  
bags or boxes  ( s& or p[d]1 ( "s k9ds (

 smoo? r\nd$ bodies ( any  %ape1 n      
sph]ical or ellipsoidal1 subject$       
p]si/5tly to unequal  pressures 9       
diff]5t directions1 & well %ak51 /able  
positions  ( equilibrium are f\nd )     
almo/ all ! p>ticles ea* t\*$ by  twelve
 o!rs4    ,"h is a curi\s subject (     
,natural ,hi/ory ?r\gh all ages till    
#ahhe1 :5 ,reynolds br\ght x into !     
prov9ce ( ,natural ,philosophy  by !    
foll[+ highly int]e/+ /atem5t3--#8,a    
well-m>k$  ph5om5on receives its        
explanation at once f ! exi/5ce (       
dilatancy 9 s&4 ,:5 ! fall+ tide leaves 
! s& firm1 z !  foot falls on x ! s&    
:it5s & appe>s mom5t>ily to dry r\nd  ! 
foot4 ,:5 ?is happ5s ! s& is full (     
wat]1 ! surface (  :i* is kept up to t (
 ! s& by capill>y attractions2 !        
pressure ( ! foot caus+ dilatation ( !  
s& m wat] is  requir$1 :i* has to be    
obta9$ ei!r by depress+ ! level (  !    
surface aga9/ ! capill>y attractions1 or
 by draw+ wat]  ?r\gh ! int]/ices ( !   
surr\nd+ s&4 ,?is latt] requires  "t to 

accompli%1 s t = ! mom5t ! capill>y =ces
 are  ov]come2 ! surface ( ! wat] is    
l[]$ bel[ t ( ! s&1  leav+ ! latt] :ite 
or dri] until a suffici5t supply has be5
  obta9$ f bel[1 :5 ! surface rises &   
wets ! s& aga94  ,on rais+ ! foot x is  
g5]ally se5 t ! s& "u ! foot  & ar\nd   
becomes mom5t>ily wet2 ?is is because1  
on ! di/ort+  =ces be+ remov$1 ! s& aga9
 contracts1 & ! excess ( wat]  f9ds     
mom5t>y relief at ! surface4'    ,?is   
proves t ! s& "u ! foot1 z well z !     
surface ar\nd  x1 mu/ be dry = a %ort "t
 aft] ! foot is press$ ^u x1  ?\gh we _c
 see x :it5$1 z ! foot is n transp>5t4  
,t  x is s has be5 v]ifi$ by ,mr4 ,alex4
 ,galt1 ,exp]im5tal ,in/ructor  9 !     
,physical ,laboratory ( ,glasg[         
,univ]sity1 by lay+ a small  squ>e (    
plate-glass on wet s& on ! sea-%ore (   
,hel5sburgh1 &  sudd5ly press+ on x by a
 /\t /ick ) ne>ly all 8 weight4  ,he    
f\nd ! s&1 bo? "u ! glass & ar\nd x 9   
contact )  ! air1 all became :ite at !  
same mom5t4 ,( all ! two hundr$  ?\s&   

million m51 wom51 & *ildr5 :o1 f !      
beg9n+  ( ! _w1 h ev] walk$ on wet s&1  
h[ _m1 prior to !  ,briti% ,association 
,meet+ at ,ab]de5 9 #ahhe1 if ask$1     
#8,is ! s&  compress$ "u y\r foot8' w\ld
 h answ]$ o!rwise ?an #8,yes6'8         
7,contra/ ) ?is ! case ( walk+ ov] a b$ 
( wet sea-we$67       #cc4 ,9 ! case (  
globes pack$ toge!r 9 close/ ord] 7&    
!re=e also 9 ! case ( ellipsoids1 if all
 simil>ly ori5t$71  our condition (     
coh]5t contact betwe5 ea* molecule &    
twelve  neighb\rs implies absolute      
rigidity ( =m & con/ancy ( bulk4  ,h5ce 
our convex solid mu/ be nei!r           
ellipsoidal nor sph]ical  9 ord] t !re  
may be ! *anges ( =m & *anges ( bulk    
:i* we h be5 consid]+ z dep5d5t on ?ree 
indep5d5t  v>iables specify+ !          
ori5tation ( ea* solid relatively to r[s
  ( ! assemblage4 ,an int]e/+ dynamical 
problem is pres5t$ by  suppos+ any      
mutual =ces1 su* z might be produc$ by  
spr+s1 to  act betwe5 ! solid molecules1
 & inve/igat+ configurations (          

equilibrium on ! supposition (          
frictionless contacts4 ,! solution  ( x 
( c\rse is t ! pot5tial en]gy ( ! spr+s 
mu/ be a  m9imum or a m9imax or a       
maximum = equilibrium1 & a m9imum =     
/able equilibrium4 ,! solution w be a   
configuration ( m9imum or  m9imax1 or   
maximum1 volume1 only 9 ! case (        
pressure equal 9 all  directions4       
#cd4 ,a purely geometrical "q1 ( no     
importance 9 respect to  ! molecul>     
tactics ( a cry/al b ( consid]able      
int]e/ 9  pure ma!matics1 is =c$ on our 
att5tion by our hav+ se5 7 #bg7  t a    
homog5e\s assemblage ( solids ( giv5    
%ape1 ea* t\*$  by twelve neighb\rs1 has
 ?ree fre$oms :i* may be conv5i5tly     
tak5 z ! ?ree angles specify+ !         
ori5tation ( ea* molecule  relatively to
 r[s ( ! assemblage z expla9$ 9  #cj4   
,consid] a solid ,s#a & ! twelve        
neighb\rs :i* t\* x1  & try if x is     
possible to cause x to t\* m ?an twelve 
(  ! bodies4 ,atta* ends ( ?ree ?ick    
flexible wires to any places  on !      

surface ( ,s#a2 c>ry ! wires ?r\gh      
int]/ices ( !  assemblage1 & atta* _!   
o!r ends at any ?ree places ( ,a1  ,b1  
,c1 respectively1 ^! be+ any ?ree ( !   
bodies outside  ! clu/] ( ,s#a & its    
twelve neighb\rs4 ,cut ! wires across   
at any *os5 positions 9 !m2 & r\nd (f ! 
cut ends1 j  leav+ contact betwe5 !     
r\nd$ ends1 :i* we %all call ff1  gg1   
hh4 ,d homog5e\sly = e o!r solid ( !    
assemblage  :at we h done = ,s#a4 ,n[   
b5d ! wires slightly s z to  sep>ate !  
pairs ( po9ts ( contact1 tak+ c>e to    
keep !m  f t\*+ any o!r bodies :i* !y   
pass ne> on _! c\rses  betwe5 ,s#a & ,a1
 ,b1 ,c respectively4 ,aft] hav+ done   
?is1  ?or\ghly rigidify all ! wires ?us 
alt]$4 ,we may n[1 hav+  ?ree indep5d5t 
v>iables at our disposal1 s *ange !     
ori5tation  ( ! molecules1 relatively to
 r[s ( ! assemblage1 z to br+  ff1 gg1 &
 hh aga9 into contact4 ,we h ?us six    
fre%  po9ts ( ,s#a2 ( :i* ?ree are f1 g1
 h2 & ! o!r  ?ree are on ! ?ree         
ext5sions ( ,s#a correspond+ to !  s+le 

ext5sions ( ,a1 ,b1 ,c respectively1 :i*
 we h be5  mak+4 ,?us we h a real       
solution ( ! int]e/+ geometrical        
problem3--,x is requir$ s to =m a       
homog5e\s assemblage ( solids  ( any    
arbitr>ily giv5 %ape t ea* solid %all be
 t\*$ by  eighte5 o!rs4 ,?is problem is 
det]m9ate1 because ! mak+ (  ! ?ree     
contacts ff1 gg1 hh1 uses up ! ?ree     
indep5d5t  v>iables left at our disposal
 aft] we h fir/ =m$ a homog5e\s         
assemblage ) twelve po9ts ( contact on  
ea* solid4 ,b our mann]  ( f9d+ a %ape =
 ea* solid :i* c all[ ! solution ( !    
problem to be real1 proves t ! solution 
is ess5tially imag9>y  = e :olly convex 
%ape4       #ce4 ,paus+ = a mom5t long] 
to consid] afre% ! geometrical  problem 
( putt+ arbitr>ily giv5 equal & simil>  
solids toge!r  to make a homog5e\s      
assemblage ( :i* ea* memb] is t\*$ by   
eighte5 o!rs1 we see imm$iately t x is  
det]m9ate 7:e!r x  has any real solution
 or n71 because :5 ! %ape ( ea* body is 
giv5 we h n9e disposables = fix+ !      

assemblage3 six = !  "* ( ! assemblage (
 ! correspond+ po9ts1 & ?ree =  !       
ori5tation ( ea* molecule relatively to 
r[s ( ! assemblage  ( correspond+ po9ts4
 ,^! n9e disposables are det]m9$ by !   
condition t ea* body has n9e pairs (    
contacts ) o!rs4    ,suppose n[ a       
homog5e\s assemblage ( ! giv5 bodies1 9 
op5  ord] ) no contacts1 to be          
arbitr>ily made accord+ to any n9e      
arbitr>ily *os5 values = ! six di/ances 
betwe5 a po9t (  ,s#a & ! correspond+   
po9ts ( its six pairs ( ne>e/ &  next   
ne>e/ neighb\rs 7 #a above71 & ! ?ree   
angles 7 #i above7  specify+ !          
ori5tation ( ea* body relatively to r[s 
( !  assemblage4 ,we may *oose 9 any n9e
 r[s ?r\gh ,s#a any n9e  pairs ( bodies 
at equal di/ances on ! two sides ( ,s#a 
f> or  ne>1 = ! eighte5 bodies :i* are  
to be 9 contact ) ,s#a4  ,h5ce !re is an
 inf9ite numb] ( solutions ( ! problem (
 :i*  only a f9ite numb] c be real4 ,e  
solution ( ! problem (  eighte5 contacts
 is imag9>y :5 ! %ape is :olly convex4  

,illu/ration3 ,,fig4 #ac4       #cf4    
,)\t = a mom5t imag9+ ! molecules (     
matt] to be  h>d solids ( convex %ape1  
we may d]ive valuable lessons 9 !       
tactics ( real cry/als by /udy+ !       
assemblage describ$ 9   #bd & #be &     
repres5t$ 9 ,figs4 #ab & #ac4 ,i mu/ = !
 pres5t  =ego ! v attractive subject ( !
 tactics pres5t$ by faces  n p>allel to 
"o or o!r ( ! f\r faces ( ! primitive   
tetrah$rons :i* we f\nd 9  #bd1 & ask y 
only to ?9k ( !  two sides ( a plate (  
cry/al p>allel to any "o ( !m1 t is to  
say1 an assemblage ( su* lay]s z ^?     
repres5t$ geometrically  9 ,fig4 #ab &  
%[n 9 /]eoscopic view 9 ,fig4 #ac4 ,if1 
z is !  case ) ! solids#aj photograph$ 9
 ,fig4 #ac1 ! "u side (  ea* solid is   
ne>ly plane b slightly convex1 & ! top  
is some:at  %>ply curv$1 we h ! k9d (   
diff]5ce betwe5 ! upp] &  "u ( ! two    
p>allel sides ( ! cry/al :i* ,i h       
already  describ$ to y 9  #ba above4 ,9 
?is case ! assemblage is =m$  by lett+ !
 solids fall d[n f above & settle 9 !   

holl[s to  :i* !y come mo/ readily1 or  
:i* give !m ! /able/ position4  ,x w\ld1
 we may suppose1 be ! holl[s p q r1 n p 
q r1 7,fig4  #ab7 t w\ld be *os52 & ?us1
 ( ! two =mations describ$ 9   #be1 we  
%\ld h t 9 :i* ! holl[s above p q r are 
occupi$ by ! comp>atively flat "u sides 
( ! molecules (  ! lay] above1 & !      
holl[s bel[ ! ap]tures p q r by !       
comp>atively %>p tops ( ! molecules ( ! 
lay]s bel[4       #cg4 ,= _m cases (    
natural cry/als ( ! :olly asymmetric    
"*1 ! true =ces betwe5 ! cry/all9e      
molecules w  det]m9e precisely ! same   
tactics ( cry/allization z w\ld be      
det]m9$ by ! influ5ce ( gravity & fluid 
viscosity 9 !  settlem5t f wat]1 ( s&   
compos$ ( uni=m molecules ( !  :olly    
unsymmetrical convex %ape repres5t$ 9   
,figs4 #ab & #ac4  ,?us we c readily    
believe t a real cry/al :i* is gr[+  by 
additions to ! face se5 9 ,fig4 #ab1    
w\ld give lay] aft]  lay] regul>ly z ,i 
h j describ$4 ,b if by "s *ange (       
circum/ances ! plate1 already gr[n to a 

?ickness ( _m lay]s 9  ?is way1 %\ld    
come to h ! side fac+ f u 9 ! diagram   
expos$ to ! "m-liquor1 or "m-gas1 & beg9
 to gr[ f  t face1 ! tactics might      
probably be t ea* molecule w\ld f9d  its
 re/+-place ) its mo/ ne>ly plane side 9
 ! wid] holl[s  "u p q r1 in/ead ( ) its
 %>pe/ corn] 9 ! n>r[]  & /eep] holl[s  
"u p q r1 z are ! molecules 9 ! lay]    
bel[ t %[n 9 ! diagram 9 ! fir/ =mation4
 ,! result  w\ld be a comp\nd cry/al    
consi/+ ( two p>ts1 ( diff]5t  cry/all9e
 quality1 coh]+ p]fectly toge!r on ! two
 sides (  an int]facial plane4 ,x seems 
probable t ?is d\ble /ructure may  be   
f\nd 9 nature1 pres5t$ by cry/als ( !   
:olly unsymmetric  class1 ?\gh x may n  
hi!rto h be5 obs]v$ or describ$ 9       
cry/allographic treatises4              
,illu/ration3 ,,fig4 #ad4       #ch4    
,?is asymmetric d\ble cry/al becomes    
simply ! well-kn[n  symmetrical         
#8tw9-cry/al'#aa 9 ! p>ticul> case 9 :i*
 ea* (  ! con/itu5t molecules is        
symmetrical on ! two sides ( a plane    

?r\gh x p>allel to ! plane ( our        
diagrams1 & also on ! two  sides ( "s   
plane p]p5dicul> to ?is plane4 ,we see1 
9 fact1 t  9 ?is case if we cut 9 two ! 
d\ble cry/al by ! plane ( ,fig4  #ad1 & 
turn "o "p ideally ?r\gh #ahj r\nd !    
int]section (  ^! two planes1 we br+ x  
into p]fect co9cid5ce ) ! o!r  "p4      
,?is we readily und]/& by look+ at ,fig4
 #ad1 9 :i* ! solid  %[n 9 outl9e may be
 ei!r an egg-%ap$ figure ( revolution1  
or  may be su* a figure flatt5$ by      
compression p]p5dicul> to !  plane ( !  
diagram4 ,! mo/ readily *os5 & ! mo/    
/able  re/+-places = ! con/itu5ts ( ea* 
successive lay] might be  ! wid] holl[s 
p q r3 & !re=e if1 f a s+le lay] to     
beg9 )1 ! assemblage 7 to gr[ by lay]   
aft] lay] add$ to  x on ea* side1 x     
might probably gr[ z a tw9-cry/al4 ,b x 
might  also be t ! pres5ce ( a molecule 
9 ! wid] holl[ p q r  on "o side1 might 
r5d] ! occupation ( ! correspond+ holl[ 
on  ! o!r side by ano!r molecule less   
probable1 or ev5 impossible4  ,h5ce1    

accord+ to ! configuration & ! molecul> 
=ces ( !  p>ticul> cry/all9e molecule 9 
natural cry/allization1 !re may  be     
necess>ily1 or almo/ necess>ily1 ! tw91 
:5 gr[? proce$s  simultane\sly on ! two 
sides3 or ! tw9 gr[? may be impossible1 
because ! fir/ occupation ( ! wid]      
holl[s on "o side1 may  compel !        
cont9uity ( ! cry/all9e quality ?r\gh\t1
 by leav+  only ! n>r[] holl[s p q r    
free = occupation by molecules  atta*+  
!mselves on ! o!r side4       #ci4 ,or !
 "* ( ! cry/all9e molecule may be su*  t
 :5 ! assemblage gr[s by ! addition (   
lay] aft] lay]  on "o side only1 ) a n v
 /rongly decid$ pref]5ce to !  wid]     
holl[s p q r1 "s *ange ( circum/ances   
may cause  ! molecules ( "o lay] to     
place !mselves 9 a holl[ p q r4  ,!     
molecules 9 ! next lay] aft] ?is w\ld   
f9d ! holl[s  p q r occupi$ on ! f>     
side1 & w\ld ?us h a bias 9  fav\r ( !  
holl[s p q r4 ,?us lay] aft] lay] might 
be add$1  con/itut+ a tw9n$ portion ( ! 
gr[?1 gr[+1 h[ev]1  ) less /rong        

security = cont9u$ homog5e\sness ?an :5 
! cry/al 0 gr[+1 z at fir/1 by          
occupation ( ! wid]  holl[s p q r4 ,a   
slight di/urbance might aga9 occur1     
caus+  ! molecules ( a fre% lay] to     
settle1 n 9 ! n>r[ holl[s  p q r1 b 9 ! 
wid] holl[s p q r1 not)/&+ !  ne>ness ( 
molecules already occupy+ ! wid] holl[s 
on !  o!r side4 ,di/urbances su* z ^!   
occurr+ irregul>ly dur+  ! gr[? ( a     
cry/al1 might produce a l>ge numb] (    
successive  tw9n+s at p>allel planes )  
irregul> int]vals betwe5 !m1  or a l>ge 
numb] ( tw9n+s 9 planes at equal        
int]vals might be  produc$ by "s regul> 
p]iodic di/urbance occurr+ = a c]ta9    
numb] ( p]iods1 & !n ceas+4 ,:e!r regul>
 & p]iodic1 or  irregul>1 ! t5d5cy w\ld 
be t ! numb] ( tw9n+s %\ld  be ev51 & t 
aft] ! di/urbances cease ! cry/al %\ld  
g on gr[+ 9 ! fir/ mann]1 because ( !   
p]man5t bias 9  fav\r ( ! wid] holl[s p 
q r4 ,^! *anges ( molecul>  tactics1 :i*
 we h be5 necess>ily l$ to by !         
consid]ation  ( ! =tuit\s conc\rse (    

molecules1 are no d\bt exemplifi$ 9  a  
l>ge v>iety ( tw9n+s & c\nt]-tw9n+s f\nd
 9 natural  m9]als4 ,9 ! artificial     
cry/allization ( *lorate ( pota% !y  are
 ( frequ5t occurr5ce1 z is prov$1 n only
 by ! tw9n+s &  c\nt]-tw9n+s readily se5
 9 ! cry/all9e =ms1 b also by  !        
brilliant iridesc5ce obs]v$ 9 _m ( !    
cry/als f\nd among  a l>ge multitude1   
:i* 0 inve/igat$ sci5tifically by ,sir  
,george  ,/okes t5 ye>s ago1 & describ$ 
9 a communication to ! ,royal  ,society 
#8,on a rem>kable ph5om5on ( cry/all9e  
reflection' 7,proc4  ,r4,s41 vol4       
xxxviii1 #ahhe1 p4 #agd74       #dj4 ,a 
v int]e/+ ph5om5on1 pres5t$ by :at 0    
orig9ally  a cle> homog5e\s cry/al (    
*lorate ( pota%1 & 0 alt]$ by  heat+ to 
ab\t #bde-bdh ,c5t41 :i* ,i am able to  
%[ y ?r\gh  ! k9dness ( ,"l ,rayleigh1 &
 ( its discov]]1 ,mr4 ,madan1  pres5ts  
ano!r v wond]ful case ( *ang+ molecul>  
tactics1  mo/ in/ructive 9 respect ( !  
molecul> con/itution ( ela/ic  solids4  
,:5 ,i hold ?is plate be=e y ) !        

p]p5dicul> to  its plane incl9$ at #aj  
or m to y\r l9e ( vision1 y see a       
t9sel-l appe>ance1 almo/ z bright z if x
 7 a plate (  poli%$ silv]1 on ?is      
little area1 :i* is a ?9 plate ( *lorate
  ( pota% cem5t$ = pres]vation betwe5   
two pieces ( glass2 &1  :5 ,i hold a    
light beh9d1 y see t ! little plate is  
almo/  p]fectly opaque l metal foil4 ,b 
n[ :5 ,i hold x ne>ly  p]p5dicul> to y\r
 l9e ( vision ! t9sel-l appe>ance is    
lo/4 ,y c see cle>ly ?r\gh ! plate1 & y 
also see t v  little light is reflect$ f
 x4 ,z a result bo? ( ,mr4 ,madan's own 
inve/igations1 & fur!r obs]vations by   
himself1 ,"l ,rayleigh came  to !       
conclusion t ! almo/ total reflection ( 
:ite light :i*  y see is due to !       
reflection ( light at _m int]facial     
planes  betwe5 successive lay]s ( tw9n$ 
& c\nt]-tw9n$ cry/al (  small irregul>  
?icknesses1 & n to any splits or        
cavities or any  o!r deviation f        
homog5e\sness ?an t pres5t$ by homog5e\s
  portions ( oppositely tw9n$-cry/als 9 

?or\gh molecul> contact  at ! int]faces4
       #da4 ,:5 ! primitive cle> cry/al 
0 fir/ heat$ v gradually  by ,madan to  
ne> its melt+-po9t 7#cei accord+ to     
,c>nelly71 x  rema9$ cle>1 & only       
acquir$ ! t9sel appe>ance aft] x _h     
cool$ to ab\t #bde or #bdhab4 ,rayleigh 
f\nd t if a cry/al  ?us alt]$ 0 aga9 &  
aga9 heat$ x always lo/ ! t9sel         
appe>ance1 & became p]fectly cle> at "s 
temp]ature consid]ably  bel[ !          
melt+-po9t1 & rega9$ x at ab\t ! same   
temp]ature  9 cool+4 ,x seems1 !re=e1   
c]ta9 t at temp]atures above  #bdh1 &   
bel[ ! melt+-po9t1 ! molecules _h s mu* 
( !rmal  motions z to keep !m hov]+ ab\t
 ! positions ( p q r1 p q  r1 ( our     
diagrams1 b n en\gh to d away ) !       
rigidity (  ! solid2 & t :5 cool$ bel[  
#bdh ! molecules 7 all[$  to settle 9 "o
 or o!r ( ! two configurations1 b )     
little  ( bias = "o 9 pref]5ce to ! o!r4
 ,x is c]ta9ly a v  rem>kable fact 9    
,natural ,hi/ory1 discov]$ by ^!        
obs]vations1  t1 :5 ! molecules come    

toge!r to =m a cry/al out ( !  wat]y    
solution1 !re %\ld be s mu* m decid$ a  
bias 9 fav\r  ( cont9u$ homog5e\sness ( 
! assemblage ?an :51 by cool+1  !y are  
all[$ to settle f _! agitations 9 a     
rigid1 b ne>ly  melt+1 solid4       #db4
 ,b ev5 9 cry/allization f wat]y        
solution ( *lorate (  pota% ! bias 9    
fav\r ( ?or\gh homog5e\sness is n 9 e   
cont+5cy decisive4 ,9 ! fir/ place1     
beg9n+1 z ! =mation  seems to beg91 f a 
s+le molecul> plane lay] su* z t        
ideally %[n 9 ,fig4 #ad1 x goes on1 n to
 make a homog5e\s cry/al  on ! two sides
 ( ?is lay]1 b probably always s z to =m
 a  tw9-cry/al on its two sides1 exactly
 z describ$ 9  #ch1 &1 if  s1 c]ta9ly = 
! reason !re /at$4 ,?is is :at ,madan   
calls  ! #8invet]ate t5d5cy to produce  
tw9s 7su* z w\ld assur$ly  drive a      
,mal?us to despair7#ac'2 & x is to ?is t
 he alludes z  #8! inevitable tw9-plate'
 9 ! passage f 8 pap] giv5 9 !          
foot-note to  #da above4       #dc4 ,9 !
 second place1 ,i mu/ tell y t _m ( !   

cry/als  produc$ f ! wat]y solution by !
 ord9>y process ( sl[  evaporation &    
cry/allization1 %[ tw9n+s & c\nt]-tw9n+s
  at irregul> int]vals 9 ! o!rwise      
homog5e\s cry/al on ei!r  "o or bo?     
sides ( ! ma9 c5tral tw9-plane1 :i*     
h5ce=?1  = brevity1 ,i %all call 7adopt+
 ! hypo!sis already expla9$1  :i* seems 
to me und\bt$ly true7 ! #8initial       
plane4' ,ea* tw9n+  is foll[$1 ,i       
believe1 by a c\nt]-tw9n+ at a v %ort   
di/ance  f x2 at all ev5ts ,"l          
,rayleigh's obs]vations#ad prove t !    
:ole numb] ( tw9n+s & c\nt]-tw9n+s 9 a  
?9 di/urb$  /ratum ( ! cry/al on "o side
 ( ! ma9 c5tral tw9-plane  is g5]ally1  
p]haps always1 ev52 s t1 except ?r\gh "s
  comp>atively v small "p or p>ts ( !   
:ole ?ickness1 !  cry/al on ei!r side ( 
! middle or initial plane is homog5e\s4 
,?is is exactly ! g5]ally regul> gr[?   
:i* ,i h describ$  to y 7 #ci7 z        
int]rupt$ occasionally or accid5tally by
 "s  unexpla9$ di/urb+ cause1 b ) an    
ess5tial bias to !  homog5e\s cont9uance

 ( ! m easy or natural "o ( ! two       
configurations4       #dd4 ,i h n[ great
 pleasure 9 %[+ y a mo/ int]e/+         
collection ( ! iridesc5t cry/als (      
*lorate ( pota%1 ea*  c>efully m\nt$ =  
pres]vation betwe5 two glass plates1 :i*
 h  be5 k9dly l5t to u = ?is ev5+ by    
,mr4 ,madan4 ,9 ,m>*1 #ahed1  ,dr4 ,w4  
,bird ,h]apa? s5t to ,pr(4 ,/okes "s    
cry/als ( *lorate  ( pota% %[+ !        
brilliant & beautiful col\rs y n[ see1  
&1 ?irty ye>s lat]1 ,pr(4 ,e4 ,j4 ,mills
 recall$ 8 att5tion to  ! subject by    
s5d+ him #8a f9e collection ( spl5didly 
col\r$  cry/als ( *lorate ( pota% (     
consid]able size1 sev]al ( !  plates    
hav+ an area ( a squ>e in* or m1 & all (
 !m  be+ ?ick en\gh to h&le )\t         
difficulty4' ,! consequ5ce  0 t ,/okes  
made a se>*+ exam9ation into ! "* (  !  
ph5om5on1 & gave ! %ort1 b spl5didly    
int]e/+1  communication to ! ,royal     
,society ( :i* ,i h already told y4  ,! 
exi/5ce ( ^! beautifully col\r$ cry/als 
_h be5 well  kn[n to *emical            

manufactur]s = a long "t1 b x does n    
appe>  t any m5tion ( !m 0 to be f\nd 9 
any sci5tific j\rnal or  treatise prior 
to ,/okes' pap] ( #ahhe4 ,he f\nd t !   
col\r 0  due to tw9n+s & c\nt]-tw9n+s 9 
a v ?9 di/urb$ /ratum  ( ! cry/al %[+   
itself by a v f9e l9e1 d>k or gli/5+1   
accord+ to ! direction ( ! incid5t light
 :5 a transv]se  section ( ! plate (    
cry/al 0 exam9$ 9 a microscope4 ,by     
comp>ison ) a spore ( lycopodium he     
e/imat$ t ! bread?  ( ?is l9e1 & !re=e !
 ?ickness ( ! di/urb$ /ratum (  !       
cry/al1 rang$ some:]e ab\t ! "o-?\s&? ( 
an in*4 ,he  f\nd t ! /ratum 0 visibly  
?ick] 9 ^? cry/als :i*  %[$ r$ col\r ?an
 9 ^? :i* %[$ blue4 ,he conclud$ t  #8! 
seat ( ! coloration is c]ta9ly a ?9     
tw9n$ /ratum' 7t  is to say1 a homog5e\s
 portion ( cry/al betwe5 a tw9n+ & a    
c\nt]-tw9n+71 & f\nd t #8a s+le         
tw9-plane does n %[  any?+ ( ! k9d4'    
#de4 ,a ye> or two lat] ,"l ,rayleigh   
ent]$ on ! subject )  an exhau/ive      
ma!matical inve/igation ( ! reflection (

 light at  a tw9-plane ( a cry/al       
7,philosophical ,magaz9e1 ,septemb]1    
#ahhh71  by ! application ( :i*1 9 a    
second pap] #8,on ! rem>kable  ph5om5on 
( ,cry/all9e ,reflection describ$ by    
,pr(4 ,/okes1'  publi%$ 9 ! same numb] (
 ! ,philosophical ,magaz9e1 he gave  :at
 seems c]ta9ly ! true explanation ( !   
results ( ,sir ,george  ,/okes'         
exp]im5tal analysis ( ^! beautiful      
ph5om5a4 ,he came  v decid$ly to !      
conclusion t ! selective quality ( !    
iridesc5t portion ( ! cry/al1 9 virtue (
 :i* x reflects  almo/ totally light    
ne>ly ( "o p>ticul> wave-l5g? = "o      
p>ticul> direction ( incid5ce 7on :i* ! 
brilliance ( !  coloration dep5ds71 _c  
be due to m]ely a s+le tw9-/ratum1  b t 
x ess5tially is due to a consid]able    
numb] ( p>allel  tw9-/rata at ne>ly     
equal di/ances4 ,! light reflect$ by ?is
  complex /ratum is1 = any p>ticul>     
direction ( incid5t &  reflect$ ray1    
*iefly t ( :i* ! wave-l5g? is equal to  
twice  ! l5g? ( ! p]iod ( ! tw9n+ &     

c\nt]-tw9n+1 on a  l9e drawn ?r\gh !    
/ratum 9 ! direction ( ei!r ! incid5t   
or ! reflect$ ray4       #df4 ,x seems  
to me probable t ea* tw9n+ is ess5tially
  foll[$ closely by a c\nt]-tw9n+4      
,probably ?ree or f\r ( ^!  tw9-/rata   
might suffice to give col\r2 b 9 any ( !
 brilliant  specim5s z _m z tw5ty or    
?irty1 or m1 might probably be  necess>y
 to give s ne>ly mono*romatic light z 0 
prov$ by  ,/okes' prismatic analysis ( !
 col\rs obs]v$ 9 _m ( 8  specim5s4 ,!   
di/urb$ /ratum ( ab\t a "o-?\s&? ( an   
in*  ?ickness1 se5 by him 9 !           
microscope1 amply suffices = ! #e1  #aj1
 or #ajj half wave-l5g?s requir$ by     
,rayleigh's !ory to acc\nt  = p]ceptible
 or brilliant coloration4 ,b :at c be ! 
cause  ( any approa* to regul>          
p]iodicity 9 ! /ructure suffici5tly     
good to give ! col\rs actually obs]v$8  
,p]iodical motion (  ! "m-liquor        
relatively to ! gr[+ cry/al might       
possibly  acc\nt = x4 ,b ,"l ,rayleigh  
tells u t he tri$ rock+ !  pan conta9+ !

 solution )\t result4 ,influ5ce ( light 
has  be5 sugge/$1 & ,i believe tri$1    
also )\t result1 by sev]al  enquir]s4   
,we "k1 by ! beautiful discov]y (       
,edmond ,becqu]el1  ( ! prismatic col\rs
 photograph$ on a prep>$ silv] plate by 
! sol> spectrum1 t #8/&+ waves' 7t is to
 say1 vibrations  ) /ation>y nodes &    
/ation>y places ( maximum vibration71   
due to co-exi/5ce ( incid5t & reflect$  
waves1 d produce su*  a p]iodic /ructure
 z t :i* ,rayleigh's !ory %[s capable  (
 giv+ a correspond+ t9t :5 illum9at$ by 
:ite light4 ,x  is difficult1 !re=e1 n  
to ?9k t light may be effective 9       
produc+ ! p]iodic /ructure 9 !          
cry/allization ( *lorate (  pota%1 to   
:i* ! iridesc5ce is due4 ,/ill1         
exp]im5tal evid5ce  seems aga9/ ?is     
tempt+ !ory1 & we mu/ p]=ce be cont5t  )
 ! "q unansw]$3--,:at c be ! cause ( #e1
 or #aj1 or  #ajj pairs ( tw9n+ &       
c\nt]-tw9n+ foll[+ "o ano!r 9  !        
cry/allization ) suffici5t regul>ity to 
give ! col\r3 &  :y1 if !re are tw9n+s &

 c\nt]-tw9n+s1 are !y n at  irregul>    
int]vals1 z ^? produc$ by ,madan's      
process1 & giv+  ! obs]v$ :ite t9sel-l  
appe>ance ) no coloration8       #dg4 ,&
 n[ ,i h sadly tax$ y\r pati5ce3 & ,i   
fe> ,i h  exhau/$ x & n exhau/$ my      
subject6 ,i feel ,i h n got  halfway    
?r\gh :at ,i hop$ ,i might be able to   
put be=e y ?is  ev5+ reg>d+ ! molecul>  
/ructure ( cry/als4 ,i p>ticul>ly       
desir$ to speak to y ( qu>tz cry/al )   
its t]n>y symmetry  & its *irality#ae2 &
 to h told y ( ! et*+#af by  hydr(luoric
 acid :i*1 z x 71 comm5ces to unbuild ! 
cry/al  by tak+ away molecule aft]      
molecule1 b n 9 ! rev]se ord] (  !      
prim>y up-build+2 & :i* ?us reveals     
diff]5ces ( tactics  9 ! alt]nate faces 
( ! six-sid$ pyramid :i* t]m9ates at    
ei!r end1 sometimes at bo? ends1 !      
six-sid$ prism con/itut+  g5]ally ! ma9 
bulk ( ! cry/al4 ,i mu/ conf9e myself to
 giv+  y a geometrical symbol = ! t]n>y 
symmetry ( ! prism & its  t]m9al        
pyramid4    ,illu/ration3 ,,fig4 #ae4   

#dh4 ,make an equilat]al equiangul>     
hexagonal prism1 )  its diagonal f edge 
to edge n9ety-five hundr$?s#ag ( its    
l5g?4 ,place a numb] ( ^! close toge!r1 
s z to make up  a hexagonal plane lay] )
 its sides p]p5dicul> to ! sides  ( !   
con/itu5t hexagonal prisms3 see ,fig4   
#ae & imag9e !  semicircles replac$ by  
_! diamet]s4 ,y see 9 ea* side ( !      
hexagonal assemblage1 edges ( !         
con/itu5t prisms1 & y see  at ea* corn] 
( ! assemblage a face 7n an edge7 ( "o (
  ! con/itu5t prisms4 ,build up a       
hexagonal prismatic assemblage  by plac+
 lay] aft] lay] ov] x ) ! con/itu5t     
prisms (  ea* lay] v]tically ov] ^? 9 ! 
lay] bel[2 & f9i% !  assemblage ) a     
six-sid$ pyramid by build+ ^u ! upp] end
  ( ! prism1 lay] aft] lay] ( dim9i%+   
hexagonal gr\ps1  ea* less by "o        
circumf]5tial r[ ?an ! lay] bel[ x4 ,y  
?us h a cry/al ( precisely ! %ape ( a   
symmetrical specim5  ( rock cry/al1 ) ! 
faces ( its t]m9al pyramid incl9$ at    
#ch #ac to ! faces ( ! prism f :i* !y   

spr+4 ,b !  assemblage ?us con/itut$ has
 #8s5>y' 7or six-ray$ symmetry74 ,to    
r$uce ?is to t]n>y symmetry1 cut a      
groove ?r\gh ! middle (  ea* alt]nate   
face ( ! prismatic molecule1 mak+ ?is   
groove  9 ! fir/ place p>allel to !     
edges3 & add a correspond+  projection1 
or fillet1 to ! middles ( ! o!r ?ree    
faces1 s  t two ( ! cyl9d]s simil>ly    
ori5t$ w\ld fit toge!r1 )  ! project+   
fillet on "o side ( "o ( !m ent]+ !     
groove  9 ! anti-correspond+ side ( !   
o!r4 ,! prismatic portion  ( !          
assemblage ?us =m$ %[s 7see ,fig4 #ae71 
on its alt]nate  edges1 faces (         
molecules ) projections & faces (       
molecules )  grooves2 & %[s only        
ori5tational diff]5ces betwe5 alt]nate  
faces1 :e!r ( ! pyramid or ( ! prism4   
,hav+ gone only s f>  f #8s5>y'         
symmetry1 we h exactly ! triple1 or     
?ree-pair1  anti-symmetry requir$ = !   
piezo-electricity ( qu>tz inve/igat$  s 
admirably by ! bro!rs ,curie#ah1 :o f\nd
 t a ?9 plate (  qu>tz cry/al cut f any 

position p]p5dicul> to a pair ( faces  (
 a symmetrical cry/al1 becomes          
positively electrifi$ on "o side  &     
negatively on ! o!r :5 pull$ 9 a        
direction p]p5dicul>  to ^? faces4 ,b   
?is assemblage has n ! *iral            
piezo-electric  quality discov]$        
!oretically by ,voigt#ai1 & exp]im5tally
 9  qu>tz & 9 t\rmal9e by himself &     
,riecke#bj1 nor ! well-kn[n  optic      
*irality ( qu>tz4    ,illu/ration3      
,,fig4 #af4    ,illu/ration3 ,,fig4 #ag4
       #di4 ,*ange n[ ! directions ( !  
grooves & fillets to ei!r (  ! oblique  
configurations %[n 9 ,fig4 #af1 :i* ,i  
call "r-h&$1  because ! directions ( !  
projections are tang5tial to ! ?reads  (
 a ?ree-?read "r-h&$ screw1 & ,fig4 #ag 
7left-h&$74 ,!  prisms ) _! grooves &   
fillets w /ill all fit toge!r if  !y are
 all "r-h&$1 or all left-h&$4           
,illu/ration3 ,,fig4 #ah4    ,fig4 #ah  
%[s ! upp] side ( a hexagonal lay] ( an 
assemblage  ?us compos$ ( ! "r-h&$      
molecule ( ,fig4 #af4 ,fig4 #ae         

un*ang$1 /ill repres5ts a horizontal    
section ?r\gh ! c5tres  ( ! molecules4  
,a prism built up ( su* lay]s1 & f9i%$  
at  ea* end ) a pyramid accord+ to !    
rule (  #dh1 has all !  qualities (     
t]n>y *iral symmetry requir$ = !        
piezo-electricity  ( qu>tz2 = !         
ori5tational diff]5ces ( ! alt]nate     
pairs  ( prismatic faces2 = ! absolute  
diff]5ce betwe5 ! alt]nate  pairs (     
faces ( ea* pyramid :i* are %[n 9 ! et*+
 by  hydr(luoric acid2 = ! m]ely        
ori5tational diff]5ce betwe5  ! p>allel 
faces ( ! two pyramids2 & = ! well-kn[n 
*iro-optic#ba prop]ty ( qu>tz4 ,look at 
two contigu\s faces ,a1  ,b ( our       
geometrical model qu>tz cry/al n[ be=e  
y1 ) its  axis v]tical4 ,y w see a      
diff]5ce betwe5 !m3 turn x upside  d[n2 
,b w be undi/+ui%able f :at ,a 01 & ,a w
 be  undi/+ui%able f :at ,b 04 ,look at 
! two t]m9al pyramids1  & y w f9d t !   
face above ,a & ! face bel[ ,b are      
id5tical 9 quality1 & t !y diff] f !    
face above ,b  & bel[ ,a4 ,?is model is 

compos$ ( ! "r-h&$ con/itu5t  molecules 
%[n 9 ,fig4 #af4 ,x is s plac$ be=e y t 
! edge (  ! prismatic "p ( ! assemblage 
ne>e/ to y %[s y fillet$  faces ( !     
prismatic molecules4 ,y see two         
pyramidal faces2 ! "o  to y\r "r h&1 ov]
 ,b1 pres5ts complicat$ projections &   
holl[s at ! corn]s ( ! con/itu5t        
molecules2 & ! pyramidal  face next y\r 
left h&1 ov] ,a1 pres5ts _! unmodifi$   
corn]s4  ,b x w be ! face next y\r left 
h& :i* w pres5t !  complex bri/l+       
corn]s1 & ! face next y\r "r h& t  w    
pres5t ! simple corn]s1 if1 = ! model   
be=e y1 y  sub/itute a model compos$ (  
left-h&$ molecules su* z ^?  %[n 9 ,fig4
 #ag4       #ej4 ,to give all !         
qualities ( symmetry & anti-symmetry (  
! pyro-electric & piezo-electric        
prop]ties ( t\rmal9e  inve/igat$        
!oretically by ,voigt#bb1 & exp]im5tally
 by himself  & ,friecke#bc1 make a holl[
 9 "o t]m9al face ( ea* ( our  con/itu5t
 prisms1 & a correspond+ projection 9   
its o!r  t]m9al face4       #ea4 ,com+  

back to qu>tz1 we c n[ und]/& p]fectly !
  two k9ds ( macl+ :i* are well kn[n to 
m9]alogi/s z be+  f\nd 9 _m natural     
specim5s ( ! cry/al1 & :i* ,i call      
respectively ! ori5tational macl+1 & !  
*iral macl+4  ,9 ! ori5tational macl+   
all ! cry/all9e molecules are  "r-h&$1  
or all left-h&$2 b ?r\gh all ( "s "p ( !
  cry/al1 ea* ( our compon5t hexagonal  
prisms is turn$ r\nd its  axis ?r\gh #fj
 f ! position x w\ld h if ! /ructure  7 
homog5e\s ?r\gh\t4 ,9 ea* ( ! two p>ts !
 /ructure  is homog5e\s1 & possesses all
 ! electric & optic prop]ties  :i* any  
homog5e\s portion ( qu>tz cry/al        
pres5ts1 & !  facial prop]ties ( natural
 uncut cry/al1 %[n 9 ! et*+ by          
hydr(luoric acid2 b !re is a            
discont9uity at ! int]face1 n  g5]ally  
plane1 betwe5 ! two p>ts1 :i* 9 our     
geometrical model  w\ld be %[n by       
non-fitt+s betwe5 ! molecules on ! two  
sides  ( ! int]face1 :ile all !         
contigu\s molecules 9 "o "p1 &  all !   
contigu\s molecules 9 ! o!r "p1 fit into

 "o ano!r  p]fectly4 ,9 *iral macl+1 :i*
 is cont9ually f\nd 9 ame?y/9e  qu>tz1 &
 sometimes 9 ord9>y cle> qu>tz cry/als1 
"s p>ts  are compos$ ( "r-h&$ molecules1
 & o!rs ( left-h&$  molecules4 ,x is n  
kn[n :e!r1 9 ?is *iral macl+1 !re is or 
!re is n also ! ori5tational macl+ on ! 
two sides ( ea*  int]face2 b we may say 
probably n2 because we "k t !           
ori5tational macl+ occurs 9 nature )\t  
any *iral macl+1 &  because !re does n  
seem reason to expect t *iral macl+ w\ld
  imply ori5tational macl+ on ! two     
sides ( ! same int]face4  ,i w\ld l to h
 spok5 to y m ( ?is mo/ int]e/+         
subject2 & to h po9t$ out to y t "s ( ! 
simple/ &  mo/ natural suppositions we c
 make z to ! *emical =ces 7or           
electrical =ces1 :i* probably means !   
same ?+7 conc]n$ 9  a s+le *emical      
molecule ( qu>tz1 ,si,o#b1 & act+ betwe5
  x & simil> neighb\r+ molecules1 w\ld  
lead ess5tially to  ^! molecules com+   
toge!r 9 triplets1 ea* necess>ily ei!r  
"r-h&$ or left-h&$1 b ) z mu*           

probability ( "o  configuration z ( !   
o!r3 & to h %[n y t ^!  triplets (      
silica #c7,si,o#b7 c =m a cry/all9e     
molecule )  all ! prop]ties ( t]n>y     
*iral symmetry1 typifi$ by our groov$   
hexagonal prisms1 & c build up a qu>tz  
cry/al by ! =tuit\s  conc\rse ( atoms4  
,i %\ld l also to h sugge/$ & expla9$  !
 possibility t a "r-h&$ cry/all9e       
molecule ?us =m$  may1 9 natural        
circum/ances ( high temp]ature1 or ev5 (
 great  pressure1 become *ang$ into a   
left-h&$ cry/al1 or vice-v]sa4  ,my     
wat*1 h[ev]1 w>ns me t ,i mu/ n ent] on 
?is subject4    ,illu/ration3 ,,fig4    
#ai4       #eb4 ,com+ back to m]e       
molecul> tactics ( cry/als1 rem>k t  our
 assemblage ( r\nd$1 ?or\ghly scal5e1   
tetrah$rons1 %[n  9 ! /]eoscopic picture
 7 #cf1 ,fig4 #ac above71 ess5tially has
  *irality because ea* con/itu5t        
tetrah$ron1 if :olly scal5e1 has        
*irality#bd4 ,i %\ld l to h expla9$ to y
 h[ a s+le or  d\ble homog5e\s          
assemblage ( po9ts has ess5tially no    

*irality1  & h[ ?ree assemblages ( s+le 
po9ts1 or a s+le assemblage (  triplets 
( po9ts1 c h *irality1 ?\gh a s+le      
triplet (  po9ts _c h *irality4 ,i %\ld 
l inde$ to h br\ght  some:at ?or\ghly   
be=e y ! geometrical !ory ( *irality2  &
 9 illu/ration to h expla9$ ! conditions
 "u :i* f\r  po9ts1 or two l9es1 or a   
l9e & two po9ts1 or a comb9ation (      
po9t1 l9e & plane1 c h *irality3 & h[ a 
homog5e\s  assemblage ( non-*iral       
objects c h *irality2 b 9 pity ,i  =be>1
 & ,i ?ank y = ! extreme pati5ce ) :i* y
 h  li/5$ to me4          ,,footnotes3  
#a ,see foot-note on  #bb bel[4    #b ,!
 holes 9 ! cyl9d]s are bor$ obliquely1 z
 %[n 9 ,fig4  #d1 :i* causes !m to rema9
 at any desir$ position on ! cord &     
all[s !m to be fre$ to move up & d[n by 
slack5+ ! cord =  a mom5t4    #c #8,on !
 ,homog5e\s ,division ( ,space1' by ,"l 
,kelv91 ,royal  ,society ,proce$+s1 vol4
 lv1 ,jan4 #ah1 #ahid4    #d ,simil>    
curves are said to be p>allel :5 !      
tang5ts to !m at  correspond+ po9ts are 

p>allel4    #e ,see foot-note to  #ab   
above4    #f #8,on ! ,division ( ,space 
) ,m9imum ,p>titional ,area1'           
,philosophical ,magaz9e1 vol4 xxiv1     
#ahhg1 p4 #ejb1 & ,acta  ,ma!matica ( ! 
same ye>4    #g ,a4 ,levy1 ,ed9burgh    
,philosophical ,j\rnal1 ,april1 #ahbb2  
,:ewell1 ,phil4 ,trans4 ,royal ,society1
 #ahbe2 ,mill]1 ,treatise on            
,cry/allography4    #h ,i call any      
geometrical figure1 or gr\p ( po9ts1    
*iral1 &  say t x has *irality1 if its  
image 9 a plane mirror1 ideally         
realiz$1 _c be br\ght to co9cide )      
itself4 ,two equal &  simil> "r h&s are 
homo*irally simil>4 ,equal & simil> "r  
& left h&s are het]o*irally simil> or   
#8allo*irally' simil>  7b het]o*irally  
is bett]74 ,^! are also call$           
#8enantiomorphs1'  aft] a usage         
introduc$1 ,i believe1 by ,g]man writ]s4
 ,any *iral  object & its image 9 a     
plane mirror are het]o*irally simil>4   
#i ,philosophical ,magaz9e1 vol4 xx1    
#ahhe1 second half ye>1 p4 #dfi1  &     

,briti% ,association ,report1 #ahhe1    
,ab]de51 p4 #hif4    #aj ,! solids ( !  
photograph are ca/+s 9 f9e pla/] ( ,p>is
  f a scal5e tetrah$ron ( p>aff9 wax1 ) 
its corn]s & edges  r\nd$1 us$ z a      
patt]n4    #aa #8,a tw9-cry/al is       
compos$ ( two cry/als jo9$ toge!r 9  su*
 a mann] t "o w\ld come into ! position 
( ! o!r by  revolv+ ?r\gh two "r angles 
r\nd an axis :i* is p]p5dicul>  to a    
plane :i* ei!r is1 or may be1 a face (  
ei!r cry/al4  ,! axis w be call$ !      
tw9-axis1 & ! plane to :i* x is         
p]p5dicul> ! tw9-plane4' ,mill]'s       
,treatise on ,cry/allography1  p4 #ajc4 
,9 ! text ! ^w #8tw9-plane1' quot$ f !  
writ+s  ( ,/okes & ,rayleigh1 is us$ to 
signify ! plane common to !  two cry/als
 9 ea* ( ! cases ref]r$ to3 & n ! plane 
p]p5dicul> to ?is plane1 9 :i* "o "p ( !
 cry/al mu/ be  rotat$ to br+ x into    
co9cid5ce ) ! o!r1 & :i* is !  tw9-plane
 z def9$ by ,mill]4    #ab #8,a cle>    
transp>5t cry/al ( potassium *lorate1 f 
:i*  ! inevitable tw9-plate _h be5 gr\nd

 away s z to r$uce x to  a s+le cry/al  
film ab\t #a mm4 9 ?ickness1 0 plac$    
betwe5  pieces ( mica & laid on a ?ick  
iron plate4 ,ab\t #c cm4 f x  0 laid a  
small bit ( potassium *lorate1 & ! heat 
( a ,buns5  burn] 0 appli$ bel[ ?is     
latt]1 s z to obta9 an indication  :5 ! 
temp]ature ( ! plate 0 approa*+ !       
fus+-po9t (  ! sub/ance 7#cei ,c accord+
 to ,pr(4 ,c>nelly74 ,! cry/al plate  0 
c>efully wat*$ dur+ ! heat+1 b no       
depreciation took  place1 & no visible  
alt]ation 0 obs]v$1 up to ! po9t at :i* 
! small s5t9el cry/al imm$iately ov] !  
burn] began to fuse4  ,! lamp 0 n[      
)drawn1 & :5 ! temp]ature _h sunk a few 
degrees a rem>kable *ange spread quickly
 & quietly ov] ! cry/al  plate1 caus+ x 
to reflect light almo/ z brilliantly z  
if a film  ( silv] _h be5 deposit$ ^u x4
 ,no fur!r alt]ation occurr$  dur+ !    
cool+2 & ! plate1 aft] be+ gr\nd &      
poli%$ on  bo? sides1 0 m\nt$ ) ,canada 
balsam betwe5 glass plates =            
exam9ation4 ,_m cry/als h be5 simil>ly  

treat$ ) precisely  simil> results2 & ! 
temp]ature at :i* ! *ange takes place1  
has be5 det]m9$ to lie betwe5 #bde &    
#bdh1 by heat+ !  plates ^u a ba? (     
melt$ t9 9 :i* a !rmomet] 0 imm]s$4  ,) 
s+le cry/al plates no decrepitation has 
ev] be5 obs]v$1  :ile ) ! ord9>y        
tw9n$-plates x always occurs m or less  
viol5tly1 ea* fragm5t %[+ ! brilliant   
reflective p[] above  notic$4'--,nature1
 ,may #bj1 #ahhf4    #ac ,nature1 ,may  
#bj1 #ahhf4    #ad ,philosophical       
,magaz9e1 #ahhh1 second half ye>1 p4    
#bfj4    #ae ,see foot-note to  #bb     
above4    #af ,widman/tt51 #ahjg4       
,leydolt 7#ahee1 ,wi54 ,akad4 ,b]4 #ae1 
#ei1 ,t4  #i1 #aj4 ,baumhau]1 ,pogg4    
,ann4 #ach1 #efc 7#ahfi72 #adj1 #bga2   
#adb1 #cbd2 #ade1  #dfj2 #aej1 #fai47 ,=
 an acc\nt ( ^! inve/igations1 see      
,mall>d1  ,trait de ,cry/allographie    
7,p>is1 #ahhd71 ,tome ,,ii1 *apitre xvi4
    #ag ,m exactly 4#iebe1 be+ #c_/d    
cot #ch #ac2 see p4 #ec4    #ah ,j4 &   
,p4 ,curie & ,c4 ,fri$el1 ,comptes      

,r5dus1 #ahhb1 #ahhc1  #ahhf1 #ahib4    
#ai ,allgeme9e ,!orie d] pizo- und      
pyroelectris*5 ,ers*e9ung5  an          
,kry/all54 ,w4 ,voigt1 ,knigl4          
,gesells*aft d] ,wiss5s*aft5 zu  ,gtt+51
 ,augu/ #b1 #ahij4    #bj ,wi$emann1    
,annal51 #ahib1 xlv1 p4 #ibc4    #ba    
,g5]ally miscall$ #8rotational4'    #bb 
,see foot-note 7#b7 to p4 #ed above4    
#bc ,see foot-note 7#c7 to p4 #ed above4
    #bd ,see foot-note to  #bb above4   
,,! ,,end      ,ox=d    ,,pr9t$ ,,at ,,!
 ,,cl>5don ,,press    ,,by ,,horace     
,,h>t1 ,,pr9t] ,,to ,,! ,,univ]sity