GSP4oQThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. Some information may be lost. The document will open as a copy.$1LComplete Graphs` a b c d e f g h i j k l m n o p q _ ,#GSP4_M_0488/Oct 29 20010#HArial    '''$'$'$'$'$'$'$'$'$'$' $' $' %' &'8'8'8'9':'nQ\aAC u@m@HQ@$@JQ@w@   , ,? Drag Me O^ <0> >0?YY@??nQ\aAt[3]M0 3b <0> B0`nQ\aAm[4]M/3  \0<0>nQ\aAm[6]M Ob S<0> B0?YY@??"nQ\aAMax sizeSc0 3  f iW<0> F0 `` $nQ\aA Graph size0 3 b |<0> B0``nQ\aAt[4] 0 1 h d^<0> H0`nQ\aAm[5] 0 1 j B<0> J0 `` `0 1  b <0> B0``nQ\aAI`~@i@  Y@(   2i~ MA hhT>,    U(nQ\aASome DiscussionUjfa%>><0>>a%{o.#((c cQThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. Some information may be lost. The document will open as a copy. $LUses` a b c d e f g h i j k l m n o p q _ ,#GSP4_M_0488/Oct 29 20010#HArial '''$'$'$'$'$'$'$'$'$'$' $' $' %'&'8'8'8'9':'>>>>>>>>>>>>>>>>>>><0>>>>>>>>>>>><0><->>>>>><0><0>>[K    [2nQ\aAComplete graphs with hair[AlQThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. Some information may be lost. The document will open as a copy.x$1#LA Visual Variation` a b c d e f g h i j k l m n o p q _ ,#GSP4_M_0488/Oct 29 20010#HArial    '''$'$'$'$'$'$'$'$'$'$' $' $' %' &'8'8'8'9':'nQ\aACz@ j@HO@@JO@Pw@   W–5jO? Drag Me O^ U<0> >0?YY@??nQ\aAt[3]MUY0 3b <0> B0`nQ\aAm[4]M/3  ~<0>nQ\aAm[6]M Ob FK<0>#p B0?YY@??"nQ\aAMax sizeFKJ[0 3  f F4<0>#p F0 `` $nQ\aA Graph size0 3 b eo<0>#p B0``nQ\aAt[4] 0 1 h R<0>#p H0`nQ\aAm[5] 0 1 j Ps<0>#p J0 `` `0 1  b J<0>#p B0``nQ\aAI¼z@*ŗ[@  >>>]8> 0     *&nQ\aA Hide Objects nQ\aAMnQ\aANV@v@,*P gP)`<?bX,@?ns such ?     t@W%@K    C !Y@0   Q1MA  > $     Y*nQ\aAShow Measurements    ^0nQ\aAPatterns in Repetition^}oQThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. Some information may be lost. The document will open as a copy.$1g<LMultiple Copies` a b c d e f g h i j k l m n o p q _ ,#GSP4_M_0488/Oct 29 20010#HArial   '''$'$'$'$'$'$'$'$'$'$' $' $' %' &'8'8'8'9':'nQ\aACz@@k@HQ@9@JQ@~@   F.b.?"Vertices O^ <0>s  >0?YY@??nQ\aAt[3]c0 3b <0> B0`nQ\aAm[4]c/3  n<0>nQ\aAm[6]c O` Z<<0>F @0?YY@??"nQ\aAMax sizeZ<^L0 3  f Z%<0> F0 `` $nQ\aA Graph size0 3 b n<0> B0``nQ\aAt[4] 0 1 h N<0> H0`nQ\aAm[5] 0 1 j Oa<0> J0 `` `0 1  b M<0> B0``nQ\aAIz@V@ TR>H@S@Pw@V@Pw@?     ?nQ\aAJ   @ Extendi   ?   -  -   > 0     &nQ\aA Show ObjectsY@@   E~ MA !nQ\aALU@v@nQ\aAMz@v@ #$@  #$ &  [$cՎ? Recede/3#$'  l6<0>.>!(!)(!*(!+(!,(!-(!.(!/(Y@`  )*+,-./0 sQq^ MA 01MA /1MA .1MA -1MA ,1MA +1MA *1MA )1MA 1MA 1    R.nQ\aAPatterns in SequencesRc(#QThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. Some information may be lost. The document will open as a copy.$#C LOverlays` a b c d e f g h i j k l m n o p q _ ,#GSP4_M_0488/Oct 29 20010#HArial   '''$'$'$'$'$'$'$'$'$'$' $' $' %' &'8'8'8'9':' nQ\aACenterx@@i@nQ\aAUnitrx@@  O^  <0> >0?YY@??nQ\aAZeror  A>@?B>@v@   ]Ҽ? Biggest/3   <0>nQ\aAm[1]s O` @9<0>F @0?YY@??"nQ\aAMax size@9DI0 3  f G<0> F0 `` ,nQ\aALargest Graph Size O^  <0>P >0?YY@??nQ\aAt[3]s   O^  <0>P >0?YY@??nQ\aAt[4]s  O^  <0>P >0?YY@??nQ\aAt[5]s  O^  <0>P >0?YY@??nQ\aAt[6]s  O^  <0>P >0?YY@??nQ\aAt[7]s 0 3b V<0> B0`0 3  ~ E}<0> ^0  ` ``  ` nQ\aAm[3]s0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[5]s0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[6]s0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[7]s0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[8]s O^  <0>F >0?YY@??nQ\aAt[8]s   O^  <0>F >0?YY@??nQ\aAt[9]s  O^  <0>F >0?YY@?? nQ\aAt[10]  ' O`  0<0>F @0?YY@??nQ\aAt[11] 0 = O`  G<0>F @0?YY@??nQ\aAt[12] G T O`  ]<0>F @0?YY@??nQ\aAt[13] ] j O`  s<0>F @0?YY@??nQ\aAt[14] s  O`  <0>F @0?YY@??nQ\aAt[15]   O`  <0>F @0?YY@??nQ\aAt[16]   O`  <0>F @0?YY@??nQ\aAt[17]   O`  <0>F @0?YY@??nQ\aAt[18]   O`  <0>F @0?YY@??nQ\aAt[19]   O`  <0>F @0?YY@?? nQ\aAt[20]   O`  <0>F @0?YY@??nQ\aAt[21]   O`  %<0>F @0?YY@??nQ\aAt[22] % 2 O`  ;<0>F @0?YY@??nQ\aAt[23] ; H O`  Q<0>F @0?YY@??nQ\aAt[24] Q ^ O`  g<0>F @0?YY@??nQ\aAt[25] g t O`  }<0>F @0?YY@??nQ\aAt[26] } 0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[9]0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[10]0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[11]0 3  ~ E.<0> ^0  ` ``  ` nQ\aAm[12]0 3  ~ EE<0> ^0  ` ``  ` nQ\aAm[13]0 3  ~ E[<0> ^0  ` ``  ` nQ\aAm[14]0 3  ~ Eq<0> ^0  ` ``  ` nQ\aAm[15]0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[16]0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[17]0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[18]0 3 ! ~ E<0> ^0  ` ``  ` nQ\aAm[19]0 3 " ~ E<0> ^0  ` ``  ` nQ\aAm[20]0 3 # ~ E<0> ^0  ` ``  ` nQ\aAm[21]0 3 $ ~ E <0> ^0  ` ``  ` nQ\aAm[22]0 3 % ~ E"<0> ^0  ` ``  ` nQ\aAm[23]0 3 & ~ E9<0> ^0  ` ``  ` nQ\aAm[24]0 3 ' ~ EO<0> ^0  ` ``  ` nQ\aAm[25]0 3 ( ~ Ee<0> ^0  ` ``  ` nQ\aAm[26]0 3 ) ~ E{<0> ^0  ` ``  ` nQ\aAm[27]0 3b <0>P B0`0 1h <0>P H0`0 1 >=b <0>P B0``0 1=>j <0>P J0 `` `?@ BA0 3b <0>P B0``Y@( D==?@ABC 2i~MA CE0 3b <0>P B0`0 1h <0>P H0`0 1 HGb <0>P B0``0 1GHj <0>P J0 `` `IJ LK0 3b <0>P B0``Y@( NGGIJKLM 2i~ MA MO0 3b <0>P B0`0 1<h <0>P H0`0 1 RQb <0>P B0``0 1Q<Rj <0>P J0 `` `ST VU0 3<b <0>P B0``Y@( XQQSTUVW 2i~MA WY0 3b <0>P B0`0 1;h <0>P H0`0 1 \[b <0>P B0``0 1[;\j <0>P J0 `` `]^ `_0 3;b <0>P B0``Y@( b[[]^_`a 2i~MA ac0 3b <0>P B0`0 1;h <0>P H0`0 1 feb <0>P B0``0 1e;fj <0>P J0 `` `gh ji0 3;b <0>P B0``Y@( leeghijk 2i~MA km0 3b <0>P B0`0 1:h <0>P H0`0 1 pob <0>P B0``0 1o:pj <0>P J0 `` `qr ts0 3:b <0>P B0``Y@( vooqrstu 2i~MA uw0 3b <0>P B0`0 19h <0>P H0`0 1 zyb <0>P B0``0 1y9zj <0>P J0 `` `{| ~}0 39b <0>P B0``Y@( yy{|}~ 2i~MA 0 3b <0>P B0`0 18h <0>P H0`0 1 b <0>P B0``0 18j <0>P J0 `` ` 0 38b <0>P B0``Y@(  2i~MA 0 3b <0>P B0`0 17h <0>P H0`0 1 b <0>P B0``0 17j <0>P J0 `` ` 0 37b <0>P B0``Y@(  2i~MA 0 3b <0>P B0`0 16h <0>P H0`0 1 b <0>P B0``0 16j <0>P J0 `` ` 0 36b <0>P B0``Y@(  2i~MA 0 3b <0>P B0`0 15h <0>P H0`0 1 b <0>P B0``0 15j <0>P J0 `` ` 0 35b <0>P B0``Y@(  2i~MA 0 3b <0>P B0`0 14h <0>P H0`0 1 b <0>P B0``0 14j <0>P J0 `` ` 0 34b <0>P B0``Y@(  2i~MA 0 3b <0>P B0`0 13h <0>P H0`0 1 b <0>P B0``0 13j <0>P J0 `` ` 0 33b <0>P B0``Y@(  2i~MA 0 3b <0>P B0`0 12h <0>P H0`0 1 b <0>P B0``0 12j <0>P J0 `` ` 0 32b <0>P B0``Y@(  2i~MA 0 3b <0>P B0`0 11h <0>P H0`0 1 b <0>P B0``0 11j <0>P J0 `` ` 0 31b <0>P B0``Y@(  2i~MA 0 3b <0>P B0`0 10h <0>P H0`0 1 b <0>P B0``0 10j <0>P J0 `` ` 0 30b <0>P B0``Y@(  2i~MA 0 3b <0>P B0`0 1/h <0>P H0`0 1 b <0>P B0``0 1/j <0>P J0 `` ` 0 3/b <0>P B0``Y@(  2i~MA 0 3b <0>P B0`0 1.h <0>P H0`0 1 b <0>P B0``0 1.j <0>P J0 `` ` 0 3.b <0>P B0``Y@(  2i~MA 0 3b <0>P B0`0 1-h <0>P H0`0 1 b <0>P B0``0 1-j <0>P J0 `` ` 0 3-b <0>P B0``Y@(  2i~MA 0 3b <0>P B0`0 1,h <0>P H0`0 1 b <0>P B0``0 1,j <0>P J0 `` ` 0 3,b <0>P B0``Y@(  2i~MA 0 3b <0>P B0`0 1+h <0>P H0`0 1 b <0>P B0``0 1+j <0>P J0 `` `   0 3+b <0>P B0``Y@(     2i~MA   0 3b <0>P B0`0 1*h <0>P H0`0 1 b <0>P B0``0 1*j <0>P J0 `` ` 0 3*b <0>P B0``Y@(  2i~MA 0 3b <0>P B0`0 1h <0>P H0`0 1 b <0>P B0``0 1j <0>P J0 `` ` 0 3b <0>P B0``Y@(  2i~MA !0 3b <0>P B0`0 1h <0>P H0`0 1 $#b <0>P B0``0 1#$j <0>P J0 `` `%& ('0 3b <0>P B0``Y@( *##%&'() 2i~MA )+0 3b <0>P B0`0 1h <0>P H0`0 1 .-b <0>P B0``0 1-.j <0>P J0 `` `/0 210 3b <0>P B0``Y@( 4--/0123 2i~MA 350 3b <0>P B0`0 1h <0>P H0`0 1 87b <0>P B0``0 178j <0>P J0 `` `9: <;0 3b <0>P B0``Y@( >779:;<= 2i~MA =?> d*+,-./0123456789:;<   >$nQ\aA Show Sizes88n%>ny~    -V0nQ\aAWhat are these curves?-Vg(#QThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. Some information may be lost. The document will open as a copy.D$MLOverlay notes` a b c d e f g h i j k l m n o p q _ ,#GSP4_M_0488/Oct 29 20010#HArial    '''$'$'$'$'$'$'$'$'$'$' $' $' %' &'8'8'8'9':' nQ\aACenteri@o@nQ\aAUnitri@Z@  O^  <0>P >0?YY@??nQ\aAZeror A(@&@B(@Pt@   9Y? Biggest/3   <0>nQ\aAm[1]s O` Y<0>F @0?YY@??"nQ\aAMax sizeYQi0 3  f J<0> F0 `` nQ\aABiggest O^  <0>Q  >0?YY@??nQ\aAt[3]s   O^  <0>Q  >0?YY@??nQ\aAt[4]s  O^  <0>Q  >0?YY@??nQ\aAt[5]s  O^  <0>Q  >0?YY@??nQ\aAt[6]s  O^  <0>Q  >0?YY@??nQ\aAt[7]s 0 3b V<0> B0`0 3  ~ E}<0> ^0  ` ``  ` nQ\aAm[3]s0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[5]s0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[6]s0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[7]s0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[8]s O^  <0>F >0?YY@??nQ\aAt[8]s   O^  <0>F >0?YY@??nQ\aAt[9]s  O^  <0>F >0?YY@?? nQ\aAt[10]  ' O`  0<0>F @0?YY@??nQ\aAt[11] 0 = O`  G<0>F @0?YY@??nQ\aAt[12] G T O`  ]<0>F @0?YY@??nQ\aAt[13] ] j O`  s<0>F @0?YY@??nQ\aAt[14] s  O`  <0>F @0?YY@??nQ\aAt[15]   O`  <0>F @0?YY@??nQ\aAt[16]   O`  <0>F @0?YY@??nQ\aAt[17]   O`  <0>F @0?YY@??nQ\aAt[18]   O`  <0>F @0?YY@??nQ\aAt[19]   O`  <0>F @0?YY@?? nQ\aAt[20]   O`  <0>F @0?YY@??nQ\aAt[21]   O`  %<0>F @0?YY@??nQ\aAt[22] % 2 O`  ;<0>F @0?YY@??nQ\aAt[23] ; H O`  Q<0>F @0?YY@??nQ\aAt[24] Q ^ O`  g<0>F @0?YY@??nQ\aAt[25] g t O`  }<0>F @0?YY@??nQ\aAt[26] } 0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[9]0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[10]0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[11]0 3  ~ E.<0> ^0  ` ``  ` nQ\aAm[12]0 3  ~ EE<0> ^0  ` ``  ` nQ\aAm[13]0 3  ~ E[<0> ^0  ` ``  ` nQ\aAm[14]0 3  ~ Eq<0> ^0  ` ``  ` nQ\aAm[15]0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[16]0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[17]0 3  ~ E<0> ^0  ` ``  ` nQ\aAm[18]0 3 ! ~ E<0> ^0  ` ``  ` nQ\aAm[19]0 3 " ~ E<0> ^0  ` ``  ` nQ\aAm[20]0 3 # ~ E<0> ^0  ` ``  ` nQ\aAm[21]0 3 $ ~ E <0> ^0  ` ``  ` nQ\aAm[22]0 3 % ~ E"<0> ^0  ` ``  ` nQ\aAm[23]0 3 & ~ E9<0> ^0  ` ``  ` nQ\aAm[24]0 3 ' ~ EO<0> ^0  ` ``  ` nQ\aAm[25]0 3 ( ~ Ee<0> ^0  ` ``  ` nQ\aAm[26]0 3 ) ~ E{<0> ^0  ` ``  ` nQ\aAm[27]0 3b <0>Q  B0`0 1h <0>Q  H0`0 1 >=b <0>Q  B0``0 1=>j <0>Q  J0 `` `?@ BA0 3b <0>Q  B0``Y@( D==?@ABC 2i~MA CE0 3b <0>Q  B0`0 1h <0>Q  H0`0 1 HGb <0>Q  B0``0 1GHj <0>Q  J0 `` `IJ LK0 3b <0>Q  B0``Y@( NGGIJKLM 2i~ MA MO0 3b <0>Q  B0`0 1<h <0>Q  H0`0 1 RQb <0>Q  B0``0 1Q<Rj <0>Q  J0 `` `ST VU0 3<b <0>Q  B0``Y@( XQQSTUVW 2i~MA WY0 3b <0>Q  B0`0 1;h <0>Q  H0`0 1 \[b <0>Q  B0``0 1[;\j <0>Q  J0 `` `]^ `_0 3;b <0>Q  B0``Y@( b[[]^_`a 2i~MA ac0 3b <0>Q  B0`0 1;h <0>Q  H0`0 1 feb <0>Q  B0``0 1e;fj <0>Q  J0 `` `gh ji0 3;b <0>Q  B0``Y@( leeghijk 2i~MA km0 3b <0>Q  B0`0 1:h <0>Q  H0`0 1 pob <0>Q  B0``0 1o:pj <0>Q  J0 `` `qr ts0 3:b <0>Q  B0``Y@( vooqrstu 2i~MA uw0 3b <0>Q  B0`0 19h <0>Q  H0`0 1 zyb <0>Q  B0``0 1y9zj <0>Q  J0 `` `{| ~}0 39b <0>Q  B0``Y@( yy{|}~ 2i~MA 0 3b <0>Q  B0`0 18h <0>Q  H0`0 1 b <0>Q  B0``0 18j <0>Q  J0 `` ` 0 38b <0>Q  B0``Y@(  2i~MA 0 3b <0>Q  B0`0 17h <0>Q  H0`0 1 b <0>Q  B0``0 17j <0>Q  J0 `` ` 0 37b <0>Q  B0``Y@(  2i~MA 0 3b <0>Q  B0`0 16h <0>Q  H0`0 1 b <0>Q  B0``0 16j <0>Q  J0 `` ` 0 36b <0>Q  B0``Y@(  2i~MA 0 3b <0>Q  B0`0 15h <0>Q  H0`0 1 b <0>Q  B0``0 15j <0>Q  J0 `` ` 0 35b <0>Q  B0``Y@(  2i~MA 0 3b <0>Q  B0`0 14h <0>Q  H0`0 1 b <0>Q  B0``0 14j <0>Q  J0 `` ` 0 34b <0>Q  B0``Y@(  2i~MA 0 3b <0>Q  B0`0 13h <0>Q  H0`0 1 b <0>Q  B0``0 13j <0>Q  J0 `` ` 0 33b <0>Q  B0``Y@(  2i~MA 0 3b <0>Q  B0`0 12h <0>Q  H0`0 1 b <0>Q  B0``0 12j <0>Q  J0 `` ` 0 32b <0>Q  B0``Y@(  2i~MA 0 3b <0>Q  B0`0 11h <0>Q  H0`0 1 b <0>Q  B0``0 11j <0>Q  J0 `` ` 0 31b <0>Q  B0``Y@(  2i~MA 0 3b <0>Q  B0`0 10h <0>Q  H0`0 1 b <0>Q  B0``0 10j <0>Q  J0 `` ` 0 30b <0>Q  B0``Y@(  2i~MA 0 3b <0>Q  B0`0 1/h <0>Q  H0`0 1 b <0>Q  B0``0 1/j <0>Q  J0 `` ` 0 3/b <0>Q  B0``Y@(  2i~MA 0 3b <0>Q  B0`0 1.h <0>Q  H0`0 1 b <0>Q  B0``0 1.j <0>Q  J0 `` ` 0 3.b <0>Q  B0``Y@(  2i~yMA 0 3b <0>Q  B0`0 1-h <0>Q  H0`0 1 b <0>Q  B0``0 1-j <0>Q  J0 `` ` 0 3-b <0>Q  B0``Y@(  2i~dMA 0 3b <0>Q  B0`0 1,h <0>Q  H0`0 1 b <0>Q  B0``0 1,j <0>Q  J0 `` ` 0 3,b <0>Q  B0``Y@(  2i~QMA 0 3b <0>Q  B0`0 1+h <0>Q  H0`0 1 b <0>Q  B0``0 1+j <0>Q  J0 `` `   0 3+b <0>Q  B0``Y@(     2i~1MA   0 3b <0>Q  B0`0 1*h <0>Q  H0`0 1 b <0>Q  B0``0 1*j <0>Q  J0 `` ` 0 3*b <0>Q  B0``Y@(  2i~@MA 0 3b <0>Q  B0`0 1h <0>Q  H0`0 1 b <0>Q  B0``0 1j <0>Q  J0 `` ` 0 3b <0>Q  B0``Y@(  2i~$MA !0 3b <0>Q  B0`0 1h <0>Q  H0`0 1 $#b <0>Q  B0``0 1#$j <0>Q  J0 `` `%& ('0 3b <0>Q  B0``Y@( *##%&'() 2i~MA )+0 3b <0>Q  B0`0 1h <0>Q  H0`0 1 .-b <0>Q  B0``0 1-.j <0>Q  J0 `` `/0 210 3b <0>Q  B0``Y@( 4--/0123 2i~$MA 350 3b <0>Q  B0`0 1h <0>Q  H0`0 1 87b <0>Q  B0``0 178j <0>Q  J0 `` `9: <;0 3b <0>Q  B0``Y@( >779:;<= 2i~MA =?> d*+,-./0123456789:;<   CQ$nQ\aA Show Sizes >>>>>>>>>>><0><0><0><0>>>>>>>>>>>>>>>>>>>> ef  C0>ZI ?nQ\aADDD ED> D$>A ף9F   .nQ\aASweep a 1:2 segment.<->>>xbl    k(nQ\aA< F   o&nQ\aA Show Segment<>  JG   ,nQ\aASweep a 1:2 segment> F   [&nQ\aA Hide Segmente>  BK   'n*nQ\aAShow Explanationn