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242 0 242 149.334 363 149.334 242 0 363 0 CPBorderColorProperty g1 version 1 CPVerticalAlignmentProperty 2 CPStrokeWidthProperty 2 dictionary 0 0 305 267 dictionary 0 0 305 267 dictionary 412 36 dictionary 0 0 305 267 dictionary 0 0 305 267 dictionary 0 0 304 266 <styles> <fill-style fill-type="none"/> <dash-style pattern="none"/> <shadow-style opacity="0" radius="0"/> </styles> • • • • • dictionary 0 0 750 490 dictionary 0 0 750 490 dictionary 0 0 750 490 CPTextProperty asd asdf asdf CPStrokePatternProperty CPCellOpacityProperty 1 CPFillProperty CPBorderOpacityProperty 1 TableNodeSnapShotProperty1 1.2 822 568 0 568 274.001 568 0 378.666 274.001 378.666 274.001 568 547.999 568 274.001 378.666 547.999 378.666 547.999 568 822 568 547.999 378.666 822 378.666 0 378.666 274.001 378.666 0 189.334 274.001 189.334 274.001 378.666 547.999 378.666 274.001 189.334 547.999 189.334 547.999 378.666 822 378.666 547.999 189.334 822 189.334 0 189.334 274.001 189.334 0 0 274.001 0 274.001 189.334 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149.334 0 0 274.001 0 274.001 149.334 547.999 149.334 274.001 0 547.999 0 547.999 149.334 822 149.334 547.999 0 822 0 CPBorderColorProperty g1 version 1 CPVerticalAlignmentProperty 2 CPStrokeWidthProperty 2 dictionary 0 0 550 338 dictionary 0 0 550 338 dictionary 824 36 dictionary 0 0 550 338 dictionary 0 0 550 338 dictionary 0 0 380 342 <styles> <fill-style fill-type="none"/> <dash-style pattern="none"/> <shadow-style opacity="0" radius="0"/> </styles> dictionary 0 0 550 338 dictionary 0 0 550 338 dictionary 0 0 550 338 CPTextProperty asd asdf asdf CPStrokePatternProperty CPCellOpacityProperty 1 CPFillProperty CPBorderOpacityProperty 1 TableNodeSnapShotProperty1 1.2 822 448 0 448 274.001 448 0 298.666 274.001 298.666 274.001 448 547.999 448 274.001 298.666 547.999 298.666 547.999 448 822 448 547.999 298.666 822 298.666 0 298.666 274.001 298.666 0 149.334 274.001 149.334 274.001 298.666 547.999 298.666 274.001 149.334 547.999 149.334 547.999 298.666 822 298.666 547.999 149.334 822 149.334 0 149.334 274.001 149.334 0 0 274.001 0 274.001 149.334 547.999 149.334 274.001 0 547.999 0 547.999 149.334 822 149.334 547.999 0 822 0 CPBorderColorProperty g1 version 1 CPVerticalAlignmentProperty 2 CPStrokeWidthProperty 2 dictionary 0 0 550 338 dictionary 0 0 550 338 dictionary 824 36 dictionary 0 0 550 338 dictionary 0 0 550 338 dictionary 0 0 380 342 <styles> <fill-style fill-type="none"/> <dash-style pattern="none"/> <shadow-style opacity="0" radius="0"/> </styles> dictionary 0 0 550 338 dictionary 0 0 550 338 dictionary 0 0 550 338 CPTextProperty asd asdf asdf CPStrokePatternProperty CPCellOpacityProperty 1 CPFillProperty CPBorderOpacityProperty 1 TableNodeSnapShotProperty1 1.2 822 448 0 448 274.001 448 0 298.666 274.001 298.666 274.001 448 547.999 448 274.001 298.666 547.999 298.666 547.999 448 822 448 547.999 298.666 822 298.666 0 298.666 274.001 298.666 0 149.334 274.001 149.334 274.001 298.666 547.999 298.666 274.001 149.334 547.999 149.334 547.999 298.666 822 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149.334 242 0 363 0 CPBorderColorProperty g1 version 1 CPVerticalAlignmentProperty 2 CPStrokeWidthProperty 2 dictionary 0 0 305 267 dictionary 0 0 305 267 dictionary 412 36 dictionary 0 0 305 267 dictionary 0 0 305 267 dictionary 0 0 304 266 <styles> <fill-style fill-type="none"/> <dash-style pattern="none"/> <shadow-style opacity="0" radius="0"/> </styles> • • • • • dictionary 0 0 305 267 dictionary 0 0 305 267 dictionary 0 0 305 267 CPTextProperty 32 pt CPStrokePatternProperty CPCellOpacityProperty 1 CPFillProperty CPBorderOpacityProperty 1 TableNodeSnapShotProperty1 1.2 363 448 0 448 121 448 0 298.666 121 298.666 121 448 242 448 121 298.666 242 298.666 242 448 363 448 242 298.666 363 298.666 0 298.666 121 298.666 0 149.334 121 149.334 121 298.666 242 298.666 121 149.334 242 149.334 242 298.666 363 298.666 242 149.334 363 149.334 0 149.334 121 149.334 0 0 121 0 121 149.334 242 149.334 121 0 242 0 242 149.334 363 149.334 242 0 363 0 CPBorderColorProperty g1 version 1 CPVerticalAlignmentProperty 2 CPStrokeWidthProperty 2 dictionary 0 0 305 267 dictionary 0 0 305 267 dictionary 412 36 dictionary 0 0 305 267 dictionary 0 0 305 267 dictionary 0 0 304 266 <styles> <fill-style fill-type="none"/> <dash-style pattern="none"/> <shadow-style opacity="0" radius="0"/> </styles> • • • • • dictionary 0 0 305 267 dictionary 0 0 305 267 dictionary 0 0 305 267 CPTextProperty 32 pt CPStrokePatternProperty CPCellOpacityProperty 1 CPFillProperty CPBorderOpacityProperty 1 TableNodeSnapShotProperty1 1.2 363 448 0 448 121 448 0 298.666 121 298.666 121 448 242 448 121 298.666 242 298.666 242 448 363 448 242 298.666 363 298.666 0 298.666 121 298.666 0 149.334 121 149.334 121 298.666 242 298.666 121 149.334 242 149.334 242 298.666 363 298.666 242 149.334 363 149.334 0 149.334 121 149.334 0 0 121 0 121 149.334 242 149.334 121 0 242 0 242 149.334 363 149.334 242 0 363 0 CPBorderColorProperty g1 version 1 CPVerticalAlignmentProperty 2 CPStrokeWidthProperty 2 dictionary 0 0 305 267 dictionary 0 0 305 267 dictionary 412 36 dictionary 0 0 305 267 dictionary 0 0 305 267 dictionary 0 0 304 266 <body visibility="hidden" vertical-alignment="tracks-master"/> </drawables> <transition-style type="inherited"/> <thumbnails> <thumbnail byte-size="6984" size="60 45" file="thumbs/st0.tiff"/> </thumbnails> <bullets> <bullet level="0" marker-type="inherited"> <content font-color="g1" paragraph-alignment="center" tab-stops="L 96" font-size="84" font-name="GillSans"> <![CDATA[Some Fractals and Fractal Dimensions]]> </content> </bullet> </bullets> </slide> <slide master-slide-id="master-slide-2"> <drawables> <body visibility="tracks-master" location="100 106" size="824 450" vertical-alignment="tracks-master"/> <title visibility="hidden" vertical-alignment="tracks-master"/> </drawables> <transition-style type="inherited"/> <thumbnails> <thumbnail byte-size="6986" size="60 45" file="thumbs/st1.tiff"/> </thumbnails> <bullets> <bullet level="0" marker-type="inherited"> <content font-color="g1" paragraph-alignment="center" tab-stops="L 96" font-size="84" font-name="GillSans"/> </bullet> <bullet level="1" marker-type="inherited"> <content font-color="g1" paragraph-head-indent="7" paragraph-first-line-indent="7" tab-stops="L 58" font-size="42" font-name="GillSans"> <![CDATA[The Cantor set: we take a line segment, and remove the middle third. For each remaining piece, we again remove the middle third, and continue indefinitely.]]> </content> </bullet> </bullets> </slide> <slide master-slide-id="master-slide-2"> <drawables> <line tail="148.207 383" head="877 385" stroke-width="5"/> <title visibility="hidden" vertical-alignment="tracks-master"/> <body visibility="hidden" vertical-alignment="tracks-master"/> </drawables> <transition-style type="inherited"/> <thumbnails> <thumbnail byte-size="6986" size="60 45" file="thumbs/st2.tiff"/> </thumbnails> <bullets> <bullet level="0" marker-type="inherited"> <content font-color="g1" paragraph-alignment="center" tab-stops="L 96" font-size="84" font-name="GillSans"/> </bullet> </bullets> </slide> <slide master-slide-id="master-slide-2"> <drawables> <line tail="634 384" head="877 384" stroke-width="5"/> <line tail="148 384" head="391 384" stroke-width="5"/> <title visibility="hidden" vertical-alignment="tracks-master"/> <body visibility="hidden" vertical-alignment="tracks-master"/> 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head="817 384" stroke-width="5"/> <line tail="850 384" head="853 384" stroke-width="5"/> <line tail="868 384" head="871 384" stroke-width="5"/> <line tail="856 384" head="859 384" stroke-width="5"/> <line tail="874 384" head="877 384" stroke-width="5"/> <title visibility="hidden" vertical-alignment="tracks-master"/> <body visibility="hidden" vertical-alignment="tracks-master"/> </drawables> <transition-style type="inherited"/> <thumbnails> <thumbnail byte-size="6986" size="60 45" file="thumbs/st7.tiff"/> </thumbnails> <bullets> <bullet level="0" marker-type="inherited"> <content font-color="g1" paragraph-alignment="center" tab-stops="L 96" font-size="84" font-name="GillSans"/> </bullet> </bullets> </slide> <slide master-slide-id="master-slide-2"> <drawables> <body visibility="tracks-master" location="100 117" size="824 450" vertical-alignment="tracks-master"/> <title visibility="hidden" vertical-alignment="tracks-master"/> </drawables> <transition-style type="inherited"/> <thumbnails> <thumbnail byte-size="6984" size="60 45" file="thumbs/st8.tiff"/> </thumbnails> <bullets> <bullet level="0" marker-type="inherited"> <content font-color="g1" paragraph-alignment="center" tab-stops="L 96" font-size="84" font-name="GillSans"/> </bullet> <bullet level="1" marker-type="inherited"> <content font-color="g1" paragraph-head-indent="7" paragraph-first-line-indent="7" tab-stops="L 58" font-size="42" font-name="GillSans"> <![CDATA[To calculate the fractal / Hausdorff / capacity / box-counting dimension, we see how many boxes (circles) of diameter 1/r^n we need to cover the set (in this case, we will use r = 3, since it fits nicely).]]> </content> </bullet> </bullets> </slide> <slide master-slide-id="master-slide-2"> <drawables> <line tail="148 384" head="151 384" stroke-width="5"/> <line tail="166 384" head="169 384" stroke-width="5"/> <line tail="154 384" head="157 384" stroke-width="5"/> <line tail="172 384" head="175 384" stroke-width="5"/> <line tail="202 384" head="205 384" 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<content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">r</content> <dict/> </plugin:element> <plugin:element tr="705" tl="704" br="708" type="text" bl="707"> <plugin:node pos="{363, 448}" ident="705"/> <plugin:node pos="{363, 373.75}" ident="708"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">N<span font-size="20">r</span> </content> <dict/> </plugin:element> <plugin:element tr="707" tl="706" br="710" type="text" bl="709"> <plugin:node pos="{0, 297.832}" ident="709"/> <plugin:node pos="{182, 297.832}" ident="710"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">1</content> <dict/> </plugin:element> <plugin:element tr="708" tl="707" br="711" type="text" bl="710"> <plugin:node pos="{363, 297.832}" ident="711"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" 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<dict/> </plugin:segment> <plugin:segment direction="horizontal" end="716" start="715"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="717" start="716"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="719" start="718"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="720" start="719"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="722" start="721"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="723" start="722"> <dict/> </plugin:segment> </plugin:table> <string key="CPVersion">1.2</string> <string key="MIMEType">application/apple-sticks</string> </plugin-data> </plugin> <body visibility="hidden" vertical-alignment="tracks-master"/> </drawables> <transition-style type="inherited"/> <thumbnails> <thumbnail byte-size="7176" size="60 45" file="thumbs/st81.tiff"/> </thumbnails> <bullets> <bullet level="0" marker-type="inherited"> <content font-color="g1" paragraph-alignment="center" tab-stops="L 96" font-size="50" font-name="GillSans">D = Lim(log(N<span font-size="30">r</span> <![CDATA[)/log(1/r)) = log(2) / log(3)]]> </content> </bullet> </bullets> </slide> <slide master-slide-id="master-slide-2"> <drawables> <body visibility="tracks-master" location="100 97" size="824 450" vertical-alignment="tracks-master"/> <title visibility="hidden" vertical-alignment="tracks-master"/> </drawables> <transition-style type="inherited"/> <thumbnails> <thumbnail byte-size="6986" size="60 45" file="thumbs/st14.tiff"/> </thumbnails> <bullets> <bullet level="0" marker-type="inherited"> <content font-color="g1" paragraph-alignment="center" tab-stops="L 96" font-size="84" font-name="GillSans"/> </bullet> <bullet level="1" marker-type="inherited"> <content font-color="g1" paragraph-head-indent="7" paragraph-first-line-indent="7" tab-stops="L 58" font-size="42" font-name="GillSans"> <![CDATA[The Koch snowflake: We start with an equilateral triangle. We duplicate the middle third of each side, forming a smaller equilateral triangle. 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size="60 45" file="thumbs/st51.tiff"/> </thumbnails> <bullets> <bullet level="0" marker-type="inherited"> <content font-color="g1" paragraph-alignment="center" tab-stops="L 96" font-size="84" font-name="GillSans"/> </bullet> </bullets> </slide> <slide master-slide-id="master-slide-2"> <drawables> <body visibility="tracks-master" location="100 117" size="824 450" vertical-alignment="tracks-master"/> <title visibility="hidden" vertical-alignment="tracks-master"/> </drawables> <transition-style type="inherited"/> <thumbnails> <thumbnail byte-size="6984" size="60 45" file="thumbs/st211.tiff"/> </thumbnails> <bullets> <bullet level="0" marker-type="inherited"> <content font-color="g1" paragraph-alignment="center" tab-stops="L 96" font-size="84" font-name="GillSans"/> </bullet> <bullet level="1" marker-type="inherited"> <content font-color="g1" paragraph-head-indent="7" paragraph-first-line-indent="7" tab-stops="L 58" font-size="42" font-name="GillSans"> <![CDATA[To calculate the fractal / 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start="730"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="734" start="731"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="735" start="732"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="736" start="733"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="737" start="734"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="738" start="735"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="739" start="736"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="740" start="737"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="741" start="738"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="742" start="739"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="743" start="740"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="744" start="741"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="725" start="724"> <dict> <reference key="CPStrokePatternProperty" id-ref="dash-style-4"/> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="horizontal" end="726" start="725"> <dict> <reference key="CPStrokePatternProperty" id-ref="dash-style-4"/> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="horizontal" end="728" start="727"> <dict> <reference key="CPStrokePatternProperty" id-ref="dash-style-4"/> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="horizontal" end="729" start="728"> <dict> <reference key="CPStrokePatternProperty" id-ref="dash-style-4"/> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="horizontal" end="731" start="730"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="732" start="731"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="734" start="733"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="735" start="734"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="737" start="736"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="738" start="737"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="740" start="739"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="741" start="740"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="743" start="742"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="744" start="743"> <dict/> </plugin:segment> </plugin:table> <string key="CPVersion">1.2</string> <string key="MIMEType">application/apple-sticks</string> </plugin-data> </plugin> <body visibility="hidden" vertical-alignment="tracks-master"/> </drawables> <transition-style type="inherited"/> <thumbnails> <thumbnail byte-size="7176" 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file="thumbs/st18.tiff"/> </thumbnails> <bullets> <bullet level="0" marker-type="inherited"> <content font-color="g1" paragraph-alignment="center" tab-stops="L 96" font-size="50" font-name="GillSans">D = Lim(log(N<span font-size="30">r</span> <![CDATA[)/log(1/r)) = log(8) / log(3)]]> </content> </bullet> </bullets> </slide> <slide master-slide-id="master-slide-2"> <drawables> <body visibility="tracks-master" location="100 82" size="824 450" vertical-alignment="tracks-master"/> <title visibility="hidden" vertical-alignment="tracks-master"/> </drawables> <transition-style type="inherited"/> <thumbnails> <thumbnail byte-size="7004" size="60 45" file="thumbs/st35.tiff"/> </thumbnails> <bullets> <bullet level="0" marker-type="inherited"> <content font-color="g1" paragraph-alignment="center" tab-stops="L 96" font-size="84" font-name="GillSans"/> </bullet> <bullet level="1" marker-type="inherited"> <content font-color="g1" paragraph-head-indent="7" paragraph-first-line-indent="7" tab-stops="L 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<plugin:table key="root" version="1.2" size="{363, 448}"> <dict/> <plugin:element tr="767" tl="766" br="770" type="text" bl="769"> <plugin:node pos="{0, 448}" ident="766"/> <plugin:node pos="{182, 448}" ident="767"/> <plugin:node pos="{0, 373.75}" ident="769"/> <plugin:node pos="{182, 373.75}" ident="770"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">r</content> <dict/> </plugin:element> <plugin:element tr="768" tl="767" br="771" type="text" bl="770"> <plugin:node pos="{363, 448}" ident="768"/> <plugin:node pos="{363, 373.75}" ident="771"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">N<span font-size="20">r</span> </content> <dict/> </plugin:element> <plugin:element tr="770" tl="769" br="773" type="text" bl="772"> <plugin:node pos="{0, 297.832}" ident="772"/> <plugin:node pos="{182, 297.832}" ident="773"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">1</content> <dict/> </plugin:element> <plugin:element tr="771" tl="770" br="774" type="text" bl="773"> <plugin:node pos="{363, 297.832}" ident="774"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">1</content> <dict/> </plugin:element> <plugin:element tr="773" tl="772" br="776" type="text" bl="775"> <plugin:node pos="{0, 223.583}" ident="775"/> <plugin:node pos="{182, 223.583}" ident="776"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">1/2</content> <dict/> </plugin:element> <plugin:element tr="774" tl="773" br="777" type="text" bl="776"> <plugin:node pos="{363, 223.583}" ident="777"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">3</content> <dict/> </plugin:element> <plugin:element tr="776" tl="775" br="779" type="text" bl="778"> <plugin:node pos="{0, 148.499}" ident="778"/> <plugin:node pos="{182, 148.499}" ident="779"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">1/2^2</content> <dict/> </plugin:element> <plugin:element tr="777" tl="776" br="780" type="text" bl="779"> <plugin:node pos="{363, 148.499}" ident="780"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">3^2</content> <dict/> </plugin:element> <plugin:element tr="779" tl="778" br="782" type="text" bl="781"> <plugin:node pos="{0, 74.2495}" ident="781"/> <plugin:node pos="{182, 74.2495}" ident="782"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">1/2^3</content> <dict/> </plugin:element> <plugin:element tr="780" tl="779" br="783" type="text" bl="782"> <plugin:node pos="{363, 74.2495}" ident="783"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">3^3</content> <dict> <fill-style key="CPCellFillProperty" fill-type="none"/> </dict> </plugin:element> <plugin:element tr="782" tl="781" br="785" type="text" bl="784"> <plugin:node pos="{0, 0}" ident="784"/> <plugin:node pos="{182, 0}" ident="785"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">1/2^n</content> <dict/> </plugin:element> <plugin:element tr="783" tl="782" br="786" type="text" bl="785"> <plugin:node pos="{363, 0}" ident="786"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">3^n</content> <dict/> </plugin:element> <plugin:segment direction="vertical" end="769" start="766"> <dict> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="vertical" end="770" start="767"> <dict> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="vertical" end="771" 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start="779"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="783" start="780"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="784" start="781"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="785" start="782"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="786" start="783"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="767" start="766"> <dict> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="horizontal" end="768" start="767"> <dict> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="horizontal" end="770" start="769"> <dict> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="horizontal" end="771" start="770"> <dict> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="horizontal" end="773" start="772"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="774" start="773"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="776" start="775"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="777" start="776"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="779" start="778"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="780" start="779"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="782" start="781"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="783" start="782"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="785" start="784"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="786" start="785"> <dict/> </plugin:segment> </plugin:table> <string key="CPVersion">1.2</string> <string key="MIMEType">application/apple-sticks</string> </plugin-data> </plugin> <body visibility="hidden" 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tr="809" tl="808" br="812" type="text" bl="811"> <plugin:node pos="{0, 448}" ident="808"/> <plugin:node pos="{182, 448}" ident="809"/> <plugin:node pos="{0, 373.75}" ident="811"/> <plugin:node pos="{182, 373.75}" ident="812"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">r</content> <dict/> </plugin:element> <plugin:element tr="810" tl="809" br="813" type="text" bl="812"> <plugin:node pos="{363, 448}" ident="810"/> <plugin:node pos="{363, 373.75}" ident="813"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">N<span font-size="20">r</span> </content> <dict/> </plugin:element> <plugin:element tr="812" tl="811" br="815" type="text" bl="814"> <plugin:node pos="{0, 297.832}" ident="814"/> <plugin:node pos="{182, 297.832}" ident="815"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">1</content> <dict/> </plugin:element> <plugin:element tr="813" tl="812" br="816" type="text" bl="815"> <plugin:node pos="{363, 297.832}" ident="816"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">1</content> <dict/> </plugin:element> <plugin:element tr="815" tl="814" br="818" type="text" bl="817"> <plugin:node pos="{0, 223.583}" ident="817"/> <plugin:node pos="{182, 223.583}" ident="818"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">1/3</content> <dict/> </plugin:element> <plugin:element tr="816" tl="815" br="819" type="text" bl="818"> <plugin:node pos="{363, 223.583}" ident="819"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">3</content> <dict/> </plugin:element> <plugin:element tr="818" tl="817" br="821" type="text" bl="820"> <plugin:node pos="{0, 148.499}" ident="820"/> <plugin:node pos="{182, 148.499}" ident="821"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">1/3^2</content> <dict/> </plugin:element> <plugin:element tr="819" tl="818" br="822" type="text" bl="821"> <plugin:node pos="{363, 148.499}" ident="822"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">3^2</content> <dict/> </plugin:element> <plugin:element tr="821" tl="820" br="824" type="text" bl="823"> <plugin:node pos="{0, 74.2495}" ident="823"/> <plugin:node pos="{182, 74.2495}" ident="824"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">1/3^3</content> <dict/> </plugin:element> <plugin:element tr="822" tl="821" br="825" type="text" bl="824"> <plugin:node pos="{363, 74.2495}" ident="825"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">3^3</content> <dict> <fill-style key="CPCellFillProperty" fill-type="none"/> </dict> </plugin:element> <plugin:element tr="824" tl="823" br="827" type="text" bl="826"> <plugin:node pos="{0, 0}" ident="826"/> <plugin:node pos="{182, 0}" ident="827"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">1/3^n</content> <dict/> </plugin:element> <plugin:element tr="825" tl="824" br="828" type="text" bl="827"> <plugin:node pos="{363, 0}" ident="828"/> <content font-color="g1" paragraph-alignment="center" tab-stops="L 84" font-size="32" font-name="GillSans">3^n</content> <dict/> </plugin:element> <plugin:segment direction="vertical" end="811" start="808"> <dict> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="vertical" end="812" start="809"> <dict> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="vertical" end="813" start="810"> <dict> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="vertical" end="814" start="811"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="815" start="812"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="816" start="813"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="817" start="814"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="818" start="815"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="819" start="816"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="820" start="817"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="821" start="818"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="822" start="819"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="823" start="820"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="824" start="821"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="825" start="822"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="826" start="823"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="827" start="824"> <dict/> </plugin:segment> <plugin:segment direction="vertical" end="828" start="825"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="809" start="808"> <dict> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="horizontal" end="810" start="809"> <dict> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="horizontal" end="812" start="811"> <dict> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="horizontal" end="813" start="812"> <dict> <number key="CPStrokeWidthProperty">6</number> </dict> </plugin:segment> <plugin:segment direction="horizontal" end="815" start="814"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="816" start="815"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="818" start="817"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="819" start="818"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="821" start="820"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="822" start="821"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="824" start="823"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="825" start="824"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="827" start="826"> <dict/> </plugin:segment> <plugin:segment direction="horizontal" end="828" start="827"> <dict/> </plugin:segment> </plugin:table> <string key="CPVersion">1.2</string> <string key="MIMEType">application/apple-sticks</string> </plugin-data> </plugin> <body visibility="hidden" vertical-alignment="tracks-master"/> </drawables> <transition-style type="inherited"/> <thumbnails> <thumbnail byte-size="7188" size="60 45" file="thumbs/st53.tiff"/> </thumbnails> <bullets> <bullet level="0" marker-type="inherited"> <content font-color="g1" paragraph-alignment="center" tab-stops="L 96" font-size="50" font-name="GillSans">D = Lim(log(N<span font-size="30">r</span> <![CDATA[)/log(1/r)) = log(3) / log(3) = 1]]> </content> </bullet> </bullets> </slide> </slide-list> <ui-state> <number key="BGShowUIStateNotesVisible">0</number> <reference key="BGShowUIStateActiveSlide" id-ref="slide-1"/> <array key="BGShowUIStateSelectedSlides"> <reference id-ref="slide-1"/> </array> <array key="BGShowUIStateSelectedBulletPoints"/> <string key="BGShowUIStateCanvasZoom">1.000000</string> <array key="BGShowUIStateSelectedDrawables"/> <string key="BGShowUIStateCanvasOrigin">{0, 0}</string> <string key="BGShowUIStateWindowSize">96 60 1184 772 0 0 1280 832 </string> </ui-state> </presentation>