発言者： 田中徹
発言日： 2019 08/20 15:20
この夏、母校のオープンキャンパスに行き
「数学がわかるとき」というお題の話を聞きました。
その中の一つに、「事柄を図示し、自分で意味づけできたとき
「わかった」となりますね。」と
初項 1/3, 公比 1/3 の等比数列の和をパワポのアニメで示されました。
私自身の35年以上前の卒業研究の一つレプタイル(自己相似拡大)で
平面を充填するの逆バージョンでできるかなと思い
beamerとzahyou環境で作成しました。
k=(自然数)^2, k=1+(自然数)^{2} は一般的にできそうですが
k=3 は特別のようです。
長いソースになりますが、骨を拾っていただければ幸いです。
%---------------------------------------------------- ソースはここから
\documentclass[dvipdfmx,12pt]{beamer}
\usepackage{etex}
\usepackage{pxjahyper}
\usepackage{minijs}
\renewcommand{\kanjifamilydefault}{\gtdefault}
%%% バッティングを避けるためのおまじない(経験則) %%%%%%%%%%
\let\labelenumi\undefined
\let\labelenumii\undefined
\let\labelenumiii\undefined
\let\endproof\undefined
\let\pushQED\undefined
\let\popQED\undefined
\let\qedhere\undefined
\let\mathqed\undefined
\let\openbox\undefined
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[notMy]{emathP}
\usepackage{multirow}
%\usepackage{ascmac}
%\usepackage{bm}
%\usetheme{AnnArbor}
%\usetheme{Antibes}
%\usetheme{Berkeley}
%\usetheme{Berlin}
%\usetheme{Copenhagen}
\usetheme{boxes}
%\usetheme{Hannover}
%\usetheme{Ilmenau}
%\usecolortheme[rgb={1,0,1}]{structure}
\setbeamertemplate{navigation symbols}{}
\def\ULen{1}
\def\MU#1{\emrlineto(0,\ULen)\edef#1{\emCurP}}
\def\ML#1{\emrlineto(-\ULen,0)\edef#1{\emCurP}}
\def\MD#1{\emrlineto(0,-\ULen)\edef#1{\emCurP}}
\def\MR#1{\emrlineto(\ULen,0)\edef#1{\emCurP}}
\makeatletter%
\def\Myemmoveto#1{%
\vecXY#1\@X\@Y%
\emmoveto(\@X,\@Y)%
}%
\makeatother%
\def\Npower#1#2{{#1\mathstrut}^{#2}}
\def\Par#1{\left(#1\right)}
\title{等比数列の和の視覚化}
%\author{田中 徹}
\date{\relax}%(2019/8/9)}
%\pagestyle{empty}
%\usefonttheme{professionalfonts}
%\mathindent=0mm
\begin{document}
\caprm{}%
\baselineskip=1.2\baselineskip
\abovedisplayskip=4mm
\belowdisplayskip=4mm
\begin{frame}[t]\frametitle{}%
\titlepage%
\end{frame}
\if0
\section*{目次}
\begin{frame}[t]\frametitle{}%
\tableofcontents%
\end{frame}
\fi
\def\OnSlide#1{%
\Addself\SNo{1}%
\onslide<\SNo->{#1}%
}%
\def\Pause{%
\Addself\SNo{1}%
\pause%
}%
\begin{frame}\frametitle{等比数列の和}%
\[
\Bunsuu{1}{k}+\Bunsuu{1}{\Npower{k}{2}}+\Bunsuu{1}{\Npower{k}{3}}+\cdots+\Bunsuu{1}{\Npower{k}{n}}
\]
\end{frame}
\section{$k=4$}
\begin{frame}[t]\frametitle{$k=4$}%
\def\SNo{0}%
%\Pause
\begin{zahyou*}[%
haiti=t,%
ul=60truemm,%
EPSclip=false,%
borderwidth=0mm,%
ueyohaku=0mm,%
sitayohaku=0mm,%
hidariyohaku=0mm,%
migiyohaku=0mm,%
xscale=1.0,%
yscale=1.0,%
gentenkigou={$O$},%
gentenhaiti={[sw]},%
yokozikukigou={$x$},%
yokozikuhaiti={[e]},%
tatezikukigou={$y$},%
tatezikuhaiti={[n]}%
]%
(-0.1,1.1)(-0.1,0.87)%
\setlinewidth{0.3pt}%
\small%
\def\A{(0,0)}%
\def\B{(1,0)}%
\Rdef[\A](1,60)\O%
\OnSlide{%
\Takakkei<linethickness=0.8pt>{\O\A\B}%
\Pause
\Put\O[nw]{面積 $1$の三角形}
}%
\Pause%
\Ifor*\Rep{1}{3}\Do{%
\def\AA{\A}%
\def\BB{\B}%
\Ifor*\Cnt{1}{7}\Do{%
\Bunten\O\AA{1}{1}\AM%
\Bunten\O\BB{1}{1}\BM%
\Bunten\AA\BB{1}{1}\AMB%
\ifnum\Rep=1%
\OnSlide{%
\emPaint<paintcolor=blue>{\AM\AA\AMB}%
\Takakkei{\AM\AA\AMB}%
%\Drawlines<dash={2pt,2pt}>{\AM\BM\AMB}%
\Dashline{0.02}{\AM\BM\AMB}%
}%
\fi%
\ifnum\Rep=2%
\OnSlide{%
\emPaint<paintcolor=red>{\BM\BB\AMB}%
}%
\fi%
\ifnum\Rep=3%
\OnSlide{
\emPaint<paintcolor=green>{\AM\AMB\BM}%
}%
\fi%
\edef\AA{\AM}%
\edef\BB{\BM}%
}%
}%
\OnSlide{%
\PutStr*{(48mm,-8mm)}(-10mm,-12mm){%
\begin{zahyou*}[%
haiti=t,%
ul=20truemm,%
EPSclip=false,%
borderwidth=0mm,%
ueyohaku=0mm,%
sitayohaku=0mm,%
hidariyohaku=0mm,%
migiyohaku=0mm,%
xscale=1.0,%
yscale=1.0,%
gentenkigou={$O$},%
gentenhaiti={[sw]},%
yokozikukigou={$x$},%
yokozikuhaiti={[e]},%
tatezikukigou={$y$},%
tatezikuhaiti={[n]}%
]%
(-0.1,1.1)(-0.1,1.1)%
\setlinewidth{0.3pt}%
\small%
\def\A{(0,0)}%
\def\B{(1,0)}%
\Rdef[\A](1,60)\O%
\Rdef[\A](0.2,240)\ALB%
\Rdef[\A](0.2,300)\ARB%
\Rdef[\B](0.2,240)\BLB%
\Rdef[\B](0.2,300)\BRB%
\Naisin\O\A\B\I%
\Kyori\I\O\RR%
\Mulself\RR{1.4}%
\Rdef[\I](\RR,270)\PosB
\OnSlide{%
\En\I\RR%
}%
\OnSlide{%
\Takakkei{\O\A\B}%
\emPaint<paintcolor=blue>{\A\ALB\ARB}%
\emPaint<paintcolor=red>{\B\BLB\BRB}%
\emPaint<paintcolor=green>{\A\ARB\BLB\B}%
}%
\OnSlide{%
\Put\PosB[s]{大きさ $\Bunsuu{1}{\Npower{4}{n}}$の領域}
}%
\OnSlide{%
\Put\PosB(-40mm,-20mm){%
$
\OnSlide{1-\Bunsuu{1}{\Npower{4}{n}}}\OnSlide{=3\times}\OnSlide{\Par{\Bunsuu{1}{4}+\Bunsuu{1}{\Npower{4}{2}}+\Bunsuu{1}{\Npower{4}{3}}+\cdots+\Bunsuu{1}{\Npower{4}{n}}}}
$
}%
}%
%\drawXYaxis%
%\zahyouMemori[g][n]%
\end{zahyou*}%
}%
to[Hankei=50mm]\O%
}%
%\drawXYaxis%
%\zahyouMemori[g][n]%
\end{zahyou*}%
\end{frame}%
\begin{frame}[t]\frametitle{もうひとつの $k=4$}%
\def\SNo{0}%
\begin{zahyou*}[%
haiti=t,%
ul=25truemm,%
EPSclip=false,%
borderwidth=0mm,%
ueyohaku=0mm,%
sitayohaku=0mm,%
hidariyohaku=0mm,%
migiyohaku=0mm,%
xscale=1.0,%
yscale=1.0,%
gentenkigou={$O$},%
gentenhaiti={[sw]},%
yokozikukigou={$x$},%
yokozikuhaiti={[e]},%
tatezikukigou={$y$},%
tatezikuhaiti={[n]}%
]%
(-0.1,2.1)(-0.1,2.1)%
\setlinewidth{0.3pt}%
\small%
\def\O{(0,0)}
\def\A{(2,0)}
\def\B{(0,2)}
\Addvec\A\B\C
\Takakkei<linethickness=0.8pt>{\O\A\C\B}
\def\ULen{1}
\def\CP{(1,1)}
\def\CurP{\CP}
\Myemmoveto\CP
\Ifor*\Cnt{1}{6}\Do{
\MU\AA
\ML\BB
\MD\CC
\MD\DD
\MR\EE
\MR\FF
\MU\GG
\ML\HH
\emPaint<paintcolor=blue>{\HH\AA\BB\CC}
\emPaint<paintcolor=green>{\HH\CC\DD\EE}
\emPaint<paintcolor=red>{\HH\EE\FF\GG}
\Mulself\ULen{0.5}
\emrmoveto(\ULen,\ULen)
\Takakkei<linethickness=0.5pt>{\HH\AA\BB\DD\FF\GG}
}
\PutStr*{(36mm,-8mm)}(-10mm,-22mm){%
\begin{zahyou*}[%
haiti=t,%
ul=20truemm,%
EPSclip=false,%
borderwidth=0mm,%
ueyohaku=0mm,%
sitayohaku=0mm,%
hidariyohaku=0mm,%
migiyohaku=0mm,%
xscale=1.0,%
yscale=1.0,%
gentenkigou={$O$},%
gentenhaiti={[sw]},%
yokozikukigou={$x$},%
yokozikuhaiti={[e]},%
tatezikukigou={$y$},%
tatezikuhaiti={[n]}%
]%
(-0.1,1.1)(-0.1,1.1)%
\setlinewidth{0.3pt}%
\small%
\def\O{(0,0)}%
\def\A{(1,0)}%
\def\B{(0,1)}%
\Addvec\A\B\C%
\Rdef[\O](0.2,180)\OAL
\Rdef[\O](0.2,270)\OBB
\Rdef[\A](0.2,270)\ACB
\Rdef[\B](0.2,180)\BCL
\Addvec\OAL\OBB\OO
\Bunten\O\C{1}{1}\I%
\Kyori\I\O\RR%
\Mulself\RR{1.5}%
\Bunten\O\I{9}{1}\I
\Rdef[\I](\RR,270)\PosB
\En\I\RR%
\Takakkei{\O\A\C\B}%
\emPaint<paintcolor=blue>{\O\B\BCL\OAL}%
\emPaint<paintcolor=red>{\O\A\ACB\OBB}%
\emPaint<paintcolor=green>{\O\OAL\OO\OBB}%
\end{zahyou*}%
}%
to[Hankei=50mm]\C%
%%%%%%%%%%%%%%%%%%%%
\end{zahyou*}%
\end{frame}%
\section{$k=2$}
\begin{frame}[t]\frametitle{ $k=2$}%
\def\SNo{0}
\begin{zahyou*}[%
haiti=b,%
ul=50truemm,%
EPSclip=false,%
borderwidth=0mm,%
ueyohaku=-50mm,%
sitayohaku=0mm,%
hidariyohaku=0mm,%
migiyohaku=0mm,%
xscale=1.0,%
yscale=1.0,%
gentenkigou={$O$},%
gentenhaiti={[sw]},%
yokozikukigou={$x$},%
yokozikuhaiti={[e]},%
tatezikukigou={$y$},%
tatezikuhaiti={[n]}%
]%
(-0.1,2.1)(-0.1,2.1)%
\setlinewidth{0.3pt}%
\small%
\def\O{(1,0)}%
\def\A{(0,0)}%
\def\B{(1,1)}%
\OnSlide{%
\Takakkei{\O\A\B}%
}%
\OnSlide{%
\Put{(0,1)}{\parbox[t]{8zw}{面積 $1$の\\直角二等辺三角形}}%
}%
\def\OO{\O}%
\def\AA{\A}%
\def\BB{\B}%
\Ifor*\Cnt{1}{7}\Do{%
\Suisen\OO\AA\BB\H%
\OnSlide{%
\emPaint<paintcolor=blue>{\OO\AA\H}
{%
\Thicklines%
\Drawlines<iro=red>{\OO\H}%
}%
}%
\edef\AA{\OO}%
\edef\OO{\H}%
}%
%%%%%%%%%%%%%%%%%%%%
\OnSlide{%
\PutStr*{(20mm,-10mm)}{$\Bunsuu{1}{\Npower{2}{n}}$の領域}to[Hankei=20mm]\B
}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\OnSlide{%
\Put\LB[se]{%
$
1-\Bunsuu{1}{\Npower{2}{n}}=\Bunsuu{1}{2}+\Bunsuu{1}{\Npower{2}{2}}+\Bunsuu{1}{\Npower{2}{3}}+\cdots+\Bunsuu{1}{\Npower{2}{n}}
$
}%
}%
\end{zahyou*}%
\end{frame}%
\section{$k=5$}
\begin{frame}[t]\frametitle{ $k=5$}%
\def\SNo{0}
\begin{zahyou*}[%
haiti=b,%
ul=50truemm,%
EPSclip=false,%
borderwidth=0mm,%
ueyohaku=-50mm,%
sitayohaku=0mm,%
hidariyohaku=0mm,%
migiyohaku=0mm,%
xscale=1.0,%
yscale=1.0,%
gentenkigou={$O$},%
gentenhaiti={[sw]},%
yokozikukigou={$x$},%
yokozikuhaiti={[e]},%
tatezikukigou={$y$},%
tatezikuhaiti={[n]}%
]%
(-0.1,2.1)(-0.1,2.1)%
\setlinewidth{0.3pt}%
\small%
\def\O{(2,0)}%
\def\A{(0,0)}%
\def\B{(2,1)}%
\OnSlide{%
\Takakkei{\O\A\B}%
}%
\OnSlide{%
\Put{(0,1)}{\parbox[t]{12zw}{面積 $1$の直角三角形\\直角を挟む辺の比 $2\::\:1$}}%
}%
\def\OO{\O}%
\def\AA{\A}%
\def\BB{\B}%
\Ifor*\Cnt{1}{7}\Do{%
\Suisen\OO\AA\BB\H%
\Bunten\OO\AA{1}{1}\PP%
\Bunten\AA\H{1}{1}\QQ%
\Bunten\H\OO{1}{1}\RR%
\OnSlide{%
{%
\Thicklines%
\Drawlines<iro=red>{\OO\H}%
}%
\Tyokkakukigou\AA\H\OO%
}%
\OnSlide{%
\emPaint<paintcolor=blue>{\OO\AA\H}
{%
\Thicklines%
\Drawlines<iro=red>{\OO\H}%
}%
\OnSlide{%
\Takakkei<iro=yellow>{\PP\QQ\RR}%
}%
}%
\edef\AA{\OO}%
\edef\OO{\H}%
}%
%%%%%%%%%%%%%%%%%%%%
\OnSlide{%
\PutStr*{(-30mm,-5mm)}(0,0)[r]{$\Bunsuu{1}{\Npower{5}{n}}$の領域\;}to[Hankei=-40mm]\B
}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\OnSlide{%
\Put\LB[se]{%
$
1-\Bunsuu{1}{\Npower{5}{n}}=4\times\Par{\Bunsuu{1}{5}+\Bunsuu{1}{\Npower{5}{2}}+\Bunsuu{1}{\Npower{5}{3}}+\cdots+\Bunsuu{1}{\Npower{5}{n}}}
$
}%
}%
\end{zahyou*}%
\end{frame}%
\section{$k=3$}
\begin{frame}[t]\frametitle{ $k=3$}%
\def\SNo{0}
\begin{zahyou*}[%
haiti=t,%
ul=50truemm,%
EPSclip=false,%
borderwidth=0mm,%
ueyohaku=-40mm,%
sitayohaku=0mm,%
hidariyohaku=0mm,%
migiyohaku=0mm,%
xscale=1.0,%
yscale=1.0,%
gentenkigou={$O$},%
gentenhaiti={[sw]},%
yokozikukigou={$x$},%
yokozikuhaiti={[e]},%
tatezikukigou={$y$},%
tatezikuhaiti={[n]}%
]%
(-0.1,2.1)(-0.1,1.8)%
\setlinewidth{0.3pt}%
\small%
\def\A{(0,0)}%
\Rdef[\A](2,30)\B%
\Suisen\B\A{(1,0)}\O%
\OnSlide{%
\Takakkei{\O\A\B}%
}%
\OnSlide{%
\Put{(0,1)}{\parbox[t]{12zw}{面積 $1$の直角三角形\\他の 2角は $30\Deg$，$60\Deg$}}%
}%
%%%----------------------------------
\Ifor*\Rep{1}{2}\Do{%
\def\OO{\O}%
\def\AA{\A}%
\def\BB{\B}%
\Ifor*\Cnt{1}{7}\Do{%
\Bunten\AA\OO{2}{1}\M%
\Bunten\AA\BB{1}{1}\H%
\IMod\Cnt{2}\Amari%
\ifnum\Rep=1%
\OnSlide{%
\Drawlines{\BB\M\H}%
\Tyokkakukigou\M\H\BB%
\ifnum\Cnt<6%
\ifnum\Amari=1%
\toukakukigou<0>{\H\BB\M;\M\BB\OO;\M\AA\H}<kakukigou=$\scriptstyle\bullet$>%
\else
\toukakukigou<0>{\M\BB\H;\OO\BB\M;\H\AA\M}<kakukigou=$\scriptstyle\bullet$>%
\fi%
\fi%
}%
\OnSlide{%
\emPaint<paintcolor=blue>{\OO\BB\M}%
%\emPaint<paintcolor=red>{\BB\M\H}%
}%
\fi%Rep=1
\ifnum\Rep=2%
\OnSlide{%
\emPaint<paintcolor=red>{\BB\M\H}%
}%
\fi%
\edef\BB{\M}%
\edef\OO{\H}%
}%
}%
%%%----------------------------------
\OnSlide{%
\PutStr*{(5mm,20mm)}(0,0)[b]{$\Bunsuu{1}{\Npower{3}{n}}$の領域\;}to[Hankei=40mm]\A
}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\OnSlide{%
\Put\LB[se]{%
$
1-\Bunsuu{1}{\Npower{3}{n}}=2\times\Par{\Bunsuu{1}{3}+\Bunsuu{1}{\Npower{3}{2}}+\Bunsuu{1}{\Npower{3}{3}}+\cdots+\Bunsuu{1}{\Npower{3}{n}}}
$
}%
}%
\end{zahyou*}%
\end{frame}%
\end{document}
▼関連発言
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└◆1476:beamerでアニメーション [田中徹] 08/20 15:20<-last