From 2893be8ffa95704aa25d51a4bd26293ab30f8475 Mon Sep 17 00:00:00 2001 From: seung6lee Date: Tue, 22 Sep 2026 22:50:45 +0900 Subject: [PATCH] add 3d printed model and refine equations --- latex/build/main.pdf | Bin 3354119 -> 4172473 bytes latex/images/cad_design.jpeg | Bin 0 -> 160926 bytes latex/images/printed.jpeg | Bin 0 -> 478098 bytes latex/main.tex | 357 ++++++++++++++++++----------------- 4 files changed, 183 insertions(+), 174 deletions(-) create mode 100644 latex/images/cad_design.jpeg create mode 100644 latex/images/printed.jpeg diff --git a/latex/build/main.pdf b/latex/build/main.pdf index 50ee0861104621b1789f2f2155023fd15277d4cb..4875ebaaaea9db6d2f84080e59789e72ca504cbd 100644 GIT binary patch delta 2227504 zcmZU)Q*16=7p`5KtF~?1wr$(yQ`>G=ZQHhO+g!EV{eJnAKiSzw^Ui%Rl5sR}jTyc@ z4cxKv1Drqx%$~lm!b}ZN?m`%3MB4g9;}{fdmvE6IB_|a$o(%xAWxf*wReVuwdRV`h7Fqv=0a-4N*%45u#-G7b}lya4;05Yo=ZUt*tY{oa@+^fB?B zkhGZ=1PC1ECxfD3Rgs7>mEtga1P>wke{CHaXj|LTwle)bY3_s!7@k12=eBJ 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zIq6c$T*bINN2MgvOK)XyGN#Z!bJ$UUy zWb(W2Q+-EUd18(oE+lmYveJ?PQJi3Vo@#Dgyvp|1UNR{d??Utf9OY1Sq&r&57H8+v57ACRKYxN{}4 zDcN{$NZFgq=_Ck3Zh84Wxjo6E-eJw;MK*=uSrc@3Dy=Ib` zE-Pe~0Ti;SCB9~20pJ|tpTdg^noQ7zydf2+wOAHONZJBI$0YP_f6jVR+FOneIi_?sDz& zjyjyIfUQD`1QgN7j{x zc0kFgMz8j5J!4(FLo4ErrR59Eh9dF7PNRW+)9D|dU=z4xVXcfvZMdAy)aVpNS z!lj(?(VLFEV0Oh)L7~vw+FxCHb6(!>Y+*n_-;5A(^`~Mkig!9~>|#Ba#@;S-!j=rZ z3D0BS@S|Z5n%XZQG1l1$KR7PDgNzP44@}b?iZer|#}?^z7BY{UCQcXuz#wCc_MoB! z{{V!2Nx5t7?7LT)8M>aM9q1mUQjZEo8YY%W1Z0zv{Xf|wr}C(5rERKPGWNy{7_K%fT4tgAls z(yku|ayG?PTpp}WI6VNx368@pFK+B*YsmDuUO9>2g5ATXqEseX`}-OXcFTjfCrrLjrA#Wc0>-J3_|2=+5DIVFyJXZ&cmYF(1^!dAQXG#DV{ zs2?fEUU=`0Xt))dSn%{J*C 0$)가 도출된다. +$b$를 상수로 두고 $a$로 미분한 $\dfrac{\partial V}{\partial a} = 0$을 정리하면 $k = a^2 + 2ab$가 되고, 대칭적으로 $a$를 상수로 두고 $b$로 미분한 $\dfrac{\partial V}{\partial b} = 0$을 정리하면 $k = 2ab + b^2$가 된다. 두 식을 연립하면 $a^2 = b^2$, 즉 $a = b$ ($a, b > 0$)가 도출된다. 이처럼 편미분을 이용한 접근은 두 방정식을 연립하여 $a = b$를 직접 이끌어 낸다. 반면 AM-GM 방법은 "분모 $(a+b)$를 최소화한다"는 기하학적 직관을 바탕으로 등호 성립 조건을 바로 읽어낸다는 장점이 있다. 두 방법 모두 같은 결론에 이르지만, AM-GM은 왜 $a = b$여야 하는지를 보다 직관적으로 설명해 준다. @@ -194,6 +178,7 @@ $a = \sqrt{\frac{k}{3}}$인 지점이 $V(a)$의 최댓값임을 알 수 있다. \begin{tikzpicture} \begin{axis}[ + width=\figwidth, height=\figheight, variable=a, domain=0:3, samples=100, @@ -201,7 +186,8 @@ $a = \sqrt{\frac{k}{3}}$인 지점이 $V(a)$의 최댓값임을 알 수 있다. xlabel=$a$, ylabel=, xtick=\empty, ytick=\empty, ymin=-2, ymax=2, - legend style={draw=none, fill=none} + legend columns=4, + legend style={at={(0.5,-0.05)}, anchor=north, draw=none, fill=none} ] \pgfmathsetmacro{\k}{5} \pgfmathsetmacro{\aopt}{sqrt(\k / 3)} @@ -241,7 +227,7 @@ $a = \sqrt{\frac{k}{3}}$인 지점이 $V(a)$의 최댓값임을 알 수 있다. \begin{equation} V_{\max} = abc = a^2 c =\left( \frac{k}{3} \right) \left( \frac{1}{2} \sqrt{\frac{k}{3}} \right) - =\frac{1}{2} \left( \frac{k}{3} \right) ^ {3/2}. + =\frac{1}{2} \left( \frac{k}{3} \right) ^ {\frac{3}{2}}. \end{equation} 이 된다. @@ -262,7 +248,7 @@ $a = \sqrt{\frac{k}{3}}$인 지점이 $V(a)$의 최댓값임을 알 수 있다. \begin{figure}[H] \centering - \includegraphics[width=0.5\textwidth]{regular_polygon.png} + \includegraphics[width=\figwidth, height=\figheight, keepaspectratio]{regular_polygon.png} \caption{윗면이 없는 정다각형 기둥} \label{fig:regular_polygon} \end{figure} @@ -275,35 +261,31 @@ $n$개의 이등변삼각형으로 분할된다. \begin{figure}[H] \centering - \includegraphics[width=0.5\textwidth]{regular_polygon_top.png} + \includegraphics[width=\figwidth, height=\figheight, keepaspectratio]{regular_polygon_top.png} \caption{정다각형 기둥 밑면} \label{fig:regular_polygon_top} \end{figure} 각 이등변삼각형의 꼭짓각은 $\frac{2 \pi}{n}$이고, 밑변의 길이는 $a$이다. 이때 꼭짓점에서 밑변에 내린 수선은 이등변삼각형을 서로 합동인 두 직각삼각형으로 나눈다. -이 직각삼각형에서 삼각함수를 이용하면 높이는 -\begin{equation*} - \frac{a}{2 \tan{(\pi / n)}} -\end{equation*} -임을 구할 수 있다. +이 직각삼각형에서 삼각함수를 이용하면 높이는 $\dfrac{a}{2 \tan\left(\frac{\pi}{n}\right)}$임을 구할 수 있다. 따라서 각 이등변삼각형의 넓이는 \begin{equation*} - \frac{1}{2} \cdot a \cdot \frac{a}{2 \tan{(\pi / n)}} - = \frac{a^2}{4 \tan{(\pi / n)}} + \frac{1}{2} \cdot a \cdot \frac{a}{2 \tan\left(\frac{\pi}{n}\right)} + = \frac{a^2}{4 \tan\left(\frac{\pi}{n}\right)} \end{equation*} 이다. 이러한 삼각형이 $n$개 존재하므로, 밑면의 전체 넓이 $A$는 \begin{equation*} - A = \frac{na^2}{4 \tan{(\pi / n)}} + A = \frac{na^2}{4 \tan\left(\frac{\pi}{n}\right)} \end{equation*} 이 된다. 이때 계산의 편의를 위해 ($t$는 $n$에 따라 결정되는 양수 상수이다) \begin{equation*} - t = \frac{n}{4 \tan(\pi / n)} + t = \frac{n}{4 \tan\left(\frac{\pi}{n}\right)} \end{equation*} 로 두면, 위 식은 다음과 같이 간단히 표현할 수 있다. \begin{equation} @@ -319,10 +301,7 @@ $n$개의 이등변삼각형으로 분할된다. \end{equation} 로 나타낼 수 있다. -기둥의 겉넓이 $S$는 윗면이 없기 때문에 밑면의 넓이 $A$와 옆면의 넓이 $Pb$의 합이다. -\begin{equation*} - S(a, b, n) = A + Pb -\end{equation*} +기둥의 겉넓이 $S$는 윗면이 없기 때문에 밑면의 넓이 $A$와 옆면의 넓이 $Pb$의 합인 $S(a, b, n) = A + Pb$이다. 따라서 제약조건은 \begin{equation} S(a, b, n) = k @@ -359,11 +338,7 @@ $b$는 더 이상 독립적으로 선택되는 변수가 아니다. 따라서 제약조건을 만족하면서 부피를 최대화하는 $(a, n)$ 쌍을 찾아야 한다. 우선 고정된 $n$에서 최대 부피가 되는 $a$를 찾는 것부터 진행한다. -즉, $n$을 $3$이상의 임의의 정수 $n_0$라고 할 때, 먼저 $n=n_0$를 상수로 두고 $a$에 대해 다음의 1변수 함수 -\[ - V_{n_0}(a):=V(a,n_0) -\] -를 최대화한다. +즉, $n$을 $3$이상의 임의의 정수 $n_0$라고 할 때, 먼저 $n=n_0$를 상수로 두고 $a$에 대한 1변수 함수 $V_{n_0}(a):=V(a,n_0)$를 최대화한다. \subsection{\texorpdfstring{$n = n_0$에서의 $a$에 대한 최적 조건}{n = n0에서의 a에 대한 최적 조건}} @@ -389,11 +364,7 @@ $A$가 $k$로만 나타내어진다는 것은 $n$의 값에 상관없이 정다 밑면의 넓이가 $\frac{k}{3}$이 되었기 때문에 이 결과를 만족한다고 할 수 있다. 이 결과는 처음에 예상하지 못한 것이었다. 밑면의 형태가 삼각형이든 정사각형이든 원이든 관계없이, 최대 부피 조건에서는 겉넓이의 정확히 $\frac{1}{3}$이 항상 밑면에 배분된다는 보편적 원리가 성립함을 발견하였다. -기둥의 높이는 식 \ref{eq:regular_polygon_b}에 구한 $A$값을 대입하면, -\begin{equation*} - b = \frac{k - A}{P} = \frac{2k}{3P} -\end{equation*} -가 된다. 따라서 최대 부피 조건에서의 밑면 넓이와 높이를 정리하면 +기둥의 높이는 식 \ref{eq:regular_polygon_b}에 구한 $A$값을 대입하면 $b = \frac{k - A}{P} = \frac{2k}{3P}$가 된다. 따라서 최대 부피 조건에서의 밑면 넓이와 높이를 정리하면 \begin{equation} \boxed{A = \frac{k}{3}, \quad b = \frac{2k}{3P}} \end{equation} @@ -408,12 +379,7 @@ $A$가 $k$로만 나타내어진다는 것은 $n$의 값에 상관없이 정다 가 된다. 여기서 $k$는 고정된 양수이며, $P > 0$이다. -따라서 $V_{\max}$는 $P$에 대하여 -\begin{equation*} - V_{\max}(P) = \frac{C}{P} - \quad \left(C = \frac{2k^2}{9} > 0\right) -\end{equation*} -와 같은 형태를 가진다. +따라서 $V_{\max}$는 $P$에 대하여 $V_{\max}(P) = \frac{C}{P}$ $\left(C = \frac{2k^2}{9} > 0\right)$와 같은 형태를 가진다. 함수 $V_{\max}(P)$는 $P > 0$에서 단조 감소 함수이므로, $P$가 작아질수록 $V_{\max}$는 증가한다. @@ -421,14 +387,14 @@ $P$가 작아질수록 $V_{\max}$는 증가한다. $P$는 식 \ref{eq:regular_polygon_P}에 $a$값을 대입하면 \begin{equation*} - P(n) = na = n \sqrt{\frac{k}{3t}} = 2 \sqrt{\frac{kn \tan{(\pi / n)}}{3}} + P(n) = na = n \sqrt{\frac{k}{3t}} = 2 \sqrt{\frac{kn \tan\left(\frac{\pi}{n}\right)}{3}} \end{equation*} 이 된다. 이때 식 \ref{eq:regular_polygon_a}에서 $a$를 $n$으로 나타내었기 때문에, $P$는 $n$에 대한 일변수 함수가 된다. -$P(n)$이 최소가 되기 위해서는 $n \tan{(\pi / n)}$이 최소가 되어야 한다. +$P(n)$이 최소가 되기 위해서는 $n \tan\left(\frac{\pi}{n}\right)$이 최소가 되어야 한다. \begin{equation} - Q(n) = n \tan{(\pi / n)} \quad (n \geq 3) + Q(n) = n \tan\left(\frac{\pi}{n}\right) \quad (n \geq 3) \end{equation} 라고 하면, $Q(n)$이 최소가 될때의 $n$이 기둥의 부피를 최대화 한다. 여기서 $n$은 정$n$각형의 변의 개수를 의미하므로 본래 정수 $n \geq 3$에 한정된다. @@ -445,11 +411,7 @@ $Q(n)$의 도함수는 곱의 미분법에 따라 Q'(n) = \tan\left(\frac{\pi}{n}\right) - \frac{\pi}{n}\sec^2\left(\frac{\pi}{n}\right) \end{equation*} 가 된다. -이때 계산을 간단히 하기 위하여 $\theta=\frac{\pi}{n}$이라 두면, -\begin{equation*} - Q'(n) = \tan{\theta} - \theta \sec^2{\theta} -\end{equation*} -로 표현할 수 있기에, +이때 계산을 간단히 하기 위하여 $\theta=\frac{\pi}{n}$이라 두면, $Q'(n) = \tan{\theta} - \theta \sec^2{\theta}$로 표현할 수 있기에, \begin{equation*} \tan{\theta} - \theta \sec^2{\theta} = 0 \quad \Longrightarrow \quad @@ -458,11 +420,7 @@ $Q(n)$의 도함수는 곱의 미분법에 따라 배각공식 $2\sin\theta\cos\theta = \sin 2\theta$에 따르면 $\sin\theta\cos\theta = \frac{\sin 2\theta}{2}$가 성립한다. -따라서 극값이 존재하기 위해서는 -\begin{equation*} - \theta = \frac{\sin 2\theta}{2} -\end{equation*} -를 만족해야 한다. +따라서 극값이 존재하기 위해서는 $\theta = \frac{\sin 2\theta}{2}$를 만족해야 한다. 이를 분석하기 위하여 다음과 같은 함수를 정의한다. \begin{equation} f(\theta) = \frac{\sin 2\theta}{2} - \theta @@ -486,18 +444,66 @@ $\theta=0$이 되려면 $n \to \infty$이어야 하며, 본래 문제에서 $n$은 정$n$각형의 변의 개수를 의미하므로 정수 $n \geq 3$에 한정된다. 따라서 $Q(n)$의 정의역은 $\{n \mid n \geq 3,\; n \in \mathbb{Z}\}$이다. -이 정의역에서 가장 작은 값은 $n=3$이므로, 좌측 경계값은 $Q(3) \simeq 5.196$이다. +이 정의역에서 가장 작은 값은 $n=3$이므로, 좌측 경계값은 $Q(3) \approx 5.196$이다. 우측 경계값을 구하기 위해서는 극한을 사용한다. -이 경우 무한대로의 극한값을 구하는 것보다 $0$으로의 극한값을 구하는게 더 쉽기 때문에 $u = \frac{1}{n}$으로 정의하면 $\lim_{n \to \infty}Q(n) = \lim_{u \to 0}Q(u)$이 된다. 이를 계산하면 +이 경우 무한대로의 극한값을 구하는 것보다 $0$으로의 극한값을 구하는게 더 쉽기 때문에 $u = \frac{1}{n}$으로 정의하면 $\lim_{n \to \infty}Q(n) = \lim_{u \to 0^+}Q\left(\frac{1}{u}\right)$이 된다. 이를 계산하면 \begin{equation} - \lim_{u \to 0}{Q(u)} = \lim_{u \to 0}{\frac{1}{u} \tan{\pi u}} - = \pi \cdot \lim_{u \to 0}{\frac{\tan{(\pi u)}}{\pi u}} - = \pi \cdot 1 - = \pi + \lim_{u \to 0^+}{Q\left(\frac{1}{u}\right)} = \lim_{u \to 0^+}{\frac{1}{u} \tan{\pi u}} + = \pi \cdot \lim_{u \to 0^+}{\frac{\tan{(\pi u)}}{\pi u}} + \label{eq:Q_limit} \end{equation} 이 된다. -$\lim_{u \to 0}{Q(u)} < Q(3)$이기 때문에 $Q(u)$의 최솟값은 $n$이 무한대일때임을 알 수 있다. +$x = \pi u > 0$이라 두면 $u \to 0^+$일 때 $x \to 0^+$이므로 $\lim_{x \to 0^+}\frac{\tan x}{x}$만 구하면 된다. + +\begin{figure}[H] + \centering + \begin{tikzpicture}[x=0.75\figheight, y=0.75\figheight] + \pgfmathsetmacro{\ang}{35} + \coordinate (O) at (0,0); + \coordinate (A) at (1,0); + \coordinate (P) at ({cos(\ang)},{sin(\ang)}); + \coordinate (H) at ({cos(\ang)},0); + \coordinate (T) at (1,{tan(\ang)}); + + \draw[->] (-0.1,0) -- (1.25,0); + \draw[->] (0,-0.1) -- (0,1.15); + \draw[gray] (1,0) arc (0:90:1); + \draw (O) -- (T); + \draw (P) -- (A); + \draw (0.18,0) arc (0:\ang:0.18); + \node at ({0.25*cos(\ang/2)},{0.25*sin(\ang/2)}) {$x$}; + + \draw[very thick, blue] (P) -- (H) node[midway, left] {$\sin x$}; + \draw[very thick, red] (A) arc (0:\ang:1); + \node[red] at ({1.07*cos(0.8*\ang)},{1.07*sin(0.8*\ang)}) {$x$}; + \draw[very thick, green!50!black] (A) -- (T) node[midway, right] {$\tan x$}; + + \node[below left] at (O) {$O$}; + \node[below right] at (A) {$A$}; + \node[above left] at (P) {$P$}; + \node[below] at (H) {$H$}; + \node[right] at (T) {$T$}; + \end{tikzpicture} + \caption{단위원에서 $\sin x$, $x$, $\tan x$의 비교} + \label{fig:unit_circle} +\end{figure} + +그림 \ref{fig:unit_circle}의 단위원에서 $0 < x < \frac{\pi}{2}$일 때 $\triangle OAP \subset \text{부채꼴 } OAP \subset \triangle OAT$이므로 넓이를 비교하면 $\frac{1}{2}\sin x < \frac{1}{2}x < \frac{1}{2}\tan x$이다. +따라서 역수를 취한 뒤 각 항에 $\sin x > 0$을 곱하면 +\begin{equation*} + \sin x < x < \tan x + \quad \Longrightarrow \quad + \frac{1}{\tan x} < \frac{1}{x} < \frac{1}{\sin x} + \quad \Longrightarrow \quad + \cos x < \frac{\sin x}{x} < 1 +\end{equation*} +이다. $x \to 0^+$일 때 $\cos x \to \cos 0 = 1$이므로 $\lim_{x \to 0^+}\frac{\sin x}{x} = 1$이고, +\begin{equation*} + \lim_{x \to 0^+}\frac{\tan x}{x} = \lim_{x \to 0^+}\frac{\sin x}{x} \cdot \frac{1}{\cos x} = 1 \cdot 1 = 1 +\end{equation*} +이다. 따라서 식 \ref{eq:Q_limit}은 $\lim_{n \to \infty}Q(n) = \pi \cdot 1 = \pi$가 된다. +$\lim_{n \to \infty}Q(n) < Q(3)$이기 때문에 $Q(n)$의 최솟값은 $n$이 무한대일때임을 알 수 있다. 이는 밑면의 변이 더 많을수록 기둥의 부피가 더 커짐을 의미한다. 정다각형에서 변의 대수가 끊임없이 늘어나면 원이 되기 때문에 부피가 최대가 되는 밑면은 원이 되며, 이는 원기둥이 된다. @@ -505,7 +511,7 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 때문에 $Q(u)$의 최솟값은 $n$이 무 \section{원기둥의 최적화} -앞서 정다각형 분석에서 $Q(n)$이 순감소하며 $V_{\max}(n)$이 단조 증가함을 보였고, 그 극한이 $\frac{k^{3/2}}{3\sqrt{3\pi}}$임을 확인하였다. 이제 극한의 도형인 원기둥의 최대 부피를 직접 계산하여 이 값과 일치하는지 검증한다. +앞서 정다각형 분석에서 $Q(n)$이 순감소하며 $V_{\max}(n)$이 단조 증가함을 보였고, 그 극한이 $\frac{k^{\frac{3}{2}}}{3\sqrt{3\pi}}$임을 확인하였다. 이제 극한의 도형인 원기둥의 최대 부피를 직접 계산하여 이 값과 일치하는지 검증한다. 원기둥(윗면 없음) 분석에서는 밑면 원의 반지름을 $a$, 기둥의 높이를 $b$라 한다. 이때, $a$와 $b$는 길이이므로 $a > 0$, $b > 0$이다. @@ -513,15 +519,12 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 때문에 $Q(u)$의 최솟값은 $n$이 무 \begin{figure}[H] \centering - \includegraphics[width=0.4\textwidth]{cylinder.png} + \includegraphics[width=\figwidth, height=\figheight, keepaspectratio]{cylinder.png} \caption{윗면이 없는 원기둥} \label{fig:cylinder} \end{figure} -윗면이 없으므로 겉넓이 $S$는 밑면과 옆면의 합이다: -\begin{equation*} - S(a, b) = \pi a^2 + 2 \pi ab -\end{equation*} +윗면이 없으므로 겉넓이 $S$는 밑면과 옆면의 합인 $S(a, b) = \pi a^2 + 2 \pi ab$이다. 겉넓이는 $k$로 고정되어 있기 때문에 제약조건은 \begin{equation} S(a, b) = k \quad \Longleftrightarrow \quad \pi a^2 + 2 \pi ab = k @@ -570,7 +573,7 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 때문에 $Q(u)$의 최솟값은 $n$이 무 최대 부피는 \begin{equation} - V_{\max} = \pi a^2 b = \pi a^3 = \pi \left( \frac{k}{3 \pi} \right)^{3/2} + V_{\max} = \pi a^2 b = \pi a^3 = \pi \left( \frac{k}{3 \pi} \right)^{\frac{3}{2}} \end{equation} 이다. @@ -581,9 +584,10 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 때문에 $Q(u)$의 최솟값은 $n$이 무 \begin{tikzpicture} \begin{axis}[ + width=\figwidth, height=\figheight, axis lines=middle, xlabel=$n$, ylabel=$V_{\max}$, - legend style={draw=none, fill=none} + legend style={at={(0.97,0.5)}, anchor=east, draw=none, fill=none} ] \pgfmathsetmacro{\k}{100} @@ -601,12 +605,12 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 때문에 $Q(u)$의 최솟값은 $n$이 무 \end{axis} \end{tikzpicture} - \caption{함수 $V_{\max}(n)$} + \caption{함수 $V_{\max}(n)$ ($k = 100$)} \label{fig:v_max_n} \end{figure} 그림 \ref{fig:v_max_n}은 정$n$각형 기둥에서 최대 부피 $V_{\max}(n)$이 변의 개수 $n$에 따라 어떻게 변화하는지를 나타낸 것이다. -이때 $k$는 일정한 값으로 고정되어 있으며, 각 $n$에 대해 앞서 도출한 최적 조건을 적용하여 계산한 값이다. +이때 $k$는 $100$으로 고정되어 있으며, 각 $n$에 대해 앞서 도출한 최적 조건을 적용하여 계산한 값이다. 그래프를 통해 다음과 같은 특징을 관찰할 수 있다. @@ -616,24 +620,22 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 때문에 $Q(u)$의 최솟값은 $n$이 무 \end{itemize} 이러한 경향은 앞서 유도한 식을 통해 이론적으로도 설명할 수 있다. -앞서 도출한 $V_{\max}(n) = \frac{2k^2}{9P(n)}$과 $P(n) = 2\sqrt{\frac{kn\tan(\pi/n)}{3}}$를 결합하면, + +앞서 도출한 $V_{\max}(n) = \frac{2k^2}{9P(n)}$과 $P(n) = 2\sqrt{\frac{kn\tan\left(\frac{\pi}{n}\right)}{3}}$를 결합하면, \begin{equation*} - V_{\max}(n) = \frac{k^{3/2}}{3\sqrt{3}} \cdot \frac{1}{\sqrt{Q(n)}} + V_{\max}(n) = \frac{k^{\frac{3}{2}}}{3\sqrt{3}} \cdot \frac{1}{\sqrt{Q(n)}} \end{equation*} 로 나타낼 수 있다. 따라서 $V_{\max}(n)$의 증감은 $Q(n) = n\tan\!\left(\frac{\pi}{n}\right)$의 변화에 의해 결정된다. 앞서 $Q(n)$이 순감소하며 $\lim_{n \to \infty} Q(n) = \pi$임을 보였다. 따라서 \begin{equation*} \lim_{n \to \infty} V_{\max}(n) - = \frac{k^{3/2}}{3 \sqrt{3\pi}} + = \frac{k^{\frac{3}{2}}}{3 \sqrt{3\pi}} \end{equation*} 가 된다. 이 값은 이후 원기둥의 최대 부피로 도출된 값과 정확히 일치한다. -이러한 일치는 정다각형이 $n \to \infty$일 때 원으로 수렴한다는 기하학적 사실의 수학적 표현이다. 탐구 초반에 세운 예상—변이 많을수록 부피가 크고 원기둥이 최적—이 극한 분석과 원기둥 직접 계산 모두에서 일관되게 검증된 셈이다. - -또한 $n$이 증가할수록 증가폭이 점점 작아지는 이유는 $\tan(\pi/n)$이 $n$이 커질수록 $\frac{\pi}{n}$에 가까워지기 때문이다. -즉, $\tan\!\left(\frac{\pi}{n}\right) \approx \frac{\pi}{n}$ ($n \to \infty$)이므로 $Q(n) \approx \pi$로 빠르게 수렴하게 되고, 이에 따라 $V_{\max}(n)$ 역시 일정 값에 점점 가까워지게 된다. +이러한 일치는 정다각형이 $n \to \infty$일 때 원으로 수렴한다는 기하학적 사실의 수학적 표현이다. 탐구 초반에 세운 예상인 ``변이 많을수록 부피가 크고 원기둥이 최적일 것이다''가 극한 분석과 원기둥 직접 계산 모두에서 검증된 것이다. 기하학적으로 해석하면, 정다각형은 변의 개수가 증가할수록 원에 가까워지며, 같은 넓이를 가질 때 둘레가 점점 감소한다. 이로 인해 동일한 겉넓이 조건에서 더 많은 면적을 밑면에 할당할 수 있게 되고, 결과적으로 부피가 증가하게 된다. @@ -649,41 +651,17 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 때문에 $Q(u)$의 최솟값은 $n$이 무 이는 3D 프린터로 제한된 필라멘트 양을 사용할 때 가장 효율적인 컨테이너 형태를 찾는 문제로부터 출발하였다. 먼저 밑면이 직사각형인 직육면체의 경우를 분석하였다. -겉넓이 $S=ab+2ac+2bc=k$의 제약조건 하에서 부피 $V=abc$를 최대화한 결과, -\[ - a=b=2c -\] -일 때 최대 부피가 됨을 도출하였다. +겉넓이 $S=ab+2ac+2bc=k$의 제약조건 하에서 부피 $V=abc$를 최대화한 결과, $a=b=2c$일 때 최대 부피가 됨을 도출하였다. 즉, 밑면은 정사각형이 되며, 높이는 밑변 길이의 절반이 된다. -또한 이때 밑면의 넓이는 항상 -\[ - A=\frac{k}{3} -\] -이 됨을 확인하였다. +또한 이때 밑면의 넓이는 항상 $A=\frac{k}{3}$이 됨을 확인하였다. 이를 일반화하여 밑면이 정$n$각형인 기둥을 분석하였다. 제약조건을 이용하여 높이를 소거하고 부피를 $(a,n)$의 함수로 표현한 뒤, 고정된 $n$에서 최대가 되는 $a$를 구하였다. -그 결과, -\[ - a=\sqrt{\frac{k}{3t}} -\] -이며, 이때 역시 밑면의 넓이는 -\[ - A=\frac{k}{3} -\] -이 됨을 확인하였다. +그 결과, $a=\sqrt{\frac{k}{3t}}$이며, 이때 역시 밑면의 넓이는 $A=\frac{k}{3}$이 됨을 확인하였다. 이는 밑면의 형태와 무관하게, 최대 부피를 가지는 경우에는 전체 겉넓이의 $\frac{1}{3}$이 밑면에, $\frac{2}{3}$이 옆면에 배분된다는 것을 의미한다. -다음으로 부피 최대화를 위해서는 둘레 $P$가 최소가 되어야 함을 보였고, 이를 위해 함수 -\[ - Q(n)=n\tan\left(\frac{\pi}{n}\right) -\] -을 분석하였다. -내부 극값이 존재하지 않음을 증명하고, 경계 분석을 통해 -\[ - \lim_{n\to\infty}Q(n)=\pi -\] -임을 도출하였다. +다음으로 부피 최대화를 위해서는 둘레 $P$가 최소가 되어야 함을 보였고, 이를 위해 함수 $Q(n)=n\tan\left(\frac{\pi}{n}\right)$을 분석하였다. +내부 극값이 존재하지 않음을 증명하고, 경계 분석을 통해 $\lim_{n\to\infty}Q(n)=\pi$임을 도출하였다. 이는 변의 개수가 증가할수록 둘레가 감소하며, 그 극한이 원의 경우와 일치함을 의미한다. 따라서 고정된 겉넓이 조건 하에서 최대 부피를 가지는 도형은 밑면이 원인 원기둥임을 알 수 있다. @@ -697,9 +675,9 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 때문에 $Q(u)$의 최솟값은 $n$이 무 \toprule 도형 & 최적 비율 & 밑면 넓이 & 최대 부피 \\ \midrule - 직육면체 & $a = b = 2c$ & $\dfrac{k}{3}$ & $\dfrac{1}{2}\!\left(\dfrac{k}{3}\right)^{\!3/2}$ \\[8pt] + 직육면체 & $a = b = 2c$ & $\dfrac{k}{3}$ & $\dfrac{1}{2}\!\left(\dfrac{k}{3}\right)^{\!\frac{3}{2}}$ \\[8pt] 정$n$각형 기둥 & $A = \dfrac{k}{3},\; b = \dfrac{2k}{3P}$ & $\dfrac{k}{3}$ & $\dfrac{2k^2}{9P(n)}$ \\[8pt] - 원기둥 & $a = b$ & $\dfrac{k}{3}$ & $\pi\!\left(\dfrac{k}{3\pi}\right)^{\!3/2}$ \\ + 원기둥 & $a = b$ & $\dfrac{k}{3}$ & $\pi\!\left(\dfrac{k}{3\pi}\right)^{\!\frac{3}{2}}$ \\ \bottomrule \end{tabular} \caption{도형별 최대 부피 조건 비교} @@ -708,49 +686,83 @@ $\lim_{u \to 0}{Q(u)} < Q(3)$이기 때문에 $Q(u)$의 최솟값은 $n$이 무 표에서 밑면 넓이가 세 경우 모두 $\frac{k}{3}$로 동일함을 한눈에 확인할 수 있다. -\subsection*{구체적 수치 예시} +\subsection*{실제 3D 프린팅 결과} -탐구의 출발점인 3D 프린팅 상황에 실제 수치를 대입하여 결과를 구체적으로 확인한다. -필라멘트 양이 겉넓이 $k = 600\,\text{cm}^2$에 해당한다고 하자. +서론에서 밝혔듯이 본 탐구는 로봇 개발용 나사와 너트를 정리할 컨테이너를, 정해진 필라멘트 양으로 최대한 크게 만들고 싶다는 동기에서 출발하였다. +따라서 도출한 결과가 실제 출력물에서도 성립하는지 확인하기 위해, 직육면체, 정육각기둥, 원기둥 컨테이너를 직접 CAD로 설계하고 3D 프린터로 출력한 뒤 물을 이용하여 부피를 측정하였다. -\textbf{원기둥:} -\begin{equation*} - a = \sqrt{\frac{k}{3\pi}} = \sqrt{\frac{600}{3\pi}} \approx 7.98\,\text{cm}, \qquad b = a \approx 7.98\,\text{cm} -\end{equation*} -\begin{equation*} - V_{\max} = \pi a^3 = \pi\!\left(\frac{600}{3\pi}\right)^{\!3/2} \approx 1596\,\text{cm}^3 -\end{equation*} +겉넓이는 $k = 90\,\text{cm}^2$로 정하고, 각 컨테이너의 바깥 치수는 앞서 도출한 최적 조건을 그대로 따랐다. +실제 출력물에서는 벽 두께를 무시할 수 없으므로 옆벽과 바닥의 두께를 모두 $d = 1\,\text{mm} = 0.1\,\text{cm}$로 하고, 벽이 바깥 치수의 안쪽을 채우도록 설계하였다. +윗면이 없으므로 안쪽 공간은 밑면 방향으로는 양쪽 벽 두께인 $2d$만큼, 높이 방향으로는 바닥 두께인 $d$만큼만 줄어든다. +예를 들어 바깥 치수가 $5 \times 5 \times 5\,\text{cm}$인 정육면체라면 안쪽 공간은 $4.8 \times 4.8 \times 4.9\,\text{cm}$가 된다. +그림 \ref{fig:cad_design}은 CAD 설계 도면이며, 그림 \ref{fig:printed}는 실제 출력물이다. -\textbf{직육면체:} -\begin{equation*} - a = b = \sqrt{\frac{k}{3}} = \sqrt{200} \approx 14.14\,\text{cm}, \qquad c = \frac{a}{2} \approx 7.07\,\text{cm} -\end{equation*} -\begin{equation*} - V_{\max} = a^2 c = \frac{1}{2}\!\left(\frac{k}{3}\right)^{\!3/2} \approx 1414\,\text{cm}^3 -\end{equation*} +\begin{figure}[H] + \centering + \begin{subfigure}[t]{0.38\textwidth} + \centering + \includegraphics[height=0.85\figheight, trim=450 204 266 170, clip]{cad_design.jpeg} + \caption{CAD 설계 도면 (단위: mm)} + \label{fig:cad_design} + \end{subfigure} + \hfill + \begin{subfigure}[t]{0.56\textwidth} + \centering + \includegraphics[height=0.85\figheight]{printed.jpeg} + \caption{출력한 컨테이너 (직육면체, 정육각기둥, 원기둥)} + \label{fig:printed} + \end{subfigure} + \caption{3D 프린팅 컨테이너의 설계와 출력물} +\end{figure} -정$n$각형 기둥의 경우 $n = 3, 4, 6$ 및 원기둥에 대한 결과를 표 \ref{tab:numerical}에 정리한다. -이때 모든 경우에서 밑면 넓이 $A = k/3 = 200\,\text{cm}^2$로 일정하다. +\subsubsection*{이론 최대 부피와 벽 두께를 고려한 부피} + +$k = 90\,\text{cm}^2$를 앞서 도출한 최적 조건에 대입하여 바깥 치수를 정하였으며(그림 \ref{fig:cad_design}), 벽 두께를 무시한 이론 최대 부피 $V_{\max}$는 표 \ref{tab:summary}의 식으로 계산하였다. + +벽 두께를 고려한 안쪽 공간의 부피 $V_{\text{벽}}$은 밑면 방향으로 양쪽 벽 두께를, 높이 방향으로 바닥 두께를 뺀 값이다. +정다각형의 경우 중심에서 변까지의 거리 $\frac{a}{2\tan\left(\frac{\pi}{n}\right)}$가 $d$만큼 줄어들므로 안쪽 한 변의 길이는 $a - 2d\tan\left(\frac{\pi}{n}\right)$가 된다($n = 4$이면 $a - 2d$). +따라서 각 도형의 $V_{\text{벽}}$은 다음과 같다. +\begin{align*} + \text{직육면체:} &\quad V_{\text{벽}} = (a - 2d)^2 (c - d) \\ + \text{정육각기둥:} &\quad V_{\text{벽}} = t\left(a - \frac{2d}{\sqrt{3}}\right)^2 (b - d) \\ + \text{원기둥:} &\quad V_{\text{벽}} = \pi (a - d)^2 (b - d) +\end{align*} + +세 도형 모두 벽이 차지하는 부피 $V_{\max} - V_{\text{벽}}$은 약 $8.7\,\text{cm}^3$로 거의 같다. +이는 벽의 부피가 대략 겉넓이와 두께의 곱인 $kd = 9.0\,\text{cm}^3$이기 때문이며, 모서리에서 벽이 서로 겹치는 부분만큼 조금 더 작게 나타난다. +벽의 부피는 곧 출력에 사용된 필라멘트의 부피이므로, 이 결과는 연구 설계에서 세운 ``필라멘트 양은 겉넓이에 비례한다''는 가정이 타당함을 보여준다. + +\subsubsection*{물을 이용한 부피 측정} + +마지막으로 출력한 세 컨테이너에 증류수를 가득 채운 뒤 담긴 물의 질량을 측정하였다. +증류수는 $1\,\text{cm}^3$당 질량이 약 $1\,\text{g}$이므로, 측정한 질량의 수치를 그대로 부피 $V_{\text{측정}}$으로 볼 수 있다. +측정 결과는 직육면체 $73\,\text{g}$, 정육각기둥 $79.2\,\text{g}$, 원기둥 $83\,\text{g}$이었다. + +오차는 $\frac{V_{\text{측정}} - V_{\text{벽}}}{V_{\text{벽}}} \times 100\,\%$로 계산하였으며, 전체 결과를 표 \ref{tab:printing}에 정리한다. \begin{table}[H] \centering - \begin{tabular}{@{}ccccc@{}} + \begin{tabular}{@{}lcccc@{}} \toprule - $n$ & 밑면 & $a\,(\text{cm})$ & $b\,(\text{cm})$ & $V_{\max}\,(\text{cm}^3)$ \\ + 도형 & $V_{\max}\,(\text{cm}^3)$ & $V_{\text{벽}}\,(\text{cm}^3)$ & $V_{\text{측정}}\,(\text{cm}^3)$ & 오차 \\ \midrule - 3 & 정삼각형 & 21.49 & 6.20 & 1241 \\ - 4 & 정사각형 & 14.14 & 7.07 & 1414 \\ - 6 & 정육각형 & 8.77 & 7.60 & 1520 \\ - $\infty$ & 원 & 7.98 & 7.98 & 1596 \\ + 직육면체 & 82.2 & 73.5 & 73.0 & $-0.7\,\%$ \\ + 정육각기둥 & 88.3 & 79.6 & 79.2 & $-0.5\,\%$ \\ + 원기둥 & 92.7 & 84.0 & 83.0 & $-1.2\,\%$ \\ \bottomrule \end{tabular} - \caption{$k = 600\,\text{cm}^2$일 때 정$n$각형 기둥의 최적 치수 및 최대 부피} - \label{tab:numerical} + \caption{$k = 90\,\text{cm}^2$일 때 이론 최대 부피, 벽 두께를 고려한 부피, 측정 부피} + \label{tab:printing} \end{table} -$n$이 증가할수록 부피가 단조 증가하여 원기둥에서 최댓값에 수렴함을 수치로 확인할 수 있다. -직육면체($n = 4$)와 원기둥을 비교하면 같은 필라멘트 양으로 약 $\frac{1596 - 1414}{1414} \approx 12.9\%$ 더 큰 부피를 얻을 수 있다. -이는 소규모 부품 정리 컨테이너를 설계할 때 밑면 형태의 선택이 실질적인 차이를 만들어냄을 보여준다. +세 컨테이너 모두 측정 부피가 벽 두께를 고려한 부피와 $1.2\,\%$ 이내로 일치하였다. +측정값이 모두 계산값보다 약간 작은 것은 물을 가장자리 끝까지 정확히 채우기 어렵다는 점, 출력 과정에서 벽이나 바닥이 설계보다 약간 두껍게 만들어질 수 있다는 점 등이 원인으로 생각된다. +또한 실온에서 증류수의 밀도는 $1\,\text{cm}^3$당 $1\,\text{g}$보다 조금 작으므로, 질량의 수치를 그대로 부피로 본 것도 작은 오차를 만든다. + +가장 중요한 점은 부피의 크기 순서인 원기둥 $>$ 정육각기둥 $>$ 직육면체가 이론값, 벽 두께를 고려한 값, 측정값 모두에서 동일하게 나타났다는 것이다. +즉, 변의 수가 많을수록 부피가 커지고 원기둥이 최적이라는 수학적 결론이 실제 출력물에서도 확인되었다. +원기둥은 직육면체보다 이론상 약 $12.8\,\%$, 측정상 약 $\frac{83 - 73}{73} \approx 13.7\,\%$ 더 큰 부피를 가졌다. +벽이 차지하는 부피는 세 도형이 거의 같으므로, 원래 부피가 작은 직육면체일수록 벽으로 잃는 비율이 커져($10.6\,\%$, 원기둥은 $9.4\,\%$) 실제 출력물에서는 원기둥의 이점이 오히려 조금 더 커진다. 본 연구를 통해 도출된 가장 중요한 통찰은 다음과 같다. @@ -760,9 +772,6 @@ $n$이 증가할수록 부피가 단조 증가하여 원기둥에서 최댓값 \item 정다각형은 변의 수가 증가할수록 원에 수렴하며, 원기둥이 최적 구조가 된다. \end{itemize} -이 결과는 기하학적 대칭성이 최적화를 유도한다는 일반적인 원리와도 연결된다. -대칭성이 높아질수록 동일한 겉넓이로 더 큰 내부 공간을 확보할 수 있다. - 다만 본 연구에는 몇 가지 한계와 확장 가능성이 있다. 첫째, 벽 두께를 무시하여 겉넓이와 필라멘트 사용량이 완전히 비례한다고 가정하였다. 실제 3D 프린팅에서는 일정한 벽 두께가 유지되므로, 도형의 모서리나 꼭짓점 처리 방식에 따라 필라멘트 사용량 계산이 달라질 수 있다. 둘째, 본 연구는 윗면이 없는 기둥에 한정되었다. 윗면이 있는 닫힌 원기둥의 경우 겉넓이 공식이 $S = 2\pi a^2 + 2\pi ab$로 바뀌며, 동일한 방법으로 최적화하면 @@ -774,6 +783,6 @@ $n$이 증가할수록 부피가 단조 증가하여 원기둥에서 최댓값 결론적으로, 고정된 겉넓이를 가지는 윗면이 없는 기둥 중 최대 부피를 가지는 도형은 원기둥이며, 이는 대칭성과 둘레 최소화의 원리에 의해 설명된다. -\printbibliography +% \printbibliography -\end{document} \ No newline at end of file +\end{document}

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