| Version 10 |
Version 9 |
| \PMlinkescapeword{divisions} |
\PMlinkescapeword{divisions} |
| \PMlinkescapeword{equivalent} |
\PMlinkescapeword{equivalent} |
| \PMlinkescapeword{unit} |
\PMlinkescapeword{unit} |
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| A Sierpinski carpet is the set of all points $(x, y)$ such that $x$ or $y$ is in |
A Sierpinski carpet is the set of all points $(x, y)$ such that $x$ or $y$ is in |
| a Cantor set. An equivalent and perhaps simpler definition is: |
a Cantor set. An equivalent and perhaps simpler definition is: |
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| Let $S_0$ be a unit square. Let $S_{n+1}$ be $S_n$, with each square divided |
Let $S_0$ be a unit square. Let $S_{n+1}$ be $S_n$, with each square divided |
| into ninths, by being divided into thirds horizontally and vertically, and the central resulting square removed, and the other resulting squares treated seperately in further divisions. The limit as $n \to \infty$ of $S_n$ is a Sierpinski carpet. |
into ninths, by being divided into thirds horizontally and vertically, and the central resulting square removed, and the other resulting squares treated seperately in further divisions. The limit as $n \to \infty$ of $S_n$ is a Sierpinski carpet. |
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| The Menger sponge is a fractal embedded in 3-dimensional space. It can be seen as a 3-d generalization of the Sierpinski carpet, which is itself a 2-dimensional generalization of the Cantor set. The Menger sponge is almost always represented as being constructed from Cantor sets using the ``middle third'' rule. |
The Menger sponge is a fractal embedded in 3-dimensional space. It can be seen as a 3-d generalization of the Sierpinski carpet, which is itself a 2-dimensional generalization of the Cantor set. The Menger sponge is almost always represented as being constructed from Cantor sets using the ``middle third'' rule. |
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| The Menger sponge consists of all points $(x, y, z)$ such that $(x, y)$, $(y,z)$, and $(x,z)$ are all in Sierpinski carpets. Each ``face'' is a Sierpinski carpet. |
The Menger sponge consists of all points $(x, y, z)$ such that $(x, y)$, $(y,z)$, and $(x,z)$ are all in Sierpinski carpets. Each ``face'' is a Sierpinski carpet. |
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| Similarily to the Sierpinski carpet the Menger sponge can be constructed in the following way: |
Similarily to the Sierpinski carpet the Menger sponge can be constructed in the following way: |
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| Start with a unit cube and split it into 27 smaller cubes of equal size. Remove the central cube and the ones joining a face with it. Then start over with the remaining smaller cubes. |
Start with a unit cube and split it into 27 smaller cubes of equal size. Remove the central cube and the ones joining a face with it. Then start over with the remaining smaller cubes. |
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| \begin{figure}[h] |
\begin{figure}[h] |
| \begin{centering} |
\begin{centering} |
| \includegraphics[scale=0.4]{menger.ps} |
\includegraphics[scale=0.4]{menger.ps} |
| \caption{An iteration of the Menger sponge -- created in Blender 2.35. |
\caption{An iteration of the Menger sponge -- created in Blender 2.35. |
| (The \PMlinktofile{Blender file}{menger.blend} for this picture.)} |
(The \PMlinktofile{Blender file}{menger.blend} for this picture.)} |
| \end{centering} |
\end{centering} |
| \end{figure} |
\end{figure} |
