|
In mathematics, the Menger sponge (also known as the Menger universal curve) is a fractal curve. It is a three-dimensional generalization of the Cantor set and Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.〔. English translation reprinted in 〕 The Menger sponge simultaneously exhibits an infinite surface area and zero volume. ==Construction== The construction of a Menger sponge can be described as follows: # Begin with a cube (''first image''). # Divide every face of the cube into 9 squares, like a Rubik's Cube. This will sub-divide the cube into 27 smaller cubes. # Remove the smaller cube in the middle of each face, and remove the smaller cube in the very center of the larger cube, leaving 20 smaller cubes (''second image''). This is a level-1 Menger sponge (resembling a Void Cube). # Repeat steps 2 and 3 for each of the remaining smaller cubes, and continue to iterate ''ad infinitum''. The second iteration will give you a level-2 sponge (''third image''), the third iteration gives a level-3 sponge (''fourth image''), and so on. The Menger sponge itself is the limit of this process after an infinite number of iterations. The following table lists the properties of sponges ''Mn'' where ''n'' is the number of iterations performed on the first cube of unit side length.〔Wolfram Demonstrations Project, ''(Volume and Surface Area of the Menger Sponge )''〕〔University of British Columbia Science and Mathematics Education Research Group, ''(Mathematics Geometry: Menger Sponge )''〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Menger sponge」の詳細全文を読む スポンサード リンク
|