Saturday, October 08, 2016

[myfnohif] Infinite cross bridges

Draw a square.  Divide it into thirds horizontally.  Take the middle rectangle and divide it into thirds vertically, yielding a small square in the center 1/9 the area of the original.  Recursively repeat infinitely on this center square.  Place electrical resistors on all segments and calculate the resistance between two corners of the original square.

Draw a square.  Draw a smaller square inside it.  Connect the corresponding vertices of the two squares.  Repeat.

Latter is equivalent to a prism with a square grid of resistors on its surface.  We can imagine other infinitely long thin geometric shapes: stack of octahedra.

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