Saturday, March 24, 2018

Table of shapes by signed dimension

Here is a table of infinite extent shapes based on their signed-dimension. None of these structures have a Hausdorff or Minkowski dimension that lets them be measured. But the signed-dimension does.

Examples signed-D Type
-3 3D lattice
-2 3D graph
of edges
-2 2D lattice

-1 3D cell complex
of faces
-1 2D path
2D graph

1.26-2 Koch snowflake tiling

Grid of Cantor dust

0 infinite
1.13-1 chain of Gosper islands

2-1.63Tile based Sierpinski hexagon (yellow)
2-1.58 Cellular automata Sierpinski triangle

The Hausdorff and Minkowski formulae only consider the positive component of the dimension, resulting an an infinite measure on all unbounded structures.

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