There is a 1.26-dimensional shape that is neither a line nor a surface
A straight line is one-dimensional, and a plane is two-dimensional. But shapes with complex, winding forms, such as the Koch curve, have an in-between dimension: they are "more complicated than one dimension, but do not fill space as completely as two dimensions." Measure how quickly the number of grid cells the shape passes through increases as the cells get smaller, and calculate the dimension yourself.
Box-counting method: Each time you reduce the grid cell size by half, measure how many times larger the number of cells the shape passes through becomes. An ordinary line increases by 2 times (one dimension), while a filled square increases by 4 times (two dimensions). If the result falls somewhere between them, the shape has a dimension between 1 and 2.
Calculated Box-Counting Dimension
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Theoretical value: -
💡 The graph on the right plots log(1/cell size) on the horizontal axis and log(number of cells crossed) on the vertical axis. The slope of this line is the dimension — a straight line gives a slope of 1, while a filled square gives a slope of 2.