# Is it possible to measure Hausdorff dimension for real world objects?

1. Sep 13, 2014

### sceptic

Is it possible practically to measure Hausdorff dimension of the surface of the Brain or the broccoli? For a broccoli Hausdorff dimension is equivalent to the so called box counting dimension, which is far more practical? I think, following the original definition Hausdorff dimension it is quite difficult to measure it directly. There are some theorems which can guarantee that some kind of simple procedure can produce the same result?

2. Sep 13, 2014

### HallsofIvy

First, what do you mean by "real world object" and, second, what do you mean by "measure"? Would you consider a cube a "real world object"? That would be easy- its "Hausdorff measure" is just its volume, height times width times depth. If you mean that as a "real world object" even a "cube" would have imperfections, ridges, etc., like a "brain" or "cauliflower" then the best you can do is approximate it- just as the best you can do with a "real world" line segment is approximate its length because it does not have exactly-defined end points.

3. Sep 13, 2014

### Doug Huffman

No, strictly speaking, it is not possible. Fractal dimensions of natural phenomena are largely heuristic and cannot be regarded rigorously as a Hausdorff dimension. (From the Wikipedia, 'Hausdorff Dimension')

"Sources of Error

It was difficult to measure the diameter of irregular pieces. We were forced to average the major and minor axes.
It was very difficult to determine where to split the broccoli as the size became smaller.
It was ever more difficult to accurately split it, some of the buds shed off when it was being handled."

http://hypertextbook.com/facts/2002/broccoli.shtml

(I am frequently amazed by the influence of Glenn Elert's Hypertextbook, not least for guiding my selection of a recumbent bicycle for 50K miles of touring.)