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The other day I was thinking (while grinding some sugar to make it easier to dissolve in water):

When you 'cut' a 3D object in multiple 3D objects:

  • the total volume remains the same
  • but the total surface area increases;

When you 'cut' a 2D object in multiple 2D objects:

  • the total area remains the same;
  • but the total perimeter increases;

When you 'cut' a 1D object in multiple 1D objects:

  • the total length remains the same;
  • but the total ????? increases;

What is the '?????' called? Obviously it is just the number of 'end points' of the lines (the 1D objects). But is there an actual term for this? It would be the equivalent of perimeter in 2D or surface area in 3D.

Nikolay Suvandzhiev
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    I think number of end point is quite ok. In that dimension you usually consider only counting measure. – Thomas Apr 08 '14 at 10:27

1 Answers1

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If you want a fancy mathematical term, you can say the $0$-dimensional Hausdorff measure. Which is the same as the counting measure, also known as the number of elements the set contains.