It’s neither of these. The first condition is not necessary for self-similarity, and the second is not sufficient.
Consider an archetypical example of a self-similar structure, the Sierpinski triangle. Looking at that picture, you can see that not every part of the triangle looks like the triangle. There are parts of the triangle that look like two triangles side by side, for instance. So it’s not necessary that every part of the whole be identical to the whole.
On the other hand, it is also not sufficient that at least one part of the whole be identical to the whole. First of all (and slightly pedantically), this would make any structure trivially self-similar, since according to the standard axioms of mereology every whole is a part of itself.
More substantively, even if you modified your definition to say “proper part” instead of just “part”, it still wouldn’t be sufficient. You can see the mathematical definition of self-similarity here. The definition is slightly opaque, but basically what it’s saying is that a bounded set is self-similar if it can be built up as a finite union of smaller-scale copies of itself. There are sets where some proper part resembles the whole, but which still cannot be built up in this way, so a proper part resembling the whole is not sufficient for self-similarity.
As an example, consider this picture. If the process it shows goes on to infinity, then there will be a proper part of the picture that will be a smaller scale version of the picture as a whole. However, the picture as a whole is not a finite union of such parts, so it is not self-similar, unlike the Sierpinski triangle.
So here’s a colloquial definition of self-similarity that captures the idea: A structure is self-similar if it can be exhaustively divided into a finite number of parts (greater than 1) such that each part exactly resembles the whole (except for scale). The Sierpinski triangle, for instance, can be exhaustively divided into three parts, all of which are smaller-scale versions of the triangle itself.
Hmm, thanks, that’s very clear. Maybe you can help me. I’m writing a philosophy paper about time, and I’d like to come up with a name for a pair of conditions on the time of, say, a change. So suppose an occurrence or situation E, and the time-interval AB (excuse the miserable notation, I can’t do anything about it):
A) there is no stretch of time CD which is a part of AB in which E does not occur.
B) there is no stretch of time CD which is a part of AB in which E occurs.
I’d especially like to come up with a way to characterize (B), and it seemed to me that ‘self-dissimilarity’ might be a good way of talking about it. But upon reading your description, I think it may just not be a close enough analogy to the geometrical case.
It’s neither of these. The first condition is not necessary for self-similarity, and the second is not sufficient.
Consider an archetypical example of a self-similar structure, the Sierpinski triangle. Looking at that picture, you can see that not every part of the triangle looks like the triangle. There are parts of the triangle that look like two triangles side by side, for instance. So it’s not necessary that every part of the whole be identical to the whole.
On the other hand, it is also not sufficient that at least one part of the whole be identical to the whole. First of all (and slightly pedantically), this would make any structure trivially self-similar, since according to the standard axioms of mereology every whole is a part of itself.
More substantively, even if you modified your definition to say “proper part” instead of just “part”, it still wouldn’t be sufficient. You can see the mathematical definition of self-similarity here. The definition is slightly opaque, but basically what it’s saying is that a bounded set is self-similar if it can be built up as a finite union of smaller-scale copies of itself. There are sets where some proper part resembles the whole, but which still cannot be built up in this way, so a proper part resembling the whole is not sufficient for self-similarity.
As an example, consider this picture. If the process it shows goes on to infinity, then there will be a proper part of the picture that will be a smaller scale version of the picture as a whole. However, the picture as a whole is not a finite union of such parts, so it is not self-similar, unlike the Sierpinski triangle.
So here’s a colloquial definition of self-similarity that captures the idea: A structure is self-similar if it can be exhaustively divided into a finite number of parts (greater than 1) such that each part exactly resembles the whole (except for scale). The Sierpinski triangle, for instance, can be exhaustively divided into three parts, all of which are smaller-scale versions of the triangle itself.
Hmm, thanks, that’s very clear. Maybe you can help me. I’m writing a philosophy paper about time, and I’d like to come up with a name for a pair of conditions on the time of, say, a change. So suppose an occurrence or situation E, and the time-interval AB (excuse the miserable notation, I can’t do anything about it):
A) there is no stretch of time CD which is a part of AB in which E does not occur.
B) there is no stretch of time CD which is a part of AB in which E occurs.
I’d especially like to come up with a way to characterize (B), and it seemed to me that ‘self-dissimilarity’ might be a good way of talking about it. But upon reading your description, I think it may just not be a close enough analogy to the geometrical case.