Is a line segment the simplest fractal?
No.
Benoit Mandelbrot actually coined the phrase fractal, and he gave this explanation of how he arrived at the name:
(“The Fractal Geometry of Nature”, which I’m trying to find an online cite for.)
A fractal has to have an element of the irregular or chaotic to it, and shouldn’t be a pattern that can be reproduced through classic geometry.
Nope. Here are several definitons. Note a line segment is excluded from the fractal ranks since its self-similarity is ‘trivial’.
While I think I see what you are getting at, this is wrong. Take a look at this more detailed discussion for pictures of some easily constructed fractals and some interesting links.
Thanks.
However, the Cantor set is a fractal which is very closely related to a line segment. I was surprised that it wasn’t mentioned at either of the links brought up already.
What made me think of this was remembering a somewhat old SciAm article discussing how antennas in the shape of a fractal are vastly superior to most other designs and can pick up all frequencies with almost equal efficiency. I was then wondering if perhaps this is why the “Straight wire” antenna is such a good standby.
PS, aint ya glad world of Mathematics is back? 
In a word, yes.