If you slice a rhombus off your octagon, you'll end up with two tiles - a rhombus and a dented octagon.
Now, here's something else: you can take four of these rhombically challenged octagons to make a bigger octagon:
You get a nice tree shape if you remove two rhombs from the original octagon.
And these two shapes can be combined in many different ways.
If you remove 3 rhombs from the octagon, you get a shape that's a combination of two squares and another rhomb - a bent spiky thing: