octo rhomb

Regular octagons cannot be used to tile by themselves - if you try, you will find there are square gaps that need to be filled.

If you slice a rhombus off your octagon, you'll end up with two tiles - a rhombus and a dented octagon.

Each of these shapes can be used to tile by themselves, or tile together. The rhomb-by-itself tiling is easy to visualize (imagine a squashed grid), here is the dented-octagon tiling:

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: