Figurate numbers

Most of us are familiar with triangular numbers, such as the arrangement of ten bowling pins in a triangle,

   .
  . .
 . . .
. . . .

or square numbers,

.  .  .  .
.  .  .  .
.  .  .  .
.  .  .  .

but we don't necessarily know how their arithmetic works, or whether we can do the same with, say, pentagons.

Can we learn something by telling a turtle how to make these numbers? Let's try.