Changes

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

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

. . . .

&nbsp;&nbsp;&nbsp;.<br>

. . .

&nbsp;&nbsp;.&nbsp;.<br>

  . .

&nbsp;.&nbsp;.&nbsp;.<br>

  .

.&nbsp;.&nbsp;.&nbsp;.

or square numbers,

or square numbers,

. . . .

.&nbsp; .&nbsp; .&nbsp; .<br>

. . . .

.&nbsp; .&nbsp; .&nbsp; .<br>

. . . .

.&nbsp; .&nbsp; .&nbsp; .<br>

. . . .

.&nbsp; .&nbsp; .&nbsp; .

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

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.

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