Pick’s Theorem 95

8.4 Amalgamation

Generalizing a bit, the inspired idea is to figure out how Pick’s formula is affected when

one adjoins one figure to another. In the general case, we want to amalgamate any kind of

figures, not just triangles and rectangles. And actually, when we formulate the issue this

way, we see that it becomes kind of easy, even though it is a little more abstract.

Key Lemma 66. Suppose that P and Q are simple polygons in the integer lattice, which

do not overlap but which are joined for a connected stretch of one or more edges on their

boundaries. Let PQ denote the amalgamated polygon obtained by joining them together.

1. If Pick’s theorem holds for P and Q separately, then it holds for their amalgamation

PQ.

2. If Pick’s theorem holds for the amalgamation PQ and for one of P or Q, then it holds

also for the other.

In other words, Pick’s theorem is preserved by the addition or removal of a simple polygon

already satisfying Pick’s theorem.

P

Q

Proof. Let us introduce the notation A

P

, A

Q

, i

P

, i

Q

and b

P

, b

Q

to refer to the area, interior,

and boundary lattice-point counts of P and Q, respectively, and similarly with A

PQ

, i

PQ

,

and b

PQ

for the amalgamated polygon. For statement (1), therefore, we assume that

A

P

= i

P

+

b

P

2

− 1

and that

A

Q

= i

Q

+

b

Q

2

− 1.