Proving Pick’s formula

Season 3 • Episode 02 • Proving Pick’s formula

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Proving Pick’s formula

Season 3

Episode 02 Time frame 2 periods

Objectives :

•

^{Find, step by step, a proof of Pik's formula.}

Materials :

•

^{Lesson : Proof of Pik's formula.}

•

^{Beamer : Proof of Pik's formula.}

•

^{Paper for posters.}

1 – Devise a plan for the proof 10 mins

A brainstorming sessionto devise a plan for the proof, moderated by the teaher.

2 – Part 1 : Prove that Pick’s formula is additive 45 mins

The aim of this rst part is to prove that if

P

and

T

are two polygons with one edge in ommon, and if Pik's formula holds for

P

and

T

, it also holds for the polygon

P T

obtained by adding

P

and

T

.

3 – Part 2 : Prove Pick’s formula for any polygon 55 mins

Followthe steps : unit square, retangle, right-angled triangle, any triangle, any onvex

Proof of Pick’s formula

Season 3

Episode 02 Document Lesson

Proposition 1

Pick’s formula is additive : if P and T are two polygons with one edge in common, and if Pick’s formula holds for P and T , it also holds for the polygon P T obtained by adding P and T . This is also true for more than two polygons.

Proof. First, it's obviousfrom the denitions that

A _{P T} = A _P + A _T

. Then, suppose that Pik'sformulaholdsforthetwopolygons,andweusethesamenotationsasintheprevious

session. Let's all

c

the number of boundary points inommon. Then we see that

I _{P T} = I _P + I _T + ( c − 2)

and also

B _{P T} = B _P + B _T − 2( c − 2) − 2 .

Wean dedue that

1

2 B _{P T} + I _{P T} − 1 = 1

2 ( B _P + B _T − 2( c − 2) − 2) + I _P + I _T + ( c − 2) − 1

= 1

2 B _P + 1

2 B _T − ( c − 2) − 1 + I _P + I _T + ( c − 2) − 1

= 1

2 B _P + I _P − 1 + 1

2 B _T + I _T − 1

= A _P + A _T

= A _{P T}

Sotheformulaalsoholds forthepolygon

P T

.It'seasytoextandthis resulttomorethan

twopolygons.

Lemma 1

^{Pik's formula is true for the unit square and for any retangle with sides}

parallel tothe axes.

Proof.Foraunitsquare

S

,we have

A _S = 1

^,

B _S = 4

^and

I _S = 0

^.As

¹ ₂ B _S + I _S − 1 = 1

^,the

formulaholds.Then,anyretangle withsidesparalleltotheaxesismadeofunitsquares,

so fromProposition 1,the formulaalsoholds for these retangles.

Lemma 2

^{Pik's formulaholds for any} right-angledtrianglewith itsperpendiular sides parallel tothe axes.

Proof. Any suhtriangle

T

isone halfof aretange

R

with sides paralleltothe axes, ut diagonally,andit'slearthat

2 A _T = A _R

.Let'sall

n

and

m

the numberofpointsoneah side of the retangleand

d

thenumberof pointsonthe diagonal.Then, asimpleanalysis gives

B _R = 2( n + m − 2)

^and

I _R = ( n − 2)( m − 2) B _T = n + m − 1 + d

and

I _T = ( n − 2)( m − 2) − d

2

Season 3 • Episode 02 • Proving Pick’s formula

²

Then we an ompute on one hand

1

2 B _R + I _R − 1 = n + m − 2 + ( n − 2)( m − 2) − 1

= n + m − 3 + ( n − 2)( m − 2)

and on the other hand

2( 1

2 B _T + I _T − 1) = B _T + 2 I _T − 2

= n + m − 1 + d + ( n − 2)( m − 2) − d − 2

= n + m − 3 + ( n − 2)( m − 2)

The two expressions are equal, and we also know from the previous Lemma that

A _R =

1

2 B _R + I _R − 1

^{, so we dedue}

2 A _T = A _R = 1

2 B _R + I _R − 1 = 2( 1

2 B _T + I _T − 1)

whih impliesthat the formulaistrue for triangle

T

:

A _T = ¹ ₂ B _T + I _T − 1

^.

Lemma 3

^{Pik's formula holds forany lattie triangle.}

Proof.Thislemmaisproved byagraphial

argument : any lattie trianglean be tur-

ned into a retangle by attahing at most

three suitable right-angled triangles and

one retangle. Sine the formula is orret

fortherighttrianglesandfortheretangle,

italsofollowsfortheoriginaltriangle.This

last step uses the fat that if the theorem

is true for the polygon PT and for the tri-

angle T, then it'salso true for P; this an

be proved in the same way as propostion

one.

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

b b b b b b b

Theorem 1

Pick’s formula is true for any lattice poly- gon.

Proof. This theorem follows from lemma

3, proposition 1 and the fat that any

lattie polygon an be ut into triangles

(triangulated).

b b b b

b

b

b