Binary Quadratic Forms (BQF)

I Binary quadratic form:

Degree 2 homogeneous polynomial in 2 variables.

f(x, y) = ax2 + bxy + cy2

I Ex. f(x, y) = 3x2 − 2xy + y2.

I Seems easy?

Binary Quadratic Forms (BQF)

I Popular with old-timey Europeans like Fermat, Lagrange,Legendre, Euler, Gauss.

I Classic questions:I Which primes are the sum of two squares?

p = x2 + y2?

I When is a BQF invariant under change of coordinates?

f(x, y) = x2 − xy + y2 = f(x− y, x).

I When are two BQF the same after change of coordinates?

What’s the Big Deal?!

Legendre writing aboutBQF in 1798.

David A. Cox writing aboutBQF in 1989.

What’s the Big Deal?!Conway writing about BQF in 1997!

Introducing: the Topograph

Introducing: the Topograph

Introducing: the Topograph

Introducing: the Topograph

Topographical FactsI No vector ever repeats.I Vectors

[ab

]and

[cd

]are adjacent if and only if

ad− bc = det

[a cb d

]= ±1.

Topographical Facts

Say[ab

]is primitive if gcd(a, b) = 1.

I Every primitive vector shows up in the topograph.

I Define the Top and Bottom operations by

T[ab

]=

[a + bb

]B[ab

]=

[a

a + b

].

I Every primitive vector can be made from[10

]or

[01

]using a

sequence of top and bottom operations.

Topographical Facts

Idea: work backwards![311

]= B3

[32

]= B3T1

[12

]= B3T1B2

[10

].

Can you see why this works? Try more examples!

Topographical Facts

Idea: work backwards![311

]= B3

[32

]= B3T1

[12

]= B3T1B2

[10

].

Can you see why this works? Try more examples!

BQF on the Topograph

f(x, y) = ax2 + bxy + cy2

I f(kx, ky) = a(kx)2 + b(kx)(ky) + c(ky)2 = k2f(x, y)

I f(−x,−y) = f(x, y)

I Enough to know the values of f(x, y) on primitive vectors.

BQF on the Topograph

BQF on the Topograph

The Arithmetic Progression Law

From (Almost) Nothing to Everything!

I We don’t even need to know the vectors!I Coordinate free way to study BQF.

From (Almost) Nothing to Everything!

From (Almost) Nothing to Everything!

From (Almost) Nothing to Everything!

From (Almost) Nothing to Everything!

Formula Found

The Climbing Lemma

The Climbing Lemma

Race to the Bottom

I We can always head downhill!

I What happens at the bottom?

Stuck in the Well

I Climbing Lemma: At most one well.I If f > 0, then it has a well and f is called positive definite.

I To tell if two positive definite forms are equivalent, go meetat the well.

Down by the River

I Climbing Lemma: At most one river!I If f 6= 0 and has a river, then f is called indefinite.

Down by the River

I Climbing Lemma: At most one river!I If f 6= 0 and has a river, then f is called indefinite.

DiscriminationIf a and c are adjacent values of f with common difference hbetween them, then the discriminant of f is

∆ = h2 − 4ac.

I Same value measured anywhere on the topograph!I Ex. ∆ = −8.

River or Well?

The Repeating River

Suppose f has a river.

I Finitely many options for a, c, h.

I Therefore the river must always repeat!

The Repeating River

f(x, y) = x2 + 3xy − 2y2

Lakes

I If f has a lake, then ∆ = h2 is a square.I One lake =⇒ f(x, y) = ax2.

Lakes

If both ± appear, there are two lakes connected by a river.

Lakes

Sometimes the river floods and the lakes merge.

Symmetries

I A symmetry of a BQF f is a linear change of coordinatesthat leaves f invariant,

f(Ax + By,Cx + Dy) = f(x, y).

I Alternatively, it is a transformation of the topograph thatpreserves the f labelling.

SymmetriesI A symmetry of a BQF f is a linear change of coordinates

that leaves f invariant,f(Ax + By,Cx + Dy) = f(x, y).

I Alternatively, it is a transformation of the topograph thatpreserves the f labelling.

SymmetriesSymmetries preserve all special features!

I Wells, rivers, and lakes, are preserved.

Another Perspective

Another Perspective

The mediant of reduced fractions ab and c

d is a + cb + d .

Another Perspective

The mediant of reduced fractions ab and c

d is a + cb + d .

Equivalence

Old Facts, New Light

Two rational numbers ab and c

d are adjacent if and only if theirdifference is an Egyptian fraction 1

m .

ab −

cd =

ad− bcbd = ± 1

bd .

Old Facts, New Light

Every reduced fraction eventually appears.I Define Top and Bottom operations on fractions by

T(ab

)=

a + bb B

(ab

)=

aa + b .

I Notice: T is “add one” and B is “flip, add one, flip.”

T(ab

)= 1 +

ab B

(ab

)=

1a+ba

=1

1 + ba

=1

1 + 1ab

.

I Applying T and B repeatedly is easy.

Tm(ab

)= m +

ab Bm

(ab

)=

1m + 1

ab

.

I (flip, add one, flip)(flip, add one, flip) = (flip, add two, flip)

Continued Fractions

I Recall that [311

]= B3T1B2

[10

].

I Check it out:

B3T1B2( 10

)=

1

3 +1

1 +1

2 +1∞

=1

3 +1

1 +12

=1

3 +23

=311

Continued Fractions

Continued fraction expansion of ab is a path from∞ to a

b !

311 =

1

3 +1

1 +1

2 +1∞

But wait, there’s more!

New model shows there are hidden cells in our topograph livingoff near the horizon.

√2 = 1+

1

2 +1

2 +1

2 +1. . .

Back to the River

f(x, y) = x2 − 2y2 =⇒ fy2 =

(xy

)2− 2

Lagrange: If d ≥ 0 and a, b, d are integers, then the continuedfraction expansion of a + b

√d is eventually periodic.

Topographical Trinitarianism

The End

Want More?

The Sensual (Quadratic) Formby John H. Conway.

Topology of Numbersby Allen Hatcher.