The flat trefoil and other oddities Joel Langer Case Western - PowerPoint PPT Presentation

The flat trefoil and other oddities Joel Langer Case Western Reserve University ICERM June, 2015

Plane curves with compact polyhedral geometry Thm Assume: C ⊂ C P 2 is an irreducible, real algebraic curve of degree d ≤ 4. Then the geometry ( C , dx 2 + dy 2 ) is compact if and only if C is a Bernoulli lemniscate ∞ .

Plane curves with compact polyhedral geometry Thm Assume: C ⊂ C P 2 is an irreducible, real algebraic curve of degree d ≤ 4. Then the geometry ( C , dx 2 + dy 2 ) is compact if and only if C is a Bernoulli lemniscate ∞ . Cor Let x ( s ) , y ( s ) parameterize an arc of C (as above) by unit speed. If x ( s ) , y ( s ) extend meromorphically to all s ∈ C , then C is a line, a circle, or ∞ .

Plane curves with compact polyhedral geometry Thm Assume: C ⊂ C P 2 is an irreducible, real algebraic curve of degree d ≤ 4. Then the geometry ( C , dx 2 + dy 2 ) is compact if and only if C is a Bernoulli lemniscate ∞ . Cor Let x ( s ) , y ( s ) parameterize an arc of C (as above) by unit speed. If x ( s ) , y ( s ) extend meromorphically to all s ∈ C , then C is a line, a circle, or ∞ . Thm ( , Singer, 2015) Let C be as above but with d < 8. Then ( C , dx 2 + dy 2 ) is compact and flat if and only if C is the sextic trefoil.

Sextics studied by Euler, Serret, Liouville and others. Rational sextics with meromorphic arc length parameterizations: √ Left: 4( x 2 + y 2 ) 3 + 54( x 2 + y 2 ) + 18 3( x 4 − y 4 ) = 27 √ Middle: ( x 2 + y 2 )(6 − 3 3 x + x 2 + y 2 ) 2 = 4 √ Right: 4( x 2 + y 2 ) 3 + (27 − 12 3 x )( x 2 + y 2 ) 2 − 12( x 2 + y 2 ) = − 1

The Euler-Serret sextic √ ( x 2 + y 2 )(6 − 3 3 x + x 2 + y 2 ) 2 = 4 √ √ √ √ 3 t 2 − 16 t 3 + 7 3 t 4 − 6 t 5 + 3 t 6 3 − 6 t + 7 √ x ( t ) = (1 + t 2 ) 2 (1 − 3 t + t 2 ) √ √ √ 3 t + 3 t 2 − 3 t 4 + 2 3 t 5 − t 6 y ( t ) = 1 − 2 3 s √ ; t = tan (1 + t 2 ) 2 (1 − 3 t + t 2 ) 6

Polyhedral geometry of a quadratic differential Q = q ( u ) du 2 = u 8 +14 u 4 +1 u 2 (1 − u 4 ) 2 du 2

Polyhedral geometry of Q = q ( u ) du 2 u has curvature K = − ∆ log λ ◮ g = | Q | = λ 2 dud ¯ . λ 2

Polyhedral geometry of Q = q ( u ) du 2 u has curvature K = − ∆ log λ ◮ g = | Q | = λ 2 dud ¯ . λ 2 ◮ For Q = ( u − u 0 ) n du 2 : � � | u − u 0 | <ǫ KdA = − n π .

Polyhedral geometry of Q = q ( u ) du 2 u has curvature K = − ∆ log λ ◮ g = | Q | = λ 2 dud ¯ . λ 2 ◮ For Q = ( u − u 0 ) n du 2 : � � | u − u 0 | <ǫ KdA = − n π . ◮ Horizontal geodesics Q > 0: q ( α ( t )) α ′ ( t ) 2 > 0. n = 1 n = − 1 n = − 2

A surface of genus 28: 4 g − 4 = Z ( Q ) − P ( Q ) Q = q ( u ) du 2 has 216 simple zeros and 54 double poles.

The Euclidean quadratic differential Q = ds 2 ◮ Isotropic coordinates: u = x + iy , v = x − iy .

The Euclidean quadratic differential Q = ds 2 ◮ Isotropic coordinates: u = x + iy , v = x − iy . ◮ Plane curve: 0 = f ( x , y ) = p ( u , v ).

The Euclidean quadratic differential Q = ds 2 ◮ Isotropic coordinates: u = x + iy , v = x − iy . ◮ Plane curve: 0 = f ( x , y ) = p ( u , v ). ◮ Q = dx 2 + dy 2 = dudv = − p u p v du 2 = − p v p u dv 2 .

The Euclidean quadratic differential Q = ds 2 ◮ Isotropic coordinates: u = x + iy , v = x − iy . ◮ Plane curve: 0 = f ( x , y ) = p ( u , v ). ◮ Q = dx 2 + dy 2 = dudv = − p u p v du 2 = − p v p u dv 2 . � � p v p u ◮ Natural equations: du dv ds = i ds = − i p u , p v

The Euclidean quadratic differential Q = ds 2 ◮ Isotropic coordinates: u = x + iy , v = x − iy . ◮ Plane curve: 0 = f ( x , y ) = p ( u , v ). ◮ Q = dx 2 + dy 2 = dudv = − p u p v du 2 = − p v p u dv 2 . � � p v p u ◮ Natural equations: du dv ds = i ds = − i p u , p v ◮ Rational natural equations: ds = kn = p 2 2 p 11 − 2 p 1 p 2 p 12 + p 2 ds = 1 du dv d τ 1 p 22 ds = τ, τ , 2 p 2 1 p 2

Unit speed ellipse

Total ellipse

Total ellipse on one sheet Decomposition of the ellipse into five Euclidean subdomains.

Circular points c ± , isotropic projections and foci ◮ Isotropic lines: u = u 0 , v = v 0

Circular points c ± , isotropic projections and foci ◮ Isotropic lines: u = u 0 , v = v 0 ◮ Zeros of Q = dudv are isotropic tangent points.

Circular points c ± , isotropic projections and foci ◮ Isotropic lines: u = u 0 , v = v 0 ◮ Zeros of Q = dudv are isotropic tangent points. ◮ Poles of Q are ideal points of C .

Circular points c ± , isotropic projections and foci ◮ Isotropic lines: u = u 0 , v = v 0 ◮ Zeros of Q = dudv are isotropic tangent points. ◮ Poles of Q are ideal points of C . ◮ Poles of order n = − 1 , − 2 , − 3 are circular ideal points of C .

Half of the unit speed Neumann quartic A totally circular rational quartic: Z − P = 4 − 8 = 4 g − 4

Biflecnodal circular point Two inflectional tangents ( u − u 1 )( u − u 2 ) = 0.

Complex Points on the Lemniscate Parallel curves projected to the real plane.

Squaring the Circle via Lemniscatic Sine

Unit speed lemniscate

Triunduloidal circular point Three unduloidal tangents T ( u ) = ( u − u 1 )( u − u 2 )( u − u 3 ) = 0.

Foci and double points for a pencil of sextics ocher-Grace locates nodes of p ( u , v ) = T ( u ) ¯ Bˆ T ( v ) − λ .

A trio of related curves parameterized by Dixon functions Fermat cubic x 3 + y 3 = 1, trihyperbola u 3 + v 3 = 1, and trefoil u 3 + v 3 = u 3 v 3 ( u = x + iy , v = x − iy ).

Equal areas in equal times 1.0 0.8 0.6 0.4 0.2 � 0.5 0.5 1.0 Centro-affine arc length along the Fermat cubic: x = sm t , y = cm t .

Group structure under Cremona transformation q r s � p � q p s r q p p � q � s Addition on the cubic H and sextic T = σ ( H ).

Conformal mapping by Dixon sine function Conformal mapping from disk to triangle.

Period parallelogram for Dixon functions sm z , cm z Lattice of inflections—zeros and poles of sm ′′ z or cm ′′ z .

Symmetries of trefoil parameterization Tiling the plane by trihexagons; subdivision of a tile into 108 subtiles (30-60-90 triangles).

The trefoil, a torus, covers the Riemann sphere 3-1 Unit speed trefoil: u = sm is , v = sm ( − is )

Uniform Subdivision of the Circle Theorem (Gauss, 1801; Wantzel, 1837) The regular n -gon may be constructed by straightedge and compass iff n = 2 j p 1 p 2 . . . p k ( p i distinct Fermat primes ) .

Uniform Subdivision of the Lemniscate Theorem (Abel, 1827) The lemniscate may be evenly subdivided by straightedge and compass for precisely the same integers n = 2 j p 1 p 2 . . . p k .

Uniform Subdivision of the Clover by Origami 1 The clover r 3 / 2 = cos( 3 2 θ ) Theorem (Cox, Shurman, 2005) The clover can be divided into n equal lengths by origami if and only if n = 2 a 3 b p 1 · · · p n where a , b ≥ 0 and p 1 , . . . , p n are distinct Pierpont primes such that p i = 5, p i = 17, or p i ≡ 1 ( mod 3).

Uniform Subdivision of the Trefoil by Origami 1 The trefoil r 3 = cos(3 θ ) Theorem ( , Singer, 2012) The trefoil can be n -subdivided by origami iff n = 2 i 3 j p 1 · · · p k (distinct Pierpont primes p m = 5 , 17, or p m ≡ 1 mod 3).