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

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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

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The flat trefoil and other oddities

Joel Langer Case Western Reserve University ICERM June, 2015

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Plane curves with compact polyhedral geometry

Thm Assume: C ⊂ CP2 is an irreducible, real algebraic curve

  • f degree d ≤ 4. Then the geometry (C, dx2 + dy2) is

compact if and only if C is a Bernoulli lemniscate ∞.

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Plane curves with compact polyhedral geometry

Thm Assume: C ⊂ CP2 is an irreducible, real algebraic curve

  • f degree d ≤ 4. Then the geometry (C, dx2 + dy2) 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 ∞.

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Plane curves with compact polyhedral geometry

Thm Assume: C ⊂ CP2 is an irreducible, real algebraic curve

  • f degree d ≤ 4. Then the geometry (C, dx2 + dy2) 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, dx2 + dy2) is compact and flat if and only if C is the sextic trefoil.

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Sextics studied by Euler, Serret, Liouville and others.

Rational sextics with meromorphic arc length parameterizations: Left: 4(x2 + y2)3 + 54(x2 + y2) + 18 √ 3(x4 − y4) = 27 Middle: (x2 + y2)(6 − 3 √ 3x + x2 + y2)2 = 4 Right: 4(x2 + y2)3 + (27 − 12 √ 3x)(x2 + y2)2 − 12(x2 + y2) = −1

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The Euler-Serret sextic

(x2 + y2)(6 − 3 √ 3x + x2 + y2)2 = 4 x(t) = √ 3 − 6t + 7 √ 3t2 − 16t3 + 7 √ 3t4 − 6t5 + √ 3t6 (1 + t2)2(1 − √ 3t + t2) y(t) = 1 − 2 √ 3t + 3t2 − 3t4 + 2 √ 3t5 − t6 (1 + t2)2(1 − √ 3t + t2) ; t = tan √ 3s 6

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Polyhedral geometry of a quadratic differential

Q = q(u)du2 = u8+14u4+1

u2(1−u4)2 du2

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Polyhedral geometry of Q = q(u)du2

◮ g = |Q| = λ2dud¯

u has curvature K = − ∆ log λ

λ2

.

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Polyhedral geometry of Q = q(u)du2

◮ g = |Q| = λ2dud¯

u has curvature K = − ∆ log λ

λ2

.

◮ For Q = (u − u0)ndu2: |u−u0|<ǫ KdA = −nπ.

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Polyhedral geometry of Q = q(u)du2

◮ g = |Q| = λ2dud¯

u has curvature K = − ∆ log λ

λ2

.

◮ For Q = (u − u0)ndu2: |u−u0|<ǫ KdA = −nπ. ◮ Horizontal geodesics Q > 0: q(α(t))α′(t)2 > 0.

n = 1 n = −1 n = −2

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A surface of genus 28: 4g − 4 = Z(Q) − P(Q)

Q = q(u)du2 has 216 simple zeros and 54 double poles.

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The Euclidean quadratic differential Q = ds2

◮ Isotropic coordinates: u = x + iy, v = x − iy.

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The Euclidean quadratic differential Q = ds2

◮ Isotropic coordinates: u = x + iy, v = x − iy. ◮ Plane curve: 0 = f (x, y) = p(u, v).

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The Euclidean quadratic differential Q = ds2

◮ Isotropic coordinates: u = x + iy, v = x − iy. ◮ Plane curve: 0 = f (x, y) = p(u, v). ◮ Q = dx2 + dy2 = dudv = − pu pv du2 = − pv pu dv2.

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The Euclidean quadratic differential Q = ds2

◮ Isotropic coordinates: u = x + iy, v = x − iy. ◮ Plane curve: 0 = f (x, y) = p(u, v). ◮ Q = dx2 + dy2 = dudv = − pu pv du2 = − pv pu dv2. ◮ Natural equations: du ds = i

  • pv

pu , dv ds = −i

  • pu

pv

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The Euclidean quadratic differential Q = ds2

◮ Isotropic coordinates: u = x + iy, v = x − iy. ◮ Plane curve: 0 = f (x, y) = p(u, v). ◮ Q = dx2 + dy2 = dudv = − pu pv du2 = − pv pu dv2. ◮ Natural equations: du ds = i

  • pv

pu , dv ds = −i

  • pu

pv ◮ Rational natural equations:

du ds = τ, dv ds = 1 τ , dτ ds = kn = p2

2p11 − 2p1p2p12 + p2 1p22

2p2

1p2

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Unit speed ellipse

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Total ellipse

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Total ellipse on one sheet

Decomposition of the ellipse into five Euclidean subdomains.

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Circular points c±, isotropic projections and foci

◮ Isotropic lines: u = u0, v = v0

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Circular points c±, isotropic projections and foci

◮ Isotropic lines: u = u0, v = v0 ◮ Zeros of Q = dudv are isotropic tangent points.

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Circular points c±, isotropic projections and foci

◮ Isotropic lines: u = u0, v = v0 ◮ Zeros of Q = dudv are isotropic tangent points. ◮ Poles of Q are ideal points of C.

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Circular points c±, isotropic projections and foci

◮ Isotropic lines: u = u0, v = v0 ◮ 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.

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Half of the unit speed Neumann quartic

A totally circular rational quartic: Z − P = 4 − 8 = 4g − 4

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Biflecnodal circular point

Two inflectional tangents (u − u1)(u − u2) = 0.

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Complex Points on the Lemniscate

Parallel curves projected to the real plane.

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Squaring the Circle via Lemniscatic Sine

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Unit speed lemniscate

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Triunduloidal circular point

Three unduloidal tangents T(u) = (u − u1)(u − u2)(u − u3) = 0.

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Foci and double points for a pencil of sextics

Bˆ

  • cher-Grace locates nodes of p(u, v) = T(u) ¯

T(v) − λ.

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A trio of related curves parameterized by Dixon functions

Fermat cubic x3 + y3 = 1, trihyperbola u3 + v3 = 1, and trefoil u3 + v3 = u3v3 (u = x + iy, v = x − iy).

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Equal areas in equal times

0.5 0.5 1.0 0.2 0.4 0.6 0.8 1.0

Centro-affine arc length along the Fermat cubic: x = sm t, y = cm t.

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Group structure under Cremona transformation

p q r spq p q r s pqs

Addition on the cubic H and sextic T = σ(H).

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Conformal mapping by Dixon sine function

Conformal mapping from disk to triangle.

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Period parallelogram for Dixon functions sm z, cm z

Lattice of inflections—zeros and poles of sm′′z or cm′′z.

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Symmetries of trefoil parameterization

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

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The trefoil, a torus, covers the Riemann sphere 3-1

Unit speed trefoil: u = sm is, v = sm(−is)

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Uniform Subdivision of the Circle

Theorem (Gauss, 1801; Wantzel, 1837) The regular n-gon may be constructed by straightedge and compass iff n = 2jp1p2 . . . pk (pi distinct Fermat primes ).

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Uniform Subdivision of the Lemniscate

Theorem (Abel, 1827) The lemniscate may be evenly subdivided by straightedge and compass for precisely the same integers n = 2jp1p2 . . . pk.

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Uniform Subdivision of the Clover by Origami

1

The clover r3/2 = cos( 3

2θ)

Theorem (Cox, Shurman, 2005) The clover can be divided into n equal lengths by origami if and

  • nly if n = 2a3bp1 · · · pn where a, b ≥ 0 and p1, . . . , pn are distinct

Pierpont primes such that pi = 5, pi = 17, or pi ≡ 1 (mod 3).

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Uniform Subdivision of the Trefoil by Origami

1

The trefoil r3 = cos(3θ) Theorem ( , Singer, 2012) The trefoil can be n-subdivided by origami iff n = 2i3jp1 · · · pk (distinct Pierpont primes pm = 5, 17, or pm ≡ 1 mod 3).