Shapes of euclidean polyhedra and hyperbolic geometry Ivan - - PowerPoint PPT Presentation

Shapes of euclidean polyhedra and hyperbolic geometry

Ivan Izmestiev

joint work with François Fillastre (University of Cergy-Pontoise), http://arxiv.org/abs/1310.1560

Second ERC Workshop Delaunay Geometry: Polytopes, Triangulations, and Spheres

Berlin, October 9, 2013

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 1 / 25

Main result

V ∶= a (positively spanning) configuration of n unit vectors in Rd. M(V) ∶= the space of (homothety classes of) polytopes with outer facet normals V.

Theorem

The space M(V) is a polyhedral ball of dimension n − d − 1. Each cell of M(V) carries a natural hyperbolic metric.

Example

6 vectors in R3 determine a 2-dimensional complex. Metrically this is a right-angled hyperbolic hexagon.

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 2 / 25

Main ingredients

Combinatorics (linear algebra):

studying the combinatorial structure of the polyhedron P(h) ∶= {Vx ≤ h} ⊂ Rd, V ∈ Rn×d fixed, h ∈ Rn variable Gale diagrams ↝ h ∈ C(V) a subfan of the secondary fan of V.

Geometry (bilinear algebra):

studying the second intrinsic volume (“quermassintegral”) vol2(h) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ area(P(h)) for d = 2

1 2 area(∂P(h))

for d = 3 ∑

dim F=2

αF(P(h))vol2(F) for d ≥ 4 Alexandrov-Fenchel inequalities for mixed volumes ↝ quadratic form of signature (+,−,...,−).

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 3 / 25

Case d = 2

Bavard, Ghys’92:

hyperbolic metric on the space of polygons with fixed edge directions ℓ1,...,ℓn ∈ C(V) ⇔

n

∑

i=1

ℓivi = 0, ℓi ≥ 0∀i Thus C(V) = Rn

≥0∩ (n − 2)-subspace

vi ℓi

Polygons up to translation = pointed (n − 2)-cone with ≤ n facets. Polygons up to similarity = (n − 3)-polytope with n − 2, n − 1, or n facets.

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 4 / 25

Facets of C(V)

ℓi = 0 ⇔ i-th edge contracts to a point

tr2(∆n−3) tr(∆n−3) ∆n−3 n facets n − 1 facets n − 2 facets v3 v2 v1 v5 v4 ℓ2 = 0 C(V) ℓ5 = 0 ℓ3 = 0 ℓ1 = 0 ℓ4 = 0

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 5 / 25

The area is a quadratic form

Consider the area of polygons with edge normals V: C(V) → R ℓ ↦ area(P(ℓ))

Theorem

area(P(ℓ)) = q(ℓ) where q is a quadratic form of signature (+,−,...,−).

p2 p3 p1 p5 p4

Why quadratic: Vertex coordinates are linear functions of ℓ area(P(ℓ)) = 1 2(det(p2 − p1,p3 − p1) +det(p3−p1,p4−p1)+det(p5−p1,p4−p1))

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 6 / 25

The signature

P(ℓ) ℓ1 ℓ2 S2 ℓ4 ℓ5 ℓ3 S3

q(ℓ) = area(S1) − area(S2) − area(S3) area(S1) = (aℓ2 + bℓ3)2 = (a′ℓ1 + b′ℓ5)2 area(S2) = (cℓ3)2, area(S3) = (dℓ5)2 After coordinate change x0 = aℓ2 + bℓ3, x1 = cℓ3, x2 = dℓ5

• btain

area(P) = x2

0 − x2 1 − x2 2

a quadratic form of signature (+,−,−).

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 7 / 25

The hyperbolic space

qh(x) = x2

0 − x2 1 − ... − x2 d

Hd ∶= {x ∣ q(x) = 1,x0 > 1} Metric of constant curvature −1. qs(x) = x2

0 + x2 1 + ... + x2 d

Sd ∶= {x ∣ q(x) = 1} Metric of constant curvature 1.

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 8 / 25

Hyperbolic polyhedra

angle of a polygon = dihedral angle of the cone = arccosq(ν1,ν2) (in the hyperbolic as in the spherical case) q(ν1,ν2) = 0 ⇔ sides form a right angle If q(x) = q′(x0,...,xd−2) − x2

d−1 − x2 d, then {xd−1 = 0} ⊥ {xd = 0}.

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 9 / 25

Space of polygons is a (truncated) orthoscheme

q has signature (+,−,...,−) ⇒ M(V) ∶= C(V) ∩ {ℓ ∣ q(ℓ) = 1} is a hyperbolic polyhedron

Theorem (Bavard,Ghys’92)

If ∣i − j∣ ≥ 2, then Fi ⊥ Fj, where Fi is the facet of M(V) corresponding to contraction of the i-th edge.

Proof.

P(ℓ) ℓ1 ℓ2 S2 ℓ4 ℓ5 ℓ3 S3

q(ℓ) = x2

0 − x2 1 − x2 2

F3 = {x1 = 0} F5 = {x2 = 0}

Example

The space of equiangular pentagons of area 1 is isometric to the regular right-angled hyperbolic pentagon.

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 10 / 25

Hyperbolic orthoschemes and complex hyperbolic

• rbifolds

Bavard, Ghys’92 Polygones du plan et polyédres hyperboliques

▸ Realization of hyperbolic orthoschemes from the Im Hof’s list. ▸ Complete list: Im Hof’90, Tumarkin’07, Kistler’11.

This was motivied by Thurston’98 Shapes of polyhedra and triangulations of the sphere

▸ Complex hyperbolic structure on the space of Euclidean metrics

• n S2 with cone angles α1,...,αn.

▸ Realization of some non-arithmetic complex Coxeter orbifolds.

We present a generalization of the Bavard-Ghys construction to higher dimensions.

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 11 / 25

From d = 2 to d > 2

Combinatorics

▸ facet normals V no more determine the combinatorial type of the

polytope P

▸ ⇒ C(V) no more a cone, but a fan (made of type cones)

Geometry

▸ What to use instead of area(P)? ▸ Is there a canonical quadratic form of signature (+,−,...,−)? ▸ Can the dihedral angles of M(V) be computed?

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 12 / 25

Type cones

V = (v1,...,vn) a positively spanning vector configuration in Rd C(V) ∶= translation classes of polytopes with facet normals V P,P′ ∈ C(V) are normally equivalent if they have the same normal fan: N(P) = N(P′) C(V) = ⊔

∆

T(∆), where T(∆) = {P ∣ N(P) = ∆}/{translations}.

Theorem

The closure of C(V) is a pointed fan with convex support.

Example

For V = {±e1,...,±ed} all P are normally equivalent. Monotypic polytopes: McMullen, Schneider, Shephard’74.

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 13 / 25

Support numbers and Gale dual

Given V ∈ Rn×d with rows vi ∈ Rd of norm 1, consider P(h) = {x ∈ Rn ∣ Vx ≤ h}, h ∈ Rn Support numbers (hi)n

i=1. We have

P(h) + p = P(h + Vp) Hence C(V) ⊂ Rn/imV.

hi p vi

Definition

π∶Rn → Rn/imV, π(ei) =∶ ¯ vi Vector configuration V = (¯ v1,..., ¯ vn) is called Gale dual to V. V ⊺V = En, rankV + rankV = n V positively spanning ⇔ V lies in an open half-space

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 14 / 25

C(V) lies in pos(V)

Theorem

P(h) ≠ ∅ ⇔ π(h) ∈ pos(V) ∶= {∑

i

λi ¯ vi ∣ λi ≥ 0}

Proof.

Note: 0 ∈ P(h) ⇔ hi ≥ 0∀i. P(h) ≠ ∅ ⇔ ∃p ∈ P(h) ⇔ 0 ∈ P(h) − p = P(h − Vp) ⇔ h − Vp ∈ Rn

≥0 ⇔ π(h) ∈ pos(¯

v1,..., ¯ vn)

Example

V are normals of the triangular bipyramid ⇒ V span a hexagonal cone

¯ v1 ¯ v2 ¯ v3 ¯ v4 ¯ v6 ¯ v5 Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 15 / 25

C(V) is the 2-core of V

The k-core of a vector configuration W ⊂ Rm is corek(W) = {x ∈ Rm ∣ ∀y s.t. ⟨y,x⟩ ≥ 0∃wi1,...,wik s.t. ⟨y,wiα⟩ ≥ 0} E.g. core1(W) = pos(W) core2(W) = ⋂

w∈W

pos(W ∖ {w})

Theorem

The closure of C(V) is core2(V).

Proof.

i-th facet non-empty ⇔ ∃p ∈ Rn such that ⟨vi,p⟩ = hi and ⟨vj,p⟩ < hj for all j ≠ i. Then use P(h − Vp) = P(h) − p etc.

Example

¯ v1 ¯ v2 ¯ v5 ¯ v6 ¯ v3 ¯ v4

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 16 / 25

Type cones are the chambers of the chamber fan

The chamber fan Ch(W) of W is the coarsest common subdivision of all cones spanned by W.

Theorem

Closures of type cones form the chamber fan of V.

Lemma

pos(Vσ) ∈ N(P(h)) ⇔ π(h) ∈ pos(V ¯

σ), where ¯

σ = {1,...,n} ∖ σ.

Proof.

Similar to the preceding two arguments.

Example

¯ v1 ¯ v2 ¯ v5 ¯ v6 ¯ v3 ¯ v4

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 17 / 25

References

All the above arguments on type cones and their arrangement appeared in

▸ McMullen’73 Representations of polytopes and polyhedral sets ▸ Shephard’71 Spherical complexes and radial projections of

polytopes From this, there is only one step to the secondary polyhedron.

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 18 / 25

Relation to the secondary polyhedron

Let P ∈ C(V), N(P) = ∆. Then its support function hP∶Rd → R is a convex conewise (wrt ∆) linear function such that h(vi) = hi. Thus polytopal fans ↔ regular triangulations support numbers ↔ weights Why is the chamber fan polytopal? Invert the construction:

▸ V is the Gale dual of V ▸ for every polytopal fan ∆ on V pick h∆ such that P(h∆) ∈ T(∆) ▸ then N(∑∆ P(h∆)) = Ch(V)

The fan Ch(V) is the normal fan of the secondary polyhedron of V, and vice versa.

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 19 / 25

Intrinsic volumes

K ⊂ Rd convex body, B ⊂ Rd unit ball vold(K + tB) =

d

∑

i=0

Wi(K)(d i )ti The coefficient Wi(K) is called the i-th quermassintegral; it is the mixed volume Wi(K) = vold(K,...,K,B,...,B) The i-th quermassintegral is proportional to the (d − i)-th intrinsic

• volume. If K = P is a polytope, then

vold−i(P) = ∑

dim F=d−i

∣NF(P)∣vold−i(F), where ∣NF(P)∣ is the normalized exterior angle at F.

Example

vold−1(P) = 1

2 vold−1(∂P),

vol0(P) = 1

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 20 / 25

The second intrinsic volume is a quadratic form

▸ mixed volumes are multilinear wrt Minkowski addition and

multiplication with positive scalars

▸ on every T(∆), Minkowski addition corresponds to addition of the

support numbers Hence for every ∆ there is a quadratic form q∆ on Rn s.t. q∆(h) = vol2(P(h)) = c vold(P(h),P(h),B,...,B) if N(P(h)) = ∆ More generally, q∆,K ∶= vold(P(h),P(h),K1,...,Kd−2)

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 21 / 25

The signature of q∆

Theorem (Alexandrov-Fenchel inequalities)

q∆(h,h′)2 ≥ q∆(h,h)q∆(h′,h′)

Corollary

The form q∆ has exactly one positive eigenvalue.

Proof.

q∆(h,h) = vol2(P(h)) > 0, det(q∆(h,h) q∆(h,h′) q∆(h′,h) q∆(h′,h′)) ≤ 0 ⇒ q∆ is indefinite on all 2-dimensional subspaces through h The equality case is known ⇒ the form has maximum possible rank.

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 22 / 25

The signature of q∆

Theorem (Alexandrov-Fenchel inequalities)

q∆(h,h′)2 ≥ q∆(h,h)q∆(h′,h′)

Corollary

The form q∆ has exactly one positive eigenvalue.

Proof.

q∆(h,h) = vol2(P(h)) > 0, det(q∆(h,h) q∆(h,h′) q∆(h′,h) q∆(h′,h′)) ≤ 0 ⇒ q∆ is indefinite on all 2-dimensional subspaces through h The equality case is known ⇒ the form has maximum possible rank. (For general q∆,K(h) the rank is unknown, since no characterization of the equality case in the AF-inequality.)

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 22 / 25

Right angles on the boundary of M(V)

Each facet of C(V) corresponds either to face truncation or to degeneration of P(h) (dimension falls).

Theorem

Facets corresponding to truncation of different vertices are orthogonal to each other.

Proof.

area(∂P(h)) = f 2

1 − f 2 2 − f 2 3

f 2

1 = area(∂Σ1) = (ah2 + bh3 + ch4 + dh5)2

f 2

3 = area(F3(Σ3)) + area(F4(Σ3))

+ area(F5(Σ3)) − area(F6(Σ3)) = (a′h3 + b′h4 + c′h5 + d′h6)2

F6 F1

Σ3 Σ2

F4 F3

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 23 / 25

From bipyramid to a right-angled hyperbolic hexagon

A priori q∆(h) ≠ q′

∆(h) and

gluing 6 hyperbolic quadrilaterals yields a cone point in the center.

q∆ q∆′

Theorem

In this case q∆ = q∆′. Thus have a right-angled hyperbolic hexagon.

Proof.

This is because q∆′(h) = q∆(h) + area(F3(Σ)) + area(F5(Σ)) − area(F2(Σ)) − area(F6(Σ)) and area(F3) + area(F5) = area(F2) + area(F6) because v3 + v5 = v2 + v6

Σ

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 24 / 25

A riddle

Continue the sequence.

?

Is “?” the polar dual of a cyclic 4-polytope on 7 vertices?

Ivan Izmestiev (FU Berlin) Shapes of euclidean polyhedra and hyperbolic geometry Berlin’2013 25 / 25