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The Computational Geometry of Congruence Testing, Part II G unter Rote Freie Universit at Berlin A B ? = G unter Rote, Freie Universit at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

∼ = ? A B

The Computational Geometry of Congruence Testing, Part II G¨ unter Rote

Freie Universit¨ at Berlin

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

∼ = ? A B

The Computational Geometry of Congruence Testing, Part II G¨ unter Rote

Freie Universit¨ at Berlin

A

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Overview

  • 1 dimension
  • 2 dimensions
  • 3 dimensions
  • 4 dimensions
  • d dimensions

     O(n log n) time O(n⌈d/3⌉ log n) time [Brass and Knauer 2002] today (joint work with Heuna Kim) O(n⌊(d+2)/2⌋/2 log n) Monte Carlo [Akutsu 1998/Matouˇ sek] ↓ O(n⌊(d+1)/2⌋/2 log n) time

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Overview

  • 1 dimension
  • 2 dimensions
  • 3 dimensions
  • 4 dimensions
  • d dimensions

     O(n log n) time O(n⌈d/3⌉ log n) time [Brass and Knauer 2002]

  • Rotations in 4-space
  • Pl¨

ucker coordinates for 2-planes in 4-space

  • The Hopf fibration of S3
  • Closest pair graph
  • 2+2 dimension reduction
  • Coxeter classification of reflection groups

today (joint work with Heuna Kim) O(n⌊(d+2)/2⌋/2 log n) Monte Carlo [Akutsu 1998/Matouˇ sek] ↓ O(n⌊(d+1)/2⌋/2 log n) time

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

4 Dimensions: Algorithm Overview

Iterative Pruning Generating Orbit Cycles Mirror Case

planes mirror symmetry n = |A| is bounded

2+2 Dimension Reduction 1+3 Dimension Reduction

edge- transitive ≤ 100|P| markers lower-dimensional components

Marking and Condensing Great Circles

|P| ≤ n/200 ≤ n

2

≤ n

2

joint work with Heuna Kim

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SLIDE 6

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

4 Dimensions: Algorithm Overview

Iterative Pruning Generating Orbit Cycles Mirror Case

planes mirror symmetry n = |A| is bounded

2+2 Dimension Reduction 1+3 Dimension Reduction

edge- transitive ≤ 100|P| markers lower-dimensional components

Marking and Condensing Great Circles

|P| ≤ n/200 ≤ n

2

≤ n

2

joint work with Heuna Kim

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Initialization: Closest-Pair Graph

2) Compute the closest pair graph G(A) = (A, { uv : u − v = δ }) where δ := the distance of the closest pair, in O(n log n) time. 1) PRUNE by distance from the origin.

  • =

⇒ we can assume that A lies on the 3-sphere S3.

  • We can assume that δ is SMALL: δ ≤ δ0 := 0.0005.

(Otherwise, |A| ≤ n0, by a packing argument.)

[ Bentley and Shamos, STOC 1976 ]

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Everything looks the same!

By the PRUNING principle, we can assume that all points look locally the same:

  • All points have congruent neighborhoods in G(A).

(The neighbors of u lie on a 2-sphere in S3; There are at most K3 = 12 neighbors.) u δ

1 2 3 4 6 5 7 8

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Everything looks the same!

By the PRUNING principle, we can assume that all points look locally the same:

  • All points have congruent neighborhoods in G(A).

(The neighbors of u lie on a 2-sphere in S3; There are at most K3 = 12 neighbors.) u δ

1 2 3 4 6 5 7 8

u v

  • Make a directed graph D from G(A)

and PRUNE its arcs uv by the joint neighborhood of u and v.

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Everything looks the same!

By the PRUNING principle, we can assume that all points look locally the same:

  • All points have congruent neighborhoods in G(A).

(The neighbors of u lie on a 2-sphere in S3; There are at most K3 = 12 neighbors.) u δ

1 2 3 4 6 5 7 8

u v

  • Make a directed graph D from G(A)

and PRUNE its arcs uv by the joint neighborhood of u and v.

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SLIDE 11

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Everything looks the same!

By the PRUNING principle, we can assume that all points look locally the same:

  • All points have congruent neighborhoods in G(A).

(The neighbors of u lie on a 2-sphere in S3; There are at most K3 = 12 neighbors.) u δ

1 2 3 4 6 5 7 8

u v

  • Make a directed graph D from G(A)

and PRUNE its arcs uv by the joint neighborhood of u and v.

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SLIDE 12

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Everything looks the same!

By the PRUNING principle, we can assume that all points look locally the same:

  • All points have congruent neighborhoods in G(A).

(The neighbors of u lie on a 2-sphere in S3; There are at most K3 = 12 neighbors.) u δ

1 2 3 4 6 5 7 8

u v

  • . . . until all arcs uv

look the same.

  • Make a directed graph D from G(A)

and PRUNE its arcs uv by the joint neighborhood of u and v.

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Algorithm Overview

Iterative Pruning Generating Orbit Cycles Mirror Case

planes mirror symmetry n = |A| is bounded

2+2 Dimension Reduction 1+3 Dimension Reduction

edge- transitive ≤ 100|P| markers lower-dimensional components

Marking and Condensing Great Circles

|P| ≤ n/200 ≤ n

2

≤ n

2

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SLIDE 14

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Algorithm Overview

Iterative Pruning Generating Orbit Cycles Mirror Case

planes mirror symmetry n = |A| is bounded

2+2 Dimension Reduction 1+3 Dimension Reduction

edge- transitive ≤ 100|P| markers lower-dimensional components

Marking and Condensing Great Circles

|P| ≤ n/200 ≤ n

2

≤ n

2

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SLIDE 15

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Algorithm Overview

Iterative Pruning Generating Orbit Cycles Mirror Case

planes mirror symmetry n = |A| is bounded

2+2 Dimension Reduction 1+3 Dimension Reduction

edge- transitive ≤ 100|P| markers lower-dimensional components

Marking and Condensing Great Circles

|P| ≤ n/200 ≤ n

2

≤ n

2

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Predecessor-Successor Figure

α u v t w s(uv) p(uv) Pick some α. s(uv) := {vw : vw ∈ E, ∠uvw = α}

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Predecessor-Successor Figure

α u v t w t′ w′ s(uv) p(uv) Pick some α. s(uv) := {vw : vw ∈ E, ∠uvw = α} τ w′ t′ torsion angle τ

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Predecessor-Successor Figure

α u v t w t′ w′ s(uv) p(uv) Pick some α. s(uv) := {vw : vw ∈ E, ∠uvw = α} τ w′ t′ torsion angle τ

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Predecessor-Successor Figure

α u v t w t′ w′ s(uv) p(uv) Pick some α. s(uv) := {vw : vw ∈ E, ∠uvw = α} τ w′ t′ torsion angle τ can PRUNE arcs from s(u, v) canonical directions

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Predecessor-Successor Figure

α u v t w t′ w′ s(uv) p(uv) Pick some α. s(uv) := {vw : vw ∈ E, ∠uvw = α} τ w′ t′ torsion angle τ can PRUNE arcs from s(u, v) τ0 For every path tuv with ∠tuv = α, ∃ vw with ∠uvw = α and torsion angle τ0. canonical directions

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Orbit cycles

p1 p0 p2 For every path pipi+1pi+2 with ∠pipi+1pi+2 = α, ∃ pi+3 with ∠pi+1pi+2pi+3 = α and torsion τ0.

!

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Orbit cycles

p1 p0 p2 p3 For every path pipi+1pi+2 with ∠pipi+1pi+2 = α, ∃ pi+3 with ∠pi+1pi+2pi+3 = α and torsion τ0.

!

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Orbit cycles

p1 p0 p2 p3 For every path pipi+1pi+2 with ∠pipi+1pi+2 = α, ∃ pi+3 with ∠pi+1pi+2pi+3 = α and torsion τ0. R(p0, p1, p2) = (p1, p2, p3) R

!

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Orbit cycles

p1 p0 p2 p3 For every path pipi+1pi+2 with ∠pipi+1pi+2 = α, ∃ pi+3 with ∠pi+1pi+2pi+3 = α and torsion τ0. p4 R(p0, p1, p2) = (p1, p2, p3) R(p0, p1, p2, p3) = (p1, p2, p3, p4) R

!

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Orbit cycles

p1 p0 p2 p3 For every path pipi+1pi+2 with ∠pipi+1pi+2 = α, ∃ pi+3 with ∠pi+1pi+2pi+3 = α and torsion τ0. p4 R(p0, p1, p2) = (p1, p2, p3) R(p0, p1, p2, p3) = (p1, p2, p3, p4) R(p1, p2, p3, p4) = (p2, p3, p4, p5) · · · Rpi = pi+1: The orbit of p0 under R, a helix R

!

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Rotations in 4 dimensions

R =     cos ϕ − sin ϕ sin ϕ cos ϕ cos ψ − sin ψ sin ψ cos ψ     = Rϕ Rψ

  • in some appropriate coordinate system.

ϕ = ±ψ: → unique decomposition R4 = P ⊕ Q into two completely orthogonal 2-dimensional axis planes P and Q ϕ = ±ψ: isoclinic rotations circle with radius r The orbit of a point p0 = (x1, y1, x2, y2) lies on a helix

  • n a flat torus Cr × Cs, with r =
  • x2

1 + y2 1, s =

  • x2

2 + y2 2

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Rotations in 4 dimensions

R =     cos ϕ − sin ϕ sin ϕ cos ϕ cos ψ − sin ψ sin ψ cos ψ     = Rϕ Rψ

  • The orbit of a point p0 = (x1, y1, x2, y2) lies on a helix
  • n a flat torus Cr × Cs, with r =
  • x2

1 + y2 1, s =

  • x2

2 + y2 2

s p0 p1 p2 p5 p6 p7 ϕ ψ Cr

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Planes in 4 dimensions

  • Every point lies on ≤ 60 orbit cycles.
  • Every orbit cycle contains ≥ 12000 points,

because δ is small.

  • Every orbit cycle generates 1 plane

(corresponding to the smaller of ϕ and ψ.) = ⇒ a collection of ≤ n/200 planes (or: great circles)

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Algorithm Overview

Iterative Pruning Generating Orbit Cycles Mirror Case

planes mirror symmetry n = |A| is bounded

2+2 Dimension Reduction 1+3 Dimension Reduction

edge- transitive ≤ 100|P| markers lower-dimensional components

Marking and Condensing Great Circles

|P| ≤ n/200 ≤ n

2

≤ n

2

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SLIDE 30

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Algorithm Overview

Iterative Pruning Generating Orbit Cycles Mirror Case

planes mirror symmetry n = |A| is bounded

2+2 Dimension Reduction 1+3 Dimension Reduction

edge- transitive ≤ 100|P| markers lower-dimensional components

Marking and Condensing Great Circles

|P| ≤ n/200 ≤ n

2

≤ n

2

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Marking Points on Great Circles

projection of another unit circle Q unit circle P

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Marking Points on Great Circles

projection of another unit circle Q unit circle P IDEA: mark those two points in P

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Marking Points on Great Circles

projection of another unit circle Q unit circle P IDEA: mark those two points in P a neighbor of P IDEA 2: Construct the closest-pair graph in the space of great circles, in O(n log n) time.

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Pl¨ ucker coordinates

planes in 4-space ⇔ great circles on S3 ⇔ a.k.a. lines in RP 3 plane through (x1, y1, x2, y2) and (x′

1, y′ 1, x′ 2, y′ 2) :

(v1, . . . , v6) =

  • x1 y1

x′

1 y′ 1

  • ,
  • x1 x2

x′

1 x′ 2

  • ,
  • x1 y2

x′

1 y′ 2

  • ,
  • y1 x2

y′

1 x′ 2

  • ,
  • y1 y2

y′

1 y′ 2

  • ,
  • x2 y2

x′

2 y′ 2

  • (v1, . . . , v6) ∈ RP 5. [Pl¨

ucker relations v1v6 − v2v5 + v3v4 = 0] Normalize: → A great circle is represented by two antipodal points on S5. This representation is geometrically meaningful: Distances on S5 are preserved under rotations of R4 / S3. (Packings of 2-planes in 4-space were considered by [Conway, Hardin and Sloane 1996], with different distances.)

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Marking Points on Great Circles

projection of another unit circle Q unit circle P IDEA: mark those two points in P a neighbor of P IDEA 2: Construct the closest-pair graph in the space of great circles, in O(n log n) time. Every plane has at most K5 ≤ 44 neighbors.

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Marking Points on Great Circles

m ≤

n 200 great circles in R4

− → m point pairs on S5 projection of another unit circle Q unit circle P IDEA: mark those two points in P At most 88 (≤ 100) points are marked on every great circle. These points replace A. → successful CONDENSATION a neighbor of P IDEA 2: Construct the closest-pair graph in the space of great circles, in O(n log n) time. Every plane has at most K5 ≤ 44 neighbors.

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Algorithm Overview

Iterative Pruning Generating Orbit Cycles Mirror Case

planes mirror symmetry n = |A| is bounded

2+2 Dimension Reduction 1+3 Dimension Reduction

edge- transitive ≤ 100|P| markers lower-dimensional components

Marking and Condensing Great Circles

|P| ≤ n/200 ≤ n

2

≤ n

2

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SLIDE 38

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Algorithm Overview

Iterative Pruning Generating Orbit Cycles Mirror Case

planes mirror symmetry n = |A| is bounded

2+2 Dimension Reduction 1+3 Dimension Reduction

edge- transitive ≤ 100|P| markers lower-dimensional components

Marking and Condensing Great Circles

|P| ≤ n/200 ≤ n

2

≤ n

2

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SLIDE 39

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Isoclinic planes

Where to mark?? projection of a neighbor Q of P unit circle P

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Isoclinic planes

Where to mark?? projection of a neighbor Q of P Problem if all closest pairs are isoclinic. unit circle P

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Isoclinic planes

Where to mark?? projection of a neighbor Q of P Problem if all closest pairs are isoclinic. unit circle P Constant distances from one circle to the other. “Clifford-parallel” ≡ isoclinic

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Clifford-parallel circles

s P :     x1 y1 x2 y2    =     cos t sin t     , Q:     r cos t r sin t s cos(α + t) s sin(α + t)     P Q r2 + s2 = 1

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Clifford-parallel circles

s P :     x1 y1 x2 y2    =     cos t sin t     , Q:     r cos t r sin t s cos(α + t) s sin(α + t)     P Q r2 + s2 = 1 h(x1, y1, x2, y2) = the right Hopf map h: S3 → S2

  • 2(x1y2 − y1x2), 2(x1x2 + y1y2), 1 − 2(x2

2 + y2 2)

  • [ Hopf 1931 ]
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SLIDE 44

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Clifford-parallel circles

s P :     x1 y1 x2 y2    =     cos t sin t     , Q:     r cos t r sin t s cos(α + t) s sin(α + t)     P Q Q′ :     r cos t r sin t s cos(α − t) s sin(α − t)     Q′ r2 + s2 = 1 h(x1, y1, x2, y2) = the right Hopf map h: S3 → S2

  • 2(x1y2 − y1x2), 2(x1x2 + y1y2), 1 − 2(x2

2 + y2 2)

  • [ Hopf 1931 ]
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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

The Hopf fibration

The fibers h−1(p) for p ∈ S2 are great circles: a Hopf bundle

http://www.geom.uiuc.edu/~banchoff/script/b3d/hypertorus.html

Every great circle belongs to a unique right Hopf bundle. Right Hopf map h: S3 → S2 Isoclinic ≡ belong to the same Hopf bundle This is a transitive relation. stereographic projection S3 → R3 (Villarceau circles)

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

The Hopf fibration

The fibers h−1(p) for p ∈ S2 are great circles: a Hopf bundle

http://www.geom.uiuc.edu/~banchoff/script/b3d/hypertorus.html

Every great circle belongs to a unique right Hopf bundle. Right Hopf map h: S3 → S2 Isoclinic ≡ belong to the same Hopf bundle This is a transitive relation. If all closest pairs are isoclinic → all great circles in a connected component of the closest-pair graph belong to the same bundle. → h maps them to points on S2. We know how to deal with S2!

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Condensing on the 2-Sphere

Equivariant condensation on the 2-sphere: Input: A ⊆ S2. Output: A′ ⊆ S2, |A′| ≤ min{|A|, 12}.

  • A′ = vertices of a regular icosahedron
  • A′ = vertices of a regular octahedron
  • A′ = vertices of a regular tetrahedron
  • A′ = two antipodal points, or
  • A′ = a single point.
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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Condensing on the 2-Sphere

Equivariant condensation on the 2-sphere: Input: A ⊆ S2. Output: A′ ⊆ S2, |A′| ≤ min{|A|, 12}.

  • A′ = vertices of a regular icosahedron
  • A′ = vertices of a regular octahedron
  • A′ = vertices of a regular tetrahedron
  • A′ = two antipodal points, or
  • A′ = a single point.

Condense each connected component of the closest-pair graph to ≤ 12 great circles. Compute closest-pair graph (on S5) from scratch. If no progress, distance between closest pairs is ≥ Dicosa → ≤ 829 great circles → 2+2 DIMENSION REDUCTION

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Algorithm Overview

Iterative Pruning Generating Orbit Cycles Mirror Case

planes mirror symmetry n = |A| is bounded

2+2 Dimension Reduction 1+3 Dimension Reduction

edge- transitive ≤ 100|P| markers lower-dimensional components

Marking and Condensing Great Circles

|P| ≤ n/200 ≤ n

2

≤ n

2

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SLIDE 50

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Algorithm Overview

Iterative Pruning Generating Orbit Cycles Mirror Case

planes mirror symmetry n = |A| is bounded

2+2 Dimension Reduction 1+3 Dimension Reduction

edge- transitive ≤ 100|P| markers lower-dimensional components

Marking and Condensing Great Circles

|P| ≤ n/200 ≤ n

2

≤ n

2

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

2+2 Dimension Reduction

We have a plane P and we know its image in B. x1 y1 x2 y2 (x1, y1, x2, y2) r2 ϕ2 ϕ1 r1 (0, 0, 0, 0) P P ⊥ “Double-polar” coordinates (r1, ϕ1, r2, ϕ2) We can change ϕ1 and ϕ2. r1 and r2 are fixed.

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

2+2 Dimension Reduction

attach (r1, r2) as a label (color) ϕ2 ϕ1

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

2+2 Dimension Reduction

attach (r1, r2) as a label (color) ϕ2 ϕ1

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

2+2 Dimension Reduction

attach (r1, r2) as a label (color) ϕ2 ϕ1 the picture for set B Are they the same up to translation on the ϕ1, ϕ2-torus?

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

2+2 Dimension Reduction

ϕ2 ϕ1 Prune without losing information: (CANONICAL SET)

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

2+2 Dimension Reduction

ϕ2 ϕ1 Prune without losing information: (CANONICAL SET) Pick a color class

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

2+2 Dimension Reduction

Prune without losing information: (CANONICAL SET) Pick a color class

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

2+2 Dimension Reduction

Prune without losing information: (CANONICAL SET) Pick a color class Compute the Voronoi diagram

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

2+2 Dimension Reduction

Prune without losing information: (CANONICAL SET) Pick a color class Compute the Voronoi diagram Assign other points to cells. Refine the coloring, based on color and relative position of assigned points, shape of Voronoi cell. Repeat. After recoloring, the reduced set has THE SAME translational symmetries as the old set.

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

2+2 Dimension Reduction

Termination: All points have the same color and the same cell shape (a modular lattice) ANY point is as good a representative as any other. CANONICAL SET c(A): move (any) representative point to (ϕ1, ϕ2) = (0, 0), or to (x1, 0, x3, 0). ∃T with TP = P and TA = B ⇐ ⇒ c(A) = c(B)

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Algorithm Overview

Iterative Pruning Generating Orbit Cycles Mirror Case

planes mirror symmetry n = |A| is bounded

2+2 Dimension Reduction 1+3 Dimension Reduction

edge- transitive ≤ 100|P| markers lower-dimensional components

Marking and Condensing Great Circles

|P| ≤ n/200 ≤ n

2

≤ n

2

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SLIDE 62

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Algorithm Overview

Iterative Pruning Generating Orbit Cycles Mirror Case

planes mirror symmetry n = |A| is bounded

2+2 Dimension Reduction 1+3 Dimension Reduction

edge- transitive ≤ 100|P| markers lower-dimensional components

Marking and Condensing Great Circles

|P| ≤ n/200 ≤ n

2

≤ n

2

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SLIDE 63

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

The Mirror Case

α u v t w t′ w′ s(uv) p(uv) Pick some α. s(uv) := {vw : vw ∈ E, ∠uvw = α} τ w′ t′ torsion angle τ can PRUNE arcs from s(u, v) τ0

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

The Mirror Case

Every edge acts like a perfect mirror of the neighborhood. → Every connnected component is the orbit of a point under a group generated by reflections. These groups have been classified. (Coxeter groups)

  • “small” components

→ condensing

  • Cartesian product of 2-dimensional groups (infinite family)

→ 2+2 dimension reduction

  • “large” components (finite family)

→ |A| ≤ n0

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SLIDE 65

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Algorithm Overview

Iterative Pruning Generating Orbit Cycles Mirror Case

planes mirror symmetry n = |A| is bounded

2+2 Dimension Reduction 1+3 Dimension Reduction

edge- transitive ≤ 100|P| markers lower-dimensional components

Marking and Condensing Great Circles

|P| ≤ n/200 ≤ n

2

≤ n

2

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SLIDE 66

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Algorithm Overview

Iterative Pruning Generating Orbit Cycles Mirror Case

planes mirror symmetry n = |A| is bounded

2+2 Dimension Reduction 1+3 Dimension Reduction

edge- transitive ≤ 100|P| markers lower-dimensional components

Marking and Condensing Great Circles

|P| ≤ n/200 ≤ n

2

≤ n

2

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Afterthoughts

  • 5 dimensions and higher
  • terrible constants
  • chimeras
  • tolerances, ≤ ε versus ≥ 10ε
  • depth of construction (→ degree of predicates)
  • Pl¨

ucker space

  • point groups in 4 dimensions
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SLIDE 68

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Symmetry groups

  • COROLLARY. The symmetry group of a finite full-dimensional

point set in 3-space (= a discrete subgroup of O(3)) is

  • the symmetry group of a Platonic solid,
  • the symmetry group of a regular prism,
  • or a subgroup of such a group.

The point groups (discrete subgroups of O(3)) are classified (Hessel’s Theorem).

[ F. Hessel 1830, M. L. Frankenheim 1826 ]

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Point groups in higher dimensions

¿The symmetry group of a finite full-dimensional point set in d-space (= a discrete subgroup of O(d)) is

  • the symmetry group of a regular d-dimensional polytope:

– a regular simplex – ∗ a hypercube (or its dual, the crosspolytope) – a regular n-gon in two dimensions – a dodecahedron (or its dual, the icosahedron) in 3 d. – a 24-cell, or a 120-cell (or its dual, the 600-cell) in 4 d.

  • the symmetry group of the Cartesian product of

lower-dimensional regular polytopes,

  • or a subgroup of such a group?

? Bold and naive CONJECTURE:

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Point groups in higher dimensions

¿The symmetry group of a finite full-dimensional point set in d-space (= a discrete subgroup of O(d)) is

  • the symmetry group of a regular d-dimensional polytope:

– a regular simplex – ∗ a hypercube (or its dual, the crosspolytope) – a regular n-gon in two dimensions – a dodecahedron (or its dual, the icosahedron) in 3 d. – a 24-cell, or a 120-cell (or its dual, the 600-cell) in 4 d.

  • the symmetry group of the Cartesian product of

lower-dimensional regular polytopes,

  • or a subgroup of such a group?

? Bold and naive CONJECTURE: Counterexample (Paco Santos, by divisibility). The symmetry groups of the root systems E6, E7, E8 in 6, 7, 8 dimensions.

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

The four-dimensional point groups

  • [ W. Threlfall and H. Seifert, Math. Annalen, 1931, 1933 ]

enumerated discrete subgroups of SO(4) (determinant +1)

  • [ J. Conway and D. Smith 2003 ]

complete enumeration of point groups 4d-rotation T ↔ pair (R, S) of 3d-rotations. (for example, via quaternions)

  • The groups generated by reflections (Coxeter groups) have

been enumerated up to 8 dimensions.

[ Norman Johnson, unpublished book manuscript ]

Goursat’s Lemma: [ ´

  • E. Goursat 1890 ]

Pairs of 3d point groups + additional information → 4d point groups

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

The four-dimensional point groups

Table 4.1. The chiral groups (groups of

  • rientation-preserving
  • rthogonal transformations)

both m and n must be odd.

[ Conway and Smith 2003 ]

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

The four-dimensional point groups

Table 4.2. The chiral groups (continued)

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

The four-dimensional point groups

Table 4.3. The achiral groups

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

The four-dimensional point groups

Table 4.3. The achiral groups

  • Project: Visualize

these groups: Schlegel diagram of a 4-polytope which has these symmetries.

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Point groups in four dimensions

¿The symmetry group of a finite full-dimensional point set in 4-space (= a discrete subgroup of O(4)) is

  • the symmetry group of a regular d-dimensional polytope:

– a regular simplex – a regular n-gon in two dimensions – a dodecahedron (or its dual, the icosahedron) in 3 d. – a 24-cell, or a 120-cell (or its dual, the 600-cell) in 4 d.

  • the symmetry group of the Cartesian product of

lower-dimensional regular polytopes,

  • or a subgroup of such a group?

? Bold and naive CONJECTURE:

slide-77
SLIDE 77

G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Point groups in four dimensions

¿The symmetry group of a finite full-dimensional point set in 4-space (= a discrete subgroup of O(4)) is

  • the symmetry group of a regular d-dimensional polytope:

– a regular simplex – a regular n-gon in two dimensions – a dodecahedron (or its dual, the icosahedron) in 3 d. – a 24-cell, or a 120-cell (or its dual, the 600-cell) in 4 d.

  • the symmetry group of the Cartesian product of

lower-dimensional regular polytopes,

  • or a subgroup of such a group?

? Bold and naive CONJECTURE: Counterexample: I × Cn (group-theoretic product, but not geometric Cartesian product) Icosahedron on S2 ⇒ 12 great circles with regular n-gons in S3

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Point groups in four dimensions

Counterexample: I × Cn (group-theoretic product, but not geometric Cartesian product) Icosahedron on S2 ⇒ 12 great circles with regular n-gons in S3

http://www.geom.uiuc.edu/~banchoff/script/b3d/hypertorus.html

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G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

Point groups in four dimensions

Counterexample: I × Cn (group-theoretic product, but not geometric Cartesian product) Icosahedron on S2 ⇒ 12 great circles with regular n-gons in S3

http://www.geom.uiuc.edu/~banchoff/script/b3d/hypertorus.html