The icosahedra of edge length 1 Daniel Robertz (j.w. K.-H. - - PowerPoint PPT Presentation

The icosahedra of edge length 1

Daniel Robertz (j.w. K.-H. Brakhage, A. Niemeyer, W. Plesken, A. Strzelczyk)

Centre for Mathematical Sciences University of Plymouth

Lancaster, 13/06/2019

Simplicial surfaces

• j. w. K.-H. Brakhage, A. Niemeyer, W. Plesken, A. Strzelczyk et al.

build surfaces from triangles belonging to very few congruence classes simplicial surfaces as combinatorial objects simplicial surfaces as Euclidean two-dim. (compact) manifolds with singularities embeddings of abstract simplicial surfaces into Euclidean 3-space

K.-H. Brakhage, A. Niemeyer, W. Plesken, A. Strzelczyk, Simplicial surfaces controlled by one triangle, 17th Int. Conference on Geometry and Graphics, 4–8 Aug. 2016, Beijing

Lancaster, 13/06/2019

Icosahedra of edge length 1

Classify embeddings of icosahedron in R3 with 12 distinct vertices admitting non-trivial symmetry

Lancaster, 13/06/2019

Icosahedra of edge length 1

Classify embeddings of icosahedron in R3 with 12 distinct vertices admitting non-trivial symmetry drop convexity, allow self-intersection of faces

Lancaster, 13/06/2019

Icosahedra of edge length 1

Classify embeddings of icosahedron in R3 with 12 distinct vertices admitting non-trivial symmetry drop convexity, allow self-intersection of faces equivalence of icosahedra: rigid transformations

Lancaster, 13/06/2019

Icosahedra of edge length 1

Classify embeddings of icosahedron in R3 with 12 distinct vertices admitting non-trivial symmetry drop convexity, allow self-intersection of faces equivalence of icosahedra: rigid transformations

• 35 inequivalent rigid icosahedra, 1 curve of flexible icosahedra

K.-H. B., A. C. N., W. P., D. R., A. S., The icosahedra of edge length 1,

• J. Algebra, in press

web page: http://algebra.data.rwth-aachen.de/Icosahedra/visualplusdata.html

Lancaster, 13/06/2019

Software

We used:

• Magma
• Maple: Involutive
• C++/Python: GINV
• Bertini

Lancaster, 13/06/2019

Software

We used:

• Magma
• Maple: Involutive
• C++/Python: GINV
• Bertini

Also under development:

• simplicial-surfaces in GAP

(M. Baumeister, A. Niemeyer)

Lancaster, 13/06/2019

Icosahedron

Combinatorial automorphism group A ∼ = C2 × A5 generated by a := (1, 2)(3, 4)(5, 7)(6, 8)(9, 11)(10, 12), b := (1, 10)(3, 9)(2, 12)(4, 11)(5, 6)(7, 8), c := (1, 7)(2, 3)(4, 11)(5, 12)(6, 8)(9, 10), d := (1, 12)(3, 9)(2, 10)(4, 11)(5, 7)(6, 8)

Lancaster, 13/06/2019

Icosahedron

Combinatorial automorphism group A ∼ = C2 × A5 generated by a := (1, 2)(3, 4)(5, 7)(6, 8)(9, 11)(10, 12), b := (1, 10)(3, 9)(2, 12)(4, 11)(5, 6)(7, 8), c := (1, 7)(2, 3)(4, 11)(5, 12)(6, 8)(9, 10), d := (1, 12)(3, 9)(2, 10)(4, 11)(5, 7)(6, 8) d generates centre of A, interchanges combinatorially opposite vertices 20 faces:

• rbit of {1, 2, 3},

30 edges:

• rbit of {1, 2},

30 diagonals of combinatorial distance 2:

• rbit of {3, 4},

6 diagonals of combinatorial distance 3:

• rbit of {1, 12}

Lancaster, 13/06/2019

Icosahedron

Combinatorial automorphism group A ∼ = C2 × A5 generated by a := (1, 2)(3, 4)(5, 7)(6, 8)(9, 11)(10, 12), b := (1, 10)(3, 9)(2, 12)(4, 11)(5, 6)(7, 8), c := (1, 7)(2, 3)(4, 11)(5, 12)(6, 8)(9, 10), d := (1, 12)(3, 9)(2, 10)(4, 11)(5, 7)(6, 8) d generates centre of A, interchanges combinatorially opposite vertices 20 faces:

• rbit of {1, 2, 3},

30 edges:

• rbit of {1, 2},

30 diagonals of combinatorial distance 2:

• rbit of {3, 4},

6 diagonals of combinatorial distance 3:

• rbit of {1, 12}

π : A → GL(12, R) natural representation of A by permutation matrices

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Theorem

The subgroups U of A with more than one element that arise as symmetry group of an icosahedron fall into 11 conjugacy classes: Automorphism group Number of U ≤ A = C2 × A5 icosahedra C2 × A5 2 C2 × D10 4 C2 × D6 2 D10 (≤ A5) 3 D6 (≤ A5) 2 C2

2

(∋ d) 1 C2

2

(∋ d, ≤ A5) 5 C2

2

(≤ A5) 1 C2 (≤ A5) 5 C2 (∋ d, ≤ A5) 10 C2 (= d) ∞

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S :=StabA Syl2(S) dG r1,G rG rf,G Trace relation C22 a, d •−

+

8 4 2 1 λ4 − 76

3 λ3 + 238λ2 − 4964 5

λ + 23767

15

C2 × A5 a, b, d 2 2 2 2 λ2 − 15λ + 45 C2 × D10 a, d −•

+

2 2 2 2 λ2 − 15λ + 269

5

C22 a, bd −+

+

2 2 2 2 λ2 − 71

5 λ + 10561 225

C2 × D10 a, d −+

+

4 2 2 2 λ4 − 18λ3 + 583

5 λ2 − 1658 5

λ + 9101

25

C2 × D6 a, d −+

+

4 4 2 2 λ4 − 26λ3 + 243λ2 − 970λ + 1397 C22 a, bd −+

+

24 10 6 3 λ12 − 5179·22

32·52 λ11 ± · · ·

C22 a, b −−

−

30 18 6 1 λ5 − 117

2 λ4 ± · · ·

C2 a − 172 48 20 5 λ43 − 73·7·11·461687

22·33·52·29·79 λ42 ± · · ·

D10 ad + 2 2 2 2 λ2 − 44

3 λ + 2131 45

D6 ad + 2 2 2 2 λ2 − 68

5 λ + 1111 25

C2 ad + 36 12 8 4 λ18 − 1106

9

λ17 ± · · · C2 ad + 168 40 24 6 λ42 − 2·719·1223

33·5·43

λ41 ± · · · D10 ad + 4 2 2 1 λ2 − 26

3 λ + 149 9 Lancaster, 13/06/2019

Classification

Choose origin as centre of mass of the 12 (equilibrated) vertices M ∈ R3×12 coordinate matrix

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Classification

Choose origin as centre of mass of the 12 (equilibrated) vertices M ∈ R3×12 coordinate matrix G := M tr M ∈ R12×12 Gram matrix

Lancaster, 13/06/2019

Classification

Choose origin as centre of mass of the 12 (equilibrated) vertices M ∈ R3×12 coordinate matrix G := M tr M ∈ R12×12 Gram matrix Gram matrices: equivalence = conjugacy by permutation matrices (g, G) − → π(g)tr G π(g) = (Gig,jg)1≤i,j≤12

Lancaster, 13/06/2019

Classification

Choose origin as centre of mass of the 12 (equilibrated) vertices M ∈ R3×12 coordinate matrix G := M tr M ∈ R12×12 Gram matrix Gram matrices: equivalence = conjugacy by permutation matrices (g, G) − → π(g)tr G π(g) = (Gig,jg)1≤i,j≤12

• Lemma. Gram matrix G with automorphism group U ≤ A. There exists

a faithful orthogonal repres. δ : U → O3(R) and M ∈ R3×12 such that δ(g) M = M π(g) for all g ∈ U, G := M tr M .

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Classification

Minimal subgroups U of A ∼ = C2 × A5 up to conjugacy: abc ∼ = C3, ac ∼ = C5, a ∼ = C2, d ∼ = C2, ad ∼ = C2

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Classification

Minimal subgroups U of A ∼ = C2 × A5 up to conjugacy: abc ∼ = C3, ac ∼ = C5, a ∼ = C2, d ∼ = C2, ad ∼ = C2

• Lemma. If a Gram matrix is fixed by an element of order 3 or 5, then its

automorphism group also contains an element of order 2.

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Classification

Minimal subgroups U of A ∼ = C2 × A5 up to conjugacy: abc ∼ = C3, ac ∼ = C5, a ∼ = C2, d ∼ = C2, ad ∼ = C2

• Lemma. If a Gram matrix is fixed by an element of order 3 or 5, then its

automorphism group also contains an element of order 2. Faithful orthogonal representations of degree 3 of C2: generator maps to   1 −1 −1   ,   −1 1 1   ,   −1 −1 −1   9 cases to consider

Lancaster, 13/06/2019

Classification

Minimal subgroups U of A ∼ = C2 × A5 up to conjugacy: abc ∼ = C3, ac ∼ = C5, a ∼ = C2, d ∼ = C2, ad ∼ = C2

• Lemma. If a Gram matrix is fixed by an element of order 3 or 5, then its

automorphism group also contains an element of order 2. Faithful orthogonal representations of degree 3 of C2: generator maps to   1 −1 −1   ,   −1 1 1   ,   −1 −1 −1   9 cases to consider Determine the U-invariant Gram matrices!

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Classification

Let e1, e2, . . . , e12 be the standard basis of R12×1, R := Q[y1, . . . , yn].

• Def. ideal

I

• gen. by

(ei − ej)tr G (ei − ej) − 1, {i, j} ∈ {1, 2}A, and 4 × 4 minors of G, where yi are the entries of G corresp. to U-orbits

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Classification

Let e1, e2, . . . , e12 be the standard basis of R12×1, R := Q[y1, . . . , yn].

• Def. ideal

I

• gen. by

(ei − ej)tr G (ei − ej) − 1, {i, j} ∈ {1, 2}A, and 4 × 4 minors of G, where yi are the entries of G corresp. to U-orbits

• Def. A maximal ideal m R associated to I is relevant if

(a) rank (Gi,j + m) ∈ (R/m)12×12 at most 3, (b) U = { g ∈ A | π(g)tr (Gi,j + m) π(g) = (Gi,j + m) }, (c) ∃ ι : R/m → R such that (ι(Gi,j + m)) is positive semidefinite.

Lancaster, 13/06/2019

Classification

Let e1, e2, . . . , e12 be the standard basis of R12×1, R := Q[y1, . . . , yn].

• Def. ideal

I

• gen. by

(ei − ej)tr G (ei − ej) − 1, {i, j} ∈ {1, 2}A, and 4 × 4 minors of G, where yi are the entries of G corresp. to U-orbits

• Def. A maximal ideal m R associated to I is relevant if

(a) rank (Gi,j + m) ∈ (R/m)12×12 at most 3, (b) U = { g ∈ A | π(g)tr (Gi,j + m) π(g) = (Gi,j + m) }, (c) ∃ ι : R/m → R such that (ι(Gi,j + m)) is positive semidefinite.

dG(m) = [R/m : Q] r1,G(m) = # real embeddings of R/m rG(m) = # relevant real embeddings of R/m rf,G(m) = # relevant real embeddings of R/m with (ι(Gi,j + m)) pw. inequivalent

Lancaster, 13/06/2019

S :=StabA Syl2(S) dG r1,G rG rf,G Trace relation C22 a, d •−

+

8 4 2 1 λ4 − 76

3 λ3 + 238λ2 − 4964 5

λ + 23767

15

C2 × A5 a, b, d 2 2 2 2 λ2 − 15λ + 45 C2 × D10 a, d −•

+

2 2 2 2 λ2 − 15λ + 269

5

C22 a, bd −+

+

2 2 2 2 λ2 − 71

5 λ + 10561 225

C2 × D10 a, d −+

+

4 2 2 2 λ4 − 18λ3 + 583

5 λ2 − 1658 5

λ + 9101

25

C2 × D6 a, d −+

+

4 4 2 2 λ4 − 26λ3 + 243λ2 − 970λ + 1397 C22 a, bd −+

+

24 10 6 3 λ12 − 5179·22

32·52 λ11 ± · · ·

C22 a, b −−

−

30 18 6 1 λ5 − 117

2 λ4 ± · · ·

C2 a − 172 48 20 5 λ43 − 73·7·11·461687

22·33·52·29·79 λ42 ± · · ·

D10 ad + 2 2 2 2 λ2 − 44

3 λ + 2131 45

D6 ad + 2 2 2 2 λ2 − 68

5 λ + 1111 25

C2 ad + 36 12 8 4 λ18 − 1106

9

λ17 ± · · · C2 ad + 168 40 24 6 λ42 − 2·719·1223

33·5·43

λ41 ± · · · D10 ad + 4 2 2 1 λ2 − 26

3 λ + 149 9 Lancaster, 13/06/2019

Example: case δ(a) = diag(1, −1, −1)

coordinate matrix M

• x1

x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 z1 z2 z3 z4 z5 z6 z7 z8 z9 z10 z11 z12

• Lancaster, 13/06/2019

Example: case δ(a) = diag(1, −1, −1)

coordinate matrix M

• x1

x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 z1 z2 z3 z4 z5 z6 z7 z8 z9 z10 z11 z12

• recall

δ(a) M = M π(a)

• x1

x1 x2 x2 x3 x4 x3 x4 x5 x6 x5 x6 y1 −y1 y2 −y2 y3 y4 −y3 −y4 y5 y6 −y5 −y6 z1 −z1 z2 −z2 z3 z4 −z3 −z4 z5 z6 −z5 −z6

• Lancaster, 13/06/2019

Example: case δ(a) = diag(1, −1, −1)

coordinate matrix M

• x1

x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 z1 z2 z3 z4 z5 z6 z7 z8 z9 z10 z11 z12

• recall

δ(a) M = M π(a)

• x1

x1 x2 x2 x3 x4 x3 x4 x5 x6 x5 x6 y1 −y1 y2 −y2 y3 y4 −y3 −y4 y5 y6 −y5 −y6 z1 −z1 z2 −z2 z3 z4 −z3 −z4 z5 z6 −z5 −z6

• 1st and 2nd column not equal,

w.l.o.g. z1 = 0, then 4 y2

1 − 1 = 0

• x1

x1 x2 x2 x3 x4 x3 x4 x5 x6 x5 x6

1 2

− 1

2

y2 −y2 y3 y4 −y3 −y4 y5 y6 −y5 −y6 z2 −z2 z3 z4 −z3 −z4 z5 z6 −z5 −z6

• Lancaster, 13/06/2019

Example: case δ(a) = diag(1, −1, −1)

{3, 4} is “orthogonal diagonal” of {1, 2}

• x1

x1 x2 x2 x3 x4 x3 x4 x5 x6 x5 x6

1 2

− 1

2

y3 y4 −y3 −y4 y5 y6 −y5 −y6 z2 −z2 z3 z4 −z3 −z4 z5 z6 −z5 −z6

• Lancaster, 13/06/2019

Example: case δ(a) = diag(1, −1, −1)

{3, 4} is “orthogonal diagonal” of {1, 2}

• x1

x1 x2 x2 x3 x4 x3 x4 x5 x6 x5 x6

1 2

− 1

2

y3 y4 −y3 −y4 y5 y6 −y5 −y6 z2 −z2 z3 z4 −z3 −z4 z5 z6 −z5 −z6

• edge {10, 12},

4y2

6 + 4z2 6 − 1 = 0

x1

x1 x2 x2 x3 x4 x3 x4 x5 x6 x5 x6

1 2

− 1

2

y3 y4 −y3 −y4 −ζ s

1 2 c

ζ s − 1

2 c

z2 −z2 z3 z4 −z3 −z4 ζ c

1 2 s

−ζ c − 1

2 s

• Lancaster, 13/06/2019

Example: case δ(a) = diag(1, −1, −1)

{3, 4} is “orthogonal diagonal” of {1, 2}

• x1

x1 x2 x2 x3 x4 x3 x4 x5 x6 x5 x6

1 2

− 1

2

y3 y4 −y3 −y4 y5 y6 −y5 −y6 z2 −z2 z3 z4 −z3 −z4 z5 z6 −z5 −z6

• edge {10, 12},

4y2

6 + 4z2 6 − 1 = 0

x1

x1 x2 x2 x3 x4 x3 x4 x5 x6 x5 x6

1 2

− 1

2

y3 y4 −y3 −y4 −ζ s

1 2 c

ζ s − 1

2 c

z2 −z2 z3 z4 −z3 −z4 ζ c

1 2 s

−ζ c − 1

2 s

• centre of mass condition
• x1

x1 x2 x2 x3 x4 x3 x4 x5 − 5

i=1 xi

x5 − 5

i=1 xi 1 2

− 1

2

y3 y4 −y3 −y4 −ζ s

1 2 c

ζ s − 1

2 c

z2 −z2 z3 z4 −z3 −z4 ζ c

1 2 s

−ζ c − 1

2 s

• 13 unknowns,

c2 + s2 − 1 = 0

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Example: case δ(a) = diag(1, −1, −1)

Primary decomposition (Magma): I =

48

• i=1

mi ∩

58

• i=49

mi ∩

62

• i=59

mi ∩

66

• i=63

mi ∩

68

• i=67

mi ∩

70

• i=69

mi ∩ m71 ∩ m72

deg 2 deg 4 deg 6 deg 8 deg 16 deg 48 deg 120 deg 688

Lancaster, 13/06/2019

Example: case δ(a) = diag(1, −1, −1)

Primary decomposition (Magma): I =

48

• i=1

mi ∩

58

• i=49

mi ∩

62

• i=59

mi ∩

66

• i=63

mi ∩

68

• i=67

mi ∩

70

• i=69

mi ∩ m71 ∩ m72

deg 2 deg 4 deg 6 deg 8 deg 16 deg 48 deg 120 deg 688

deg 2: 20 maximal ideals with 12 distinct vertices 5 non-equivalent Gram matrices with automorphism group C2 × A5, C2 × D10, C2 × D10, C2

2, C2 2

deg 4: 8 maximal ideals with 12 distinct vertices no positive semidefinite Gram matrix deg 6: R/m has no real embeddings

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Example: case δ(a) = diag(1, −1, −1)

deg 8: 4 maximal ideals with 12 distinct vertices 2 non-equivalent Gram matrices with automorphism group C2 × D10 deg 16: 2 maximal ideals with 12 distinct vertices 2 of the 4 real embeddings yield positive semidefinite Gram matrices 2 non-equivalent Gram matrices with automorphism group C2 × D6 deg 48: 2 maximal ideals with 12 distinct vertices 6 of the 10 real embeddings yield positive semidefinite Gram matrices 3 non-equivalent Gram matrices with automorphism group C2

2

• deg. 120: 6 of the 18 real embeddings yield pos. semidef. Gram matrices

1 Gram matrix with automorphism group C2

2

• deg. 688: 20 of the 48 real embeddings yield pos. semidef. Gram matrices

5 non-equivalent Gram matrices with automorphism group C2

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Classification

• Proposition. There exist ε > 0 and a non-constant real analytic map

Φ : [0, ε) → R3×12 such that Φ(t) is the coordinate matrix of a d-invariant icosahedron for all but finitely many t ∈ [0, ε). There exist infinitely many isometry types of d-invariant icosahedra.

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Classification

• Proposition. There exist ε > 0 and a non-constant real analytic map

Φ : [0, ε) → R3×12 such that Φ(t) is the coordinate matrix of a d-invariant icosahedron for all but finitely many t ∈ [0, ε). There exist infinitely many isometry types of d-invariant icosahedra. Proof. p1 = 0, p2 = 0, . . . , p15 = 0 (lin. indep.) quadratic equations for entries y1, y2, . . . , y16 of coord. matrix M corresp. to d-symmetry τ(y1, . . . , y16) :=

• det Dp|1, . . . , (−1)i−1 det Dp|i, . . . , − det Dp|16
• Solution to initial value problem

φ′(t) = τ(φ1(t), . . . , φ16(t)), φ(0) = y0. for τ(y0) = 0 gives curve of (possibly degenerate) icosahedra. Rule out degeneracies. . .

• Lancaster, 13/06/2019

References

• D. J. Bates, J. D. Hauenstein, A. J. Sommese, C. W. Wampler.

Bertini: Software for Numerical Algebraic Geometry.

• Y. A. Blinkov, C. F. Cid, V. P. Gerdt, W. Plesken, D. Robertz. The

MAPLE Package “Janet”: I. Polynomial Systems. In Proc. of Computer Algebra in Scientific Computing CASC 2003, 31–40. Garching, Germany,

• 2003. http://algebra.data.rwth-aachen.de/software/Janet
• Y. A. Blinkov, V. P. Gerdt. The specialised computer algebra system
• GINV. Programmirovanie 34 (2), 67–80, 2008. http://invo.jinr.ru
• W. Bosma, J. Cannon, C. Playoust. The Magma algebra system. I. The

user language, J. Symbolic Comput. 24 (3-4), 235–265, 1997.

• B. Schulze. Symmetry as a sufficient condition for a finite flex. SIAM J.

Discrete Math. 24 (4), 1291–1312, 2010.

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References

K.-H. Brakhage, A. Niemeyer, W. Plesken, A. Strzelczyk. Simplicial surfaces controlled by one triangle, 17th Int. Conference on Geometry and Graphics, 4–8 Aug. 2016, Beijing.

• A. Strzelczyk. Simpliziale Fl¨

achen aus kongruenten Dreiecken: kombinatorische Grundlagen und geometrische Beispiele, PhD thesis, RWTH Aachen University, 2019. http://publications.rwth-aachen.de/record/756346

• R. Sauer, Differenzengeometrie, Springer, 1970.

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Linear algebra

• Lemma. Let G ∈ Rn×n be symmetric, positive semidefinite, of rank k.

(a) There exists M ∈ Rk×n such that M tr M = G. (b) G and M have the same row space. (c) If L ∈ Rk×n also satisfies Ltr L = G, then ∃! g ∈ Rk×k such that L = g M. Moreover, g is orthogonal.

Proof.

(a) (E1, . . . , En) orthogonal basis of eigenrows of G with eigenvalues λ1 > 0, . . . , λk > 0, λk+1 = . . . = λn = 0. Rows

1 √λi Ei define M

(because Etr

i Ei repres. orthogonal projection of R1×n onto Ei).

(b) Obvious. (c) Clear from (b).

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