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 R 3 with 12 distinct vertices admitting non-trivial symmetry Lancaster, 13/06/2019
Icosahedra of edge length 1 Classify embeddings of icosahedron in R 3 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 R 3 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 R 3 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 ∼ = C 2 × A 5 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 ∼ = C 2 × A 5 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: orbit of { 1 , 2 , 3 } , 30 edges: orbit of { 1 , 2 } , 30 diagonals of combinatorial distance 2: orbit of { 3 , 4 } , 6 diagonals of combinatorial distance 3: orbit of { 1 , 12 } Lancaster, 13/06/2019
Icosahedron Combinatorial automorphism group A ∼ = C 2 × A 5 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: orbit of { 1 , 2 , 3 } , 30 edges: orbit of { 1 , 2 } , 30 diagonals of combinatorial distance 2: orbit of { 3 , 4 } , 6 diagonals of combinatorial distance 3: orbit of { 1 , 12 } π : A → GL(12 , R ) natural representation of A by permutation matrices Lancaster, 13/06/2019
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 = C 2 × A 5 icosahedra C 2 × A 5 2 C 2 × D 10 4 C 2 × D 6 2 D 10 ( �≤ A 5 ) 3 D 6 ( �≤ A 5 ) 2 2 C 2 ( ∋ d ) 1 2 C 2 ( �∋ d, �≤ A 5 ) 5 2 C 2 ( ≤ A 5 ) 1 C 2 ( ≤ A 5 ) 5 C 2 ( �∋ d, �≤ A 5 ) 10 C 2 (= � d � ) ∞ Lancaster, 13/06/2019
S :=Stab A Syl 2 ( S ) d G r 1 ,G r G r f,G Trace relation λ 4 − 76 3 λ 3 + 238 λ 2 − 4964 C 22 λ + 23767 � a, d � •− 8 4 2 1 + 5 15 λ 2 − 15 λ + 45 C 2 × A 5 � a, b, d � 2 2 2 2 λ 2 − 15 λ + 269 C 2 × D 10 � a, d � −• 2 2 2 2 + 5 λ 2 − 71 C 22 � a, bd � − + 5 λ + 10561 2 2 2 2 + 225 λ 4 − 18 λ 3 + 583 5 λ 2 − 1658 � a, d � − + λ + 9101 C 2 × D 10 4 2 2 2 + 5 25 λ 4 − 26 λ 3 + 243 λ 2 − 970 λ + 1397 � a, d � − + C 2 × D 6 4 4 2 2 + λ 12 − 5179 · 2 2 3 2 · 5 2 λ 11 ± · · · C 22 � a, bd � − + 24 10 6 3 + λ 5 − 117 2 λ 4 ± · · · C 22 � a, b � −− 30 18 6 1 − λ 43 − 73 · 7 · 11 · 461687 2 2 · 3 3 · 5 2 · 29 · 79 λ 42 ± · · · C 2 � a � − 172 48 20 5 λ 2 − 44 3 λ + 2131 D 10 � ad � + 2 2 2 2 45 λ 2 − 68 5 λ + 1111 D 6 � ad � + 2 2 2 2 25 λ 18 − 1106 λ 17 ± · · · C 2 � ad � + 36 12 8 4 9 λ 42 − 2 · 719 · 1223 λ 41 ± · · · C 2 � ad � + 168 40 24 6 3 3 · 5 · 43 λ 2 − 26 3 λ + 149 D 10 � ad � + 4 2 2 1 9 Lancaster, 13/06/2019
Classification Choose origin as centre of mass of the 12 (equilibrated) vertices M ∈ R 3 × 12 coordinate matrix Lancaster, 13/06/2019
Classification Choose origin as centre of mass of the 12 (equilibrated) vertices M ∈ R 3 × 12 coordinate matrix G := M tr M ∈ R 12 × 12 Gram matrix Lancaster, 13/06/2019
Classification Choose origin as centre of mass of the 12 (equilibrated) vertices M ∈ R 3 × 12 coordinate matrix G := M tr M ∈ R 12 × 12 Gram matrix Gram matrices: equivalence = conjugacy by permutation matrices → π ( g ) tr G π ( g ) = ( G ig,jg ) 1 ≤ i,j ≤ 12 ( g, G ) �− Lancaster, 13/06/2019
Classification Choose origin as centre of mass of the 12 (equilibrated) vertices M ∈ R 3 × 12 coordinate matrix G := M tr M ∈ R 12 × 12 Gram matrix Gram matrices: equivalence = conjugacy by permutation matrices → π ( g ) tr G π ( g ) = ( G ig,jg ) 1 ≤ i,j ≤ 12 ( g, G ) �− Lemma. Gram matrix G with automorphism group U ≤ A . There exists a faithful orthogonal repres. δ : U → O 3 ( R ) and M ∈ R 3 × 12 such that G := M tr M . δ ( g ) M = M π ( g ) for all g ∈ U, Lancaster, 13/06/2019
Classification Minimal subgroups U of A ∼ = C 2 × A 5 up to conjugacy: � abc � ∼ � ac � ∼ � a � ∼ � d � ∼ � ad � ∼ = C 3 , = C 5 , = C 2 , = C 2 , = C 2 Lancaster, 13/06/2019
Classification Minimal subgroups U of A ∼ = C 2 × A 5 up to conjugacy: � abc � ∼ � ac � ∼ � a � ∼ � d � ∼ � ad � ∼ = C 3 , = C 5 , = C 2 , = C 2 , = C 2 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 . Lancaster, 13/06/2019
Classification Minimal subgroups U of A ∼ = C 2 × A 5 up to conjugacy: � abc � ∼ � ac � ∼ � a � ∼ � d � ∼ � ad � ∼ = C 3 , = C 5 , = C 2 , = C 2 , = C 2 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 C 2 : generator maps to 1 0 0 − 1 0 0 − 1 0 0 , , 0 − 1 0 0 1 0 0 − 1 0 0 0 − 1 0 0 1 0 0 − 1 9 cases to consider Lancaster, 13/06/2019
Classification Minimal subgroups U of A ∼ = C 2 × A 5 up to conjugacy: � abc � ∼ � ac � ∼ � a � ∼ � d � ∼ � ad � ∼ = C 3 , = C 5 , = C 2 , = C 2 , = C 2 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 C 2 : generator maps to 1 0 0 − 1 0 0 − 1 0 0 , , 0 − 1 0 0 1 0 0 − 1 0 0 0 − 1 0 0 1 0 0 − 1 9 cases to consider Determine the U -invariant Gram matrices! Lancaster, 13/06/2019
Classification Let e 1 , e 2 , . . . , e 12 be the standard basis of R 12 × 1 , R := Q [ y 1 , . . . , y n ] . ( e i − e j ) tr G ( e i − e j ) − 1 , { i, j } ∈ { 1 , 2 } A , Def. ideal I gen. by and 4 × 4 minors of G , where y i are the entries of G corresp. to U -orbits Lancaster, 13/06/2019
Classification Let e 1 , e 2 , . . . , e 12 be the standard basis of R 12 × 1 , R := Q [ y 1 , . . . , y n ] . ( e i − e j ) tr G ( e i − e j ) − 1 , { i, j } ∈ { 1 , 2 } A , Def. ideal I gen. by and 4 × 4 minors of G , where y i are the entries of G corresp. to U -orbits Def. A maximal ideal m � R associated to I is relevant if (a) rank ( G i,j + m ) ∈ ( R/m ) 12 × 12 at most 3 , (b) U = { g ∈ A | π ( g ) tr ( G i,j + m ) π ( g ) = ( G i,j + m ) } , (c) ∃ ι : R/m → R such that ( ι ( G i,j + m )) is positive semidefinite. Lancaster, 13/06/2019
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