Head to tail compositions Oleg Viro November 27, 2014 1 / 23 - PowerPoint PPT Presentation

Relations Theorem. Any relation among reflections in lines follow from relations l = 1 and R l ◦ R m = R l ′ ◦ R m ′ , where l, m, l ′ , m ′ are as above. R 2 Lemma. A composition of any 4 reflections in lines can be transformed by these relations to a composition of 2 reflections in lines. Let 1 � 2 and 3 � 4 . 2 3 4 1 5 / 23

Relations Theorem. Any relation among reflections in lines follow from relations l = 1 and R l ◦ R m = R l ′ ◦ R m ′ , where l, m, l ′ , m ′ are as above. R 2 Lemma. A composition of any 4 reflections in lines can be transformed by these relations to a composition of 2 reflections in lines. Let 1 � 2 and 3 � 4 . 3 2 1 4 5 / 23

Relations Theorem. Any relation among reflections in lines follow from relations l = 1 and R l ◦ R m = R l ′ ◦ R m ′ , where l, m, l ′ , m ′ are as above. R 2 Lemma. A composition of any 4 reflections in lines can be transformed by these relations to a composition of 2 reflections in lines. Let 1 � 2 and 3 � 4 . 3= 2 1 4 5 / 23

Relations Theorem. Any relation among reflections in lines follow from relations l = 1 and R l ◦ R m = R l ′ ◦ R m ′ , where l, m, l ′ , m ′ are as above. R 2 Lemma. A composition of any 4 reflections in lines can be transformed by these relations to a composition of 2 reflections in lines. Let 1 � 2 and 3 � 4 . 1 4 5 / 23

Relations Theorem. Any relation among reflections in lines follow from relations l = 1 and R l ◦ R m = R l ′ ◦ R m ′ , where l, m, l ′ , m ′ are as above. R 2 Lemma. A composition of any 4 reflections in lines can be transformed by these relations to a composition of 2 reflections in lines. � 5 / 23

Relations Theorem. Any relation among reflections in lines follow from relations l = 1 and R l ◦ R m = R l ′ ◦ R m ′ , where l, m, l ′ , m ′ are as above. R 2 Lemma. A composition of any 4 reflections in lines can be transformed by these relations to a composition of 2 reflections in lines. � Proof of Theorem. By Lemma, any relation can be reduced to a relation of length ≤ 3 . 5 / 23

Relations Theorem. Any relation among reflections in lines follow from relations l = 1 and R l ◦ R m = R l ′ ◦ R m ′ , where l, m, l ′ , m ′ are as above. R 2 Lemma. A composition of any 4 reflections in lines can be transformed by these relations to a composition of 2 reflections in lines. � Proof of Theorem. By Lemma, any relation can be reduced to a relation of length ≤ 3 . A composition of odd number of reflections reverses orientation and cannot be id . 5 / 23

Relations Theorem. Any relation among reflections in lines follow from relations l = 1 and R l ◦ R m = R l ′ ◦ R m ′ , where l, m, l ′ , m ′ are as above. R 2 � Lemma. A composition of any 4 reflections in lines can be transformed by these relations to a composition of 2 reflections in lines. � Proof of Theorem. By Lemma, any relation can be reduced to a relation of length ≤ 3 . A composition of odd number of reflections reverses orientation and cannot be id . � A composition of two different reflections is not identity. 5 / 23

Relations Theorem. Any relation among reflections in lines follow from relations l = 1 and R l ◦ R m = R l ′ ◦ R m ′ , where l, m, l ′ , m ′ are as above. R 2 � Lemma. A composition of any 4 reflections in lines can be transformed by these relations to a composition of 2 reflections in lines. � Generalization of Lemma. In R n , a composition of any n + 2 reflections in hyperplanes is a composition of n reflections in hyperplanes. 5 / 23

Relations Theorem. Any relation among reflections in lines follow from relations l = 1 and R l ◦ R m = R l ′ ◦ R m ′ , where l, m, l ′ , m ′ are as above. R 2 � Lemma. A composition of any 4 reflections in lines can be transformed by these relations to a composition of 2 reflections in lines. � Generalization of Lemma. In R n , a composition of any n + 2 reflections in hyperplanes is a composition of n reflections in hyperplanes. Generalization of Theorem. Any relation among reflections in hyperplanes of R n follow from relations R 2 l = 1 and R l ◦ R m = R l ′ ◦ R m ′ . 5 / 23

Flips and flippers Generalize reflections! 6 / 23

Flips and flippers Generalize reflections! A flip is an isometry which is an involution (i.e., has period 2) and is determined by its fixed point set. 6 / 23

Flips and flippers Generalize reflections! A flip is an isometry which is an involution (i.e., has period 2) and is determined by its fixed point set. Flipper is the fixed point set of a flip. 6 / 23

Flips and flippers Generalize reflections! A flip is an isometry which is an involution (i.e., has period 2) and is determined by its fixed point set. Flipper is the fixed point set of a flip. Key example: R → R : x �→ 2 a − x , the reflection of R in a point a . 6 / 23

Flips and flippers Generalize reflections! A flip is an isometry which is an involution (i.e., has period 2) and is determined by its fixed point set. Flipper is the fixed point set of a flip. Key example: R → R : x �→ 2 a − x , the reflection of R in a point a . a �→ 2 a − a = a 6 / 23

Flips and flippers Generalize reflections! A flip is an isometry which is an involution (i.e., has period 2) and is determined by its fixed point set. Flipper is the fixed point set of a flip. Key example: R → R : x �→ 2 a − x , the reflection of R in a point a . Generalization: a symmetry of R n in a k -subspace. 6 / 23

Flips and flippers Generalize reflections! A flip is an isometry which is an involution (i.e., has period 2) and is determined by its fixed point set. Flipper is the fixed point set of a flip. Key example: R → R : x �→ 2 a − x , the reflection of R in a point a . Generalization: a symmetry of R n in a k -subspace. Further examples in hyperbolic spaces, spheres, projective spaces and other symmetric spaces. 6 / 23

Flips and flippers Generalize reflections! A flip is an isometry which is an involution (i.e., has period 2) and is determined by its fixed point set. Flipper is the fixed point set of a flip. Key example: R → R : x �→ 2 a − x , the reflection of R in a point a . Generalization: a symmetry of R n in a k -subspace. Further examples in hyperbolic spaces, spheres, projective spaces and other symmetric spaces. Correspondence Flipper S ← → Flip in S is the shortest connection between simple static geometric objects - flippers - and isometries. 6 / 23

Symmetry about a point is a flip. 7 / 23

Symmetry about a point is a flip. Composition of flips in points A B 7 / 23

Symmetry about a point is a flip. Composition of flips in points A B 7 / 23

Symmetry about a point is a flip. Composition of flips in points A B 7 / 23

Symmetry about a point is a flip. Composition of flips in points A B 7 / 23

Symmetry about a point is a flip. Composition of flips in points is a translation: R B ( R A ( X )) X A B R A ( X ) 7 / 23

Symmetry about a point is a flip. Composition of flips in points is a translation: R B ( R A ( X )) X A B R A ( X ) − − − − − − − − − → − → AB = 1 X R B ( R A ( X ) 2 7 / 23

Symmetry about a point is a flip. Composition of flips in points is a translation: R B ( R A ( X )) X A B R A ( X ) − − − − − − − − − → − → AB = 1 X R B ( R A ( X ) 2 − → AB is half the arrow representing R B ◦ R A . 7 / 23

Head to tail Compare the head to tail addition − AB + − → BC = − − → → AC to ( R C ◦ R B ) ◦ ( R B ◦ R A ) = R C ◦ R 2 B ◦ R A = R C ◦ R A . 8 / 23

Flip-flop decomposition Which isometries are compositions of two flips? 9 / 23

Flip-flop decomposition Which isometries are compositions of two flips? Any isometry of R n . 9 / 23

Flip-flop decomposition Which isometries are compositions of two flips? Any isometry of R n . Djokoviˇ c Theorem. Any isometry of a non-degenerate inner product space over any field can be presented as a composition of two involutions isomteries. ( Product of two involutions, Arch. Math. 18 (1967), 582-584.) 9 / 23

Flip-flop decomposition Which isometries are compositions of two flips? Any isometry of R n . Djokoviˇ c Theorem. Any isometry of a non-degenerate inner product space over any field can be presented as a composition of two involutions isomteries. ( Product of two involutions, Arch. Math. 18 (1967), 582-584.) Corollary. Any isometry of an affine space with a non-degenerate bilinear form can be presented as a composition of two flips. 9 / 23

Flip-flop decomposition Which isometries are compositions of two flips? Any isometry of R n . Djokoviˇ c Theorem. Any isometry of a non-degenerate inner product space over any field can be presented as a composition of two involutions isomteries. ( Product of two involutions, Arch. Math. 18 (1967), 582-584.) Corollary. Any isometry of an affine space with a non-degenerate bilinear form can be presented as a composition of two flips. Corollary. Any isometry of a hyperbolic space, sphere, projective space, etc. is a composition of two flips. A flip-flop decomposition. 9 / 23

Biflippers An ordered pair of flippers ( A, B ) is a biflipper, an analogue for an arrow representing a translation. 10 / 23

Biflippers An ordered pair of flippers ( A, B ) is a biflipper, an analogue for an arrow representing a translation. If A ∩ B = ∅ , then we connect A and B with the shortest arrow. 10 / 23

Biflippers An ordered pair of flippers ( A, B ) is a biflipper, an analogue for an arrow representing a translation. If A ∩ B = ∅ , then we connect A and B with the shortest arrow. B A 10 / 23

Biflippers An ordered pair of flippers ( A, B ) is a biflipper, an analogue for an arrow representing a translation. If A ∩ B = ∅ , then we connect A and B with the shortest arrow. B A If A ∩ B � = ∅ , B A 10 / 23

Biflippers An ordered pair of flippers ( A, B ) is a biflipper, an analogue for an arrow representing a translation. If A ∩ B = ∅ , then we connect A and B with the shortest arrow. B A If A ∩ B � = ∅ , B A 10 / 23

Biflippers An ordered pair of flippers ( A, B ) is a biflipper, an analogue for an arrow representing a translation. If A ∩ B = ∅ , then we connect A and B with the shortest arrow. B A If A ∩ B � = ∅ , B A To what extent are the representations non-unique? 10 / 23

Biflippers An ordered pair of flippers ( A, B ) is a biflipper, an analogue for an arrow representing a translation. If A ∩ B = ∅ , then we connect A and B with the shortest arrow. B A If A ∩ B � = ∅ , B A To what extent are the representations non-unique? Equivalence relation: R B ◦ R A = R B ′ ◦ R A ′ . ( A, B ) ∼ ( A ′ , B ′ ) if 10 / 23

Biflippers An ordered pair of flippers ( A, B ) is a biflipper, an analogue for an arrow representing a translation. If A ∩ B = ∅ , then we connect A and B with the shortest arrow. B A If A ∩ B � = ∅ , B A To what extent are the representations non-unique? Equivalence relation: R B ◦ R A = R B ′ ◦ R A ′ . ( A, B ) ∼ ( A ′ , B ′ ) if Problem. Find an explicit description for the equivalence. 10 / 23

Biflippers for a rotation 11 / 23

Biflippers for a rotation an ordered pair of lines. R l ◦ R m ( x ) l R m ( x ) β β α α m x 11 / 23

Biflippers for a rotation an ordered pair of lines. The lines intersect at the center of rotation. R l ◦ R m ( x ) l R m ( x ) β β α α m x 11 / 23

Biflippers for a rotation an ordered pair of lines. The lines intersect at the center of rotation. The angle between the lines is half the rotation angle. R l ◦ R m ( x ) l R m ( x ) β β α α m x 11 / 23

Biflippers for a rotation an ordered pair of lines. The lines intersect at the center of rotation. The angle between the lines is half the rotation angle. On a picture the order of lines is shown by an oriented arc. 11 / 23

Biflippers for a rotation an ordered pair of lines. The lines intersect at the center of rotation. The angle between the lines is half the rotation angle. On a picture the order of lines is shown by an oriented arc. Equivalent biflippers are obtained by rotations about the vertex. 11 / 23

Biflippers for a rotation an ordered pair of lines. The lines intersect at the center of rotation. The angle between the lines is half the rotation angle. On a picture the order of lines is shown by an oriented arc. Equivalent biflippers are obtained by rotations about the vertex. 11 / 23

Biflippers for a rotation an ordered pair of lines. The lines intersect at the center of rotation. The angle between the lines is half the rotation angle. On a picture the order of lines is shown by an oriented arc. Equivalent biflippers are obtained by rotations about the vertex. 11 / 23

Biflippers for a rotation an ordered pair of lines. The lines intersect at the center of rotation. The angle between the lines is half the rotation angle. On a picture the order of lines is shown by an oriented arc. Equivalent biflippers are obtained by rotations about the vertex. 11 / 23

Head to tail for rotations 12 / 23

Head to tail for rotations Given two rotations, present them by biflippers. m 2 m 1 B A l 1 n 2 12 / 23

Head to tail for rotations Given two rotations, present them by biflippers. By rotating the biflippers, make the second line in the first biflipper to coincide with the first line in the second, so that the biflippers are ( l, m ) and ( m, n ) . m 2 m 1 B A l 1 n 2 12 / 23

Head to tail for rotations Given two rotations, present them by biflippers. By rotating the biflippers, make the second line in the first biflipper to coincide with the first line in the second, so that the biflippers are ( l, m ) and ( m, n ) . m B A l n 12 / 23

Head to tail for rotations Given two rotations, present them by biflippers. By rotating the biflippers, make the second line in the first biflipper to coincide with the first line in the second, so that the biflippers are ( l, m ) and ( m, n ) . Erase m and draw an oriented arc from l to n , i.e., form the ordered angle ( l, n ) . m B A l n 12 / 23

Head to tail for rotations Given two rotations, present them by biflippers. By rotating the biflippers, make the second line in the first biflipper to coincide with the first line in the second, so that the biflippers are ( l, m ) and ( m, n ) . Erase m and draw an oriented arc from l to n , i.e., form the ordered angle ( l, n ) . l n C 12 / 23

Composing reflections in line and point O m 13 / 23

Composing reflections in line and point O m 13 / 23

Composing reflections in line and point O m 13 / 23