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Geometric Generation of Permutation Sequences MathViz Champaign-Urbana March 28 2009 Geometric generation of permutation sequences Dennis Roseman Permutahedron Geometric generation of permutation sequences Change Ringing Bouncing Dennis Roseman Problem List Cell Structure University of Iowa coloring edges coloring facets [email protected] Braids Beam March 26, 2009 Calculation Edges in layers Tiling Overture Geometric generation of permutation sequences Dennis Music Roseman Original motivation: apply mathematics to the composition of Permutahedron music. Change Ringing Bouncing Mathematics Focus: Problem List Some geometry of the n-dimensional permutahedron. Cell Structure coloring edges coloring facets Visualization Focus Braids Higher dimensional visualization including braids used as a Beam Calculation visualization tool. Edges in layers Tiling From Abstract Mathematics to Musical Composition Geometric generation of permutation sequences Dennis Roseman Permutahedron Change Two very different musical examples: Ringing change ringing Bouncing Problem List Nomos Alpha of Xenankis Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling From Change Ringing by Wilfrid G. Wilson Geometric generation of permutation sequences Dennis Roseman Permutahedron Change Ringing Bouncing Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling From Change Ringing by Wilfrid G. Wilson Geometric generation of permutation sequences Dennis Roseman Permutahedron Change Ringing Bouncing Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling From Change Ringing by Wilfrid G. Wilson Geometric generation of permutation sequences Dennis Roseman Permutahedron Change Ringing Bouncing Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling From Change Ringing by Wilfrid G. Wilson Geometric generation of permutation sequences Dennis Roseman Permutahedron Change Ringing Bouncing Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling From Change Ringing by Wilfrid G. Wilson Geometric generation of permutation sequences Dennis Roseman Permutahedron Change Ringing Bouncing Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling From Formal Music by I. Xenakis|Nomos Alpha Geometric generation of permutation sequences Dennis Roseman Permutahedron Change Ringing Bouncing Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling From Formal Music by I. Xenakis|Nomos Alpha Geometric generation of permutation sequences Dennis Roseman Permutahedron Change Ringing Bouncing Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling From Formal Music by I. Xenakis|Nomos Alpha Geometric generation of permutation sequences Dennis Roseman Permutahedron Change Ringing Bouncing Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling From Formal Music by I. Xenakis|Nomos Alpha Geometric generation of permutation sequences Dennis Roseman Permutahedron Change Ringing Bouncing Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling Music and Mathematics Geometric generation of permutation sequences Definition Dennis Roseman A musical composition is a family of sequences of related musical events. Permutahedron Change Ringing Time and voices Bouncing Progression in time is related to succession in a sequence; each Problem List Cell Structure sequence represents a \voice". coloring edges coloring facets Braids The mathematical objects we chose are permutations. Beam Calculation Construct families of sequences length k of permutations of Edges in layers order n , where k and n are independent. Tiling 2 They are n-tuples|plot them as points in Rn. 3 Take the convex hull. 4 The resulting polytope is the permutahedron P(n) Permutations Geometrically: the Permutahedron Geometric generation of permutation sequences Dennis Roseman The Permutahedron Permutahedron 1 Change Take the n! permutations Sn to be all permutations of Ringing (1; 2;:::; n) Bouncing Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling 2 They are n-tuples|plot them as points in Rn. 3 Take the convex hull. 4 The resulting polytope is the permutahedron P(n) Permutations Geometrically: the Permutahedron Geometric generation of permutation sequences Dennis Roseman The Permutahedron Permutahedron 1 Change Take the n! permutations Sn to be all permutations of Ringing (1; 2;:::; n) Bouncing Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling 3 Take the convex hull. 4 The resulting polytope is the permutahedron P(n) Permutations Geometrically: the Permutahedron Geometric generation of permutation sequences Dennis Roseman The Permutahedron Permutahedron 1 Change Take the n! permutations Sn to be all permutations of Ringing (1; 2;:::; n) Bouncing 2 They are n-tuples|plot them as points in Rn. Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling 3 Take the convex hull. 4 The resulting polytope is the permutahedron P(n) Permutations Geometrically: the Permutahedron Geometric generation of permutation sequences Dennis Roseman The Permutahedron Permutahedron 1 Change Take the n! permutations Sn to be all permutations of Ringing (1; 2;:::; n) Bouncing 2 They are n-tuples|plot them as points in Rn. Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling 4 The resulting polytope is the permutahedron P(n) Permutations Geometrically: the Permutahedron Geometric generation of permutation sequences Dennis Roseman The Permutahedron Permutahedron 1 Change Take the n! permutations Sn to be all permutations of Ringing (1; 2;:::; n) Bouncing 2 They are n-tuples|plot them as points in Rn. Problem List Cell Structure 3 Take the convex hull. coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling 4 The resulting polytope is the permutahedron P(n) Permutations Geometrically: the Permutahedron Geometric generation of permutation sequences Dennis Roseman The Permutahedron Permutahedron 1 Change Take the n! permutations Sn to be all permutations of Ringing (1; 2;:::; n) Bouncing 2 They are n-tuples|plot them as points in Rn. Problem List Cell Structure 3 Take the convex hull. coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling Permutations Geometrically: the Permutahedron Geometric generation of permutation sequences Dennis Roseman The Permutahedron Permutahedron 1 Change Take the n! permutations Sn to be all permutations of Ringing (1; 2;:::; n) Bouncing 2 They are n-tuples|plot them as points in Rn. Problem List Cell Structure 3 Take the convex hull. coloring edges coloring facets 4 The resulting polytope is the permutahedron P(n) Braids Beam Calculation Edges in layers Tiling 2 P(3) is a hexagon in R3 in the plane . 3 P(4) is a truncated octahedron in R3 . Low Dimensional Cases Geometric generation of permutation sequences Dennis Roseman Permutahedron Some Examples: Change 1 2 Ringing P(2) is the line segment in R with endpoints (1; 2) and Bouncing (2; 1) . Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling 2 P(3) is a hexagon in R3 in the plane . 3 P(4) is a truncated octahedron in R3 . Low Dimensional Cases Geometric generation of permutation sequences Dennis Roseman Permutahedron Some Examples: Change 1 2 Ringing P(2) is the line segment in R with endpoints (1; 2) and Bouncing (2; 1) , a subset of the line x + y = 1 + 2. Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling 2 P(3) is a hexagon in R3 in the plane . 3 P(4) is a truncated octahedron in R3 . Low Dimensional Cases Geometric generation of permutation sequences Dennis Roseman Permutahedron Some Examples: Change 1 2 Ringing P(2) is the line segment in R with endpoints (1; 2) and Bouncing (2; 1) , a subset of the line x + y = 3. Problem List Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling 3 P(4) is a truncated octahedron in R3 . Low Dimensional Cases Geometric generation of permutation sequences Dennis Roseman Permutahedron Some Examples: Change 1 2 Ringing P(2) is the line segment in R with endpoints (1; 2) and Bouncing (2; 1) . Problem List 2 P(3) is a hexagon in R3 in the plane . Cell Structure coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling 3 P(4) is a truncated octahedron in R3 . Low Dimensional Cases Geometric generation of permutation sequences Dennis Roseman Some Examples: Permutahedron 2 Change 1 P(2) is the line segment in R with endpoints (1; 2) and Ringing (2; 1) . Bouncing 2 3 Problem List P(3) is a hexagon in R in the plane subset of the plane Cell Structure x + y + z = 6. coloring edges coloring facets Braids Beam Calculation Edges in layers Tiling Low Dimensional Cases Geometric generation of permutation sequences Dennis Roseman Permutahedron Some Examples: Change 1 2 Ringing P(2) is the line segment in R with endpoints (1; 2) and Bouncing (2; 1) . Problem List 2 P(3) is a hexagon in R3 in the plane . Cell Structure 3 3 coloring edges P(4) is a truncated octahedron in R . coloring facets Braids Beam
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