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 Calculation
Edges in layers
Tiling 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 . Cell Structure 3 P(4) is a truncated octahedron in R3 subset of the coloring edges coloring facets hyperplane x + y + z + w = 10. Braids
Beam Calculation
Edges in layers
Tiling The Permutahedron of order 2
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron
Change Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation 2 Edges in layers Figure: The two permutations (1, 2) and (2, 1) : a line segment in R Tiling The Permutahedron of order 3
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron
Change Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation Figure: Hexagon in R3 of the six permutations of order 3: Edges in layers (1, 2, 3), (2, 1, 3), (3, 1, 2), (3, 2, 1), (2, 3, 1), (1, 3, 2), (1, 2, 3) Tiling The Permutahedron of order 4
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron
Change Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation Figure: The 24 permutations of order 4 determine a truncated Edges in layers octahedron in R4 which we show in R3 Tiling Change Ringing n bells
Geometric generation of permutation sequences
Dennis Roseman Definition
Permutahedron A sequence Σ of permutations is a change ringing Change composition if Ringing
Bouncing Σ begins and ends with the identity permutation of Sn Problem List Otherwise each of the n! order n permutations occurs Cell Structure exactly one time coloring edges coloring facets Two consecutive permutations of Σ differ by switching Braids two consecutive integers. Beam Calculation
Edges in layers
Tiling Ringing Changes on Three Bells
Geometric generation of permutation sequences
Dennis Roseman 1 2 3 2 1 3 Permutahedron 3 1 2 Change Ringing 3 2 1 Bouncing 2 3 1 Problem List 1 3 2 Cell Structure 1 2 3 coloring edges coloring facets Braids Table: One way to ring changes on 3 bells; the second reverses the
Beam order. Calculation
Edges in layers
Tiling Ringing Changes Geometrically
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 Figure: The change Double Canterbury Pleasure Minimus Tiling corresponds to a Hamiltonian path in the edge set of P(4) There is no relationship between one change and another Each permutation is treated equally. Musically one expects to make choices. There is a fixed length to a ring of changes The difficulty of calculation increases rapidly with n
Change Ringing
Geometric generation of permutation sequences
Dennis Roseman Critique of Change Ringing
Permutahedron Change ringing is very limited—hard to get non-trivial
Change examples. Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling Each permutation is treated equally. Musically one expects to make choices. There is a fixed length to a ring of changes The difficulty of calculation increases rapidly with n
Change Ringing
Geometric generation of permutation sequences
Dennis Roseman Critique of Change Ringing
Permutahedron Change ringing is very limited—hard to get non-trivial
Change examples. Ringing
Bouncing There is no relationship between one change and another
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling There is a fixed length to a ring of changes The difficulty of calculation increases rapidly with n
Change Ringing
Geometric generation of permutation sequences
Dennis Roseman Critique of Change Ringing
Permutahedron Change ringing is very limited—hard to get non-trivial
Change examples. Ringing
Bouncing There is no relationship between one change and another Problem List Each permutation is treated equally. Musically one expects Cell Structure to make choices. coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling The difficulty of calculation increases rapidly with n
Change Ringing
Geometric generation of permutation sequences
Dennis Roseman Critique of Change Ringing
Permutahedron Change ringing is very limited—hard to get non-trivial
Change examples. Ringing
Bouncing There is no relationship between one change and another Problem List Each permutation is treated equally. Musically one expects Cell Structure to make choices. coloring edges coloring facets There is a fixed length to a ring of changes Braids
Beam Calculation
Edges in layers
Tiling Change Ringing
Geometric generation of permutation sequences
Dennis Roseman Critique of Change Ringing
Permutahedron Change ringing is very limited—hard to get non-trivial
Change examples. Ringing
Bouncing There is no relationship between one change and another Problem List Each permutation is treated equally. Musically one expects Cell Structure to make choices. coloring edges coloring facets There is a fixed length to a ring of changes Braids The difficulty of calculation increases rapidly with n Beam Calculation
Edges in layers
Tiling Build room in the shape of P(4) with all walls made of mirror. From inside the room shine a “generic” laser beam from point x0 in direction λ0. The beam as it reflects will hit successive walls giving a sequence of points x1, x2,.... Since the beam is generic there will be a unique vertex (permutation) πi of P(4) nearest to xi . Thus we generate our sequence of permutations S(x0, λ0) = (π1, π2,...)
A new way to get a sequences of permutations
Geometric generation of permutation sequences Bouncing a light in a mirrored P(4) Dennis Roseman
Permutahedron
Change Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling From inside the room shine a “generic” laser beam from point x0 in direction λ0. The beam as it reflects will hit successive walls giving a sequence of points x1, x2,.... Since the beam is generic there will be a unique vertex (permutation) πi of P(4) nearest to xi . Thus we generate our sequence of permutations S(x0, λ0) = (π1, π2,...)
A new way to get a sequences of permutations
Geometric generation of permutation sequences Bouncing a light in a mirrored P(4) Dennis Roseman Build room in the shape of P(4) with all walls made of
Permutahedron mirror.
Change Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling The beam as it reflects will hit successive walls giving a sequence of points x1, x2,.... Since the beam is generic there will be a unique vertex (permutation) πi of P(4) nearest to xi . Thus we generate our sequence of permutations S(x0, λ0) = (π1, π2,...)
A new way to get a sequences of permutations
Geometric generation of permutation sequences Bouncing a light in a mirrored P(4) Dennis Roseman Build room in the shape of P(4) with all walls made of
Permutahedron mirror. Change From inside the room shine a “generic” laser beam from Ringing point x0 in direction λ0. Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling Since the beam is generic there will be a unique vertex (permutation) πi of P(4) nearest to xi . Thus we generate our sequence of permutations S(x0, λ0) = (π1, π2,...)
A new way to get a sequences of permutations
Geometric generation of permutation sequences Bouncing a light in a mirrored P(4) Dennis Roseman Build room in the shape of P(4) with all walls made of
Permutahedron mirror. Change From inside the room shine a “generic” laser beam from Ringing point x0 in direction λ0. Bouncing
Problem List The beam as it reflects will hit successive walls giving a Cell Structure sequence of points x1, x2,.... coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling Thus we generate our sequence of permutations S(x0, λ0) = (π1, π2,...)
A new way to get a sequences of permutations
Geometric generation of permutation sequences Bouncing a light in a mirrored P(4) Dennis Roseman Build room in the shape of P(4) with all walls made of
Permutahedron mirror. Change From inside the room shine a “generic” laser beam from Ringing point x0 in direction λ0. Bouncing
Problem List The beam as it reflects will hit successive walls giving a Cell Structure sequence of points x1, x2,.... coloring edges coloring facets Since the beam is generic there will be a unique vertex Braids (permutation) πi of P(4) nearest to xi . Beam Calculation
Edges in layers
Tiling A new way to get a sequences of permutations
Geometric generation of permutation sequences Bouncing a light in a mirrored P(4) Dennis Roseman Build room in the shape of P(4) with all walls made of
Permutahedron mirror. Change From inside the room shine a “generic” laser beam from Ringing point x0 in direction λ0. Bouncing
Problem List The beam as it reflects will hit successive walls giving a Cell Structure sequence of points x1, x2,.... coloring edges coloring facets Since the beam is generic there will be a unique vertex Braids (permutation) πi of P(4) nearest to xi . Beam Calculation Thus we generate our sequence of permutations Edges in layers S(x0, λ0) = (π1, π2,...) Tiling Bounce points
Geometric generation of permutation sequences
Dennis Roseman Definition Permutahedron The points x1, x2,... are called either intersection points Change Ringing (they are calculated as an intersection of a ray and ∂P(n)) or
Bouncing bounce points (since our beam bounces there).
Problem List
Cell Structure coloring edges Bouncing for P(n) coloring facets Braids Cleary we can define this process for permutations of order n.
Beam Calculation
Edges in layers
Tiling Lets bounce
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 Figure: An example of 16 bounces Corresponding Sequence of Permutations
Geometric 2 1 4 3 generation of 1 4 2 3 permutation 2 1 4 3 3 4 2 1 sequences 1 3 2 4 Dennis 4 2 1 3 Roseman 2 4 3 1 1 2 4 3 3 1 2 4 Permutahedron 1 3 4 2 3 2 1 4 Change 3 2 4 1 Ringing 2 3 1 4 4 1 3 2 Bouncing 2 4 1 3 3 1 2 4 Problem List 2 3 4 1 3 1 2 4 Cell Structure 4 2 1 3 coloring edges 2 4 3 1 coloring facets 3 4 1 2 Braids 1 4 2 3 3 4 1 2 Beam 1 4 2 3 Calculation 1 3 4 2 2 1 3 4 Edges in layers 1 3 4 2 2 1 3 4 Tiling 4 1 3 2 4 1 3 2 1 2 3 4 2 4 3 1 1 4 2 3 Lets bounce
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 Figure: Another example of 16 bounces The numbers we use are not necessarily pitches
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 Figure: Any knob or input/output on the Control Panel of this Moog
Tiling corresponds to a number. Photo by Kevin Lightner Bouncing, Ringing
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron
Change Ringing Comparing Bouncing to Change Ringing Bouncing Change ringing: finite number of possibilities Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling Bouncing, Ringing
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron
Change Ringing Comparing Bouncing to Change Ringing Bouncing Bouncing:infinite number of possibilities Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling Bouncing, Ringing
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron Change Comparing Bouncing to Change Ringing Ringing
Bouncing Bouncing:infinite number of possibilities Problem List Change ringing is hard is it to calculate. Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling Bouncing, Ringing
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron Change Comparing Bouncing to Change Ringing Ringing
Bouncing Bouncing:infinite number of possibilities Problem List Bouncing is easy to calculate Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling Bouncing, Ringing
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron Change Comparing Bouncing to Change Ringing Ringing
Bouncing Bouncing:infinite number of possibilities Problem List Bouncing is easy to calculate as we will see Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling Bouncing, Ringing
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron Comparing Bouncing to Change Ringing Change Ringing Bouncing:infinite number of possibilities Bouncing Bouncing is easy to calculate Problem List Change Ringing—no relationship between one change Cell Structure coloring edges and another coloring facets Braids
Beam Calculation
Edges in layers
Tiling Bouncing, Ringing
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron Comparing Bouncing to Change Ringing Change Ringing Bouncing:infinite number of possibilities Bouncing Bouncing is easy to calculate Problem List Bouncing: if one varies x and λ one gets related Cell Structure coloring edges permutation sequences S(x, λ) coloring facets Braids
Beam Calculation
Edges in layers
Tiling Bouncing, Ringing
Geometric generation of permutation sequences
Dennis Roseman Comparing Bouncing to Change Ringing Permutahedron
Change Bouncing:infinite number of possibilities Ringing Bouncing is easy to calculate Bouncing
Problem List Bouncing: if one varies x and λ one gets related
Cell Structure permutation sequences S(x, λ) coloring edges coloring facets Change Ringing: each permutation is treated equally Braids
Beam Calculation
Edges in layers
Tiling Bouncing, Ringing
Geometric generation of permutation sequences
Dennis Roseman Comparing Bouncing to Change Ringing Permutahedron
Change Bouncing:infinite number of possibilities Ringing Bouncing is easy to calculate Bouncing
Problem List Bouncing: if one varies x and λ one gets related
Cell Structure permutation sequences S(x, λ) coloring edges coloring facets Bouncing: distinct sequences have distinct characteristics Braids
Beam Calculation
Edges in layers
Tiling Bouncing, Ringing
Geometric generation of permutation sequences
Dennis Roseman Comparing Bouncing to Change Ringing
Permutahedron Bouncing:infinite number of possibilities Change Bouncing is easy to calculate Ringing Bouncing Bouncing: if one varies x and λ one gets related Problem List permutation sequences S(x, λ) Cell Structure coloring edges Bouncing: distinct sequences have distinct characteristics coloring facets Change Ringing: difficulty of calculation increases rapidly Braids
Beam with n Calculation
Edges in layers
Tiling Bouncing, Ringing
Geometric generation of permutation sequences
Dennis Roseman Comparing Bouncing to Change Ringing Permutahedron Bouncing:infinite number of possibilities Change Ringing Bouncing is easy to calculate Bouncing Bouncing: if one varies x and λ one gets related Problem List permutation sequences S(x, λ) Cell Structure coloring edges Bouncing: distinct sequences have distinct characteristics coloring facets Braids Bouncing: calculation is quadratic with respect to n Beam Calculation
Edges in layers
Tiling The rest of the talk: mathematics and visualization
Geometric generation of permutation sequences
Dennis Roseman Questions we now address 1 How fast can we calculate the bouncing path for fairly Permutahedron high orders—8, 12, 16, 32? Change Ringing 2 How can we visualize the calculational process and the Bouncing results? Problem List 3 Cell Structure What is the geometry of a high dimensional coloring edges coloring facets permutahedron? Braids 4 How does the geometry of the permutahedron change Beam with n? Calculation
Edges in layers
Tiling Basic approach
Geometric generation of permutation sequences Change Ringing Dennis Roseman 1 Construct a special polygonal path in the wireframe of a
Permutahedron permutahedron. Change 2 The sequence of vertices on that path is the desired Ringing permutation sequence. Bouncing
Problem List
Cell Structure Bouncing, an alternative coloring edges n coloring facets 1 Take a generically generated generic path in R that Braids avoids the wireframe Beam 2 Calculation Obtain the sequence of permutations by “digitizing” to
Edges in layers permutations near the path.
Tiling Vertices of the Permutahedron
Geometric generation of permutation sequences
Dennis Theorem Roseman There are n! vertices in P(n). Permutahedron
Change All vertices of P(n) all lie on an (n − 2)-sphere with center Cn, Ringing the centroid of P(n), and radius ρn. Bouncing
Problem List Cell Structure Definition coloring edges coloring facets This sphere is the permutahedral sphere of order n and ρn Braids the permutahedrdal radius. The distance between Cn and the Beam Calculation centroid of Yα is the inner permutahedrdal radius. Edges in layers
Tiling Generators of the Symmetric Group
Geometric generation of permutation sequences
Dennis Roseman Definition
Permutahedron An elementary transposition is a permutation that
Change interchanges consecutive integers, Ringing
Bouncing Problem List Note: Cell Structure coloring edges This is interchange of consecutive integers (wherever they are) coloring facets not interchange of integers in consecutive positions (whatever Braids the integers are). Beam Calculation
Edges in layers
Tiling Two permutations are connected by an edge if and only if coordinates differ by a switch of two coordinates of consecutive value. Thus any edge corresponds to an elementary transposition. √ Thus all edges have length 2. The order of any vertex is (n − 1).
Basic general edge facts
Edges of the Permutahedron
Geometric generation of Definition permutation sequences The union of edges of P(n) is called the wireframe of P(n) Dennis Roseman Example Permutahedron The four edges from (1, 2, 3, 4, 5) go to (2, 1, 3, 4, 5), Change Ringing (1, 3, 2, 4, 5), (1, 2, 4, 3, 5), and (1, 2, 3, 5, 4). Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling Thus any edge corresponds to an elementary transposition. √ Thus all edges have length 2. The order of any vertex is (n − 1).
Edges of the Permutahedron
Geometric generation of Definition permutation sequences The union of edges of P(n) is called the wireframe of P(n) Dennis Roseman Example Permutahedron The four edges from (1, 2, 3, 4, 5) go to (2, 1, 3, 4, 5), Change Ringing (1, 3, 2, 4, 5), (1, 2, 4, 3, 5), and (1, 2, 3, 5, 4). Bouncing Problem List Basic general edge facts Cell Structure coloring edges Two permutations are connected by an edge if and only if coloring facets coordinates differ by a switch of two coordinates of Braids consecutive value. Beam Calculation
Edges in layers
Tiling √ Thus all edges have length 2. The order of any vertex is (n − 1).
Edges of the Permutahedron
Geometric generation of Definition permutation sequences The union of edges of P(n) is called the wireframe of P(n) Dennis Roseman Example Permutahedron The four edges from (1, 2, 3, 4, 5) go to (2, 1, 3, 4, 5), Change Ringing (1, 3, 2, 4, 5), (1, 2, 4, 3, 5), and (1, 2, 3, 5, 4). Bouncing Problem List Basic general edge facts Cell Structure coloring edges Two permutations are connected by an edge if and only if coloring facets coordinates differ by a switch of two coordinates of Braids consecutive value. Beam Calculation Thus any edge corresponds to an elementary transposition. Edges in layers
Tiling The order of any vertex is (n − 1).
Edges of the Permutahedron
Geometric generation of Definition permutation sequences The union of edges of P(n) is called the wireframe of P(n) Dennis Roseman Example Permutahedron The four edges from (1, 2, 3, 4, 5) go to (2, 1, 3, 4, 5), Change Ringing (1, 3, 2, 4, 5), (1, 2, 4, 3, 5), and (1, 2, 3, 5, 4). Bouncing Problem List Basic general edge facts Cell Structure coloring edges Two permutations are connected by an edge if and only if coloring facets coordinates differ by a switch of two coordinates of Braids consecutive value. Beam Calculation Thus any edge corresponds to an elementary transposition. Edges in layers √ Tiling Thus all edges have length 2. Edges of the Permutahedron
Geometric generation of Definition permutation sequences The union of edges of P(n) is called the wireframe of P(n) Dennis Roseman Example Permutahedron The four edges from (1, 2, 3, 4, 5) go to (2, 1, 3, 4, 5), Change Ringing (1, 3, 2, 4, 5), (1, 2, 4, 3, 5), and (1, 2, 3, 5, 4). Bouncing Problem List Basic general edge facts Cell Structure coloring edges Two permutations are connected by an edge if and only if coloring facets coordinates differ by a switch of two coordinates of Braids consecutive value. Beam Calculation Thus any edge corresponds to an elementary transposition. Edges in layers √ Tiling Thus all edges have length 2. The order of any vertex is (n − 1). Visualizing the edges
Geometric generation of permutation sequences Rotate and project to low dimensions Dennis Roseman We can generically rotate a wireframe of any order 3 Permutahedron permutahedron then project homemorphically into R .
Change Ringing We can generically rotate a wireframe of any order 2 Bouncing permutahedron then project non-homemorphically into R and Problem List still get a meaningful image. Cell Structure coloring edges coloring facets Braids Color the edges Beam We can use (n − 1) colors on the edges to code the Calculation corresponding transpositions. Edges in layers
Tiling Coloring Edges: Order 4
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 Coloring Edges: Order 5
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 Coloring Edges: Order 6
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 Cells of the Permutahedron
Geometric generation of permutation sequences Proposition Dennis An k-cell of P(n) is either subgroup which is a product of k Roseman symmetric groups or a coset of one of such subgroup. Here Permutahedron P(0) = {1}. Change Ringing Bouncing Definition Problem List
Cell Structure Let Yα = {(x1,..., xn) ∈ Sn : x1 = 1} and coloring edges Y = {(x ,..., x ) ∈ S : x = n}. We call Y the first Young coloring facets ω 1 n n n α Braids subgroup of Sn,Yω the last Young subgroup of Sn.
Beam Calculation Remark Edges in layers
Tiling Yα and Yω are isomorphic to Sn−1 Re-examining P(4)
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron
Change Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Figure: The edges of one color are all the cosets of a Young subgroup Calculation of order two. The hexagons are all cosets of the two Young subgroups Edges in layers isomorphic to P(3). The squares are cosets of P(2) × P(2). Tiling Re-examining P(4)
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron
Change Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Figure: The edges of one color are all the cosets of a Young subgroup Calculation of order two. The hexagons are all cosets of the two Young subgroups Edges in layers isomorphic to P(3). The squares are cosets of P(2) × P(2). Tiling Re-examining P(4)
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron
Change Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Figure: The edges of one color are all the cosets of a Young subgroup Calculation of order two. The hexagons are all cosets of the two Young subgroups Edges in layers isomorphic to P(3). The squares are cosets of P(2) × P(2). Tiling Facets of the Permutahedron
Geometric generation of permutation sequences Definition Dennis The facets are the the (n − 2)-cells of P(n). Roseman
Permutahedron Proposition Change Ringing P(n) has 2n − 2 facets Bouncing Problem List Example Cell Structure coloring edges So P(8) has 254 facets and P(12) has 4094. coloring facets Braids Beam Implication Calculation
Edges in layers The number of facets is exponential in n. Our light beam
Tiling calculation should not be based on examination of all facets. A duality of facets and edges
Geometric generation of permutation sequences
Dennis Roseman Transposition colors for facets Permutahedron
Change At a vertex v of facet F you see (n − 1) edges all of distinct Ringing colors. Bouncing
Problem List One of these colors is not an edge of F .
Cell Structure coloring edges This color will identify our corresponding elementary coloring facets transposition. Braids
Beam Calculation
Edges in layers
Tiling The facet color is the unique color not an edge color of the facet.
Coloring the facets
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 Figure: Here we color the three generators: σ1 σ2 σ3 Tiling The facet color is the unique color not an edge color of the facet.
Coloring the facets
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 Figure: Here we color the three generators: red green blue Tiling The facet color is the unique color not an edge color of the facet.
Coloring the facets
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 Figure: Here we color the three generators: σ1 σ2 σ3 Tiling Coloring the facets
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 Figure: Here we color the three generators: σ1 σ2 σ3 The facet Tiling color is the unique color not an edge color of the facet. Two group presentations:
Geometric generation of permutation Presentation of Order n Braid Group sequences
Dennis Generators: σ1, . . . , σn−1 Roseman Relations: Permutahedron σi σj = σj σi if j 6= i ± 1 Change Ringing σi σi+1σi = σi+1σi σi+1 Bouncing Problem List Presentation of Order n Symmetric Group Cell Structure coloring edges Generators: σ1, . . . , σn−1 coloring facets Relations: Braids −1 Beam σi = σi Calculation σ σ = σ σ if j 6= i ± 1 Edges in layers i j j i Tiling σi σi+1σi = σi+1σi σi+1 Inverses and elementary transpositions
Geometric generation of permutation sequences
Dennis From the symmetric group to the braid group Roseman A finite sequence of elementary transpositions τ1, τ2, . . . , τn Permutahedron corresponds to a word in the symmetric group: Change Ringing τ τ ··· τ . Bouncing 1 2 n
Problem List
Cell Structure coloring edges But if (somehow) we can distinguish elementary transpositions coloring facets from their inverses we would obtain a word in the braid group: Braids
Beam 1 2 n Calculation τ1 τ2 ··· τn . Edges in layers
Tiling Signs for transpositions: one of many methods
Geometric generation of permutation sequences Definition Dennis Roseman A braid sign convention is a function that associates to any
Permutahedron bounce point xi of any bouncing path (xi ) = ±1. Change Ringing Example Bouncing −→ Problem List Let N be the vector from the identity permutation to the Cell Structure reverse of the identity. Define the sign at x to be the sign of coloring edges i coloring facets −−−→ −→ the dot product xi−1xi · N . Think of the identity as the “south Braids Beam pole ”. Positive means we were heading north before we Calculation “bounced”; negative means heading “south”. Edges in layers
Tiling A bouncing path braid
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron Definition
Change Given a bouncing path x0, x1, x2, x3 ... and a braid sign Ringing convention we obtain a bounce path braid— the braid given Bouncing
Problem List by the word in the braid group:
Cell Structure 1 2 3 coloring edges σ(x1) σ(x2) σ(x3) ··· coloring facets Braids
Beam Calculation
Edges in layers
Tiling Example of Bounce Braid for Order 4
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 Example of Bounce Braid for Order 8
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 Example of Bounce Braid for Order 12
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 Example of Bounce Braid for Order 16
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 Bounce Braids
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron Braid as an Aid Change Ringing In general we need to look at the bounce permutations
Bouncing together with the bounce braid. Problem List The bounce path indicates where the bounce occurs, the braid Cell Structure coloring edges tells us something about how the “type” of bounce. coloring facets Braids
Beam Calculation
Edges in layers
Tiling The nearest permutation to a point
Geometric generation of permutation Definition sequences n A generic point z = (z1,..., zn) of R will have n distinct Dennis Roseman coordinate values.
Permutahedron The rank of zi , denoted r(zi ), is one plus the number of Change coordinates of z smaller than z . Ringing i
Bouncing
Problem List Definition
Cell Structure The rank vector ρ(z) = (r(z1),..., r(zn)) coloring edges coloring facets Braids In other words: Beam Calculation Simply Put: The rank of z is the closest permutation to z. Edges in layers Or not: A generic point is mapped to a chamber of the real Tiling n-braid arrangement Finding the intersection of a ray and ∂P(n)
Geometric generation of permutation sequences
Dennis Roseman The key Permutahedron
Change Focus on the plane P that contains the three points Ringing 1 the centroid C of P(n), Bouncing 2 Problem List the initial point x0 Cell Structure 3 the tip of our vector x0 + λ. coloring edges coloring facets We then project the wireframe of P(n) onto this plane. Braids
Beam Calculation
Edges in layers
Tiling Projection wireframe P(5)
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 Projection wireframe P(5)
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 Projection wireframe P(5)
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 Projection wireframe P(5)
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 Projection wireframe P(5)
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 Projection wireframe P(5)
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 Projection wireframe P(6)
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 Projection wireframe P(6)
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 Projection wireframe P(7)
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 Projection wireframe P(7)
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 Projection wireframe P(7)
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 Projection wireframe P(7)
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 Projection wireframe P(7)
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 Projection wireframe P(7)
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 (This is important since if we do our bouncing inside a round ball, the path will be planar) There seems to be some structure there that is evident from the projections. (Some will be clearer with a colored wireframe) The “central” portion of figures is hard to understand but the area around the edge is much clearer. (In fact the bounding polygonal path of the projection is the projection of a simple closed polygonal path of P(n) edges )
Observations
Geometric generation of permutation sequences The higher permutahedra are not round. Dennis Roseman
Permutahedron
Change Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling There seems to be some structure there that is evident from the projections. (Some will be clearer with a colored wireframe) The “central” portion of figures is hard to understand but the area around the edge is much clearer. (In fact the bounding polygonal path of the projection is the projection of a simple closed polygonal path of P(n) edges )
Observations
Geometric generation of permutation sequences The higher permutahedra are not round. Dennis Roseman (This is important since if we do our bouncing inside a
Permutahedron round ball, the path will be planar)
Change Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling (Some will be clearer with a colored wireframe) The “central” portion of figures is hard to understand but the area around the edge is much clearer. (In fact the bounding polygonal path of the projection is the projection of a simple closed polygonal path of P(n) edges )
Observations
Geometric generation of permutation sequences The higher permutahedra are not round. Dennis Roseman (This is important since if we do our bouncing inside a
Permutahedron round ball, the path will be planar) Change There seems to be some structure there that is evident Ringing from the projections. Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling The “central” portion of figures is hard to understand but the area around the edge is much clearer. (In fact the bounding polygonal path of the projection is the projection of a simple closed polygonal path of P(n) edges )
Observations
Geometric generation of permutation sequences The higher permutahedra are not round. Dennis Roseman (This is important since if we do our bouncing inside a
Permutahedron round ball, the path will be planar) Change There seems to be some structure there that is evident Ringing from the projections. Bouncing
Problem List (Some will be clearer with a colored wireframe)
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling (In fact the bounding polygonal path of the projection is the projection of a simple closed polygonal path of P(n) edges )
Observations
Geometric generation of permutation sequences The higher permutahedra are not round. Dennis Roseman (This is important since if we do our bouncing inside a
Permutahedron round ball, the path will be planar) Change There seems to be some structure there that is evident Ringing from the projections. Bouncing
Problem List (Some will be clearer with a colored wireframe) Cell Structure The “central” portion of figures is hard to understand but coloring edges coloring facets the area around the edge is much clearer. Braids
Beam Calculation
Edges in layers
Tiling Observations
Geometric generation of permutation sequences The higher permutahedra are not round. Dennis Roseman (This is important since if we do our bouncing inside a
Permutahedron round ball, the path will be planar) Change There seems to be some structure there that is evident Ringing from the projections. Bouncing
Problem List (Some will be clearer with a colored wireframe) Cell Structure The “central” portion of figures is hard to understand but coloring edges coloring facets the area around the edge is much clearer. Braids (In fact the bounding polygonal path of the projection is Beam the projection of a simple closed polygonal path of P(n) Calculation
Edges in layers edges )
Tiling Creating great path: projection to the plane P
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 Same as previous figure after rotation in R3
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 2 Let π0 = ρ(x0). 3 Consider the projection φ of the wireframe W of P(n) onto P. Let D be the convex hull of φ(W ). 4 Each edge of ∂D is a projection of a single edge of W . 5 A union of these edges which form a polygonal arc in P(n) is called a great path.
6 From π0, follow this great path in the general direction of λ obtaining vertex sequence p0, p1,....
7 At each pi find the intersection point of the ray with hyperplanes determined by facets at pi . 8 By convexity of P(n) the closest such intersection point is our bounce point.
A Great Path Method
Geometric generation of 1 Let P be the plane through points: x0, x0 + λ and the permutation sequences centroid of P(n). Dennis Roseman
Permutahedron
Change Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling 3 Consider the projection φ of the wireframe W of P(n) onto P. Let D be the convex hull of φ(W ). 4 Each edge of ∂D is a projection of a single edge of W . 5 A union of these edges which form a polygonal arc in P(n) is called a great path.
6 From π0, follow this great path in the general direction of λ obtaining vertex sequence p0, p1,....
7 At each pi find the intersection point of the ray with hyperplanes determined by facets at pi . 8 By convexity of P(n) the closest such intersection point is our bounce point.
A Great Path Method
Geometric generation of 1 Let P be the plane through points: x0, x0 + λ and the permutation sequences centroid of P(n).
Dennis 2 Roseman Let π0 = ρ(x0).
Permutahedron
Change Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling 4 Each edge of ∂D is a projection of a single edge of W . 5 A union of these edges which form a polygonal arc in P(n) is called a great path.
6 From π0, follow this great path in the general direction of λ obtaining vertex sequence p0, p1,....
7 At each pi find the intersection point of the ray with hyperplanes determined by facets at pi . 8 By convexity of P(n) the closest such intersection point is our bounce point.
A Great Path Method
Geometric generation of 1 Let P be the plane through points: x0, x0 + λ and the permutation sequences centroid of P(n).
Dennis 2 Roseman Let π0 = ρ(x0). 3 Consider the projection φ of the wireframe W of P(n) Permutahedron onto P. Let D be the convex hull of φ(W ). Change Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling 5 A union of these edges which form a polygonal arc in P(n) is called a great path.
6 From π0, follow this great path in the general direction of λ obtaining vertex sequence p0, p1,....
7 At each pi find the intersection point of the ray with hyperplanes determined by facets at pi . 8 By convexity of P(n) the closest such intersection point is our bounce point.
A Great Path Method
Geometric generation of 1 Let P be the plane through points: x0, x0 + λ and the permutation sequences centroid of P(n).
Dennis 2 Roseman Let π0 = ρ(x0). 3 Consider the projection φ of the wireframe W of P(n) Permutahedron onto P. Let D be the convex hull of φ(W ). Change Ringing 4 Each edge of ∂D is a projection of a single edge of W . Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling 6 From π0, follow this great path in the general direction of λ obtaining vertex sequence p0, p1,....
7 At each pi find the intersection point of the ray with hyperplanes determined by facets at pi . 8 By convexity of P(n) the closest such intersection point is our bounce point.
A Great Path Method
Geometric generation of 1 Let P be the plane through points: x0, x0 + λ and the permutation sequences centroid of P(n).
Dennis 2 Roseman Let π0 = ρ(x0). 3 Consider the projection φ of the wireframe W of P(n) Permutahedron onto P. Let D be the convex hull of φ(W ). Change Ringing 4 Each edge of ∂D is a projection of a single edge of W . Bouncing 5 A union of these edges which form a polygonal arc in P(n) Problem List
Cell Structure is called a great path. coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling 7 At each pi find the intersection point of the ray with hyperplanes determined by facets at pi . 8 By convexity of P(n) the closest such intersection point is our bounce point.
A Great Path Method
Geometric generation of 1 Let P be the plane through points: x0, x0 + λ and the permutation sequences centroid of P(n).
Dennis 2 Roseman Let π0 = ρ(x0). 3 Consider the projection φ of the wireframe W of P(n) Permutahedron onto P. Let D be the convex hull of φ(W ). Change Ringing 4 Each edge of ∂D is a projection of a single edge of W . Bouncing 5 A union of these edges which form a polygonal arc in P(n) Problem List
Cell Structure is called a great path. coloring edges 6 coloring facets From π0, follow this great path in the general direction of Braids λ obtaining vertex sequence p0, p1,.... Beam Calculation
Edges in layers
Tiling 8 By convexity of P(n) the closest such intersection point is our bounce point.
A Great Path Method
Geometric generation of 1 Let P be the plane through points: x0, x0 + λ and the permutation sequences centroid of P(n).
Dennis 2 Roseman Let π0 = ρ(x0). 3 Consider the projection φ of the wireframe W of P(n) Permutahedron onto P. Let D be the convex hull of φ(W ). Change Ringing 4 Each edge of ∂D is a projection of a single edge of W . Bouncing 5 A union of these edges which form a polygonal arc in P(n) Problem List
Cell Structure is called a great path. coloring edges 6 coloring facets From π0, follow this great path in the general direction of Braids λ obtaining vertex sequence p0, p1,....
Beam 7 Calculation At each pi find the intersection point of the ray with
Edges in layers hyperplanes determined by facets at pi .
Tiling A Great Path Method
Geometric generation of 1 Let P be the plane through points: x0, x0 + λ and the permutation sequences centroid of P(n).
Dennis 2 Roseman Let π0 = ρ(x0). 3 Consider the projection φ of the wireframe W of P(n) Permutahedron onto P. Let D be the convex hull of φ(W ). Change Ringing 4 Each edge of ∂D is a projection of a single edge of W . Bouncing 5 A union of these edges which form a polygonal arc in P(n) Problem List
Cell Structure is called a great path. coloring edges 6 coloring facets From π0, follow this great path in the general direction of Braids λ obtaining vertex sequence p0, p1,....
Beam 7 Calculation At each pi find the intersection point of the ray with
Edges in layers hyperplanes determined by facets at pi . Tiling 8 By convexity of P(n) the closest such intersection point is our bounce point. Yet more braids!
Geometric generation of permutation sequences
Dennis Roseman A braid from edges, not facets Permutahedron 1 Put together the great paths used in calculating a bounce Change Ringing sequence. Bouncing 2 This sequence of edges of ∂P(n) gives a sequence of Problem List elementary transpositions. Cell Structure coloring edges 3 There are ways to define signs to this sequence giving yet coloring facets Braids more braids.
Beam Calculation
Edges in layers
Tiling Very briefly—one way to get signs
Geometric generation of permutation sequences Example Dennis Roseman 1 Consider the polygonal path β = P ∩ ∂P(n) Permutahedron 2 orient β using λ Change Ringing 3 orient the (n − 3)-cells of ∂P(n) Bouncing 4 use intersection numbers of the β with those cells to get Problem List our sign Cell Structure coloring edges coloring facets Note: Braids
Beam There is a quick indirect way to calculate this from the Calculation construction of the great path. Edges in layers
Tiling Sorting networks
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron
Change A connection to topic in computer science Ringing There is a relationship between a colored great path which Bouncing joins two antipodal permutations and the concept of a sorting Problem List network. Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling A different bouncing braid
Geometric generation of permutation sequences Dennis A braid based on edges near bounce Roseman 1 Take bounce points x , x ,... x where x lies in facet F Permutahedron 1 2 n i i Change 2 Let ei be the edge of Fi closest to xi with associated an Ringing elementary transposition Σ(xi ) Bouncing 3 Problem List There are a number of ways to assign a braid sign
Cell Structure convention i coloring edges coloring facets 4 This gives a braid word Braids 1 2 n Beam Σ(x1) Σ(x2) ... Σ(xn) Calculation
Edges in layers
Tiling A different bouncing braid
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron Change There is not time to go into detail . . . Ringing
Bouncing To give a sense that the color of an edge tells something about
Problem List the nature of the bounce consider the following graphics.
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling A projection of labeled wireframe of P(5)
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 A projection of labeled wireframe of P(5)
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 A projection of labeled wireframe of P(6)
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 A projection of labeled wireframe of P(7)
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 An alternative to bouncing: permutahedral tiles
Geometric generation of permutation sequences
Dennis Roseman
Permutahedron
Change Ringing Theorem Bouncing Rn can be tiled with translated copies P(n) Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling A black and white tiling of the plane by hexagons
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 Fitting two adjacent P(4)s
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 Fitting three adjacent P(4)s
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 Fitting four adjacent P(4)s
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 A “black and white tiling of R3 by P(4)s
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 A “black and white tiling of R3 by P(4)s
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 Quadratic, not exponential.
Projections, color code, braids.
Related to Young subgroups and cosets
It does not become rounder. In fact in some directions it “flattens out”.
and some answers 1 How fast can we calculate the bouncing path for fairly high orders—8, 12, 16, 32? 2 How can we visualize the calculational process and the results? 3 What is the geometry of a high dimensional permutahedron? . 4 How does the geometry of the permutahedron change with n?
Review
Geometric generation of permutation sequences
Dennis Our Questions: Roseman
Permutahedron
Change Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling Quadratic, not exponential.
Projections, color code, braids.
Related to Young subgroups and cosets
It does not become rounder. In fact in some directions it “flattens out”.
1 How fast can we calculate the bouncing path for fairly high orders—8, 12, 16, 32? 2 How can we visualize the calculational process and the results? 3 What is the geometry of a high dimensional permutahedron? . 4 How does the geometry of the permutahedron change with n?
Review
Geometric generation of permutation sequences
Dennis Our Questions: and some answers Roseman
Permutahedron
Change Ringing
Bouncing
Problem List
Cell Structure coloring edges coloring facets Braids
Beam Calculation
Edges in layers
Tiling Quadratic, not exponential.
Projections, color code, braids.
Related to Young subgroups and cosets
It does not become rounder. In fact in some directions it “flattens out”.
Review
Geometric generation of permutation sequences
Dennis Our Questions: and some answers Roseman 1 How fast can we calculate the bouncing path for fairly Permutahedron high orders—8, 12, 16, 32? Change Ringing 2 How can we visualize the calculational process and the Bouncing results? Problem List 3 What is the geometry of a high dimensional Cell Structure coloring edges permutahedron? . coloring facets 4 Braids How does the geometry of the permutahedron change
Beam with n? Calculation
Edges in layers
Tiling Projections, color code, braids.
Related to Young subgroups and cosets
It does not become rounder. In fact in some directions it “flattens out”.
Review
Geometric generation of permutation sequences
Dennis Our Questions: and some answers Roseman 1 How fast can we calculate the bouncing path for fairly Permutahedron high orders—8, 12, 16, 32? Quadratic, not exponential. Change Ringing 2 How can we visualize the calculational process and the Bouncing results? Problem List 3 What is the geometry of a high dimensional Cell Structure coloring edges permutahedron? . coloring facets 4 Braids How does the geometry of the permutahedron change
Beam with n? Calculation
Edges in layers
Tiling Related to Young subgroups and cosets
It does not become rounder. In fact in some directions it “flattens out”.
Review
Geometric generation of permutation sequences
Dennis Our Questions: and some answers Roseman 1 How fast can we calculate the bouncing path for fairly Permutahedron high orders—8, 12, 16, 32? Quadratic, not exponential. Change Ringing 2 How can we visualize the calculational process and the Bouncing results? Projections, color code, braids. Problem List 3 What is the geometry of a high dimensional Cell Structure coloring edges permutahedron? . coloring facets 4 Braids How does the geometry of the permutahedron change
Beam with n? Calculation
Edges in layers
Tiling It does not become rounder. In fact in some directions it “flattens out”.
Review
Geometric generation of permutation sequences
Dennis Our Questions: and some answers Roseman 1 How fast can we calculate the bouncing path for fairly Permutahedron high orders—8, 12, 16, 32? Quadratic, not exponential. Change Ringing 2 How can we visualize the calculational process and the Bouncing results? Projections, color code, braids. Problem List 3 What is the geometry of a high dimensional Cell Structure coloring edges permutahedron? Related to Young subgroups and cosets. coloring facets 4 Braids How does the geometry of the permutahedron change
Beam with n? Calculation
Edges in layers
Tiling Review
Geometric generation of permutation sequences
Dennis Our Questions: and some answers Roseman 1 How fast can we calculate the bouncing path for fairly Permutahedron high orders—8, 12, 16, 32? Quadratic, not exponential. Change Ringing 2 How can we visualize the calculational process and the Bouncing results? Projections, color code, braids. Problem List 3 What is the geometry of a high dimensional Cell Structure coloring edges permutahedron? Related to Young subgroups and cosets. coloring facets 4 Braids How does the geometry of the permutahedron change
Beam with n? It does not become rounder. In fact in some Calculation directions it “flattens out”. Edges in layers
Tiling Thank you
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