[math-fun] Sodalite (is it sugar-free?)
As a schoolboy (I know --- it's difficult to imagine!) I became interested in a Euclidean d-space polytope I'll call TKD_d : the convex hull of points assigned the same value of multinomial coefficient, in some layer of the obvious generalisation of Pascal's triangle from binomials to d+1 variables (assuming most general position). Another way to envisage its vertices and edges is the Cayley graph of permutations on n+1 symbols, generated by adjacent transpositions. TKD_d has (d+1)! vertices, 2^(d+1) - 2 facets (if d > 1 , at any rate). TKD_1 is a just line segment; TKD_2 is a regular hexagon; TKD_3 is an Archimedean tetrakaidecahedron (truncated octahedron) with 6 square and 8 hexagonal faces; TKD_4 has 120 equal tetrahedral vertices, 180 edges, 90 faces, 30 cells comprising 10 tetrakaidecahedra and 20 hexagonal prisms. Two questions: For what d does TKD_d tile d-space? (It surely does so for d = 1,2,3 --- what about d = 4 ?) Is (the edge-skeleton of) such a tiling O'Keeffe's "d-dimensional sodalite" net, described cryptically as "holes in the A_d^* lattice" on page 2387 of J. H. Conway, N. J. A. Sloane (1997) "Low–dimensional lattices. VII. Coordination sequences" http://neilsloane.com/doc/Me220.pdf ? Fred Lunnon
A.k.a. permutahedron. It tiles d-space for all d. It's the Voronoi cell of either the A_d lattice or its dual. Scott On Sun, Feb 8, 2015 at 12:11 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
As a schoolboy (I know --- it's difficult to imagine!) I became interested in a Euclidean d-space polytope I'll call TKD_d : the convex hull of points assigned the same value of multinomial coefficient, in some layer of the obvious generalisation of Pascal's triangle from binomials to d+1 variables (assuming most general position).
Another way to envisage its vertices and edges is the Cayley graph of permutations on n+1 symbols, generated by adjacent transpositions.
TKD_d has (d+1)! vertices, 2^(d+1) - 2 facets (if d > 1 , at any rate). TKD_1 is a just line segment; TKD_2 is a regular hexagon; TKD_3 is an Archimedean tetrakaidecahedron (truncated octahedron) with 6 square and 8 hexagonal faces; TKD_4 has 120 equal tetrahedral vertices, 180 edges, 90 faces, 30 cells comprising 10 tetrakaidecahedra and 20 hexagonal prisms.
Two questions:
For what d does TKD_d tile d-space? (It surely does so for d = 1,2,3 --- what about d = 4 ?)
Is (the edge-skeleton of) such a tiling O'Keeffe's "d-dimensional sodalite" net, described cryptically as "holes in the A_d^* lattice" on page 2387 of J. H. Conway, N. J. A. Sloane (1997) "Low–dimensional lattices. VII. Coordination sequences" http://neilsloane.com/doc/Me220.pdf ?
Fred Lunnon
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There's a Wiki article about the permutohedron: < http://en.wikipedia.org/wiki/Permutohedron >. I think you can think of this as the convex hull of its vertices -- e.g., the ordered n-tuples comprising all permutations of {1, 2, 3, . . ., n} in R^n. Call this convex hull, the n-permutohedron, P(n). The symmetry in its definition proves that all n! permutations of the vertices extend to isometries of the polytope. According to Wikipedia, the n-permutohedron tiles n-space for all n: Each permutohedron P(n) lies in the affine hyperplane Q of R^n defined as Q := {(x_1,...,x_n) in R^n | x_1 + ... + x_n = n}. Let lattice L in R^n be defined by L := {(k_1,...,k_n) in Z^n | k_1+...+k_n = 0 AND all k_i-k_j == 0 mod n} Then L acts as a group of translations on Q, and the translations {gP(n) | g in L} of P(n) tile R^n. (I became interested in P(n) because any two of these tiles gP(n), hP(n) are either disjoint or intersect on an (n-1)-face, making this tiling a good setting for n-dimensional Hex* on, when n is even.) --Dan ____________________________________________________________________ * A game I invented in grad school. I tried to market it, but there were manufacturing problems.
On Feb 8, 2015, at 12:42 PM, Scott Huddleston <c.scott.huddleston@gmail.com> wrote:
A.k.a. permutahedron. It tiles d-space for all d. It's the Voronoi cell of either the A_d lattice or its dual. Scott
On Sun, Feb 8, 2015 at 12:11 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
As a schoolboy (I know --- it's difficult to imagine!) I became interested in a Euclidean d-space polytope I'll call TKD_d : the convex hull of points assigned the same value of multinomial coefficient, in some layer of the obvious generalisation of Pascal's triangle from binomials to d+1 variables (assuming most general position).
Another way to envisage its vertices and edges is the Cayley graph of permutations on n+1 symbols, generated by adjacent transpositions.
TKD_d has (d+1)! vertices, 2^(d+1) - 2 facets (if d > 1 , at any rate). TKD_1 is a just line segment; TKD_2 is a regular hexagon; TKD_3 is an Archimedean tetrakaidecahedron (truncated octahedron) with 6 square and 8 hexagonal faces; TKD_4 has 120 equal tetrahedral vertices, 180 edges, 90 faces, 30 cells comprising 10 tetrakaidecahedra and 20 hexagonal prisms.
Two questions:
For what d does TKD_d tile d-space? (It surely does so for d = 1,2,3 --- what about d = 4 ?)
Is (the edge-skeleton of) such a tiling O'Keeffe's "d-dimensional sodalite" net, described cryptically as "holes in the A_d^* lattice" on page 2387 of J. H. Conway, N. J. A. Sloane (1997) "Low–dimensional lattices. VII. Coordination sequences" http://neilsloane.com/doc/Me220.pdf ?
Fred Lunnon
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Despite my earlier casual dismissal of the alternative construction for the 3-space honeycomb of truncated octahedra, more leisurely contemplation suggests I may have been a little hard on myself (such modesty!). In fact, "omni-truncating" the hypercube tiling of d-space leads to another sequence, which coincides with tiling permutohedra at d = 3 . For d = 2,3,4, these are resp. the square tiling, sodalite, and (I think) the regular honeycomb of 24-cells meeting in tesseract vertices. Do this sequence or some associated lattice have recognised names? Have the coordination sequences been conjectured or proved? Fred Lunnon On 2/8/15, Scott Huddleston <c.scott.huddleston@gmail.com> wrote:
A.k.a. permutahedron. It tiles d-space for all d. It's the Voronoi cell of either the A_d lattice or its dual. Scott
On Sun, Feb 8, 2015 at 12:11 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
As a schoolboy (I know --- it's difficult to imagine!) I became interested in a Euclidean d-space polytope I'll call TKD_d : the convex hull of points assigned the same value of multinomial coefficient, in some layer of the obvious generalisation of Pascal's triangle from binomials to d+1 variables (assuming most general position).
Another way to envisage its vertices and edges is the Cayley graph of permutations on n+1 symbols, generated by adjacent transpositions.
TKD_d has (d+1)! vertices, 2^(d+1) - 2 facets (if d > 1 , at any rate). TKD_1 is a just line segment; TKD_2 is a regular hexagon; TKD_3 is an Archimedean tetrakaidecahedron (truncated octahedron) with 6 square and 8 hexagonal faces; TKD_4 has 120 equal tetrahedral vertices, 180 edges, 90 faces, 30 cells comprising 10 tetrakaidecahedra and 20 hexagonal prisms.
Two questions:
For what d does TKD_d tile d-space? (It surely does so for d = 1,2,3 --- what about d = 4 ?)
Is (the edge-skeleton of) such a tiling O'Keeffe's "d-dimensional sodalite" net, described cryptically as "holes in the A_d^* lattice" on page 2387 of J. H. Conway, N. J. A. Sloane (1997) "Low–dimensional lattices. VII. Coordination sequences" http://neilsloane.com/doc/Me220.pdf ?
Fred Lunnon
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Sounds like the D_n root lattices. See Ch 4 of Sphere Packings, Lattices and Groups. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Sat, Feb 14, 2015 at 11:11 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Despite my earlier casual dismissal of the alternative construction for the 3-space honeycomb of truncated octahedra, more leisurely contemplation suggests I may have been a little hard on myself (such modesty!).
In fact, "omni-truncating" the hypercube tiling of d-space leads to another sequence, which coincides with tiling permutohedra at d = 3 . For d = 2,3,4, these are resp. the square tiling, sodalite, and (I think) the regular honeycomb of 24-cells meeting in tesseract vertices. Do this sequence or some associated lattice have recognised names? Have the coordination sequences been conjectured or proved?
Fred Lunnon
On 2/8/15, Scott Huddleston <c.scott.huddleston@gmail.com> wrote:
A.k.a. permutahedron. It tiles d-space for all d. It's the Voronoi cell of either the A_d lattice or its dual. Scott
On Sun, Feb 8, 2015 at 12:11 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
As a schoolboy (I know --- it's difficult to imagine!) I became interested in a Euclidean d-space polytope I'll call TKD_d : the convex hull of points assigned the same value of multinomial coefficient, in some layer of the obvious generalisation of Pascal's triangle from binomials to d+1 variables (assuming most general position).
Another way to envisage its vertices and edges is the Cayley graph of permutations on n+1 symbols, generated by adjacent transpositions.
TKD_d has (d+1)! vertices, 2^(d+1) - 2 facets (if d > 1 , at any rate). TKD_1 is a just line segment; TKD_2 is a regular hexagon; TKD_3 is an Archimedean tetrakaidecahedron (truncated octahedron) with 6 square and 8 hexagonal faces; TKD_4 has 120 equal tetrahedral vertices, 180 edges, 90 faces, 30 cells comprising 10 tetrakaidecahedra and 20 hexagonal prisms.
Two questions:
For what d does TKD_d tile d-space? (It surely does so for d = 1,2,3 --- what about d = 4 ?)
Is (the edge-skeleton of) such a tiling O'Keeffe's "d-dimensional sodalite" net, described cryptically as "holes in the A_d^* lattice" on page 2387 of J. H. Conway, N. J. A. Sloane (1997) "Low–dimensional lattices. VII. Coordination sequences" http://neilsloane.com/doc/Me220.pdf ?
Fred Lunnon
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But D_n or D*_n does not coincide with sodalite when d = 3 . Anyway, it's still unclear to me just how far to truncate the hypercube tiling in general, in order to get a sensible sequence ... more rumination required! WFL On 2/14/15, Neil Sloane <njasloane@gmail.com> wrote:
Sounds like the D_n root lattices. See Ch 4 of Sphere Packings, Lattices and Groups.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Sat, Feb 14, 2015 at 11:11 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Despite my earlier casual dismissal of the alternative construction for the 3-space honeycomb of truncated octahedra, more leisurely contemplation suggests I may have been a little hard on myself (such modesty!).
In fact, "omni-truncating" the hypercube tiling of d-space leads to another sequence, which coincides with tiling permutohedra at d = 3 . For d = 2,3,4, these are resp. the square tiling, sodalite, and (I think) the regular honeycomb of 24-cells meeting in tesseract vertices. Do this sequence or some associated lattice have recognised names? Have the coordination sequences been conjectured or proved?
Fred Lunnon
On 2/8/15, Scott Huddleston <c.scott.huddleston@gmail.com> wrote:
A.k.a. permutahedron. It tiles d-space for all d. It's the Voronoi cell of either the A_d lattice or its dual. Scott
On Sun, Feb 8, 2015 at 12:11 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
As a schoolboy (I know --- it's difficult to imagine!) I became interested in a Euclidean d-space polytope I'll call TKD_d : the convex hull of points assigned the same value of multinomial coefficient, in some layer of the obvious generalisation of Pascal's triangle from binomials to d+1 variables (assuming most general position).
Another way to envisage its vertices and edges is the Cayley graph of permutations on n+1 symbols, generated by adjacent transpositions.
TKD_d has (d+1)! vertices, 2^(d+1) - 2 facets (if d > 1 , at any rate). TKD_1 is a just line segment; TKD_2 is a regular hexagon; TKD_3 is an Archimedean tetrakaidecahedron (truncated octahedron) with 6 square and 8 hexagonal faces; TKD_4 has 120 equal tetrahedral vertices, 180 edges, 90 faces, 30 cells comprising 10 tetrakaidecahedra and 20 hexagonal prisms.
Two questions:
For what d does TKD_d tile d-space? (It surely does so for d = 1,2,3 --- what about d = 4 ?)
Is (the edge-skeleton of) such a tiling O'Keeffe's "d-dimensional sodalite" net, described cryptically as "holes in the A_d^* lattice" on page 2387 of J. H. Conway, N. J. A. Sloane (1997) "Low–dimensional lattices. VII. Coordination sequences" http://neilsloane.com/doc/Me220.pdf ?
Fred Lunnon
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I was right first time --- doesn't often happen! Starting with a tiling of d-space by hypercubes, we can progressively truncate at vertices until only centre points of original hypercube cells remain: at this stage the vertices have swelled to hypercubes, and we have the dual tiling (which is of course congruent to the original one). Now suppose instead that we had stoppped half-way: at this stage, swollen vertices would have been congruent to the shrunken cells, and the resulting tiling --- again --- transitive on cells and vertices. This is the sequence of uniform tilings I had in mind: what initially confused me was that the procedure behaves slightly differently in even from odd dimension. As remarked earlier, the cells in d = 1,2,3,4 -space are segments (doubled?), squares, tetrakaidecahedra (truncated octahedra), and icositetrachora (regular 24-cells). Does this sequence of "semi-dual hypercube" tilings have a moniker; and what is known about it? Fred Lunnon On 2/15/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
But D_n or D*_n does not coincide with sodalite when d = 3 .
Anyway, it's still unclear to me just how far to truncate the hypercube tiling in general, in order to get a sensible sequence ... more rumination required!
WFL
On 2/14/15, Neil Sloane <njasloane@gmail.com> wrote:
Sounds like the D_n root lattices. See Ch 4 of Sphere Packings, Lattices and Groups.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Sat, Feb 14, 2015 at 11:11 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Despite my earlier casual dismissal of the alternative construction for the 3-space honeycomb of truncated octahedra, more leisurely contemplation suggests I may have been a little hard on myself (such modesty!).
In fact, "omni-truncating" the hypercube tiling of d-space leads to another sequence, which coincides with tiling permutohedra at d = 3 . For d = 2,3,4, these are resp. the square tiling, sodalite, and (I think) the regular honeycomb of 24-cells meeting in tesseract vertices. Do this sequence or some associated lattice have recognised names? Have the coordination sequences been conjectured or proved?
Fred Lunnon
On 2/8/15, Scott Huddleston <c.scott.huddleston@gmail.com> wrote:
A.k.a. permutahedron. It tiles d-space for all d. It's the Voronoi cell of either the A_d lattice or its dual. Scott
On Sun, Feb 8, 2015 at 12:11 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
As a schoolboy (I know --- it's difficult to imagine!) I became interested in a Euclidean d-space polytope I'll call TKD_d : the convex hull of points assigned the same value of multinomial coefficient, in some layer of the obvious generalisation of Pascal's triangle from binomials to d+1 variables (assuming most general position).
Another way to envisage its vertices and edges is the Cayley graph of permutations on n+1 symbols, generated by adjacent transpositions.
TKD_d has (d+1)! vertices, 2^(d+1) - 2 facets (if d > 1 , at any rate). TKD_1 is a just line segment; TKD_2 is a regular hexagon; TKD_3 is an Archimedean tetrakaidecahedron (truncated octahedron) with 6 square and 8 hexagonal faces; TKD_4 has 120 equal tetrahedral vertices, 180 edges, 90 faces, 30 cells comprising 10 tetrakaidecahedra and 20 hexagonal prisms.
Two questions:
For what d does TKD_d tile d-space? (It surely does so for d = 1,2,3 --- what about d = 4 ?)
Is (the edge-skeleton of) such a tiling O'Keeffe's "d-dimensional sodalite" net, described cryptically as "holes in the A_d^* lattice" on page 2387 of J. H. Conway, N. J. A. Sloane (1997) "Low–dimensional lattices. VII. Coordination sequences" http://neilsloane.com/doc/Me220.pdf ?
Fred Lunnon
My favorite way to understand these uniform tessellations you're looking at.is via reflection groups and ringed Dynkin diagrams. On Mon, Feb 16, 2015 at 8:36 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I was right first time --- doesn't often happen!
Starting with a tiling of d-space by hypercubes, we can progressively truncate at vertices until only centre points of original hypercube cells remain: at this stage the vertices have swelled to hypercubes, and we have the dual tiling (which is of course congruent to the original one).
Now suppose instead that we had stoppped half-way: at this stage, swollen vertices would have been congruent to the shrunken cells, and the resulting tiling --- again --- transitive on cells and vertices. This is the sequence of uniform tilings I had in mind: what initially confused me was that the procedure behaves slightly differently in even from odd dimension.
As remarked earlier, the cells in d = 1,2,3,4 -space are segments (doubled?), squares, tetrakaidecahedra (truncated octahedra), and icositetrachora (regular 24-cells).
Does this sequence of "semi-dual hypercube" tilings have a moniker; and what is known about it?
Fred Lunnon
On 2/15/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
But D_n or D*_n does not coincide with sodalite when d = 3 .
Anyway, it's still unclear to me just how far to truncate the hypercube tiling in general, in order to get a sensible sequence ... more rumination required!
WFL
On 2/14/15, Neil Sloane <njasloane@gmail.com> wrote:
Sounds like the D_n root lattices. See Ch 4 of Sphere Packings, Lattices and Groups.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Sat, Feb 14, 2015 at 11:11 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Despite my earlier casual dismissal of the alternative construction for the 3-space honeycomb of truncated octahedra, more leisurely contemplation suggests I may have been a little hard on myself (such modesty!).
In fact, "omni-truncating" the hypercube tiling of d-space leads to another sequence, which coincides with tiling permutohedra at d = 3 . For d = 2,3,4, these are resp. the square tiling, sodalite, and (I think) the regular honeycomb of 24-cells meeting in tesseract vertices. Do this sequence or some associated lattice have recognised names? Have the coordination sequences been conjectured or proved?
Fred Lunnon
On 2/8/15, Scott Huddleston <c.scott.huddleston@gmail.com> wrote:
A.k.a. permutahedron. It tiles d-space for all d. It's the Voronoi cell of either the A_d lattice or its dual. Scott
On Sun, Feb 8, 2015 at 12:11 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
As a schoolboy (I know --- it's difficult to imagine!) I became interested in a Euclidean d-space polytope I'll call TKD_d : the convex hull of points assigned the same value of multinomial coefficient, in some layer of the obvious generalisation of Pascal's triangle from binomials to d+1 variables (assuming most general position).
Another way to envisage its vertices and edges is the Cayley graph of permutations on n+1 symbols, generated by adjacent transpositions.
TKD_d has (d+1)! vertices, 2^(d+1) - 2 facets (if d > 1 , at any rate). TKD_1 is a just line segment; TKD_2 is a regular hexagon; TKD_3 is an Archimedean tetrakaidecahedron (truncated octahedron) with 6 square and 8 hexagonal faces; TKD_4 has 120 equal tetrahedral vertices, 180 edges, 90 faces, 30 cells comprising 10 tetrakaidecahedra and 20 hexagonal prisms.
Two questions:
For what d does TKD_d tile d-space? (It surely does so for d = 1,2,3 --- what about d = 4 ?)
Is (the edge-skeleton of) such a tiling O'Keeffe's "d-dimensional sodalite" net, described cryptically as "holes in the A_d^* lattice" on page 2387 of J. H. Conway, N. J. A. Sloane (1997) "Low–dimensional lattices. VII. Coordination sequences" http://neilsloane.com/doc/Me220.pdf ?
Fred Lunnon
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Yes and yes then, thanks to Dan and Scott. The webpage http://en.wikipedia.org/wiki/Permutohedron and Dan remark on a simple and elegant coordinatisation for the d-space tiling, embedded in the hyperplane x_1 + ... + x_(d+1) = 0 . I failed to find this, having been wickedly sidetracked by another d = 3 embedding, into 3-space, which (multi-dimensionally speaking) plainly leads nowhere :--- (x,y,z) is a vertex of the 3-space tiling just when these 6 congruences all fail: x +/- y , y +/- z , z +/- x <> 0 (mod 4) . Dan: << (I became interested in P(n) because any two of these tiles gP(n), hP(n) are either disjoint or intersect on an (n-1)-face, making this tiling a good setting for n-dimensional Hex* on, when n is even.) >> Alright, I'll buy it --- why only when n is even? (or should that have read "odd"?) Fred Lunnon On 2/8/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
As a schoolboy (I know --- it's difficult to imagine!) I became interested in a Euclidean d-space polytope I'll call TKD_d : the convex hull of points assigned the same value of multinomial coefficient, in some layer of the obvious generalisation of Pascal's triangle from binomials to d+1 variables (assuming most general position).
Another way to envisage its vertices and edges is the Cayley graph of permutations on n+1 symbols, generated by adjacent transpositions.
TKD_d has (d+1)! vertices, 2^(d+1) - 2 facets (if d > 1 , at any rate). TKD_1 is a just line segment; TKD_2 is a regular hexagon; TKD_3 is an Archimedean tetrakaidecahedron (truncated octahedron) with 6 square and 8 hexagonal faces; TKD_4 has 120 equal tetrahedral vertices, 180 edges, 90 faces, 30 cells comprising 10 tetrakaidecahedra and 20 hexagonal prisms.
Two questions:
For what d does TKD_d tile d-space? (It surely does so for d = 1,2,3 --- what about d = 4 ?)
Is (the edge-skeleton of) such a tiling O'Keeffe's "d-dimensional sodalite" net, described cryptically as "holes in the A_d^* lattice" on page 2387 of J. H. Conway, N. J. A. Sloane (1997) "Low–dimensional lattices. VII. Coordination sequences" http://neilsloane.com/doc/Me220.pdf ?
Fred Lunnon
participants (4)
-
Dan Asimov -
Fred Lunnon -
Neil Sloane -
Scott Huddleston