Wythoff polytope (no ontology)
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A Wythoff polytope is one that might be derived by a mirror-edge construction on a symmetry group that has a Coxeter-Dynkin Symbol. This is a set of ''position polytopes'' for which vector algebra can be consistantly applied. | A Wythoff polytope is one that might be derived by a mirror-edge construction on a symmetry group that has a Coxeter-Dynkin Symbol. This is a set of ''position polytopes'' for which vector algebra can be consistantly applied. | ||
| - | + | == Some theory == | |
| - | == Some | + | |
| - | + | ||
Although Mrs A. B. Stott described the process in expansions and contractions from a unit platonic figure, it is easier to understand it in terms of contracting edges of a fully expanded polyhedron. For the octahedral-mirror polyhedra, the figure in question is the '''truncated cuboctahedron'''. | Although Mrs A. B. Stott described the process in expansions and contractions from a unit platonic figure, it is easier to understand it in terms of contracting edges of a fully expanded polyhedron. For the octahedral-mirror polyhedra, the figure in question is the '''truncated cuboctahedron'''. | ||
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The seven octahedral-mirror polyhedra are then derived by setting the vertex, edge or face edges to 0 or 1, as follows. The numbers in brackets represent the edge of the (vertex, edge, face) edges. | The seven octahedral-mirror polyhedra are then derived by setting the vertex, edge or face edges to 0 or 1, as follows. The numbers in brackets represent the edge of the (vertex, edge, face) edges. | ||
| - | * The '''Cube''' (1,0,0) derives by keeping the ''vertex'' edges, but setting the other two to zero. | + | *The '''Cube''' (1,0,0) derives by keeping the ''vertex'' edges, but setting the other two to zero. |
| - | + | *The '''Cuboctahedron''' (0,1,0) derives from keeping the ''edge'' edges at unity, but reducing the vertex and face edges to zero. | |
| - | * The '''Cuboctahedron''' (0,1,0) derives from keeping the ''edge'' edges at unity, but reducing the vertex and face edges to zero. | + | *The '''Octahedron''' (0,0,1) comes from keeping the face-edges, but setting the vertex and edge edges to zero. |
| - | + | *The '''Truncated Octahedron''' (0,1,1) comes from setting just the vertex-edges to zero. | |
| - | * The '''Octahedron''' (0,0,1) comes from keeping the face-edges, but setting the vertex and edge edges to zero. | + | *The '''Truncated Cube''' (1,1,0) comes from sett the face-edges to zero |
| - | + | *The '''Rhombo-Cuboctahedron''' (1,0,1) comes from setting the edge-edge to zero. | |
| - | * The '''Truncated Octahedron''' (0,1,1) comes from setting just the vertex-edges to zero. | + | *The '''Truncated Cuboctahedron''' (1,1,1) is the complete figure itself. |
| - | + | ||
| - | * The '''Truncated Cube''' (1,1,0) comes from sett the face-edges to zero | + | |
| - | + | ||
| - | * The '''Rhombo-Cuboctahedron''' (1,0,1) comes from setting the edge-edge to zero. | + | |
| - | + | ||
| - | * The '''Truncated Cuboctahedron''' (1,1,1) is the complete figure itself. | + | |
One is not restricted to keeping these at 0 or 1. In practice, they can assume any value. | One is not restricted to keeping these at 0 or 1. In practice, they can assume any value. | ||
| - | == From | + | == From edge-length to vectors == |
| - | + | ||
Although the lengths represent the height of the vertex from the opposite mirror, one can represent it as a vector radiating from the centre of the polytope: the polytope (1,1,1) is the ''sum'' of polytopes (1,1,0) and (0,0,1). | Although the lengths represent the height of the vertex from the opposite mirror, one can represent it as a vector radiating from the centre of the polytope: the polytope (1,1,1) is the ''sum'' of polytopes (1,1,0) and (0,0,1). | ||
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The vectors are generally in an oblique coordinate system. The three vectors of the octahedral-mirrors are | The vectors are generally in an oblique coordinate system. The three vectors of the octahedral-mirrors are | ||
| - | * | + | *vertex - (1,1,1) gives the vertices of a cube of edge 2. |
| - | + | *edge - (√2,√2,0) gives the vertices of an cuboctahedron, of edge 2. | |
| - | * | + | *face - (√2,0,0) gives the vertices of an octahedron of edge 2. |
| - | + | ||
| - | * | + | |
Just as the ordinary rectangular coordinates are given by ''all changes of sign'', the resulting points of this system are reflected by ''all permutations, all change of sign''. | Just as the ordinary rectangular coordinates are given by ''all changes of sign'', the resulting points of this system are reflected by ''all permutations, all change of sign''. | ||
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One can preform the dot product of vectors of this set, not so much by converting them to an orthogonal set, but by the method of ''matrix dot''. The matrix-dot means that the one vector is first multiplied by a matrix, and then the dot product is taken. | One can preform the dot product of vectors of this set, not so much by converting them to an orthogonal set, but by the method of ''matrix dot''. The matrix-dot means that the one vector is first multiplied by a matrix, and then the dot product is taken. | ||
| - | Mathematically, it equates to < | + | Mathematically, it equates to A<sub>''i'',''j''</sub> v<sub>''i''</sub> w<sub>''j''</sub>, where ''v'' and ''w'' are vectors. The matrix A<sub>''i'',''j''</sub> is made of the dot product of vectors like (1,0,0) and (0,1,0). |
This is the '''Stott''' matrix, because vectors suggest motion, and the variations along each axis here are exactly the Stott operators. | This is the '''Stott''' matrix, because vectors suggest motion, and the variations along each axis here are exactly the Stott operators. | ||
| - | + | {|class='matbox' style='text-align: center;' | |
| - | + | | ||class='m t l'| ||3||2√2||√2||class='m t r'| || ||class='m t l'| ||2||-√2||0||class='m t r'| | |
| - | + | |- | |
| - | + | |style='padding-left: 1em; padding-right: 0.5em;'|A<sub>''ij''</sub> =||class='m l'| ||2√2||4||2||class='m r'| ||style='padding-left: 3em; padding-right: 0.5em;'|D<sub>''ij''</sub> = {{Over|1|2}}||class='m l'| ||-√2||2||-1||class='m r'| | |
| - | + | |- | |
| - | + | | ||class='m b l'| ||√2||2||2||class='m b r'| || ||class='m b l'| ||0||-1||2||class='m b r'| | |
| - | + | |- | |
| + | | | ||
| + | |- | ||
| + | | ||colspan='5'|Stott Matrix|| ||colspan='5'|Dynkin Matrix | ||
| + | |} | ||
When one calculates from the Stott matrix, a vector that is orthogonal to the plane of the other two vectors (ie a vector normal to v=0), one gets three vectors V, E, F. These vectors are perpendicular to the mirror planes, and the angle they make between each other is the supplement ( 180 deg- x) of the angles between the mirrors. | When one calculates from the Stott matrix, a vector that is orthogonal to the plane of the other two vectors (ie a vector normal to v=0), one gets three vectors V, E, F. These vectors are perpendicular to the mirror planes, and the angle they make between each other is the supplement ( 180 deg- x) of the angles between the mirrors. | ||
| - | This is, in effect, the dynkin symbol of the group, written in matrix form. The entries < | + | This is, in effect, the dynkin symbol of the group, written in matrix form. The entries D<sub>''ij''</sub> are the negative cosines of the angles between mirrors ''i' and ''j''. The diagonal represents the nodes ''i''. |
In practice, it is easier to create the dynkin matrix, directly from the dynkin symbol, and take its inverse to get the stott-matrix. | In practice, it is easier to create the dynkin matrix, directly from the dynkin symbol, and take its inverse to get the stott-matrix. | ||
== Applications == | == Applications == | ||
| - | |||
Apart from finding the diameter of a wythoff figure, these matrices find use in finding the distance between any two points (such as the top and bottom of a segmentotope), from which the square of the height might be found from subtracting it from the square of the lacing-length. | Apart from finding the diameter of a wythoff figure, these matrices find use in finding the distance between any two points (such as the top and bottom of a segmentotope), from which the square of the height might be found from subtracting it from the square of the lacing-length. | ||
Segmentotopes occur as sections (or segments), between rows of vertices of uniform polychora. | Segmentotopes occur as sections (or segments), between rows of vertices of uniform polychora. | ||
Revision as of 21:28, 12 February 2014
A Wythoff polytope is one that might be derived by a mirror-edge construction on a symmetry group that has a Coxeter-Dynkin Symbol. This is a set of position polytopes for which vector algebra can be consistantly applied.
Some theory
Although Mrs A. B. Stott described the process in expansions and contractions from a unit platonic figure, it is easier to understand it in terms of contracting edges of a fully expanded polyhedron. For the octahedral-mirror polyhedra, the figure in question is the truncated cuboctahedron.
This figure has three kinds of edge, which we might designate as vertex between the octagons and squares, edge between the octagons and hexagons, and face, between the hexagons and squares. Were one to increase the length of the vertex edges, the hexagons would remain the same size, but move out radially from the centre. The octagons and squares would generally become irregular.
The seven octahedral-mirror polyhedra are then derived by setting the vertex, edge or face edges to 0 or 1, as follows. The numbers in brackets represent the edge of the (vertex, edge, face) edges.
- The Cube (1,0,0) derives by keeping the vertex edges, but setting the other two to zero.
- The Cuboctahedron (0,1,0) derives from keeping the edge edges at unity, but reducing the vertex and face edges to zero.
- The Octahedron (0,0,1) comes from keeping the face-edges, but setting the vertex and edge edges to zero.
- The Truncated Octahedron (0,1,1) comes from setting just the vertex-edges to zero.
- The Truncated Cube (1,1,0) comes from sett the face-edges to zero
- The Rhombo-Cuboctahedron (1,0,1) comes from setting the edge-edge to zero.
- The Truncated Cuboctahedron (1,1,1) is the complete figure itself.
One is not restricted to keeping these at 0 or 1. In practice, they can assume any value.
From edge-length to vectors
Although the lengths represent the height of the vertex from the opposite mirror, one can represent it as a vector radiating from the centre of the polytope: the polytope (1,1,1) is the sum of polytopes (1,1,0) and (0,0,1).
The base vectors are as in the ordinary vector analysis, the vectors v = (1,0,0), e = (0,1,0), and f = (0,0,1). The edge is the perpendicular of (1,0,0) to the plane v=0, but the vector (1,0,0) points from the centre of the polyhedron to that point.
In vector analysis, each point on the euclidean space represents a position vector, from (0,0,0) to (x,y,z). Now it represents a position polytope, by dropping perpendiculars to each of the planes v=0, e=0, f=0, and reflecting.
The vectors are generally in an oblique coordinate system. The three vectors of the octahedral-mirrors are
- vertex - (1,1,1) gives the vertices of a cube of edge 2.
- edge - (√2,√2,0) gives the vertices of an cuboctahedron, of edge 2.
- face - (√2,0,0) gives the vertices of an octahedron of edge 2.
Just as the ordinary rectangular coordinates are given by all changes of sign, the resulting points of this system are reflected by all permutations, all change of sign.
One can preform the dot product of vectors of this set, not so much by converting them to an orthogonal set, but by the method of matrix dot. The matrix-dot means that the one vector is first multiplied by a matrix, and then the dot product is taken.
Mathematically, it equates to Ai,j vi wj, where v and w are vectors. The matrix Ai,j is made of the dot product of vectors like (1,0,0) and (0,1,0).
This is the Stott matrix, because vectors suggest motion, and the variations along each axis here are exactly the Stott operators.
| 3 | 2√2 | √2 | 2 | -√2 | 0 | ||||||
| Aij = | 2√2 | 4 | 2 | Dij = 1∕2 | -√2 | 2 | -1 | ||||
| √2 | 2 | 2 | 0 | -1 | 2 | ||||||
| Stott Matrix | Dynkin Matrix | ||||||||||
When one calculates from the Stott matrix, a vector that is orthogonal to the plane of the other two vectors (ie a vector normal to v=0), one gets three vectors V, E, F. These vectors are perpendicular to the mirror planes, and the angle they make between each other is the supplement ( 180 deg- x) of the angles between the mirrors.
This is, in effect, the dynkin symbol of the group, written in matrix form. The entries Dij are the negative cosines of the angles between mirrors i' and j. The diagonal represents the nodes i.
In practice, it is easier to create the dynkin matrix, directly from the dynkin symbol, and take its inverse to get the stott-matrix.
Applications
Apart from finding the diameter of a wythoff figure, these matrices find use in finding the distance between any two points (such as the top and bottom of a segmentotope), from which the square of the height might be found from subtracting it from the square of the lacing-length.
Segmentotopes occur as sections (or segments), between rows of vertices of uniform polychora.
