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http://math.ucr.edu/home/baez/week186.html dynkin diagram and q polynomial

Coxeter group Dn and hemicubes

Hemicubeoctahedron is half-cube (on the inside) and half-orthogon (on the outside). It is thus without a center (?) and without a volume (?)

Note that it wants to choose half of the orthogon. But which half? this would depend on the coordinates. But we are working coordinate-free. Thus the natural answer is ambiguity, as in a mixed quantum state. It is the same ambiguity that is in the internal tension of the original center, "those things are which show themselves to be" - "to be or not to be?" So it is modeling that ambiguity. And that ambiguity is modeled locally at all points. And that is how the simplex is adorned by local growth.

Dn demihypercube construction

Half measure polytope

• Also known as a demihypercube in n-space, n>2. The uniform polytope constructed by completely truncating the alternate vertices of a measure polytope, that is, truncating half its vertices by hyperplanes that pass through a truncated vertex’s edge-neighboring vertices. If n=2, this truncation of a square leaves only the square’s diagonal, which is not a polygon. But when n=3, this truncation of a cube produces a regular tetrahedron, and when n=4, it produces a regular hexadecachoron from a tesseract. For n>4, demihypercubes are no longer regular, only uniform, with two different kinds of facets: 2n (n–1)-dimensional demihypercubes and 2n–1 (n–1)-dimensional simplexes. The names of the demihypercubes are constructed by prefixing demi- to the name of a measure polytope: demipenteract, demihexeract, demihepteract, and so forth. As noted above, a demicube is a regular tetrahedron, and a demitesseract is a regular hexadecachoron.
• The vertex figure of a half measure polytope H in n-space is a rectified (n–1)-dimensional simplex (the simplex has n vertices and n facets). The facets of the rectified simplex are (1) n (n–2)-dimensional simplexes, which are the vertex figures of the (n–1)-dimensional simplex facets of H, and (2) n rectified (n–2)-dimensional simplexes, which are the vertex figures of the (n–1)-dimensional demihypercube facets of H. Thus, for example, the vertex figure of the demicube is a rectified triangle, which is a smaller triangle, the vertex figure of a tetrahedron (the demicube). The vertex figure of the demitesseract is a rectified tetrahedron, which is an octahedron, the vertex figure of a regular hexadecachoron (the demitesseract). And so on.
• Euclidean n-space can always be uniformly honeycombed by half measure polytopes and cross polytopes: In the regular honeycomb of n-space by measure polytopes, remove alternate vertices, thereby transforming each measure polytope into a half measure polytope, and fill the gaps with n-dimensional cross polytopes (the vertex figures of the measure polytope honeycomb). In the plane, this changes the checkerboard tiling into another checkerboard tiling rotated 45° to the original; in 3-space this produces the uniform honeycomb of regular tetrahedra and octahedra; and in 4-space this produces the regular honeycomb of hexadecachora
• Vertices: Take those coordinates (of a cube) which have an even number of minus signs.
• Edges: ?

The polytopes are irregular because the Dynkin diagram is not a path but has a branch. Thus there should be not one center or volume but rather the "generation" of the polytope should derive from two perspectives. These two perspectives may relate to the bipartition of the demicube which itself is a bipartition of the cube. The two perspectives are related by double edges.

There should be two different, complementary, flags of faces. And they should relate to the cross polytopes - one flag should be like the cross polytopes and the other flag should be the opposite - and together they should form a cube.

The pascal triangle can be explained by noting that we have the fusion of two paths in the Dynkin diagram, one of length 3 and one that grows without bound. The first nontrivial example is the 16 cell.

The purpose is to split the perspective. The human perspective is given by the short path.

The Higher Dimensional Hemicubeoctahedron Daniel Pellicer, in: Symmetries of Graphs, Maps and Polytopes. 2014.

http://www.sciencedirect.com/science/article/pii/S0001870809001017 Homology representations arising from the half cube

http://math.ucr.edu/home/baez/week187.html The Coxeter group for Dn is the subgroup of the symmetries of the n-dimensional cube generated by permutations of the coordinate axes and reflections along pairs of coordinate axes.

http://link.springer.com/chapter/10.1007/978-3-642-04295-9_1 geometry of cuts and metrics

http://arxiv.org/abs/1506.06702 paper about 16 cube

nim and sierpinski demihypercube

John Baez on demicubes

http://link.springer.com/chapter/10.1007/978-3-642-21590-2_15 nth roots of pitch class inversion

Coxeter groups and polytopes

• dynkin diagram coxeter group An reflections are encoded by the sequence 1, q2, q3, ..., q*(n-1)
• symmetry group is Sn
• Koch snowflake is an illustration of An for all n.
• Bn hypercube construction given by having the volume grow by having the volume be conceived as a mirror from which two mirror images arise on either sides. Thus mirror points are linked by edges.
• Cn cross-polytope construction given by having two points extend from the center in opposite directions. Each of the points links to all of the existing vertices. The old volume gives way to a new volume of one dimension higher. The initial C is given by two unconnected points. And these two points came from the center, which is why they are unconnected.
• Thus we have duality of adding two opposing vertices or adding a hyperplane with two reflections, of growing the center or the volume. We have the duality of vertices and facets.

Symmetry groups

• An: symmetric group
• Bn: A deck of n cards where card j has j written on one side and -j on the other side. All possible arrangements of stacks of cards with orientation.
• Dn: corresponds to all arrangements of stacks of n cards with orientation with an even number of cards turned over (pairs of reflections).

Exceptional groups

Also this suggests that the exceptional groups are modeling interactions of limited human perspectives.

Generative center

Centras: apibendrinimas. Išrašus jį atsiranda matas.

Projective space. All lines that go through the origin. Natural origin = "center" of simplex. Natural infinity = fold out the next point = so vertices are halfway between Center and Infinity.

center is what gives 3 rather than 2. center is what links involutions in the dynkin diagram.

the linked involutions, the edges, are numbered and are added as the normal form which is preserved.

An add the center along with edges from the center to the other vertices. Then you can revision in a higher dimension.

Q-analogue

q-characteristic

0=q=infinity jų požiūriu

Visa šeima įvairių q - iš esmės panašūs - bet mes esame keisti q=1

0=1=infinity mes esame keisti.

All of the nodes chip in a weight "q" to allow the new weight to be distinguished from the rest "q^k". They all go down by one (temporarily?) to stay distinguished, to meet their previous obligations.

Young diagrams - q-analogue Chu

q-analogue of a simplex is a projective space over a finite field homology

Theological interpretation

Would like to believe that God is good. Eternal life, eternal learning is driven by the building up that God inspires. And all learning is dependent. But at any height there is the dual: we can unlearn (or tear down) whatever we learn (or build up). Unlearning is independent like the cube. But in between is the space of eternal life. It is based on the idea that God doesn't have to be good, life doesn't have to be fair. We can unlearn in order to learn. And ultimately we hope that truly God is good, that truly we can learn without unlearning. That is what we are demonstrating. So it is related to John Harland's quest about learning.

Readings

• A Leisurely Introduction to Simplicial Sets

Coxeter groups

Lecture notes by Federico Ardila

Videos

• Lecture 10 properties of Bruhat order
• Lecture 15 Mobius function Eulerian characteristic Eulerian posets
• Lecture 20 Dihedral group D5
• Lecture 25 Contragradient action Coxeter groups are automatic Root poset
• Lecture 30 Root systems for An and Bn
• Lecture 31 Bn. Root system Coxeter group
• Lecture 32 Crystallographic root systems Lie groups Cartan matrix
• Lecture 33 Coxeter matrix mij and aij aji
• Lecture 34 Rank 2 Dynkin diagrams
• Lecture 35 Coxeter group reflection group bipartite
• Lecture 36 Weyl group is finite if bilinear form is positive definite
• Lecture 37 Group representation sum of irreducibles
• Lecture 38 Weyl group finite iff bilinear form is positive definite
• Lecture 39 Classification of finite Coxeter groups
• Lecture 40 Proof
• Lecture 41 Proof
• Lecture 42 Classification of regular polytopes

https://ecommons.cornell.edu/handle/1813/3206 catalan and coxeter

https://ecommons.cornell.edu/handle/1813/17339 coxeter dynkin interview

https://en.wikipedia.org/wiki/Conway_polyhedron_notation

polyhedron operators https://en.wikipedia.org/wiki/Alternation_(geometry)

Think of -1-cell as the center (of all things), the spirit. And think of 0-cell not simply as a point but as a 0-dimensional open arc (the point shell) with regard to that center (the spirit). The point shells are glued onto the spirit, and similarly, open arcs are glued onto point shells, and so on, inductively.

Simplexes observe coordinate systems.

Greg Friedman. An elementary illustrated introduction to simplicial sets.