The alternating group is given by products of even numbers of transpositions and so it is relevant for Dn. And it is generated by three-cycles so it is relevant for the threesome. An can be thought of as PSL(F1).
In the letter that I am writing I am describing about a Center which generates four families of polytopes An, Bn, Cn, Dn. The Center is the -1 simplex, which is to say, it is the unexplained unique -1 simplex which should be accompanying as an "empty set" every k-simplex made up of 1-dimensional vertices, 2-dimensional edges, 3-dimensional faces and such in k-dimensions. https://en.wikipedia.org/wiki/Simplex Look for the table of simplexes, see how it relates to Pascal's triangle, and note the unexplained -1 dimensional simplex. I note that the -1 simplex can be conceived as the unexpressed unique center which we can associate with each simplex. For example, it is the center of a point, the center of an edge, the center of a triangle, the center of a tetrahedron, and so on. We can also imagine the Totality of the simplex, which for a 2-dimensional triangle would be the area, and for a 3-dimensional tetrahedron would be the volume, whatever is defined by all of the vertices. So we can imagine that first the Center is all by itself, and then it generates vertices along with the Totality of these vertices, so that the Center expresses itself but is never expressed, always moving on as the new Center of the vertices it has created. The Center is never expressed because each vertex it creates is distinct from the rest and so the Center is ever new.
Well, it turns out that this is just one of four "theological" models of how the Center can behave. We can think of the Center as modeling God and the Totality as modeling Everything. And this particular process generates the Simplexes, which are known as polytope family An because the symmetries of these simplexes are given by the Symmetric group. Which is to say, every vertex is linked to every other vertex, and the only symmetry is that the vertices could be relabelled or renumbered in any possible way. The lack of any additional symmetry also means that each vertex is uniquely "positive" as Kirby keeps saying. So this is a model that "God is good" in every direction.
But there are four families of such polytopes: An: n-dimensional simplexes Bn: n-dimensional cubes Cn: n-dimensional cross polytopes (such as the octahedron) Dn: n-dimensional demicubes (demihypercubes) which I find are best thought of as half-cubes (hemicubes) with double edges.
The symmetries of these four families are captured by their symmetry groups (Coxeter groups of reflections), which are the Weyl groups of the root systems, which are vector bases for the Lie algebras, whose vector "addition" is the analogue (by way of the exponential function) of the matrix "multiplication" in the Lie groups, which are the continuous groups, which are restricted by the possible, allowable "short-cuts" for inverting (undoing) a continuous action, so that there could be a continuous geometry in which nothing gets ripped apart. In other words, I think that the symmetries of the four families of polytopes dictate four geometries, such as Steve distinguishes. And it is interesting what geometries are yielded by the 5 exceptional Lie groups.
There is a paper by Victor Kac http://arxiv.org/pdf/math/9912235.pdf "Classification of infinite-dimensional simple groups of supersymmetries and quantum field theory" where in his final section (page 20) on "Speculations and Visions" he notes that "Each of the four types W, S, H, K of simple primitive Lie algebras (L, L0) correspond to the four most important types of geometries of manifolds: all manifolds, oriented manifolds, symplectic and contact manifolds." I found this quote through a link by John Baez https://golem.ph.utexas.edu/category/2007/10/geometric_representation_theor_3.html about a video lecture given by his co-teacher James Dolan as part of their "Geometric Representation Theory Seminar" http://math.ucr.edu/home/baez/qg-fall2007/ which discusses the relevance of the polytopes to logic and geometry.
John Baez has a series of essays about these geometries which I suppose starts here: http://math.ucr.edu/home/baez/week181.html Note, however, that he swaps the letters Cn and Dn from the way I and many others use them. An yields projective geometry Bn yields conformal geometry Cn yields conformal geometry Dn yields symplectic conformal geometry. Basically, symplectic is relevant when two quantities "position" and "momentum" are related, as by time, energy. Symplectic is, I think, what happens when possiblities tracked in the "complex" quantum reality get manifested (through a natural or human "measurement") as a "real" actuality. Dn is where I imagine circle-folding is relevant, again.
The Center and the Totality distinguish An, Bn, Cn, Dn in the following way. The An simplexes have both a Center and a Totality. For example, a tetrahedron has 1 Center (the center), 4 vertices, 6 edges, 4 faces and 1 Totality (the volume). Well, that's a row in Pascal's triangle. And we see that in each row there is always 1 Center and always 1 Totality at the opposite ends of Pascal's triangle.
Let's consider next the "pascal triangle" which counts the pieces of the cross-polytopes (the orthoplexes, the octahedrons). Here in each row there is a power of two that keeps increasing as follows. An octahedron has 6 vertices, 12 edges and 8 faces. So the pascal triangle is:
1 x 2^0 3 x 2^1 3 x 2^2 1 x 2^3
In other words, it has 1 center, 3 x 2 vertices, 3 x 4 edges and 1 x 8 faces. And it has no Totality, no volume!
This makes sense if we consider what the Center is doing here. It is creating two vertices at a time. We can think of them as "implicit opposites" because there is no way of telling them apart. However, later on, we will see how Dn labels them as "positive" and "negative". Anyways, each new dimension yields 2 new vertices that are connected to all of the previous vertices to keep them all distinct. The Center yields first two unconnected points, then a square, then the octahedron, then the tetracross (the 16-cell). So each vertex is directly connected to every vertex except for its opposite. If you were to connect all of the opposites, then you would have a simplex (but the dimension would be twice as large because the simplex is generated one vertex at at time). Anyways, when all the vertices are related to each other, (except for their opposites!), then we have the 8 faces of the octahedron. That's why the octahedron has, by definition, as we can see from Pascal's triangle, no concept of totality, no volume. It can't because the opposites aren't allowed to be related. They don't need to be related because they are, by definition, opposite.
So that's a rather typical example of where physical reality can be misleading or not the most helpful model. I first came to that conclusion when I tried to understand electricity in terms of "running water". Finally, a teacher explained to me that it can't be thought of that way because the relationship between voltage, amperage, resistance, etc. is simply different and the analogy fails spectacularly. Similarly, the spin of an electron doesn't relate to our physical models. And that seems to be the case with most of modern physics. In general, the Nunez/Lakoff view that mathematical thinking arose to match the experience of the human body seems to me very foreign. Instead, it seems natural to me that mathematical structures reflect our internal, spiritual modeling. Steve's focus on vision seems fruitful and then there arises the question as to whether our visual evolution was driven by our spiritual inner life or our practical outer life. And I'm curious what mathematical thinking is like for the blind and for those conscious in the womb. But I think the latter could be "playing" with the type of philosophical Center and Totality issues that I'm discussing. It's enough to abstractly think two or three abstract "things" and the various ways they may relate.
Now the cubes Bn are just like the cross-polytopes but with the rows of the "pascal triangle" reversed. A cube has 8 points, 12 edges, 6 faces. This means, by analogy above, that is has a Totality but it has no Center! So I will explain what seems to be happening.
Let us start with a Totality and imagine it as an unfolding mirror. The Totality is itself the one initial mirror. We can think of it as defining two opposite directions. Now imagine that mirror dividing itself into two and moving out into those two opposite directions. Imagine that it is thereby opening up a mirror in the perpendicular direction. Now we have a second dimension which also defines opposite directions. Now imagine the totality of all the mirrors dividing to reveal a new dimension with a new mirror. As this process continues, we get 2^N quadrants, that is, N pairs of "implicit opposites". I see that we really don't get vertices, edges, faces. Indeed, the quadrants keeps getting refined at each step. We end up drawing them as "vertices" but it's not like the simplex or octahedron where we add new vertices at each step. Rather, we multiply each quadrant by 2 at each step. Also, in counting the "pieces", we have to be mindful of the different "paths" or "sequences" by which, for example, the faces of a cube arose. We have a Totality, but there isn't any Center because, frankly, there aren't any vertices, either. It's just pairs of mirrors sliding out (to infinity?) and opening up more pairs of mirrors.
Now, theologically, we can imagine that we have the issues that Kirby raises which I suppose come up amongst his Martian and Earthling theologians. Namely, the Martians suppose that in every direction, "God is good". There is no concept of "bad", of negative. But Earthlings have polar valuations. At this point there isn't any moral distinction. It's like "up" and "down", "right" and "left", "forwards" and "backwards", without any moral preference (ignoring connotations!) In the case of the octahedrons (cross-polytopes), each dimension has a single, independent polarity and the Center keeps adding new dimensions. In the case of the cubes, the polarities build on each other, so that each "quadrant" participates in all of the polarities. We could flip our thinking around in the last couple of sentences because these are dual structures, that is, we could say that each triangular face in an octahedron involves a choice from the three polarities (dimensions) and thus defines a "quadrant". In every way these two families are dual to each other. The difference is that the Center is building bottom-up whereas the Totality is building top-down. This distinction appears in the Simplex where the Center creates vertices whereas the Totality could be thought of as getting rid of vertices, in the opposite direction.
Finally, we can explain the Dn polytopes. I have found that these are actually poorly or inconsistently defined. How could that be for such an important object? Well, partly, it seems that this math seems "peripheral", too concrete and simple yet messy for those mathematicians and physicists trying to make big discoveries without having to imagine a "big picture". But also, there is a great book by Imre Latakos, "Proofs and refutations", which shows how in his dialogue about Euler's characteristic formula how mathematicians make up their definitions as they go along, and then remake them as they feel they need, quite arbitrarily, in fact. So I've found inconsistent descriptions of these polytopes, especially as to whether they have single edges or double edges. It's just a philosophical distinction because the symmetry group is the same in each case, which is what matters to them. So I offer a philosophical answer.
Typically, the Dn polytopes are defined as the demicubes (demihypercubes) which are gotten by starting with a cube and taking half of its vertices. In three dimensions, this means that a cube's vertices can be split into the vertices of two tetrahedrons, for example, if we color alternating vertices white and black. This can be done for an N-dimensional cube as well, in which case we take half of the vertices, say, the black ones, and relink them by adding new edges between "second nearest neighbors". The object we get will not be a regular polytope, that is, not all the faces will be the same, but there will be at least two different kinds, as with the cuboctahedron. However, every vertex will look the same, nevertheless, and so it will be a uniform polytope. And the symmetry group will be almost the same as for the cube. The symmetry group for the N-cube includes the symmetric group SN which relabels all of the "vertices" but also the group of reflections Z2^N because you can reflect the cube (or octahedron) in each dimension (they are built of opposites!) which is not true for the simplex (no sense of opposites). Well, for the Dn polytopes we only allow even reflection sequences of reflections. If you reflect once, then you have to reflect in some direction yet again, because we've trashed every other vertex, or so to say, we've prohibited odd sequences of reflections.
Let's instead think of folding the cube! (or sphere!) In order to do this, we have to distinguish a "vertex" (a quadrant) in our cube. And we will identify it with the vertex which is opposite to it in every way, in every dimension. Now they are one and the same, and likewise, we identify in this direction each pair of vertices. So we get a spiky construct that looks like, and is, a coordinate system. Each edge is actually a double edge. Indeed, from the "pascal triangle" for the Dn polytopes, it is most elegant to have double edges instead of single edges. Then this pascal triangle is actually the sum of two pascal triangles, one for bottom-up simplexes and the other for top-down cubes. These spiky constructs are similar to the "half cubes" (hemicubes). However, we have to add a double edge between the tips of each "vector" in the coordinate system. This makes each coordinate system into a simplex (all vertices are related!) with double edges and a distinguished "origin". The angles look 90-45-45, though, because the origin is special. What are the the pieces of our coordinate system? There are two kinds because we have a double perspective:
Instead of imagining this as a "coordinate system" with the vector tips related (which reminds me of Kirby's "closing the lid" operator), we can also imagine these as cubes with double edges and also "reinforcements" on every 2-dimensional face which connects the opposite corners, making an X of double edges on each face. That seems to yield the same construct.
Even as I described it, it's messier than the others. I found an open access paper on it by R.M.Green, "Homology representations arising from the half cube" http://www.sciencedirect.com/science/article/pii/S0001870809001017 and there is also a paper by Daniel Pellicer on a very much related construct, "The Higher Dimensional Hemicubeoctahedron": https://books.google.lt/books?id=HarWCwAAQBAJ&printsec=frontcover#v=onepage&q&f=false The latter construct is very similar: Imagine the vertices of an n-dimensional cube and make squares out of every pair of opposite edges (carving up the middle). Look at the "cube" as if it were an octahedron and place, alternating, half of the simplexes that you would need to cover it all. So there is a "cubic" inside and a "simplex" outside.
Philosophically and even theologically, I think that Dn is the whole point of An, Bn, Cn. We started with An having both a Center and a Totality. We can think of it has having arisen from an initial tension in the original "implicit opposites", Center and Totality, before they found expression. Then we saw that instead, if we created in terms of such "opposites", we could generate a series Cn with a Center but no Totality, as with the octahedrons. Or we could generate a series Bn with a Totality but no Center. Now, understandably, Dn is the series with no Center and no Totality. That is, it has an Anti-Center and an Anti-Totality. Each pair of opposite vertices can be fused together to create a Coordinate System, an Origin which is an Anti-Center. With regard to that coordinate system, suddenly every polarity becomes explicit. One direction is with the vectors of the coordinate system, and one direct is against. So we have "good" and "bad" made plain in every direction. But there is one direction which is above it all, namely the two fused vertices who formed the Origin. Apparently, they chose one corner, and from that corner everything emmanates as "good" towards the other corner, but "bad" in the opposite direction. Or as we say in math, "positive" and "negative". And, apparently, you can't try to get out of choosing, you can't just meet in the center, because you would then lose a dimension. The Totality insists on the space being full, one way or the other, and all of the N-cube's cells likewise must assert which way is the "good" way. Of course, which direction is "good" is completely arbitrary. But the Center is above it all. I expect that the point of this all is to model what I think is the big truth, which is that, "God doesn't have to be good. Life doesn't have to be fair."
A related way to think about this all is in terms of Christopher Alexander's books "The Nature of Order". Here is a picture of a sequence of "wholeness preserving transformations" which very much bring to my mind what I've described for An, Bn, Cn...
Well, suppose we already live in a built environment. Then the question is, how can we restructure it? And that is what I think Dn is all about.
So the Dn is a kind of "sphere folding". But I suppose what's important is also "sphere unfolding"? That is, abandoning our particular Anti-Center, our particular key dimension for orienting ourselves as to what is "good" and what is "bad". I think that is what I mean when I say: God doesn't have to be "good".
I started writing this letter to share my experiences about circle folding. I hope I've at least suggested that I have reason to believe it can be informative about the most central issues in math and life. I've expressed that in ideas and language that I personally am more familiar with. I've ended up writing here about my own explorations. For my own sake, I'm encouraged that the kinds of models that I think are most basic for life seem to be at the very heart of mathematics and help us sense how and why it all gets generated.
I wish to offer some pictures at some point.
Thank you to all of us for giving us our various worlds.
I look forward to sharing some great thoughts about this as regards the binomial expansion of the simplexes https://en.wikipedia.org/wiki/Simplex I will explain the missing diagonal "-1". It's basically the imagined center of the simplex, from which we keep creating a new point. We can expand (unlabelled + labelled)**N and when they are all unlabelled, we get the center. When we treat the center as a new point, then we get the next simplex.
I'm thinking that will relate somehow to the "field with one element", an imaginary object in mathematics: https://en.wikipedia.org/wiki/Field_with_one_element
Also, the simplexes seem to be very much related to the unitary groups An, the most important of the Lie groups: https://en.wikipedia.org/wiki/Coxeter%E2%80%93Dynkin_diagram
All of this to confirm the importance of the "tetrahedral" point of view.