Classification of a Digon

[Higher_Order]

Is a digon a simplex (it Does have n+1 hypercells and n+1 vertices (vertices are the hypercells)), or is it a hypercube, it has those properties as well.
Is it the only figure classified as both, or is it for this reason classified as neither?

[wendy]

A digon is actually a polygon, ie a 2dimensional thing. One of the features it has is an across symetry.

The diteelon in 1 dimension is the first power of the crind (sphere-making), tegum (orþotope-making), prism products, and the second power of the pyramid product. It is all these things.

[Higher_Order]

Ok, thanks, it occurred to me a while ago, and I've been wondering if for classification that was possible.

[Keiji]

I haven't previously classified the digon as a simplex, however if you define the vertices of an n-simplex to have the positions of (n+1) attracting spheres in n dimensions, then a digon would indeed be a 1-simplex (and a point a 0-simplex).

The digon is definitely a 1-hypercube though, and the nullframe can be considered a 1-hypersphere (two points).

[raumaan]

By the way, i think it might be classified as a 1D hypercube because:
http://www.youtube.com/watch?v=-x4P65EKjt0

[wendy]

The digon is a polygon, not a line-polytope. There is a significant difference.

The thing that corresponds to the first power of the prism, tegum and crind products, and the second power of the pyramid product, is a ditelon, or a a dyad. The surface was correctly described in the article, but the wrong name is used in the title.

A digon is a polygon. This means that it has two edges and two vertices. You can imagine that a 'fat digon' in the form of the intersection of two circles. The normal S mirror is the line that connects the two circle-centres, and reflects one vertex to another. The essential 'A' mirror (which the ditelon does not have), connects the two vertices, and reflects 'across' the digon. The fat digon shrinks to a normal polygon.

The important distinction here is that it has a pair of orthogonal mirrors, the 'truncated digon' is a rectangle. It is the lead member of zero-height prisms that occur in many different dynkin graphs. Coxeter speaks of these as nullitopes and hosotopes (ie {p,2} and {2,p} respectively.

[broken850]

we can use to work with 4D; anyone can posit some arbitrary set of equations that will produce some kind of universe in 4D. The question is whether the result is of any interest in giving us a better understanding of 4D.

[wendy]

Its relevance is that the digon brings into focus that shapes that are not solid have an 'around-space' or things happening around but not in, the space they live. In short, it's like the shape's 'hyper-space'. Specifically, a line-segment has only an order-2 symmetry, which reflects end to end. A digon has an arround-symmetry that leaves the thing undisturbed, but reflects its hyperspace (ie the 2d space it lives in). The order of symmetry is then the product of the surround symmetry (ie the 1d line reflected end to end), and its around symmetry (which leaves the line undisturbed). Now, 2x2=4, and that's the order of a digon.

[Eric B]

I think of a true digon as the non-Euclidian joining of two curved lines on both ends. In Euclidian space, it can only be two line segments occupying the same space, and thus really only one line segment.

[wendy]

A line segment can be classified as a simplex (ie point ^2), or as the first power of a cross or measure polytope. In all cases, the symmetry gives 2.

A digon is a kind of polygon with two sides. Its symmetry is four. A digon exists in a 2d space, and the truncated digon is in general a rectangle, in in particular, a square. The importance is that in symmetries like [p,q,r], there are polytopes that have r copies of p, q at each vertices. When one of these is 2, then it 'disappears', to get r copies of say, p. But the digonal symmetry of crossing mirrors is still there.

[Eric B]

I don't quite understand that definition. A digon would mean a closed figure with a total of two angles and two sides. That can exist in a positively curved space, but collapses to a single line in flat Euclidean space. (The two "angles" are 0°) A rectangle or square is a tetragon: four angles and four sides. I guess if you smoothed two opposite angles into curves, then you would have the non-Euclidean digon. (I call it the "lentil shape").

I also think of it (in Euclidean space) as parallel lines, with the joining angles hypothetically at infinity. Like a d2 6-gon (hexagram) is two triangles, the d3 6-gon (made by extending the lines until they meet), would consist of three sets of parallel lines ("meeting" at infinity. Of course, in positively curved geometry, these lines would meet finitely, producing a "star" made of three "lentil" shape digons).

[Klitzing]

Your observation is completely correct.

You could even use that shape to tile a sphere: like an orange has several slices (lunes), you would get a spherical tiling with m lunes, all meeting at the poles.

Then you could consider this shape to be truncated. That introduces 2 additional m-gons at the poles, while the former digons get truncated into rectangles each. Kind a shape like a spherical version of an m-prism.

Finally, you could do that truncation deeper and deeper, until the rectangles would have zero heights, the digons get reduced to nothing. Then the m-gons would cover a hemi-sphere each. The total fuigure then just consists of 2 faces only, i.e. becomes a di-hedron.

(This is why the related 3D symmetry group for all these mentioned 3D shapes would be called the dihedral one: D_m, having the symmetries of a 2D m-gon plus the orthogonal reflection, which interchanges those poles.)

For sure you could boyle that back to flat-faced polahedra within euclidean 3D (instead of tiles on S2). The prism then still remains non-degenerate. But none of the extremes: that orange-shape degenerates into an m times covered edge, and the dihedron degenerates into a double-cover of the m-gon. That's it.

--- rk