## The Schläfli Series

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

### The Schläfli Series

A triangle, formed by two sides of a polygon, has as its third side the 'shortchord'. The shortchord is the 'vertex-figure' of the polygon. We designate its square by the lower case letter, so the shortchord2 of P is p.

The diameter of the circum-circle than has d² = 4/(4-a²) or 4=v²+4/d² = 4/d² = 4-v²

The vertex-figure of a regular polyhedron {P,Q} has v² = 4p²/(4-q²), which is a pQo (ie a Q-gon, of side p.

We find its corresponding diameter2 at 4/d² = 4 - 4p²/(4-q²), or multoplying through by 4-q²/4, 4/d² = ((4-q²) - p² )/(4-q²) -> d² = 4(4-q²)/(4-p²-q²).

If v² represents the vertex-figure, and w² represents the edge-figure, that is vertex-of-vertex, then we get the vertex-figure becomes p²v², and the new vertex-figure denominator is 2v²-p²w², where the numerator is 2q²v²

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we use v² = 1, 2, Ø² amd 3 for {3.4,5,6} respectively.

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`  (1)   1    always  (2)   2    always  (3)  3-b   4-a²   = 2(2) - a²(1)  (4)  4-2b     2(3-b)-1(2)    1 = sc(3)  (5)  5-3b    2(4-2b)-1(3-b)  1 = sc(3)  `

Here we find the Schlafli function for {p,3,3,3...}, where a²=1+b, and b is the third chord (first parallel to the edge).

When P=2, 3, 4, 5, this gives (for 2d+), 2n-2, n, 4, and 2-(n-2)/Ø. In the hexagon, b=2, so the 3d polytope at (4-2b) gives 0.

The diameter² of any regular polytope is then 2sch(v)/sch(f), where v is the vertex-figure symmetry and f is the figure symmetry. It does not even touch the polytope.

The value of sch(ab) = sch(a)sch(b), so this allows us to find the diameter of any marked node figure, and we can deal with branching groups.
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