Cutting pizza and Lazy Caterer's sequence generalizations!?

Question

Here's a fun one: it is possible to divide a (circular) pizza with n cuts into more than 2n pieces (not necessarily equal, and the cuts do not all go through the center or bisect the pizza).

- the ordinary way, n cuts all through the center gives 2n uniform pieces and it is trivial to make them equal size.

However the MAXIMUM number of pieces p(n) for n cuts is called the lazy caterer's sequence and goes like this: 1, 2, 4, 7, 11, 16, 22, 29...
See e.g. the Wikipedia picture of the pancake cut into 7 pieces by 3 cuts (the middle piece is a (possibly equilateral) triangle)
http://en.wikipedia.org/wiki/Lazy_caterer%27s_sequence
Note: "equal-area" does NOT imply "uniform"!

So anyway my questions are these:
a) for n cuts, define the general algorithm to make the cuts
b) what is the ratio r(n) of largest to smallest pieces area?
Try e.g. computing r(3) and r(4)
c) is there a more useful sequence of cuts such that 2n

Addenda (I ran out of room...):
- the cuts must be straight-line, no corners allowed!
- the general lazy-caterer number of pieces is
p(n) = (n²+n+2)/2 = =n(n+1)/2 +1 = C(n+2,2)
; see Wikipedia for proof and justification

c) is there a more useful sequence such that 2n
A property of the ordinary (symmetric) cutting scheme which we seem to lose with lazy-caterer is that all symmetric cutting schemes can further be subdivided, e.g. 3 cuts give us 6 equal-area pieces, but using those 3 cuts, 3 further cuts can then give us 12 equal-area pieces, 6 more cuts to 24 pieces etc.
So again, can you find a more useful sequence of cuts such that 2n but preserving some of the ability to subdivide? Must a cut-sequence necessarily have some sort of (radial) symmetry to preserve this subdivision property?

For diagrams, see
http://www.research.att.com/~njas/sequences/A000124.gif
which has diagrams of p(3)→7 cuts
p(4)→11 and p(5)→16 cuts.
Note that we can make the 7-cut case radially symmetric, but not the higher-order cases. So, can we say anything at all useful about e.g. the 11-cut case? i.e. some constraints on those annoying subdivided triangles? How would you numerically optimize the 11-cut case?

Can anyone find a closed-form for the areas of the three different types of pieces in the 7-cut case (radially symmetric version)? requires a little integration.

[Finally, if we relax the requirement for cuts to be purely straight lines e.g. we allow one corner (of some angle), we can achieve equal-area pieces, at least in the 7-cut case. But that kind of makes it a different problem?]

Answer 1

If you really want the expression for the area of the largest pieces in the p(3) case, then if x = distance from center to chord in a unit circle, it's:

(1/3)(π - 3√3 x² - 3(Arcsin((1/2)(√(1-x²) - √3 x)) -(1/2)(√(1-x²) - √3 x)(√(1-(1/4)(1-x²) - √3 x)²)) + (√3)(1/4)(1-x²) - √3 x)²)

A total mess, for sure. For 4 of the smaller areas to be roughly equal, it works out to 4 areas of about 0.200258 and 3 areas of about 0.780188, and I imagine that's the ratio r(3) that you seek, 0.780188 / 0.200258 = 3.8959. Forget about r(4), that's too much work.

The procedure of the cuts can be imagined to be performed on 2 perpendicular axes, as in the following sequence: (1, 5), (2, 4), (3,3), (4,2), (5,1). As an aside, if this is taken to the infinite limit, the envelope is the Astroid plane curve. (See Trammel of Archimedes). Once this is understood, the method for using curved cuts in the circle to create equal sized pieces can be worked out. Instead of straight cuts, make the cuts along lattice lines, so that each cut has 1 corner. Then you do a conformal mapping of this figure onto a circle, and with a little further refinement, all the cut pieces can have equal areas. See link on conformal mapping, as well as on the astroid.

Another matter regards symmetry of the cuts. Obviously, any n cuts yielding the maximum number of pieces can be arranged with 1 axis of mirror symmetry, as described above. What about rotational symmetry, under rotations of 1/3, 1/4, 1/5, etc of the circle the cuts look the same? Yes, it is still possible to arrange 3n cuts way that rotational symmetry is met under 1/3 rotations, even though many of the resulting pieces can be extremely tiny. Unfortunately there's not enough room here to explain this in detail.

Is there a "more useful [scheme] of cuts"? Let me get back to you on that.

Answer 2

That would truthfully provide you the quality feasible threat to get the proper pizza you desired. Trying to reserve it over the cellphone from anybody to whom English is a 3rd language is a dicey proposition...