General method - counting pieces of a divided 3D figure?

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Le Douanier

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A three dimensional figure is divided by a straight edged tool such as a knife (I mean to say that the cuts are straight), and the figure remains in its original position. Given a certain figure and a certain number of cuts, how can one find the maximum number of pieces that the figure can be divided into?

Taking it a step further, how can the maximum number of pieces of the outer surface be counted? For example, given a rubber ball (sphere) and a certain number of divisions, the objective would be to count the maximum number of pieces the outer surface of the ball could be divided into.

I've experimented with oranges and such, but it seems like there should be a more elegant or at least vaguely mathematical way to approach this.

Thanks for any help.
 
It SHOULD help you...
what we see there is a 2D (circle), with 7 pieces after 3 cuts.

Now make that the top view of a 3D cylindrical piece of cheese;
instead of the 3rd cut being as shown, make it an horizontal cut, say cutting the cylinder in half;
you'll then have 8 pieces instead of 7.

I was more giving you this site as a hint.
 
Denis said:
… what we see there is a 2D (circle), with 7 pieces after 3 cuts.

Now make that the top view of a 3D cylindrical piece of cheese;
instead of the 3rd cut being as shown, make it an horizontal cut,
say cutting the cylinder in half;

you'll then have 8 pieces instead of 7 …
Click to expand...


And, one might think, if you angle all three cuts, such that the center triangle is the base of a pyramid=shaped piece with its tip located halfway down the cylinder, then the bottom half of the cylinder will also be cut into seven pieces, for a total of 14 pieces from three cuts.

If one were to think that, then would they be correct?

 
Denis said:
It SHOULD help you...
what we see there is a 2D (circle), with 7 pieces after 3 cuts.

Now make that the top view of a 3D cylindrical piece of cheese;
instead of the 3rd cut being as shown, make it an horizontal cut, say cutting the cylinder in half;
you'll then have 8 pieces instead of 7.

I was more giving you this site as a hint.
Click to expand...
I can play around with various combinations endlessly, but I can't see a way to definitively say, "This is the maximum number," instead of, "There are at least ___ pieces..."

I don't know if a generalized formula that works for various polyhedra can be derived. This is just a problem I've been thinking about on my own, so I can't be sure. The cylinder is simpler than, say, a sphere, because it can be thought of as a circle with depth. With simple figures it's not so hard to maximize the pieces through trial and error, but many-sided figures are too complicated to work without a more efficient approach.

Should I be thinking about the geometric properties of the particular shape I'm working with, or angles formed by line segments, or something else?
 
Le Douanier said:
Should I be thinking about the geometric properties of the particular shape I'm working with, or angles formed by line segments, or something else?
Click to expand...
...something else...like inventing a cutting machine :!:

Seriously, I see no "sensible" solutions unless all cuts are vertical or horizontal. Mark?
 
Denis said:
Le Douanier said:
Should I be thinking about the geometric properties of the particular shape I'm working with, or angles formed by line segments, or something else?
Click to expand...
...something else...like inventing a cutting machine :!:

Seriously, I see no "sensible" solutions unless all cuts are vertical or horizontal. Mark?
Click to expand...

Well, if a program can be made to generate the max pieces for a circle... maybe figuring that out and applying the circle method by framing other shapes in terms of circles :?:
 
Le Douanier said:
Well, if a program can be made to generate the max pieces for a circle... maybe figuring that out and applying the circle method by framing other shapes in terms of circles :?:
Click to expand...
YA!! And that's why I gave you that link ... now get going :idea:
 
Ok, so I see the logic and understand the solution. Now, to solve the second part of my problem, that of counting the maximum number of pieces the outer surface can be divided into. I have a feeling it'll be: max pieces - number not touching the outer surface = max outer surface pieces. But there's always the case of the same number of pieces being reached by multiple patterns that may differ in number of outer/inner pieces. Maybe the ratio of surface area to total area has something to do with it...

hm.
 
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