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Cube Division by Planes
Cube division by planes is the problem of determining the regions into which planes divide a cube. For the average below, draw 
planes independently
from the motion-invariant measure on planes,
restricted to those intersecting the cube and normalized
to a probability measure. In terms of a plane's distance 
from a fixed origin and unit
normal vector 
,
the measure is proportional to 
, where 
is spherical area measure
(Santaló 1945). The average number of regions is then
 |
(Finch 2003, p. 482).
The maximum number of regions is the same as for space division by planes, namely
 |
(Yaglom and Yaglom 1987, pp. 102-106). An arrangement of planes in general position
can be scaled and translated so that every region intersects the cube.
For 
,
2, ... planes, the maximum has the values 2, 4, 8, 15,
26, 42, ... (OEIS A000125), a sequence
whose values are sometimes called cake numbers because
of their relation to the cake cutting problem.
See also
Cake Cutting, Cake Number, Cylinder Cutting, Space Division by Planes, Square Division by LinesExplore with Wolfram|Alpha
References
Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.Santaló, L. A. "Valor medio del número de regiones en que un cuerpo del espacio es dividido por
planos arbitrarios." Rev. Unión Mat. Argentina 10, 101-108,
1945. https://inmabb.criba.edu.ar/revuma/pdf/v10n4/p101-108.pdf.Sloane,
N. J. A. Sequence A000125/M1100
in "The On-Line Encyclopedia of Integer Sequences."Yaglom,
A. M. and Yaglom, I. M. Challenging
Mathematical Problems with Elementary Solutions, Vol. 1. New York: Dover,
1987.Referenced on Wolfram|Alpha
Cube Division by PlanesCite this as:
Weisstein, Eric W. "Cube Division by Planes." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CubeDivisionbyPlanes.html