| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
n
Given a space isomorphic to R containing a "hypercheese", and
such a cheese in which it is possible to inscribe n mutually
perpendicular line segments, and moreover, the cheese is of
such a shape that no straight line can pass through the cheese
more than once: Suppose that we may cut this cheese with m
(n-1)-planes, i.e. surfaces of the general equation:
n
--
\
/ a x = 0
-- i i
i <> r
for some r in {1,...,n}
Let M:(Z+ x Z+) --> Z+ be the function mapping m and n to the
maximum number pieces that can be produced using any m cuts
(note that this also varies with n). Prove that:
m
Given any m>0, Lim M(m,n) = 2
n -> +infinity
SDC.
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 481.1 | ENGINE::ROTH | Sat May 03 1986 11:59 | 16 | ||
If the (convex) piece of cheese satisfies the conditions in .0, you
may let it expand arbitrarily in size without changing the number of
pieces it's sliced into. So the problem really depends on how many
pieces can one subdivide n dimensional Euclidean space into with m
hyperplanes.
Clearly, for any m <= n, the number of subdivisions of n-space will
be 2^m, since you will need m > n to ever completely enclose any finite
extents of n-space between hyperplanes (eg, with m = n+1 you enclose
an n-dimensional simplex).
An interesting combinatorial problem arises with n finite:
How many pieces can you subdivide n-space into with m > n hyperplanes?
- Jim
| |||||