__Splitting rectangular prisms____:__ Consider
a rectangular prism in (d+1)-dimensional Euclidean space (d>0),
with one vertex labelled O. This vertex has d+1 orthogonal edges
emanating from it. The lengths of these edges in ascending order
are L_{1}, L_{2}, ... ,L_{d+1}, with L_{d+1}
being arbitrarily set equal to 1. The shape of this prism can
be defined by d quantities involving ratios of consecutive edge-lengths
from this ordered list, namely by the d-dimensional vector **Y**=(Y_{1},Y_{2},
... ,Y_{d}), where Y_{n} := L_{n} / L_{n+1}.
Thus each component of shape is a number in [0,1].

The prism is to be split by a randomly-positioned hyperplane of dimension d orthogonal to an edge. The cutting plane is an equally-likely choice from the space of all such planes. Having made the cut, the prism containing O is retained, whilst the other prism is discarded.

The retained prism is rescaled so that its longest edge equals
1 and the process then iterated, thus forming a Markov process
of shape vectors. In paper #67 I
investigate the equilibrium conditions and equilibrium distribution
for this Markov process. After solving an equation which may well
have Guiness-Book-of-Record claims as the longest to ever consume
journal space, I found that the limiting distribution on [0,1]^{d}
exists and has pdf

This joint pdf is very easy to manipulate to obtain marginals, conditionals and moments.

** Variations on a theme for d=1:** The planar case
is studied #68, with many variations
on the random splitting method and on the way of selecting the
rectangle to be retained. An abundance of new probability distributions
on [0,1] are obtained as limiting distributions of shape.

** Other dynamics for random sequences of planar rectangles:**
One is bounded only by one's imagination in dreaming up random
rules to generate a sequence of rectangles. Francis Chen and I
have looked extensively at many such rules. Finding limiting shape
densities has been our focus. In paper #70,
we present four of the most appealing dynamical schemes. We name
them after famous mathematicians of the past, because each problem
captures some of the flavour of their work.

(a) __D. G. Kendall:__ A point
P undergoes Brownian motion in the plane commencing at the origin
O. The Brownian motion has no drift whilst the variance in direction
x is b times that in direction y. At any time t, we consider the
rectangle with sides parallel to the x and y axes and having diagonal
OP. The shape of this rectangle has, for all t>0, a pdf

(b) __Crofton:__
Given a rectangle R, draw an isotropic uniformly distributed
random transect T. This is a
line uniformly sampled from the space of all lines which intersect
R. We create, as the next rectangle in sequence, the unique rectangular
subset of R having (i) sides parallel to those of R and (ii) the
intersection of T and R as diagonal.
The big surprise here is that, for any starting shape, the distributions
of all subsequent shapes are independent and identically-distributed!
The limit is reached after just one step! The limiting pdf is

(c)__ Renyi and Sulanke:__ For a starting rectangle R, place n>1
iid points uniformly distributed on the long side and m>1 iid
points uniformly distributed on the short side. Form the next
rectangle in sequence as the Cartesian product of the ranges of
these two samples of points. Provided m > n, limiting distributions
exist. A host of differing forms occur, depending on the relationship
between m and n. For example, if m=3 and n=2, the limiting pdf
is

whilst, for any n, the result of taking m to infinity, yields

(d) __Blaschke:__ Start with
R. Take a congruent copy of R and place it randomly in the plane
in a manner such that the intersection of the copy with R itself
is a rectangle; this intersection becomes the next rectangle in
sequence. The randomness is uniform over all positions and orientations
which achieve this rectangular intersection. We have a series
solution to the limiting shape distribution. Note that squares
have positive probability mass in the limit distribution -- numerically
the mass is 0.161587. A nicer analytic solution eludes us (see
unsolved problems #1).