  We show that the sequence of thinned uniform random counting measures
converges weakly to the Poisson random measure. We call such measures
'orthogonal dice' due to their natural connection with certain counting
problems and link the sizes of orthogonal dice with the primes by constructing
the largest known prime orthogonal die. We give many examples of possible use
of the construct of orthogonal dice in various areas of applications including
gambling, cosmology, random matrices, approximation theory and circuits. The
gambling example reveals for instance that fair six-sided die numbered
$\{1,2,3,4,5,6\}$ generate negative covariance in point representations of
hands across players, in contrast to seven-sided dice numbered
$\{1,2,3,4,5,6,7\}$ that generate zero covariance and are orthogonal. The
cosmology example suggests that a 'supermassive' orthogonal die underlies the
galaxy point patterns of the Universe. The random matrix application reveals
that the variance of the spectral gap of the Gaussian orthogonal ensemble
exhibits non-monotone scaling with thinning for the Dirac (empirical) random
measure, in contrast to the orthogonal dice where the spectral gap variance is
monotone. Finally, the approximation application shows that the discrete
Legendre polynomials converge to the Charlier polynomials in $L^p$ for integers
$p\ge1$ whereas the electronic circuit/shot noise application identifies
electric current as the Ornstein-Uhlenbeck process driven by an orthogonal die
random measure.
