Cellular Automaton Shading for Building Envelopes - Cellular Automata in Image Processing and Geometry - page 216

be desirable and feasible especially considering the number conserving cellular au-
tomata (NCCA) [27] [19]. However, here for simplicity, the CA pattern is expected
to become still as soon as the propagation reaches the boundary of BE.

11.4.1 Triangular Cellular Automata

Although the most commonly used lattice for CA regardless of the dimension is an
orthogonal grid, several studies have been carried out to examine the properties of
triangular cellular automata (TCA), e.g. life-like rules on r1 Moore's neighborhood
in a triangular lattice have been studied in [4], the computational universality of
an 8-state triangular reversible partitioned CA has been demonstrated in [26]. The
effect of simple memory on a particular reversible, structurally dynamic CA in the
triangular tessellation has been demonstrated in [1]. TCAs has also been proposed
for BE shading in [63] and the applications of totalistic and semi-totalistic [21]
TCAs have been further studied in [25]. Figure 11.16 shows an example of the
evolution of a semi-totalistic triangular CA (stTCA) on a regular triangular lattice
with voids. For an illustrative corresponding demonstration see [61]. The history
of evolution of a 2D CA can be conveniently shown as a column of consecutive
time steps, where unfolded lists of cells are shown as single rows, as shown in
Fig. 11.17. However, unlike the 1D case, the information about the adjacencies is
partially lost. From such diagrams some fundamental information for CASS can be
derived, namely: whether a certain automaton stabilizes, at how many time steps
and with what final opacity rate (FOR). Figure 11.18 shows all stable stTCAs. As
Fig. 11.18 indicates, there are eight 2D2Cr1stTCAs suitable for CASS: rules: 50,
114, 118, 178, 242, 246, 250 and 254. After stabilization, the average opacities,
called here FOR are: 0.51, 0.66, 0.63, 0,55, 0.77, 1, 0.85 and 1, respectively. For

Fig. 11.16 stTCA # 250 at selected time steps starting from a single black cell. The pattern
stabilizes at the 29th time step. White, gray and white facets indicate transparent facets, voids
in the mesh and opaque facets, respectively.