  We construct models hosting classical fractal spin liquids on two realistic
three-dimensional (3D) lattices of corner-sharing triangles: trillium and
hyperhyperkagome (HHK). Both models involve the same form of three-spin Ising
interactions on triangular plaquettes as the Newman-Moore (NM) model on the 2D
triangular lattice. However, in contrast to the NM model and its 3D
generalizations, their degenerate ground states and low-lying excitations
cannot be described in terms of scalar cellular automata (CA), because the
corresponding fractal structures lack a simplifying algebraic property, often
termed the 'Freshman's dream'. By identifying a link to matrix CAs -- that
makes essential use of the crystallographic structure -- we show that both
models exhibit fractal symmetries of a distinct class to the NM-type models. We
devise a procedure to explicitly construct low-energy excitations consisting of
finite sets of immobile defects or "fractons", by flipping arbitrarily large
self-similar subsets of spins, whose fractal dimensions we compute
analytically. We show that these excitations are associated with energetic
barriers which increase logarithmically with system size, leading to "fragile"
glassy dynamics, whose existence we confirm via classical Monte Carlo
simulations. We also discuss consequences for spontaneous fractal symmetry
breaking when quantum fluctuations are introduced by a transverse magnetic
field, and propose multi-spin correlation function diagnostics for such
transitions. Our findings suggest that matrix CAs may provide a fruitful route
to identifying fractal symmetries and fracton-like behaviour in lattice models,
with possible implications for the study of fracton topological order.
