“… a countable family of closed sets \(\mathcal{T}=\{T_1,T_2,\ldots\}\) which covers the plane without gaps or overlaps. More explicitly, the union of the sets \(T_1,T_2,\ldots\) (which are known as the tiles of \(\mathcal{T}\)) is to be the whole plane, and the interiors of the sets \(T_i\) are to be pairwise disjoint” (Grünbaum and Shephard 1987, page 16)
… a GIS coverage!
Symmetry in mathematical tiling theory
The symmetries of a tiling map tiles on to other tiles.
Sets of tiles mapped onto one another like this are transitivity groups, of which an isohedral tiling has only one.
Cartographically it probably doesn’t work like that…
Symmetry in cartographic tiling theory
Directionality or orientation is irrelevant to mathematicians.
However, it is important for mapping: we need a concept of cartographic transitivity groups where different tile orientations ‘count’.
This work is an exploration of the combined effect of pattern, orientation, texture, and colour as visual ‘multi-variables’. Along the way we’ve seen some things that appear promising:
Contrasting variables by applying the same colour scheme to multiple attributes
Emergent multivariate regions (with uncertain boundaries)
(Related) an overall sense of complex landscapes
Challenges
There is plenty to think about
Using colour well is hard at the best of times—using several colours at once is even harder!
