The earliest abstract illustrations are maps. Maps as navigation tools
are rooted in a metric space, meaning that there is a relationship between
the physical distance between two locations in the world and the physical
distance between the representations of the locations on paper. Yet maps
by their nature abstract out detail. Roads are often represented as straight
lines, rather than pairs of curved lines. Coastlines are abbreviated. The
concept of mapping extends to the field of medical illustration, where
the terrain of the human body is abstracted and mapped.
Geometric diagrams also have a long history. Manuscripts containing
proofs with diagrams of the Pythagorean theorem exist from the time of
the ancient Greeks. These diagrams are also based on a concept of metric
space, intended to prove theorems concerning distance in Euclidean space.
Descartes' Cartesian coordinate system in the 17th century rationalized
the use of graphs in analytic geometry. And scientific discoveries in physics
and chemistry resulted in increasingly abstract diagrams.
The history of topological diagrams is difficult to trace. Tree hierarchies show up in the middle ages as a way of documenting lineage. Religion and cosmology of the middle ages and renaissance made great use of diagrams, including graphs, to show the relationships between concepts. Gardener (1982) and Yates (1982) document the combinatorial diagrams of Raymond Lull in the 13th century. In the Renaissance, the rediscovery of Greek thought resulted in many diagrams of philosophical and scientific concepts, as in the following illustration.
Figure 2.1. Sixteenth century representation of the four basic elements.
The qualities the elements share are labeled in the edges. Opposites are
joined by the unlabeled edges. From Charles de Bouelles, Liber de generatione.
Reproduced from Heninger (1977).
In the 19th century, the work of Boole inspired the invention of Venn diagrams. Also in that period, Charles Peirce proposed extensions to these diagrams and invented his own system known as existential graphs. Notwithstanding the 18th century graph-theoretic discoveries of Euler, it has been in this century that the explosion of abstract mathematics has resulted in the proliferation of tree and graph representations. Now, even subway routes are represented as topological diagrams rather than metric maps.