The traveling map exhibit MAPS: FINDING OUR PLACE IN THE WORLD at the Walter’s in Baltimore, Maryland (via the Field Museum, Chicago) wrapped up last weekend. While looking at John Adam’s Road distance map of England and Wales I was put in mind of how map projections try to preserve several of:

- Area
- Distance
- Shape, and
- Direction/angle

Then the question arose: Might information (thematic) topology now be interchangable with purely geographical topology?

John Adam’s map from 1680 places towns in relative (not absolute) geographic lat-long coordinates in a rough framework that preserves their orientation to one another and in the rough-shape of their original geography. But the primary purpose of this map is to emphasize the relationship between towns and intra-town distances. Below by Royal Geographic Society.

This topological focus (of NODES and EDGES in math-speak) is perfectly represented in Adam’s map. Circles (nodes) are inscribed with town names and straight lines (edges) connecting town circles are annotated with road distance (not straight-line geographic distance).

Modern scientific cartography, with an emphasis on visualization, might finally be loosening the geographic straight jacket to the point where purely lat-long geography doesn’t matter so much but the inter-connection (edges) of said features (nodes) gains emphases and is preserved.

I believe such thematic topology maps are “geographically” accurate and employ projection just like conventional cartography but these “projections” are as of now ad hoc and not properly defined or formalized and are often created manually. Efficient and effective mathematical formulas should be devised and listed along conventional map projections in publications like Snyder’s (USGS) Map Projections: A Working Manual.

The nearest we come to topologic maps are subway maps and cartograms. More on cartograms below.

This subway map of New York City is a topological map where the island area of Manhattan is relatively small geographically but is significantly exaggerated to accommodate the “accurate” display of the topological nodes and edges of subway stations and subway lines. (Dorling cartogram example below by Zach Johnson.)

Cartograms are a good example of topological maps:

- Area of symbol represents the NODE weight alone.
- Distance is based on EDGE weight first and and geographic distance second (trying to approximate the “relatedness” between each, eg close countries close, far countries far).
- Direction is approximated.
- Shape is approximated.

Zach Johnson has a good post on this topic on his blog (cartograms are the focus of his Masters Thesis).

Below a New York Times map showing the weighted electoral votes of the 48 contiguous states as the topological area of each state.

Let us examine a map of water flow in a stream network (Kelso and Araya):

One usually sees these maps with a conic projection to preserve equal-land-area. But the river segments are drawn exaggerated to their geographic width to represent the EDGE weight between nodes in the true geography space.

The map is dispensing with equal-land-area between nodes (the overall area and shape are preserved) and instead focusing on DISTANCE and DIRECTION between each node. The edges are “preserved” by exaggerating the stream centerlines to preserve the thematic variable. Overall SHAPE is preserved, but local land AREA is not.

Such topological maps are not diagrams because they are still rooted in land-geography; the placement of the nodes is guided by land geography but shift accordingly to best show the interrelationships between nodes. Ignoring the land-geography by listing the nodes and edges in a chart or table is not a map. A topological map takes a complex n-dimensioned space and represents that topology in a 2d dimensioned space.

Some precipitation maps use “gridded” tightly spaced, regularized nodes and edges (above: Swiss Atlas, 2.0). The “weight” of rain and snow fall is indicated by color. Because of the spacing of the nodes and the hyper-localness of the mapped theme, this “topological map” manages to preserve both the topology and the geography.

The above example from the 8th Edition National Geographic Atlas of the World focuses on the quantity difference between nodes and represents that with height spikes (3d). If this were a topologic projection that needed to show the contents of each node (not the inter-relationship between nodes) the spike height would be flattened out into area alone (2.5d), leading to a grossly exaggerated land-area map but correct population-area cartogram such as:

Above from the Dutch company Mapping Worlds via Zach Johnson.

Tom Patterson (above) uses this 2.5d term to talk about relief shading of land elevation. But I think it can be used to represent any map that is a representation of more than simply 2 variables (lat and long). Really, much of thematic cartography is 2.5d when it tries to represent complex datasets (like precipitation) with color and other visual variables.

So visualization / modern scientific cartography is focused on examining and preserving / projecting topological relationships. Often these are closely related to geographic space, but not always. That is why I am so fascinated by cartograms

How do we measure the “error” and “conformal”-ness in a topological map?

- Area: Does this “view” of the topology preserve the node and edge weights?
- Distance: Does this “view” preserve the inter-relations between nodes?
- Direction: Both topological between nodes and geographically.
- Shape: Purely geographical. This is what sets some cartograms above others.

For topologic shape:

Projecting a n-dimensioned topology onto a 2d surface has one or more points tangent to the 2d surface. An ideal solution shows all nodes and edges shown flattened out but this would likely require using an interrupted projection with dashed linkage lines between like-lobes content (I have seen this somewhere, need example).

For geographic shape and direction:

We are concerned with local shape (direct neighbors in the topology) and global shape. In the England example above for the Dorling cartogram the north-south direction axis tilts left in the topology. A “best” solution preserves this geographic orientation by rotating the topology network until it “conforms” more to the geography.

Finally, we can visualize this with a modified cartography cube from Zach Johnson: