## Tile maps on a trip

##### Jun 21, 2023

My friend Ray sent me to a recent blog about tile maps. Typical tile maps use squares or hexagons, although in theory many other shapes will do. Unsurprisingly, the field follows the latest development of math researchers who study the space packing problem. The space packing problem concerns how to pack a space with objects. The study of tesselations is to pack space with one or a few shapes.

It was an open question until recently whether there exists an "aperiodic monotile," that is to say, a single shape that can cover space in a non-repeating manner. We all know that we can use squares to cover a space, which creates the familiar grid of squares, but in that case, a pattern repeats itself all over the space.

Now, some researchers have found an elusive aperiodic monotile, which they dubbed the Einstein monotile. Below is a tesselation using these tiles:

Within this design, one cannot find a set of contiguous tiles that repeats itself.

The blogger then made a tile map using this new tesselation. Here's one:

It doesn't matter what this is illustrating. The blog author cites a coworker, who said: "I can think of no proper cartographic use for Penrose binning, but it’s fun to look at, and so that’s good enough for me." Penrose tiles is another mathematical invention that can be used in a tesselation. The story is still the same: there is no benefit from using these strange-looking shapes. Other than the curiosity factor.

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Let's review the pros and cons of using tile maps.

Compare a typical choropleth map of the United States (by state) and a tile map by state. The former has the well-known problem that states with the largest areas usually have the lowest population densities, and thus, if we plot demographic data on such maps, the states that catch the most attention are the ones that don't weigh as much - by contrast, the densely populated states in New England barely show up.

The tile map removes this area bias, thus resolving this problem. Every state is represented by equal area.

While the tesselated design is frequently better, it's not always. In many data visualization, we do intend to convey the message that not all states are equal!

The grid arrangement of the state tiles also makes it easier to find regional patterns. A regional pattern is defined here as a set of neighboring states that share similar data (encoded in the color of the tiles). Note that the area of each state is of zero interest here, and thus, the accurate descriptions of relative areas found on the usual map is a distractor.

However, on the tile map, these regional patterns are conceptual. One must not read anything into the shape of the aggregated region, or its boundaries. Indeed, if we use strange-looking shapes like Einstein tiles, the boundaries are completely meaningless, and even misleading.

There also usually is some distortion of the spatial coordinates on a tile map because we'd like to pack the squares or hexagons into a lattice-like structure.

Lastly, the tile map is not scalable. We haven't seen a tile map of the U.S. by county or precinct but we have enjoyed many choropleth maps displaying county- or precinct-level data, e.g. the famous Purple Map of America. There is a reason for this.

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Here is an old post that contains links to various other posts I've written about tile maps.