Map of an Empire

Lewis Carroll first posed the thought experiment in 1893:

“That’s another thing we’ve learned from your Nation,” said Mein Herr, “map-making. But we’ve carried it much further than you. What do you consider the largest map that would be really useful?”

“About six inches to the mile.”

“Only six inches!” exclaimed Mein Herr. “We very soon got to six yards to the mile. Then we tried a hundred yards to the mile. And then came the grandest idea of all! We actually made a map of the country, on the scale of a mile to the mile!”

“Have you used it much?” I enquired.

“It has never been spread out, yet,” said Mein Herr: “the farmers objected: they said it would cover the whole country, and shut out the sunlight ! So we now use the country itself, as its own map, and I assure you it does nearly as well.”

Borges expounded in 1946:

…In that Empire, the Art of Cartography attained such Perfection that the map of a single Province occupied the entirety of a City, and the map of the Empire, the entirety of a Province. In time, those Unconscionable Maps no longer satisfied, and the Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and which coincided point for point with it. The following Generations, who were not so fond of the Study of Cartography as their Forebears had been, saw that that vast Map was Useless, and not without some Pitilessness was it, that they delivered it up to the Inclemencies of Sun and Winters. In the Deserts of the West, still today, there are Tattered Ruins of that Map, inhabited by Animals and Beggars; in all the Land there is no other Relic of the Disciplines of Geography.

… which Eco took seriously in 1963, in an analysis as follows:

1. Requirements for a 1:1 Map, in which he establishes six requirements for a map truly to be considered a 1:1 map of an empire;
2. Methods of Production of the Map, in which he explores various characteristics of such a map, e.g., suspended, transparent, etc.;
3. Paradox of the Normal Map, in which he analyzes the logical consequences of the construction of such a map.

But Eco’s requirements are imprecisely specified; properly stated, they admit a theoretical solution, albeit with practical difficulties. Let’s go through his requirements.

1. That the map be, in fact, one to one, and therefore coextensive with the territory of the empire.

A natural requirement; but we will return to this in the end.

2. That it be a map and not a plaster cast; in other words, dismissing the possibility of covering the surface of the empire with a malleable material reproducing every relief, even minimal. In this case, the project would be considered, not cartography, but rather the packaging or paving of the empire, and it would thus be more appropriate legally to decree the empire a map of itself, with all the consequence semiotic paradoxes.

Here we immediately have a problem: Eco begs the question. The map should be a map; fine, but what is a map? He’s clear it’s not a plaster cast (or cast of any other material), but just insists, tautologically, that it should be a map. Perhaps the crux of this requirement is that the map should be a representation, i.e., not the empire itself but rather a depiction of some kind.

3. That the empire in question be that X than which nihil maius cogitari possit, and hence that the map cannot be produced and spread out in a desert area of a second, separate empire X₂ such that X₂ > X (as if a 1:1 map of the Principality of Monaco were to be spread out in the Sahara). In this case the project would lose all theoretical interest.

There are two problems with this requirement: first, the pompous Latin, and second, the ambiguous reference to “theoretical interest”. We can dispense both at once - ex ungue leonem - by inferring that Eco means that the map must be entirely contained within the empire. (Ignore, for the moment, the seemingly natural conclusion that the map must be coexistent with the empire; in dubio pro reo).

Why? Nihil maius cogitari possit clearly suggests that nothing should be greater, i.e., outside, of the territory of empire $X$, but does not make any claims on any sub-empires with $X$. In his example of a “second, separate” $X_2$, it is unclear whether “separate” indicates that $X_2$ is not contained in $X$, or simply that $X_2$ is a clopen subset of $X$, which would decompose X into separate subsets $X_2$ and $X \setminus X_2$.

And what of “theoretical interest”? One might naively assume that this refers to the tantalizing notion that a point on the map would refer to a point in the empire exactly where that point on the map is located, because the map is situated inside the empire itself; that is, ein Fixpunkt.

But Eco’s Latin reference clearly alludes to the Vatican, entirely enclaved within Eco’s own Italy, a country for which Brouwer’s fixed point theorem canonically fails! Not to mention that the presence of the Vatican, seat of the Sanctus Papa, implies something greater than Italy, and indeed, all of humanity. In fact, in ipso enim vivimus et movemur et sumus, putting the two together: a continuous map of Italy with no fixed point would rotate around the Vatican.

Therefore Eco must not refer to Brouwer’s fixed point theorem. It must simply mean that the map should be contained within the empire, no more.

4. That the map be faithful, depicting not only the natural reliefs of the empire but also its artifacts, as well as the totality of the empire’s subjects (this last is an ideal condition, which may be discarded in the production of an impoverished map).

A natural requirement, although the “ideal condition” indicates that there’s a spectrum, from total faithfulness (do bugs count as empire’s subjects?) to a somewhat more “impoverished” representation. We will return to this in more detail later.

5. That it be a map and not an atlas with partial pages. In theory there is nothing to prevent the realization, over a reasonable amount of time, of a series of partial projections on separate sheets, to be used individually for reference to different portions of the territory. The map may be produced on separate sheets, but only on condition that they be sutured in such a way as to construct the overall map of the entire territory of the empire.

And here’s where we need to split hairs. If separate sheets may be “sutured” together, but an atlas with partial pages is insufficient, what specifically do we mean by “sutured”? Clearly a folded map is acceptable:

… It is necessary, then for the map to be capable of being folded and then reopened with a different orientation…

and so there is some leeway for the capabilities of the “sutures” and moreso for the “separate sheets”. Eco explores all sorts of properties of potential materials - transparency, permeable, extended, adjustable - but does not consider even more extreme physical properties. For Eco, “permeable” implies some sort of physicality, akin to gauze - e.g.,

… they would cause tangles in the very fine membrane above them, resulting in serious discomfort and once more making the map unfaithful.

But what about a material that is infinitely thin so as to be essentially physically inconsequential? That can adjust its electrostatic properties upon a touch of a button or some other signal from the cartographer?

Exploring “separate sheets” even further: what if one could hold a “primary” sheet in one’s hand with the other “secondary” sheets invisible and incorporeal, but electromagnetically attached to the primary? By manipulating the primary sheet, one could explore the neighboring secondary sheets, in effect transferring the cartographic representations to the corporeal sheet in the hand. One could, in essence, have immediate access to the entirety of the sutured atlas through simple manipulation of a single handheld sheet.

Moreover: such an advance in material science would have a profound effect on the utility and usability of the map; no longer static (or requiring enormous effort to fold and unfold), it would become mobile, a sort of mobile map, allowing the cartographer to move throughout the empire as they consult the map.

6. That the map, finally, be a semiotic tool - that it be capable, in other words, of signifying the empire or of allowing references to the empire, especially in those instances when the empire is not otherwise perceptible. This last condition means that the map cannot be a transparent sheet in any way fixed over the territory on which the reliefs of the territory itself are projected point by point; for in that case any extrapolation carried out on the map would be carried out at the same time on the territory beneath it, and the map would lose its function as maximum existential graph.

We come now to the crux of the analysis: what is a map? Let us also recall Requirement 4, “that the map be faithful, depicting not only the natural reliefs of the empire but also its artifacts, as well as the totality of the empire’s subjects…”

Summarizing these two requirements, we see that the map must depict the empire precisely but must not be the empire. Eco is clearly concerned that the map remain faithful; a map that no longer depicts the empire is no longer valuable as a map!

Here we naturally must consult Michelson and Morley: if we place geostationary satellites above the empire, each with a large lens focusing real-time images of the empire, what is the latency with which we could detect changes to the empire itself before propagating to the map? Considering a satellite with an altitude of approx. 35000 km, the signal from this satellite must take 120 ms to reach our mobile map. Of course, we could do better: a network of low-Earth orbit satellites that in sequence precisely cover the domain of our empire, improving latency by a hundredfold with a tremendous increase in complexity.

We must then, of course, turn this signal into an appropriate semiotic representation of the empire (is a rendering of the satellite image appropriately semiotic for Eco? Surely yes, since such a rendering is not the empire itself.), but such a translation surely can be performed in significantly less time than the latency of the signal itself.

What of the resolution of the map? Eco repeatedly makes reference to the “subjects” of the empire, so our map must resolve at least 1 m on the ground. For a geostationary camera, the Rayleigh criterion implies that the size of the lens must be at least 20 m, posing some significant engineering challenges; a low-Earth orbit satellite could attain similar resolution with a dramatically smaller lens.

Now, how evolved of a semiotic tool must this map be? One can imagine Eco requesting a number of features: labels or iconographs representing key points of interest in the empire; or filters, groupings, indexing, etc.

Of course, the cartographer may simply build a “Map, Very Plain”, or “MVP”, and then iteratively augment it to support the most critical requests of those consulting the map.

Finally, Eco summarizes his analysis with the “Paradox of the Normal Map”:

When the map is installed over all the territory (whether suspended or not), the territory of the empire has the characteristic of being a territory entirely covered by a map. The map does not take account of this characteristic, which would have to be presented on another map that depicted the territory plus the lower map. But such a process would be infinite (the “third man” argument). In any case, if the process stops, a final map is produced that represents all of the maps between itself and the territory, but does not represent itself. We call this map the Normal Map.

Although presented as a logical paradox, this problem is merely an engineering challenge about resolution: if a map resolves at some minimum level $D$ and the map is $r:1$ for some $r<1$, then the map-within-the-map must resolve at level $rD < D$, details the original map cannot display.

But recall the very first requirement:

1. That the map be, in fact, one to one, and therefore coextensive with the territory of the empire.

Eco requires that $r=1$! As long as the mapping ratio remains fixed at 1:1, each successive nested map resolves at the same ratio as its parent, so each map can happily contain within it another map. Therefore, there is no specific “Normal Map”, just slightly delayed maps-within-maps due to run-of-the-mill electrostatic recording and transmission latency.

Would these nested maps remain faithful? Certainly the empire does not substantively change from one second to the next; we would have to consult Zeno to determine if this means that the empire is static forever. This requires further research.