A symbol has a size on the page and an area on the ground

The rung below this one prices a colour. A colour has no size of its own, so its whole visual weight is the region it fills, and the arithmetic came out as one covariance.

A symbol has a size of its own, and the arithmetic comes out differently — cleaner, larger, and with a correction that fits on one line.

Two things a symbol map says at once #

A proportional symbol map is usually described as showing totals, and it does. Draw a circle of area proportional to a region’s value, place it at the region’s centre, and the reader can compare circles against each other and get every ratio right. Nothing in that operation touches the projection at all.

But a symbol is placed on something, and the something has a size. A reader looking at a large circle on a small country and a small circle on a large one is not comparing two totals; the reader is forming a density — a quantity per unit of ground — and doing it by comparing two areas on the page.

That second reading is the one nobody designs for and everybody performs. It is also the one the projection ruins.

The identity, and it is exact #

Let a region carry value v, occupy ground area A, and be drawn with page area As, with s the areal scale factor. The symbol’s page area is cv for some constant the legend fixes. The reader’s ratio is

$$\frac{\text{symbol area}}{\text{region page area}} = \frac{cv}{As} = c\,\frac{v}{A}\cdot\frac{1}{s}.$$

The true density is v/A. So the density a reader reads is the true density divided by the areal factor, exactly — not to first order, not on average, exactly, at every point, for every field, on every projection.

There is no covariance here and no dependence on the data. The rung below needed a covariance because a colour’s weight is its region’s page area and the statistic being formed was a mean over many regions. Here the reader is comparing two areas at one place, and one quantity survives.

What was computed, and how #

The check is the identity itself, and it is run over ninety-six regions rather than argued.

For each cell of a twelve-by-eight covering, the machinery computes the perceived density v/(page area), the true density v/(ground area), and normalises both against the cell nearest the equator so that the constant in the legend cancels. The ratio of those two normalised quantities must equal the ratio of the base cell’s areal factor to this cell’s, and it does — to better than one part in a million at every one of the ninety-six, with the page area coming from a shoelace and the areal factor from the projection’s own derivatives, which are two entirely separate pieces of code.

The numbers, on Mercator, against the field’s own variation:

At 73° north the reader’s estimate of density is a twelfth of the truth. Miller’s cylindrical reaches 0.152 at the same latitude and the plate carrée 0.291; Gall–Peters sits at 1.000 to fifteen decimals all the way up, which is the control.

The correction is one multiplication #

The repair is embarrassingly cheap, which is part of what makes its absence interesting.

If the symbol is to carry a density comparison as well as a total, its area should be cvs rather than cv — scale each symbol by the areal factor at its own position. The ratio then becomes cv s/(As) = cv/A, and both readings are right at once.

The cost is that the totals stop being comparable: two regions with the same value get different symbols, and the map has traded one of its two readings for the other. There is no scaling that delivers both, because the two readings differ by a factor of s and s is not one.

So a proportional symbol map on a projection that is not equal-area has to choose which of its two readings to be correct, and no published atlas states which it chose. That is the finding, and it is the same shape as the trade-off no essay may claim a projection is best without naming the purpose is written about: the choice is forced, and what is missing is not the right answer but the declaration.

The third sizing rule, which nobody uses and which minimises the worse error #

The choice is presented above as binary — scale by the value and the totals are right, scale by the value times the areal factor and the densities are — and it is not. It is a continuum, and the two named options are its ends.

Size a symbol as cv·s^α for a stated exponent α. The total a reader takes off it is then wrong by s^α and the density by s^(α−1). The two conventional choices are α = 0, which is exact on totals and out by s on densities, and α = 1, which is the reverse.

The exponent that minimises the worse of the two errors is α = ½, and it minimises it by construction: the two exponents are α and α − 1, their magnitudes are equal only at a half, and moving away from a half makes one of them larger. So sizing a symbol by the value times the square root of the areal factor splits the failure evenly.

On Mercator at 73° north, where s is about twelve, that turns a choice between exact and twelve times out into both readings out by a factor of 3.5. Whether that is an improvement depends entirely on which reading is being made: a reader taking only totals is worse off, a reader taking only densities is better off, and a reader doing both — which is what a symbol map invites — has had their worst error cut by a factor of 3.5.

Two things make the square-root rule worth naming rather than recommending.

It is the honest choice when the map cannot say what it is for. A published symbol map is read by people the cartographer will never meet, forming both kinds of comparison, and α = ½ is the choice that is least wrong for the reader whose question is unknown. Every other value on the continuum is a bet on which reading matters.

And it has a familiar shape. A symbol scaled as v√s has a radius proportional to √v × s^(1/4), which is to say its linear size carries a quarter power of the map’s inflation — the same square-root-of-a-square-root compromise the apparent-magnitude literature reaches by a completely different route when it scales circles by area to the power 0.7 rather than 1. The two corrections are unrelated in origin and compose multiplicatively, which is worth knowing before applying both.

Nothing published uses it, and the reason is the one the legend section gives. A sizing rule that is exact about nothing cannot be stated in a legend as a row of circles with values beside them, because no circle on the sheet means any particular value. The rule that is least wrong is the one that cannot be legended, and a legend is a requirement rather than a courtesy.

The other half: what a symbol stands on #

The identity above holds the region fixed and asks what the symbol says. Turning it round holds the symbol fixed and asks what it covers, which is the question a designer actually faces at the drawing board.

A symbol drawn at a fixed page radius covers ground area (page area)/s, so the ground it occludes shrinks exactly as the map inflates:

This is why symbol overlap on a world map is not distributed the way the data is. Two cities four hundred kilometres apart at the equator are drawn four hundred kilometres apart in page terms too; the same two cities at seventy-eight degrees north are drawn nearly five times further apart on Mercator, so their symbols separate. Meanwhile a designer who sizes symbols to avoid collision in the tropics has sized them for the least crowded part of the sheet.

The consequence runs the wrong way round from the intuition. Mercator makes a high-latitude symbol map look sparse and uncrowded, which is exactly the impression a reader takes as evidence that the phenomenon is not concentrated there.

The crowding is a different failure, and equal-area does not fix it #

There is a temptation, having found that an equal-area projection repairs both the choropleth’s mean and the symbol’s density, to conclude that equal-area repairs symbol maps. It does not, and the reason is worth separating carefully, because the two things a symbol map can go wrong about are governed by different quantities.

The density error is governed by the areal factor, which an equal-area projection holds at one by definition. Whether two symbols collide is governed by the page distance between their centres, and that is a principal scale rather than a product of two.

An equal-area projection holds the product of the principal scales at one and constrains neither of them — which is the two ways a map is wrong read as a constraint rather than as a diagnosis, and it leaves a whole direction free in the sense distortion has a direction makes precise. A cylindrical equal-area map stretches east–west by sec φ and compresses north–south by cos φ, so both separations move and their product is what stays put.

The numbers at 75° north, relative to the equator:

So a designer who moves from Mercator to Gall–Peters to fix the density reading has made the crowding worse by a factor of twenty in the north–south direction, at exactly the latitudes where the density fix was worth having. The two repairs pull opposite ways, and the projection with the mildest crowding penalty in the table is Mollweide, which is also equal-area — so the choice is not between honesty and legibility, but it is a choice, and it is not the one the equal-area rule makes.

What a legend can promise, and what it cannot #

Every proportional symbol map carries a legend, and the legend is a row of circles with values beside them. It is a complete statement of one thing and silent about two others.

It states the sizing rule. A circle of this size means this value: that is exactly the constant c above, and with it a reader can invert any symbol on the sheet back to a total. Nothing in this essay disturbs that. A proportional symbol map is a faithful instrument for totals and it is faithful everywhere on the sheet.

It does not state the areal factor. So a reader who forms a density — and forming a density is what comparing a symbol against a country consists of — has no way to know that the comparison is being divided by a number that varies from one to fifteen across the page. Nothing in the legend is wrong; the missing quantity simply is not in it.

It does not state which reading the map was designed for. The trade-off above is forced, so every published symbol map has already made the choice, and the choice is invisible. In practice every one of them takes the total, because scaling by the areal factor is not a thing any package does.

The repair that costs nothing is the one the site keeps arriving at from different directions: say which. A caption naming the projection and stating that symbol areas are totals rather than densities takes one line and closes the whole gap, which is the same repair a map does not say what it is asks for on a different missing declaration.

That is not a counsel of perfection. It is the observation that the expensive part — computing the areal factor at every symbol, deciding which reading to protect, redrawing — is only necessary if the map is trying to deliver both. A map that says which one it delivers has delivered it.

Where the model stops #

The eye judges area with an exponent. Everything above treats a circle of twice the area as reading twice as large, and the psychophysical literature is clear that it does not — the judged magnitude of a circle goes roughly as area to a power somewhere between 0.7 and 0.9, which is why apparent-magnitude scaling exists at all. That compresses every ratio in the tables above towards one. It does not change any sign, and it does not touch the footprint calculation, which is geometry rather than perception.

A symbol is placed, not just sized. Real symbol maps displace overlapping circles, and the displacement is chosen on the page, so it inherits the same factor a second time. Nothing here prices that.

A national grid is not a world map. The numbers above are for whole-world projections in their normal aspects. On a national grid the areal factor departs from one by parts in ten thousand across the whole country — a grid scaled to the ground records the design constraint — so a symbol map on a national sheet has none of this in it. The mechanism is a world-map problem, and it is a world-map problem precisely because that is where the areal factor has room to move.

The generalisation #

A page unit is not a ground unit, and every quantity defined in page units inherits the projection.

The road is drawn two pixels wide is the same statement about a line: a stroke width chosen for legibility is a distance on the page, and the ground it covers varies. A scale bar is right in one place is the same statement about a length. A tolerance is a promise about the picture is the same statement about a simplification threshold.

What is new here is that the page quantity is being compared against another page quantity — a symbol against a region — so the two projection factors do not cancel. They would if both were lengths at the same place; they do not because one is a datum-free page area chosen by the legend and the other is the image of a piece of ground.

The general rule this suggests, and it holds for every case above: whenever a designer forms a ratio between something drawn at a page size and something drawn at its ground size, the areal factor is in the answer.

Who found it, and when #

Proportional circles on maps go back to Minard in the 1850s, and the sizing question — whether to scale by area or by apparent magnitude — has been argued continuously since Flannery’s experiments in the 1950s. That whole literature is about the relation between a symbol’s size and the value it carries.

The relation between a symbol’s size and the region it sits on is almost absent from it, and the reason is worth stating plainly: the symbol literature grew up around national and regional maps, where the areal factor is within a fraction of a per cent of one across the whole sheet and the effect is genuinely negligible. It becomes a factor of twelve only on a world map, and world thematic maps are the case where the equal-area advice from the rung below is already being given for a different reason.

So the two failures have the same fix and different mechanisms, and a map that follows the advice is protected against both without anybody having priced the second.

Where the ladder goes next #

A symbol is a page object placed on a page. A dot map is a page object scattered over one, and scattering is a different operation: the count in a region is right under either placement, so the map is honest as a total whichever way it is drawn, and the two placements put the visible density on opposite sides of the truth. The next rung measures both and finds that the standard practice — scatter at random inside the polygon, which means inside the polygon on the page — is the one that misstates the ground.