This chapter read drawings. The guide below gathers the questions that help read them with the same care with which one reads a formula.

Five questions for a diagram

  1. What does the drawing copy? A diagram is an icon of some relations, never of all. In a graph, only the connections; in a Venn diagram, only the inclusions. Find out what matters and ignore the rest.
  2. What is convention? The scale, where the axes start, the colours, the shape of the box. These choices do not appear as statements, and they change what the eye concludes.
  3. Length, area or angle? The eye compares lengths well, areas badly and angles worse. Ask which of them carries the information and whether it was drawn in the right proportion.
  4. Does the drawing prove or illustrate? If every relation the eye uses can be guaranteed by an argument, the drawing proves. If not, it illustrates, and it may deceive.
  5. What was left out? An extra dimension, a variable that did not fit, a value the axis cut off. Every drawing is a choice, and the choice says something.

A complete reading of a newspaper chart

A newspaper shows, in a line chart, the price of a product that went from 10.00 to 10.50 in a year, with the vertical axis running from 9.90 to 10.60. The line crosses the whole chart from bottom to top.

  1. What it copies: the order of the prices in time. That is right.
  2. The convention: the axis does not start at zero. In a line chart this can be legitimate, but height no longer measures size.
  3. What carries the information: the slope. It looks huge because the box was stretched vertically.
  4. Proves or illustrates: the chart illustrates a rise of $5\%$ and suggests a surge.
  5. What was left out: inflation over the period. If inflation was $6\%$, the price fell in real terms, and the chart tells the story backwards.

Glossary

  • Diagram. A sign that represents relations by the arrangement of figures in the plane.
  • Icon of relations. For Peirce, a sign that resembles its object not in the things, but in the structure of the relations between them.
  • Euler diagram. Circles positioned according to the known relations between sets: inside, outside or crossing.
  • Venn diagram. Circles that always cross, showing all $2^n$ regions, with shading for the empty ones.
  • Latitude and longitude (Oresme). The intensity of a quality (vertical) along the extension (horizontal). Ancestors of the graph of a function.
  • Logarithmic scale. An axis on which equal steps multiply by the same factor. Turns exponential growth into a straight line.
  • Truncated axis. An axis that does not start at zero. Exaggerates differences in bar charts.
  • Complex (Argand) plane. Representation of $a+bi$ as the point $(a,b)$. Multiplying by $i$ is turning by $90°$.
  • Graph. Points (vertices) and connections (edges), in which only who connects to whom matters.
  • Planar graph. A graph that can be drawn without crossings.
  • Proof without words. A diagram that makes a proof visible, with little or no explanation.
  • Projection. The shadow of an object in a space of lower dimension, like the drawing of a cube on paper.
Exercise: apply the guide

A pie chart shows market shares of 23%, 25%, 26% and 26% for four companies. What does the guide say?

The information is in angles, which the eye compares badly: the four slices look equal. If the question is “who leads?”, a bar chart, which compares lengths, would show the difference. If the question is “is the market divided?”, the pie answers well, precisely because it hides the small differences.

Exercise: what the graph ignores

On a metro map, two stations appear three stops apart. Could they be five minutes’ walk from each other?

Yes. The metro map is a graph: it copies the order of the stations and the interchanges, not the distances. Two stations on different lines can be three stops apart by rail and one block apart by road. The question “how far do I walk?” calls for the city map, not the diagram.

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