Case Study 1: Tufte's Train Schedule as the Ur-Small-Multiple

Case Study 1: Tufte's Train Schedule as the Ur-Small-Multiple

In 1880, a French railway engineer named E.J. Marey created a schedule chart that was decades ahead of its time. A century later, Edward Tufte rediscovered it, put it in a book, and used it to define what small multiples should look like. Every modern data-journalism small multiple traces its lineage back to this chart.

The Situation

In the late 19th century, European railway networks were expanding rapidly, and the problem of designing efficient timetables was becoming genuinely difficult. A major line like the Paris-Lyon-Marseille route (the PLM, one of France's most important railways) had dozens of trains running daily in both directions, each with its own schedule, each stopping at some combination of stations along the route, each needing to avoid collisions with other trains on the same tracks.

The standard way to present a train schedule, then as now, was a table: columns for each train, rows for each station, with departure and arrival times filled into the cells. Tables are efficient for looking up a single fact — "what time does the 8:15 from Lyon arrive in Marseille?" — but they are terrible for seeing the overall structure of the schedule. A reader looking at a PLM timetable could not easily tell which trains were fastest, where the bottlenecks were, how trains in opposite directions related to each other, or whether there were periods of the day with too many trains on the same track.

E.J. Marey, a French physiologist and engineer best known for his early work in chronophotography, proposed a different representation. Instead of a table, he drew a chart: time on the horizontal axis, stations on the vertical axis, with each train drawn as a diagonal line from its departure station (top-left) down to its arrival station (bottom-right). A steep line meant a fast train; a shallow line meant a slow train; a horizontal segment meant the train was stopped at a station. The direction of the line (down-right or up-right) indicated the direction of travel.

Marey's chart was not a single chart — it was many lines on a single grid, each line representing one train, with the whole network's daily schedule visible as a pattern of overlapping diagonal lines. The chart made the structure of the schedule visible at a glance. A reader could see which trains were fastest (steepest lines), where trains met or crossed (intersections of lines), and which periods of the day were busy or quiet (density of lines). The chart was, in effect, a visual representation of the schedule's complete structure — information that a table could technically contain but could not show.

Edward Tufte reproduced Marey's chart in The Visual Display of Quantitative Information (1983) and Envisioning Information (1990), using it as an example of what visual information design could accomplish. Marey's chart was not a small multiple in the strict modern sense — it was a single integrated chart with many lines — but Tufte's treatment of it, combined with related examples in his books, set the template for how modern designers think about displaying "many of the same thing" in a single coherent visual. The intellectual lineage from Marey to Tufte to modern small multiples is real and traceable, and it is the reason this case study is worth studying.

The Data

Marey's chart displayed the daily operating schedule of the PLM line between Paris and Lyon (approximately 500 kilometers). For each train on the line, the underlying data included:

Train identification: Name or number of the train.
Origin station: Where the train started (typically Paris or Lyon, though some trains started at intermediate stations).
Destination station: Where the train ended.
Direction of travel: Northbound (toward Paris) or southbound (toward Lyon).
Stops: A list of stations at which the train stopped, with arrival and departure times at each stop.
Schedule times: Departure and arrival times at every station along the route.

A typical schedule included twenty to thirty daily trains in each direction, each with its own pattern of stops and its own total travel time. Some trains were express services that stopped only at major stations (Dijon, Mâcon) and made the full Paris-to-Lyon run in about five hours. Others were local services that stopped at every station and took eight or nine hours for the same route. Some trains ran only on certain days of the week or only during certain seasons.

The full schedule was complex — not impossibly so, but complex enough that a table representation made it nearly impossible to see the structure of the network's operations. A reader with a tabular schedule could look up individual trains but could not see how the trains interacted with each other as a system.

The Visualization

Marey's chart was a single two-dimensional grid:

Horizontal axis (x): Time of day, running from midnight on the left to midnight on the right (or from 4 AM to 4 AM the following day, depending on the version). Hours were marked at regular intervals.
Vertical axis (y): Distance along the route, with Paris at the top and Lyon at the bottom. Intermediate stations (Melun, Sens, Tonnerre, Dijon, Mâcon, Villefranche, and others) were marked along the y-axis at distances proportional to their actual locations along the track.

On this grid, each train was drawn as a diagonal line:

A southbound train (Paris to Lyon) started in the top-left area of the chart (Paris, early in the day) and sloped down and to the right (arriving at Lyon, later in the day). The steeper the slope, the faster the train.
A northbound train (Lyon to Paris) started in the bottom-left area (Lyon, early in the day) and sloped up and to the right (arriving at Paris, later in the day).
A stop at a station was drawn as a short horizontal segment where the line paused at a station's y-coordinate for the duration of the stop.
A change in speed was drawn as a change in slope where the train accelerated or decelerated.

With twenty or thirty daily trains drawn on the same grid, the chart became a lattice of diagonal lines. Express trains appeared as steep, nearly-straight lines crossing the grid from one corner to the other. Local trains appeared as shallower lines with visible stops. Trains in opposite directions crossed each other at various points on the grid — and importantly, the points where they crossed were where the trains met in real space along the track, revealing the physical encounters between trains that the table could not show.

The beauty of Marey's chart was that it made the complete schedule visible as a pattern. A reader could see at a glance:

Which trains were fastest (steepest lines).
Which trains were slowest (shallowest lines, often with many stops).
Which periods of the day were busy (dense regions of the grid with many overlapping lines).
Which periods were quiet (sparse regions with few lines).
Where trains met (intersections between northbound and southbound lines).
Where bottlenecks might occur (places where many lines converged).
How the schedule as a whole was structured (the overall pattern of the lattice).

None of this was visible in a table. All of it was visible in Marey's chart within a few seconds of looking at it.

The Impact (and the Lineage)

Marey's chart was influential within French railway engineering in the late 19th century but did not become widely known outside that context. The chart was reproduced in engineering handbooks and used as a practical scheduling tool by railway operators for decades. It was not celebrated as a visualization innovation because the visualization community as such did not exist yet — the study of information design as a discipline is a 20th-century development.

The chart's modern prominence begins with Edward Tufte. In The Visual Display of Quantitative Information (1983), Tufte reproduced Marey's chart and wrote about it with the same reverence he applied to Minard's 1869 Napoleon march map. Tufte argued that Marey's chart was an example of what data visualization could do when the designer was willing to invent a chart type to match the questions being asked — the same principle that Case Study 1 in Chapter 5 identified in Minard's work. Marey had needed to show train schedules in a way that tables could not, and he had invented a visual form that did the job. Tufte celebrated the chart, explained its design principles, and used it as a teaching example for generations of readers.

In Envisioning Information (1990), Tufte extended his treatment of the chart and placed it in a broader discussion of "small multiples" as a design principle. Tufte's formulation of small multiples was not identical to Marey's chart — Marey's chart was a single integrated chart with many lines, while Tufte's small multiples were typically multiple separate panels showing related data — but the underlying idea was the same: show many instances of the same kind of relationship in a single coherent visual, so that the reader can see the pattern across instances.

The influence of Tufte's treatment has been enormous. Modern data journalism, scientific figure design, and dashboard design all trace part of their lineage to Tufte's small-multiples principle, and Tufte's treatment traces directly to Marey's chart. The chain of influence goes something like this:

1880: Marey draws the PLM schedule chart.
1983: Tufte reproduces and praises the chart in Visual Display.
1990: Tufte extends the principle in Envisioning Information.
2000s: Leland Wilkinson's The Grammar of Graphics formalizes the small-multiple concept as "facets" in a systematic chart-building framework.
2005: Hadley Wickham's ggplot2 (for R) implements faceting as a first-class feature, making it easy to produce small multiples with a single line of code.
2010s: matplotlib adds subplot support, seaborn adds FacetGrid, and every modern visualization library supports some form of faceting.
Today: Small multiples are the default answer to "how do I show the same relationship across many groups?" in virtually every data visualization context.

The direct technical lineage runs from Marey to Tufte to Wilkinson to Wickham to every modern library. The conceptual debt is shared, but the starting point is Marey's chart. Every time a modern designer makes a small multiple of 50 state trajectories or 30 product trends, they are working in a tradition that began with a 19th-century French engineer trying to display a railway schedule.

Why It Worked: A Theoretical Analysis

Marey's chart — and by extension the small-multiple tradition it inspired — succeeded because it applied principles that this chapter has been building.

1. It enabled comparison across many instances in a single view. Tables of train schedules let the reader look up one train at a time but not compare trains as a group. Marey's chart let the reader compare all trains simultaneously — see which were fast, which were slow, which met where — without any explicit computation. The comparison was built into the visual structure.

2. It used consistent encoding for every instance. Every train, whatever its specifics, was drawn as a line on the same grid, with slope encoding speed, direction encoding travel direction, and stops encoding pauses. The consistency meant the reader could read every train the same way. This is the similarity principle from Section 8.2 applied before the principle was formally named.

3. It used spatial alignment to encode the physical reality of the track. Stations on the y-axis were positioned in proportion to their actual distances along the track. A reader could trace a train's line downward and see the train's progress in real physical space, not just abstract "station N of 15." This is an early example of making the chart's spatial structure carry meaning — a technique that would become central to good data visualization.

4. It made patterns visible at the chart level, not just at the individual data level. The density of lines in different regions of the grid showed busy and quiet periods. The overall pattern of the lattice showed the structure of the schedule. These were emergent properties of the visual representation — patterns the reader could see in the whole chart that were not directly encoded in any single line. This is what small multiples and related techniques do: they let the reader see patterns across instances that no individual instance could reveal.

5. It replaced a lookup-oriented representation with a structure-oriented one. The table was good for lookup ("what time does train N arrive?"). The chart was good for structure ("how is the schedule organized?"). These are different jobs, and different representations serve different jobs. Marey recognized that the table was insufficient for the structural question and invented a representation for that specific purpose. This is the question-first discipline from Chapter 5 applied to representation design.

Complications and Limits

Marey's chart, and the Tufte treatment of it, are celebrated but not universally praised. A few legitimate critiques:

The chart assumes a linear route. Marey's chart worked because the PLM line was essentially one track from Paris to Lyon. Stations could be placed in a clean order along the y-axis. A chart of a branching railway network — with multiple lines diverging from Paris to several destinations — would not work as a single Marey-style grid. The chart is optimized for linear or near-linear systems and does not generalize to arbitrary network topologies. This is a real limitation, not a failure, but it means the chart type is not universally applicable.

The chart requires the reader to learn the encoding. A viewer seeing the chart for the first time has to figure out that slopes encode speed, that crossings indicate meetings, that horizontal segments indicate stops. This learning cost is low for an engineer or a regular railway user, but it is higher for a casual viewer. Modern small multiples have a lower learning cost because each panel is a familiar chart type (line, bar, scatter) and only the data varies. Marey's chart invented its own chart type, which is more demanding.

The chart cannot show detailed timetable information. The chart is great for structure but bad for specifics. A reader who wants to know exactly what time the 8:15 from Lyon arrives in Dijon has to squint at the chart and estimate. A table gives the answer to the minute. The two representations are complementary, not interchangeable — and Marey's chart was not designed to replace the table, but to supplement it with structural information the table hid.

Tufte's enthusiasm may overstate the chart's novelty. Time-distance charts had been used for various purposes before Marey, including in military logistics and marine navigation. Marey's contribution was the specific application to railway scheduling and the clarity of his design, not the invention of the form from nothing. This is a mild critique of the popular narrative rather than a substantive critique of the chart itself.

Lessons for Modern Practice

Marey's chart is 145 years old, and the lessons it offers are directly applicable to modern visualization practice.

When tables fail at structure, invent a chart. If the reader needs to see the structure of a system and a table representation is hiding the structure, consider whether a visual representation would work. The visual does not have to replace the table — it can supplement it — but for structural questions, the visual is almost always better.

Consistent encoding is the precondition for pattern recognition. Marey's chart showed twenty or thirty trains with the same encoding, and the reader could see patterns across trains because the encoding was consistent. Modern small multiples work the same way: shared chart type, shared scales, shared design, with only the data varying. The consistency is what enables the comparison. Violating the consistency defeats the purpose.

Spatial alignment carries meaning. Marey positioned stations along the y-axis in proportion to their real distances. The reader could read the chart spatially and see real-world relationships. When you design a chart, consider whether the spatial arrangement of elements can carry meaning beyond their individual values. Sometimes the positioning itself is an encoding.

The question drives the chart type. Marey needed to show structure. A table cannot show structure. He invented a chart type that could. The lesson from Chapter 5 applies here: the question determines the chart type, not the data. If the standard chart types do not serve your specific question, be willing to invent — or adapt — a chart type that does.

Small multiples are an inheritance, not an innovation. Every modern small-multiple figure stands on a lineage going back at least to Marey and through Tufte, Wilkinson, Wickham, and countless others. When you produce a small multiple, you are working in a tradition with specific conventions, specific strengths, and specific limitations. Study the tradition. The best modern small multiples are the ones whose designers understood where the technique came from and what it was originally trying to do.

Good visualization principles transcend their original context. Marey was designing for 19th-century railway engineers. The principles he applied — consistent encoding, spatial alignment, structure over lookup — work equally well for 21st-century climate scientists, business analysts, and data journalists. The specific application is contextual; the principles are durable.

Tables and charts are complements, not substitutes. Marey's chart did not replace the PLM timetable. It supplemented it. The table is still better for lookup; the chart is still better for structure. When you produce a visualization, ask whether your audience also needs a tabular view for lookup tasks — and if so, provide both. The two representations serve different questions, and presenting both is often more useful than choosing between them.

Discussion Questions

On the chart type as invention. Marey invented a chart type that was custom-built for the schedule problem. Modern designers usually work within a fixed vocabulary of chart types (bar, line, scatter, etc.). When is inventing a new chart type justified, and when is it excessive? What are the costs of a custom chart type compared to using a standard one?
On the relationship between tables and charts. Marey's chart supplemented the PLM timetable rather than replacing it. In your own work, when do you need both a table and a chart? How do you decide what each is for?
On the learning cost of unfamiliar chart types. Marey's chart required the reader to learn that slopes encode speed and crossings indicate meetings. Modern small multiples are easier because each panel is a familiar chart type. Is the lower learning cost of modern small multiples an advantage, or does it come at the cost of the visual creativity Marey demonstrated?
On the small-multiple lineage. Does it matter that modern small multiples trace their lineage through Tufte to Marey? Would you design your small multiples differently if you did not know the history? Is the tradition just context, or is it a set of constraints that good designers should honor?
On spatial encoding. Marey's y-axis encoded station distances along the track. Modern small multiples rarely use such "meaningful" spatial arrangements — panels are usually arranged in a regular grid based on alphabetical or ranked order, not on any spatial relationship among the groups. Is this a loss? When could you use meaningful spatial arrangement in your own work?
On what visualization cannot do. Marey's chart was excellent for showing structure but bad at showing exact times. Every chart has trade-offs: what it enables, what it hides. In your own work, what are the things your charts enable that tables cannot show? What are the things your charts hide that tables can show?

Marey's chart is 145 years old. The principles it embodies — consistent encoding, meaningful spatial arrangement, structural visibility, the invention of forms to match questions — are still the principles of good multi-instance visualization today. When you make a small multiple of COVID trajectories or revenue trends, you are practicing a craft that began with a French engineer trying to visualize a railway schedule. The craft has evolved, the tools have changed, but the underlying discipline — show the structure, not just the lookup — is the same. Marey's chart is a reminder that good visualization practice is older than you think, and that the basic questions have answers that have been refined over generations.