# Escaping Flatland

#### Synopsis

Displays of data are necessarily produced on a two-dimensional surface– paper or screen. Yet these are often at worst, misleading, or at best, incomplete. The representation of multidimensional phenomena on a two-dimensional surface was, and remains, graphics greatest challenge. In this chapter we discuss and illustrate some of the approaches that were used to communicate multidimensional phenomena within the practical limitations that we are always faced with.

#### Chapter contents

• Contour Maps
• Three-Dimensional Plots
• Going Forward

## Selected Figures

### Figure 8.1: Picturing a higher-dimensional plane in 2D

Karl Pearson tries to show a solution to the problem of fitting a plane of closest fit. In 2D, the fitted plane appears as a line.
Source: Karl Pearson, “LIII. On Lines and Planes of Closest Fit to Systems of Points in Space,” London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2:11 (1901), 559–572, page 560.

### Figure 8.2: Contour map of magnetic declination

Edmund Halley drew lines of equal magnetic declination on a map, possibly the first contour map of a data-based variable. The figure shows the map for the Atlantic Ocean. The curve are the isogonal lines, with the degree of magnetic declination given as numbers along each. The thick line is the agonic line of no variation where the compass reading is true; the dashed line with ships shows the track of Halley’s second voyage.
Source: Edmond Halley, A New and Correct Chart Shewing the Variations of the Compass in the Western & Southern Oceans as Observed in ye Year 1700 by his Maties Command, 1701.

### Figure 8.3: Detail showing Halley’s observations

This figure shows the central portion of Halley’s map with the locations of his observations. The triangles show the locations of observations from the first voyage; circles show those from the second voyage.
Source: Detail from Lori L. Murray and David R. Bellhouse, “How Was Edmond Halley’s Map of Magnetic Declination (1701) Constructed?,” Imago Mundi, 69 (2017):1, 72–84, Plate 10.

### Figure 8.4: Contour map of a bivariate table

The graph shows the level curves of recordings of soil temperature measured over time, for months of one year (horizontal axis) by hours of the day. The maximum temperature occurs in early July, around 3:00 PM.
Source: L. F. Kaemtz, Cours complet de météorologie. Paris: Paulin, 1845, Appendix figure 2.

### Figure 8.5: Population density of Paris

Louis-Léger Vauthier showed the population density of Paris by many contour levels representing densities of 200 to 1,200 people per unit area.
Source: Louis-Léger Vauthier, 1874.

### Figure 8.6: Axonometric projection of a 3D surface

The labeled points and connecting lines are meant to illustrate how surfaces and lines appear when projected onto the planes formed by the coordinate axes.
Source: Gustav Zeuner, Abhandlungen aus der mathematischen statistik. Leipzig: Verlag von Arthur Felix, 1869, fig. 4.

### Figure 8.7: 3D population pyramid

Luigi Perozzo showed the age distributions of the population of Sweden from 1750 to 1875 as a three-dimensional surface. Census years go from left to right, age is shown front (old) to back (young), and the height of the surface represents the count of people of that age.
Source: Luigi Perozzo, “Stereogrammi Demografici – Seconda memo- ria dell’Ingegnere Luigi Perozzo. (Tav. V),” Annali di Statistica, Serie 2, Vol. 22, (Ministero d’Agricoltura, Industria e Commercio, Direzione di Statistica), 1881, pp. 1–20.

### Figure 8.8: 3D statistical sculpture

Perozzo created this 3D model of the population data as a tangible object, perhaps the first statistical sculpture.
Source: Centre Pompidou.

### Plate P.15: Galton’s isochronic chart of travel time

Isolevel contours of equal travel time from London are depicted by shading.
Source: Reproduction courtesy of Royal Geographical Society, S0011891.