13.4 Visualization

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13.4

Visualization

It seems almost impossible to overestimate the power of visualization, as a mode

of knowledge representation, to influence the interpretation of data. ¹⁰ In this regard,

supremacy belongs to Cartesian coördinates, perhaps the most important mathemat-

ical invention of all time. Two-dimensional representations that can be drawn on

paper (or viewed on a screen) are particularly significant. As already mentioned,

one of the main motivations of PCA (Sect. 13.2.2) is to enable a complex dataset

to be represented on paper. This applies equally well to dynamical representations

of evolving systems, in which phase portraits (state diagrams in phase space; cf.

Fig. 12.3) of a dynamical system such as a living cell can be very influential.

Another kind of visualization consists in generating images from binary expan-

sions. ¹¹ On paper, both the actual decimal digits of the irrational numberpiπ and those

of the rational approximation 22/7 look random; when their binary expansions are

drawn as rows of light (corresponding to 0) and dark (corresponding to 1) squares,

pattern (or its absence) is immediately discernible (Fig. 13.2).

More generally, visualization should be considered as part of the overall process

of accumulating convincing evidence for the validity of a proposition. It should not,

therefore, be merely an alternative to a written or verbal representation, but should

transcend the limitations of those other types of representations.

Fig. 13.2 The binary expansion of the first 1600 decimal digits (mod 2) ofpiπ (left) and22 divided by 722/7 (right),

represented as an array of light (0) and dark (1) squares, to be viewed left to right, top to bottom

10 Cf. Sect. 31.3.

11 It is said that Leibniz was the first to raise this possibility in a letter to one of the Bernoulli

brothers, in which he wondered whether it might be possible to discern a pattern in the binary

expansion ofpiπ.