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(SNP) has consistently come in third or second place (by 2011 it was placed first
in Scotland), but the third major British party is still in the reckoning and has
quite a different pattern of support.
The abstract creation of an electoral tetrahedron is quite simple. The points,
representing the electoral divisions in which the vote is counted, are placed so
that their distance from the four apexes is in exact proportion to the share of the
four-party vote represented by those apexes. To stick to convention, looking from
above, the Conservatives have the right hand apex, Labour the left, the Scottish
Liberal Democrat (SLD) party the top (still on the plane) and the Nationalists
have the apex in the centre (now above the other three).
The problems are familiar when we attempt to project the three-dimensional
structure of voting on to a two-dimensional plane for drawing. Here the method
involves taking slides from an animation and showing those slices from particular
angles. However, another method has also been developed in this case. That is
to unfold a net of the tetrahedron as four equilateral triangles (Figure 9.15).
A point is drawn on the side of the tetrahedron it was originally closest to.
As the centre of the political battleground tends to be relatively empty (due
perhaps to tactical voting), the unfolded tetrahedron technique does not create
results that are too ambiguous. 9 In fact, each triangle in the net contains only
those divisions where a particular one of the four parties came last. The net
can actually be further subdivided into areas in which the exact order of the
parties is known. Such an arrangement makes interpretation of a complex three-
dimensional situation considerably simpler (Figure 9.16), although in explicitly
using two dimensions something has to be lost - in this case the fortunes of the
party coming fourth in each constituency. 10 The four triangles show the positions
of the parties coming first, second and third. Along the six edges are shown the
relatives positions of the parties that come first and second. It is clear that the
most common result is Labour first, SNP second.
significantly. In particular, straight fights between Labour and the Conservatives, which were again
the most common, are now relatively rare. The increase in Conservative v Labour v SNP contests
is a direct function of the larger number of SNP candidates. This also explains why, despite the
fall in the number of SLD candidates, the proportion of four-way contests reached a high point of
twenty-three per cent of contests in 1988' (Bochel and Denver, 1988, p. v).
9 Movement to the apexes of the electoral tetrahedron would indicate that the following had
occurred: 'This apparent consolidation of strength in the parties' own territories is an interesting
phenomenon; it is unclear on the available evidence whether incumbency of itself gives an advantage
or whether parties successfully targeted their campaign effort to exploit existing support' (Bochel
and Denver, 1990, p. vi). In hindsight what was really happening was voting polarisation following
social polarisation.
10 How reliable are our visual and mental abilities when dealing with this complexity?: 'The use
of such a system poses interesting theoretical questions: Is exploring data by looking at projections
“safe” - if you look at enough different projections of structureless data, will you find structure
by chance? If it is safe, is it “effective”? - in what sense can the information in a d-dimensional
point cloud be extracted from a few of its 3-dimensional projections? The method, properly applied,
appears to be both safe and effective, even allowing for the fact that we do not know the statistical
properties of the eye as pattern detector' (Donoho et al ., 1988, p. 119).
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