The polyhedron metaphor

“The verb phrase can begin with any of the nodes and continues from there to the right” [Mindt 1995, p.14], where ”the final element of the verb phrase”

[ibid.] is situated. The topology of the basic polyhedron – in combination with the meta-rule “leftward movement prohibited” – allows for 15 different paths, including the “path” mv, which is of length zero. Table A.8 lists these paths

Table A.8: Possible paths in the prototypical polyhedron

Structure Mindt’s example

mv I go

mod-mv you can forget all that aux-mv she has left your house cat-mv he wanted to say something do-mv I don’t sleep in the afternoons mod-aux-mv the women must have come far cat-aux-mv he seemed to have solved a problem aux-cat-mv he had wanted to rush out

mod-cat-mv I might want to stay on

do-cat-mv don’t you want to kiss my lips?

mod-aux-cat-mv she must have hoped to find him

aux-cat-aux-mv D. was not believed to have been a man of means mod-cat-aux-mv he must pray to be spared

do-cat-aux-mv he did not wish to be stopped for speeding mod-aux-cat-aux-mv it can be seen to have been inevitable

ordered by length. Mindt classified these structures by looking at the faces, i.e. the circuits of edges, of the basic polyhedron and named these polygons

“verbal triangles”. The topology of the basic polyhedron allows for five verbal triangles³⁰: mod-cat-mv,mod-aux-mv, cat-aux-mv, mod-aux-cat, and do-cat-mv. Themod-cat-mvtriangle is considered basic because it does not contain nodes of the typeauxordo. It “accounts for three types of verb phrase”

[Mindt 1995, p.15], i.e. paths in the basic polyhedron, which are labelled by the formulae mod-mv, cat-mv and mod-cat-mv.

30NB The order of the nodes in the names of the verbal triangles incorporates the meta-rule of imperative rightward movement within the basic polyhedron.

Figure A.2: Mindt’s verbal triangles

The basic polyhedron The mod-cat-mv triangle

The mod-aux-mv triangle Thecat-aux-mv triangle

The mod-aux-cat triangle Thedo-cat-mv triangle

The three triangles that contain anauxnode form a group that is called ‘the auxtriangles’. They “operate within themod-cat-mvtriangle” [Mindt 1995, p.16] because all three of them share an edge with the basicmod-cat-mv trian-gle: themod-aux-mvtriangle shares themod-mvedge with the basic triangle;

the cat-aux-mv triangle shares the cat-mv edge with the basic triangle; and the mod-aux-cat triangle shares the mod-cat edge with the basic triangle.

The property of sharing an edge with the basic triangle is seen as a propensity for modifying the connection that is represented by the pertinent edge. The aux triangles classify the formulae labelling verb phrases containing an auxiliary as follows: the mod-aux-mv triangle accounts for aux-mv and mod-aux-mv; the cat-aux-mv triangle accounts for cat-aux-mvand mod-cat-aux-mv; the mod-aux-cat³¹ triangle accounts for aux-cat-mv, aux-cat-aux-mv, mod-aux-cat-mv, and mod-aux-cat-aux-mv.

Naturally, thedo-cat-mv triangle accounts for the verb phrases containing do (= auxiliary do), namely do-mv, do-cat-mv and do-cat-aux-mv. It is worth mentioning that the modals’ lack of do-periphrasis is reflected by the topology of the basic polyhedron. The do-cat-mv triangle and the mod-cat-mv triangle correspond in the second and the third node but there is no connection between their first nodes, i.e. “the first elements of the two triangles are mutually exclusive” [Mindt 1995, p.19].

The structure of the basic polyhedron – as discussed so far – needs to be enhanced to incorporate a description of the interplay of perf,prog and pass because none of the formulae listed in table A.8 contains two auxiliaries next to each other. Mindt’s solution to this problem posits that bothauxnodes of the basic polyhedron allow for internal structure: auxis seen as a placeholder for up to three auxiliary nodes, which are simply numbered consecutively from 1 to 3 (aux1:=perf³²,aux2 :=prog,aux3:=pass). The three auxiliary nodes and a fourth node labelledcat/mvare pairwise connected to form a tetrahedron.

31“Themod-aux-cat triangle does not occur on its own, but is always followed by either a main verb or a combination of auxiliary + main verb” [Mindt1995, p.18]. This property – in combination with the topology of the basic polyhedron – explains the fact that the longest possible path, which comprises five nodes, is accounted for by this verbal triangle.

32Mindtuses the wordperfectiveinstead ofperfectwhen referring to the auxiliary: “aux1 (perfective)” [Mindt1995, p.22]. When specifying infinitives, he uses the word perfect, though: “perfect infinitive” [Mindt1995, p.12].

The cat/mv node is a placeholder for the final element, i.e. the catenative or the main verb “which is premodified by aux” [Mindt 1995, p.24]. The nested bracketing of the auxiliary phrase is made explicit by labelling the interconnecting line segments. The six labels specify whether the first or the second participle is needed. Additional arrows emphasize the meta-rule of imperative rightward movement. There are seven possible paths within theauxtetrahedron (cf. figure

Figure A.3: The internal structure of the components of the aux position

A.3). The formulae labelling these paths are: aux1-cat/mv,aux2-cat/mv, aux3-cat/mv, aux1-aux2-cat/mv,aux1-aux3-cat/mv, aux2-aux3-cat/mv and aux1-aux2-aux3-cat/mv. There is a one-to-one correspon-dence between this notation and the [perf, prog, pass]-notation, e.g. aux2-cat/mv ↔ [perf: ∅, prog: +, pass: ∅] or aux1-aux3-cat/mv ↔ [perf: +, prog: ∅, pass: +]. If both aux nodes have internal structure, then structures containing the string aux-cat-auxmight be expanded consid-erably. Theoretically,7² = 49combinations are possible. The extreme case would be aux1-aux2-aux3-cat-aux1-aux2-aux3-mv, e.g.^??we have been being begged to have been being loved – admittedly a somewhat constructed example.

Still, 9 out of the 49 formulae are needed to specify which auxiliaries are used in aux-cat-aux, even if there is just one auxiliary in each node. A few more of

the remaining 40 formulae might be needed to accommodate constructions such as she has been trying to be accepted (aux1-aux2-cat-aux3-mv). Complex aux-cat-aux constructions, i.e. those where both aux nodes have more than one auxiliary, are rare for several reasons. Firstly, the aux2-aux3-combination as such is rare (awkwardness ofbe(en) being). Secondly, the catenative verb and the second auxiliary phrase have to ‘fit semantically’³³. Thirdly, some catenative verbs are followed by a second participle, which cannot be expanded³⁴.

Double aux constructions with internal structure are not the only way to expand the verb phrase. A more flexible way is the combination of two or more catenative verbs. The internal structure of cat can be cat-cat³⁵ or cat-aux-cat³⁶. Figure A.4 represents this. “The components of the verb phrase which precede or follow thecatposition are disregarded in this diagram”

[Mindt 1995, p.26]. Contrary to the auxtetrahedron, the cattriangle allows

Figure A.4: The internal structure of the components of the cat position

for further nesting because each of itscatvertices can have an internal structure on its own. Theoretically, there is no upper limit to the length of strings that combine cat and aux. A fairly long example containing four catenative verbs would be they might have seemed to have been going to be allowed to beg to be loved (mod-aux-cat-aux-cat-cat-cat-aux-mv).

To recap, the basic polyhedron – in combination with the optional integration of a tetrahedron and a (series of nested) triangle(s) representing the internal structures of the aux node and the cat node – provides a powerful ordering

33E.g.seem to have been stealing but^?want to have been loved.

34E.g.get married but*get been married,*get had married and*get been marrying.

35E.g. “the gendarme appeared to beginto understand” [Mindt1995, p.26].

36E.g. “can be seen to have startedhappening then” [Mindt1995, p.26].

scheme for classifying English verb forms. Oriented paths along the edges of the basic polyhedron determine the linear order of the components of the verb phrase. The topology, i.e. the arrangement of the nodes and edges, of the basic polyhedron accounts for the great variety³⁷ of English verb forms while ensuring that all paths that are possible represent grammatical verb forms.

One major drawback of the polyhedron metaphor is the fact that it is not immediately comprehensible. At first, the topology of the basic polyhedron seems to be counterintuitive. An above-average capacity to think in three dimensions is needed because the basic polyhedron is asymmetric. Most people will need a blueprint of the basic polyhedron in front of them while working with this metaphor. Furthermore, one needs to have grasped the meta-rule of imperative rightward movement to understand this metaphor. All that does not detract from its academic value but makes it difficult to use it in teaching. One reviewer of Mindt’s book put it like this: “These 3-D models [. . . ] are certainly a novel approach, but their pedagogical value and necessity are not immediately apparent” [Wynne 1997, p.2]. The eight³⁸ paths of the aux tetrahedron can be mapped onto the eight corners of the unit cube. This yields an alternative visualization of the interplay of perf,prog andpass, which is presented in the next section.