286 K. Worden et al. Fig. 34.1 Fraternal (non-identical) twins (publicity still from 1988 filmTwins) the question is essentially of generalisation; having learned from model data, can one say something meaningful about the structure twinned with the model? The use of the work ‘twin’ in the context of modelling is actually interesting semantically. Clearly the idea is to suggest a one-to-one relationship or identity between a structure and a model; however, this is unjustified if one refers to biology. According to recent statistics,2 roughly one in 65 births results in twins; of these, the vast majority are fraternal—or nonidentical twins (see Fig. 34.1). Furthermore, opposite-sex twin pairs make up roughly 33% of fraternal twins. In fact, identical twins result from only one in 285 births, and even identical twins have distinct teeth marks and fingerprints. Finally, twins may look identical, but behave incompletely different ways. In order to establish an over-arching mathematical framework, one will need to be precise and meaningful in one’s terminology. The use of the term ‘twin’ is inconsistent with this goal for two reasons discussed above; the first is that there is already widespread and disparate use of the term in the engineering community; the second is that it doesn’t really make sense as an analogy anyway. The view taken in this paper, will be that a more meaningful term is provided by the word mirror. A mirror is an instrument that faithfully reflects reality in terms of the aspects of an object that are mirror-facing; it provides no ‘information’ about aspects that are not mirror-facing. The idea of ‘mirror-facing’ will be formalised in the following as a context. Finally, if the object moves, the movement will be reflected perfectly, in the mirror—at least as far as those aspects that are mirror-facing. This paper then, will attempt to motivate a mathematical basis for understanding mirrors.3 As such, it will have the opportunity to develop independently of current conceptions as to what a ‘digital twin’ is, but leaving the possibility for engineers to adopt the technology in developing whatever their favoured definition of a digital twin actually is.4 Enough of levity; it is important to remember that everything here is motivated by the desire to construct meaningful validated models of structures and systems; if one were to do nothing more than rearrange the terminology and dress the problem in pretty mathematical trappings, then that would be ultimately empty. This paper is motivated by the belief that a general mathematical theory of models and their validation will be of value; however, the current paper will not be able to go beyond development of the basic terminology and theory and some attempts to convince the reader of the ultimate possibilities. One might argue that general frameworks have already been proposed in terms of the formulation and evaluation of models, and that there is no need to propose another one until the existing ones have been fairly evaluated. This is a fair Furthermore, a lot of the motivation for digital twins comes from industry, and it is not conceivable that a profit-driven enterprise would require a model to function outside the immediate context of interest if that extended functionality came at an increased cost. 2Twin statistics from http://www.twinsuk.co.uk—accessed 30th June 2018. 3The termdigital mirror is already in use to define an item of technology; the items being exactly what one might imagine them to be. One could use the term with complete confidence that the two meanings are unlikely to be confused; however, for simplicity the objects of interest will just be referred to as ‘mirrors’, although different kinds of mirrors will be introduced. 4As a final observation on biological twins, there is an interesting link between twins and mirrors. Twenty five percent of identical twins are mirror-image twins; their hair falls in different directions, their fingerprints are mirror images of each other and, if one of them is right-handed, the other will be left-handed.
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