Multi-dimensional space concept

Romance of Many Dimensions" by Edwin A. Abbott. While Abbott s work tries to introduce the reader to the concept of the multi-dimensional space, it chooses fewer dimensions than three as starting point. By doing so, Abbott came up with imaginary laws of nature that apply in one and two dimensions. Although these laws, which for instance explain how rain is experienced in two dimensions, are unrealistic, they impressively illustrate the mystery of lower dimensionalities. 

Using the equation that each "higher" dimension is perpendicular to the one "below" it (these are three-dimensional concepts, but the principle transcends its description), we can say that consciousness and time are perpendicular to space,and are therefore multi-dimensional phenomena. This brings us back to the Shamanic model of the psyche, which we can now see as a more precisely differentiated description of human awareness than any provided by Western psychology. 

Although all of the separation problems involving binary mixtures may be solved by use of the general methods presented in subsequent chapters for multi-component mixtures, it is, nevertheless, rewarding to consider the special case of the separation of binary mixtures because this separation may be represented graphically in two-dimensional space. Many of the concepts of distillation may be illustrated by the graphical method of design proposed by McCabe and Thiele.9... 

In traditional phonetics, phones were seen as a set of distinct, discrete, separable units. This notion is no longer defensible because we now know that in reality the phonetic space is a multi-dimensional continuum (describable with either articulatory or acoustic dimensions), and there are no bottom-up divisions within it. The concept of using phones should now be seen as an idealised division of phonetic space and this can be justifiable as a handy notational device it is cumbersome to describe every sound in terms of continuous dimensions (of, say, tongue height). It is important to realise, however, that phones really are a discrete abstraction of a continuous space no procedure or process for objectively defining these units can be devised. 

The Flamelet Generated Manifolds (FGM) approach aims to provide a tool for simplified treatment of the chemical system for flames where also transport in terms of diffusion is important. This is particularly the case for premixed flames where there are also less reactive parts on one side of the flame zone from which radicals diffuse towards the flame zone. The concept is based on two simplification procedures one is the flamelet approach where a multi-dimensional flame is considered as an ensemble of one-dimensional flames, and another which is the consideration of a low-dimensional manifold in composition space. 

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