Geometry, Euclidean

Exact calculations have already been carried out for simple one and two dimensional Euclidean geometries by exploiting properties of polynomials (chapter 5.2.1) and circulant matrices (chapter 5.2.2) over the finite field J-[q, q p wherep is prime. We will here rely instead on the theory of input-free modular systems, which is more suitable for dealing with the dynamics of completely arbitrary lattices. 

Vaisman, Foundations of Three-Dimensional Euclidean Geometry (1980)... 

In this introductory treatment of the applications of group theory to chemistry, all mathemahcal tools are introduced and developed as they are needed. Familiarity is assumed with only the basic ideas of Euclidean geometry, trigonometry and complex numbers. 

The Fourth Dimension and Non-Euclidean Geometry in Modern Art. Princeton Univ. Press, 1983. 

Sommerville, D. M. Y. Bibliography of Non-Euclidean Geometry. Univ. of St. Andrews,... 

Note that dOd(l> is the natural surface area coming from the Euclidean geometry of the space in which the two-sphere sits. 

Complex scalar products arise naturally in quantum mechanics because there is an experimental interpretation for the complex scalar product of two wave functions (as we saw in Section 1.2). Students of physics should note that the traditional brac-ket notation is consistent with our complex scalar product notation—just put a bar in place of the comma. The physical importance of the bracket will allow us to apply our intuition about Euclidean geometry (such as orthogonality) to states of quantum systems. 

Scientific theories themselves can be distinguished as deductive or inductive in nature, according to the underlying character of their premises. In a deductive theory, the fundamental premises are axioms or postulates that are neither questionable nor explainable within the theory itself. Outstanding examples of deductive theories include Euclidean geometry (based on Euclid s five axioms) and quantum mechanics (based on Schrodinger s prescription for converting classical trajectory equations into wave equations). An inductive theory, on the other hand, is based on universal laws of experience that express what has always been found to be true in the past, and may therefore be reasonably expected to hold in the future. Thermodynamics is the pre-eminent example of an inductive theory. 

To introduce the notation and concepts to be used below, let us first briefly recall some elementary aspects of the Euclidean geometry of a triangle of points V, V2, V3 in ordinary three-dimensional physical space. Each point Vi can be represented by a column vector vt (denoted with a single underbar) whose entries are the coordinates in a chosen Cartesian axis system at the origin of coordinates ... 

It is also possible to adapt the general matrix-algebraic operations (9.8)—(9.11) to describe the Euclidean geometry of (9.2)-(9.6). To do so, we note that each column vector y i can be identified as a matrix of one column (nc = 1), so that (9.3) becomes a special case of (9.10) to define a space of column vectors. We can now create an associated space of row vectors by defining, for any given column vector v,... 

Euclidean geometry was originally deduced from Euclid s five axioms. However, it is now known that necessary and sufficient criteria for Euclidean spatial structure can be stated succinctly in terms of distances, angles, and triangles, or, alternatively, the scalar product of the space. We can express these criteria by employing Dirac notation for abstract ket vectors R ) of a given space M with scalar product (R R7). 

The criteria (9.23), (9.24), and (9.26) are all rather obvious properties of Euclidean geometry. All of these properties can be traced back to mathematical properties of the scalar product (R Ry), the key structure-maker of a metric space. We therefore wish to determine whether a proposed definition of scalar product satisfies these criteria, and thus guarantees that M is a Euclidean space. 

In short, the theorems, terminology, and working methods of Euclidean geometry can be carried over intact into this abstract thermodynamic domain. 

It is a remarkable fact that properties (13.4a-c) are necessary and sufficient to give Euclidean geometry. In other words, if any rule can be found that associates a number (say, (X Y)) with each pair of abstract objects ( vectors X), Y)) of the manifold in a way that satisfies (13.4a-c), then the manifold is isomorphic to a corresponding Euclidean vector space. We introduced a rather unconventional rule for the scalar products (X Y) [recognizing that (13.4a-c) are guaranteed by the laws of thermodynamics] to construct the abstract Euclidean metric space Ms for an equilibrium state of a system S, characterized by a metric matrix M. This geometry allows the thermodynamic state description to be considerably simplified, as demonstrated in Chapters 9-12. 

Riemann s lecture indeed shook geometry to its foundations. He was the first to propose extending Euclidean geometry concepts beyond three dimensions. More importantly, Riemann showed how one could entirely reject Euclid s fifth postulate ( through a point... 

It is quite conceivable that the geometry of space in the very small does not satisfy the axioms of [Euclidean] geometry. .. The properties which distinguish space from other conceivable triply-extended magnitudes are only to be deduced from experience. 

Cox98] H. S. M. Coxeter, Non-Euclidean Geometry, 6th edition, MAA Spectrum. Mathematical Association of America, 1998. 

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