3D Euclidean Space
3D Euclidean space refers to the three-dimensional space described by the Euclidean geometry, where the position of a point is specified by three coordinates. In physics, it is commonly used to represent the physical space we live in and to describe the position and movement of objects. This space is characterized by the three perpendicular axes - x, y, and z.
6 Key excerpts on "3D Euclidean Space"
eBook - ePub- Henri Poincaré(Author)
- 2012(Publication Date)
- Dover Publications(Publisher)
...Beings whose minds were made as ours, and with senses like ours, but without any preliminary education, might receive from a suitably-chosen external world impressions which would lead them to construct a geometry other than that of Euclid, and to localise the phenomena of this external world in a non-Euclidean space, or even in space of four dimensions. As for us, whose education has been made by our actual world, if we were suddenly transported into this new world, we should have no difficulty in referring phenomena to our Euclidean space. Perhaps somebody may appear on the scene some day who will devote his life to it, and be able to represent to himself the fourth dimension. Geometrical Space and Representative Space.—It is often said that the images we form of external objects are localised in space, and even that they can only be formed on this condition. It is also said that this space, which thus serves as a kind of framework ready prepared for our sensations and representations, is identical with the space of the geometers, having all the properties of that space. To all clear-headed men who think in this way, the preceding statement might well appear extraordinary; but it is as well to see if they are not the victims of some illusion which closer analysis may be able to dissipate. In the first place, what are the properties of space properly so called? I mean of that space which is the object of geometry, and which I shall call geometrical space. The following are some of the more essential:— 1st, it is continuous; 2nd, it is infinite; 3rd, it is of three dimensions; 4th, it is homogeneous—that is to say, all its points are identical one with another; 5th, it is isotropic. Compare this now with the framework of our representations and sensations, which I may call representative space. Visual Space.—First of all let us consider a purely visual impression, due to an image formed on the back of the retina...
eBook - ePub- Rudolf Carnap(Author)
- 2012(Publication Date)
- Dover Publications(Publisher)
...In Kantian terms, it is indeed both analytic and a priori. But it is not possible to say that it is also synthetic. It is simply a deductive system based on certain axioms that do not have to be interpreted by reference to any existing world. This can be demonstrated in many different ways, one of which is given in Bertrand Russell’s early book, The Principles of Mathematics (not to be confused with the later Principia Mathematica). 15 Russell shows how it is possible to define Euclidean space entirely as a system of primitive relations for which certain structural properties are assumed; for example, one relation is symmetric and transitive, another is asymmetric, and so on. On the basis of these assumptions it is possible to derive logically a set of theorems for Euclidean space, theorems that comprise the whole of Euclidean geometry. This geometry says nothing at all about the world. It says only that, if a certain system of relations has certain structural properties, the system will have certain other characteristics that follow logically from the assumed structure. Mathematical geometry is a theory of logical structure. It is completely independent of scientific investigations; concerned solely with the logical implications of a given set of axioms. Physical geometry, on the other hand, is concerned with the application of pure geometry to the world. Here the terms of Euclidean geometry have their ordinary meaning. A point is an actual position in physical space. Of course we cannot observe a geometrical point, but we can approximate it by making, say, a tiny spot of ink on a sheet of paper. In a similar way, we can observe and work with approximations of lines, planes, cubes, and so on. These words refer to actual structures in the physical space we inhabit and are also part of the language of pure or mathematical geometry; here, then, lies a primary source of nineteenth-century confusion about geometry...
eBook - ePub- Michael Harrison, Patrick Waldron(Authors)
- 2011(Publication Date)
- Routledge(Publisher)
...Vectors and vector spaces DOI: 10.4324/9780203829998-7 5.1 Introduction It is customary to think of vectors as entities with magnitudes and directions, and spaces as like the two-dimensional space we write in and the three-dimensional space we live in and move around in. Vectors are therefore distinct from scalars, which have magnitudes only. Our aim in this chapter is to develop a collection of results that apply to such vector entities in real n -dimensional space, or simply n -space. Our approach will be both geometric and analytic. The vector geometry will provide fresh insights into what we have already encountered in our algebraic study of n × 1 and 1 × n matrices, while the analysis will echo the matrix algebra itself. However, as we are familiar with one, two and three spatial dimensions, and can visualize more easily in these cases, we begin with vectors in 2-space (ℝ 2) and 3-space (ℝ 3) in order to fix the main ideas intuitively. It will quickly be seen that the vectors in these cases may readily be associated with 2 × 1 and 3 × 1 matrices, respectively. However, later generalization is intended not only to take us from 2- and 3-space to n -space and n × 1 matrices, but also to abstract the main properties of vectors in n -space so that they apply as well to kinds of objects other than real row or column matrices. 5.2 Vectors in 2-space and 3-space 5.2.1 Vector geometry In 2-space, also known as the plane or Euclidean plane or Cartesian plane, a simple geometric approach is to represent vectors by arrows, where the length of the arrow represents the magnitude of the vector and the direction of the arrow, relative to some arbitrary datum in the plane, represents the direction of the vector. To draw a vector in the plane, we must know not only its magnitude and direction, but also its location. Thus three vectors, v, w and z, may be depicted as in Figure 5.1...
eBook - ePub- Barry Dainton(Author)
- 2016(Publication Date)
- Routledge(Publisher)
...13 Curved space 13.1 New angles on old problems We may have pursued the classical debate on space and motion a good way beyond its original boundaries, but there are further modifications to the classical world-view of a still more fundamental kind that we have yet to explore. Viewing motion as a process unfolding in spacetime does not require or involve the abandonment of the classical conception of space as a three-dimensional Euclidean structure, and the various spacetimes we have considered thus far retained this conception of space: there are no spatial distances over time in neo-Newtonian worlds, but the spaces-at-times that remain are entirely Euclidean. The spacetime perspective came into its own when Minkowski gave a spacetime interpretation of Einstein's special theory of relativity in 1908, but significant advances in the understanding of space – advances that play a crucial role in Einstein's general theory of relativity – had already occurred. By the middle of the nineteenth century it had become clear to mathematicians that space can take different forms and that the structure of Euclidean space is just one spatial structure among many. This discovery gave rise to new questions: what sort of space do we live in and how can we find out? It also transforms the debate between substantivalists and relationists, as we shall see in Chapter 14. However, before entering these debates we need to know something about these strange non-classical, non-Euclidean curved spaces, and this chapter is devoted to this end (and so can safely be skipped by those already familiar with the basics). 13.2 Flat and curved spaces In §9.2 we encountered Flatland, a two-dimensional plane inhabited by two-dimensional (but nonetheless intelligent) creatures. This imaginary world was used to undermine a deep-seated assumption that we naturally make and never normally have reason to question: that space, even nothingness itself, is necessarily three-dimensional...
eBook - ePub- J. R. Lucas(Author)
- 2018(Publication Date)
- Routledge(Publisher)
...Nevertheless, the entrenched position that Euclidean geometry holds in our affections is not, as is often made out, mere prejudice or intellectual conservatism, a carry-over from the classical education to which we were subjected in our schooldays. The same sort of arguments which we adduced to establish the topological properties of space (in its fourth sense) can be carried further to show why when we come to consider the geometrical properties of space (in its third sense) we should prefer to have it a Euclidean space if we reasonably can. [ 1 ] Republic, VII, 533c: (“Where the Starting point is something one does not know, and the conclusion and the intervening steps are fabricated out of things one does not know, how on earth can this sort of entailment ever become knowledge?”) [ 2 ] H. Poincaré, La Science et l ’Hypothese (Paris, 1902), ch. III, IV, V, pp. 49–109, esp. pp. 66–7, 79–80, 83; and La Valeur de la Science (Paris, 1914), ch. III, pp. 59–95; tr. G. B. Halsted (New York, 1958). [ 3 ] Perhaps Kant did. See Gedanken von der wahren Schatzung der lebendigen Kräfte, § 10 (124); and G. Martin, Kant’s Metaphysics and Theory of Science, tr. P. G. Lucas (Manchester, 1955), ch. I, esp. p. 18. § 34 The measurement of space Euclidean geometry is a metrical geometry. Length and angle are concepts fundamental to it, and, as we shall see (§ 39, pp. 185–6), in Pythagoras’ proposition they are related in the simplest feasible fashion. If, therefore, we can give good reason for having space Euclidean, we have thereby made it metrical. Normally, however, the order of argument is reversed. First a metric is established, and then the claims of the different metrical geometries are considered. There is a danger in this. We do not have a uniform theory or method of measurement...
eBook - ePub- Albert Einstein(Author)
- 2013(Publication Date)
- Routledge(Publisher)
...In the physics of measurement this is attained by the application of the Cartesian system of co-ordinates. This consists of three plane surfaces perpendicular to each other and rigidly attached to a rigid body. Referred to as a system of co-ordinates, the scene of any event will be determined (for the main part) by the specification of the lengths of the three perpendiculars or co-ordinates (x, y, z) which can be dropped from the scene of the event to those three plane surfaces. The lengths of these three perpendiculars can be determined by a series of manipulations with rigid measuring-rods performed according to the rules and methods laid down by Euclidean geometry. In practice, the rigid surfaces which constitute the system of co-ordinates are generally not available; furthermore, the magnitudes of the co-ordinates are not actually determined by constructions with rigid rods, but by indirect means. If the results of physics and astronomy are to maintain their clearness, the physical meaning of specifications of position must always be sought in accordance with the above considerations. 1 1 A refinement and modification of these views does not become necessary until we come to deal with the general theory of relativity, treated in the second part of this book. We thus obtain the following result: Every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for “distances,” the “distance” being represented physically by means of the convention of two marks on a rigid body. 3 Space and Time in Classical Mechanics DOI: 10.4324/9781315886749-3 The purpose of mechanics is to describe how bodies change their position in space with “time.” I should load my conscience with grave sins against the sacred spirit of lucidity were I to formulate the aims of mechanics in this way, without serious reflection and detailed explanations...
