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Classifications of Space
Engineer Ahmed Rafique Aziz
B.Sc Engineer,Electrical and Electronics Engineering
Professor ( Assessed),Researcher (Active),Euclidean Plane Geometry,Geometry,Pure Mathematics
Khulna University of Engineering and Technology ( KUET),Khulna-9203,Bangladesh.
Telephone:+88-01748324836
E-mail :azizahmedrafique@gmail.com
Definition of Space :
Space is the boundless three-dimensional extent in which objects and events have relative position
and direction.[1]
Physical space is often conceived in three linear dimensions, although modern physicists
usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime.
In mathematics, "spaces" are examined with different numbers of dimensions and with different
underlying structures. The concept of space is considered to be of fundamental importance to an
understanding of the physical universe. However, disagreement continues between philosophers over
whether it is itself an entity, a relationship between entities, or part of a conceptual framework.
Debates concerning the nature, essence and the mode of existence of space date back to antiquity;
namely, to treatises like the Timaeus of Plato, or Socrates in his reflections on what the Greeks called
khora (i.e. "space"), or in the Physics of Aristotle (Book IV, Delta) in the definition of topos (i.e. place), or
even in the later "geometrical conception of place" as "space qua extension" in the Discourse on Place
(Qawl fi al-Makan) of the 11th-century Arab polymath Alhazen.
[2]
Many of these classical philosophical
questions were discussed in the Renaissance and then reformulated in the 17th century, particularly
during the early development of classical mechanics. In Isaac Newton's view, space was absolute—in the
sense that it existed permanently and independently of whether there were any matter in the space.
[3]
Other natural philosophers, notably Gottfried Leibniz, thought instead that space was in fact a collection
of relations between objects, given by their distance and direction from one another. In the 18th century,
the philosopher and theologian George Berkeley attempted to refute the "visibility of spatial depth" in his
Essay Towards a New Theory of Vision. Later, the metaphysician Immanuel Kant said neither space nor
time can be empirically perceived, they are elements of a systematic framework that humans use to
structure all experiences. Kant referred to "space" in his Critique of Pure Reason as being: a subjective
"pure a priori form of intuition", hence it is an unavoidable contribution of our human faculties.
In the 19th and 20th centuries mathematicians began to examine non-Euclidean geometries, in which
space can be said to be curved, rather than flat. According to Albert Einstein's theory of general relativity,
space around gravitational fields deviates from Euclidean space.
[4]
Experimental tests of general relativity
have confirmed that non-Euclidean space provides a better model for the shape of space.
Philosophy of Space :
Leibnez and Newton :
In the seventeenth century, the philosophy of space and time emerged as a central issue in epistemology
and metaphysics. At its heart, Gottfried Leibniz, the German philosopher-mathematician, and Isaac
Newton, the English physicist-mathematician, set out two opposing theories of what space is. Rather than
being an entity that independently exists over and above other matter, Leibniz held that space is no more
than the collection of spatial relations between objects in the world: "space is that which results from

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places taken together".[5]
Unoccupied regions are those that could have objects in them, and thus spatial
relations with other places. For Leibniz, then, space was an idealised abstraction from the relations
between individual entities or their possible locations and therefore could not be continuous but must be
discrete.[6]
Space could be thought of in a similar way to the relations between family members. Although
people in the family are related to one another, the relations do not exist independently of the people.[7]
Leibniz argued that space could not exist independently of objects in the world because that implies a
difference between two universes exactly alike except for the location of the material world in each
universe. But since there would be no observational way of telling these universes apart then, according
to the identity of indiscernibles, there would be no real difference between them. According to the
principle of sufficient reason, any theory of space that implied that there could be these two possible
universes, must therefore be wrong.
Newton took space to be more than relations between material objects and based his position on
observation and experimentation. For a relationist there can be no real difference between inertial motion,
in which the object travels with constant velocity, and non-inertial motion, in which the velocity changes
with time, since all spatial measurements are relative to other objects and their motions. But Newton
argued that since non-inertial motion generates forces, it must be absolute.[9]
He used the example of
water in a spinning bucket to demonstrate his argument. Water in a bucket is hung from a rope and set to
spin, starts with a flat surface. After a while, as the bucket continues to spin, the surface of the water
becomes concave. If the bucket's spinning is stopped then the surface of the water remains concave as it
continues to spin. The concave surface is therefore apparently not the result of relative motion between
the bucket and the water.
[10]
Instead, Newton argued, it must be a result of non-inertial motion relative to
space itself. For several centuries the bucket argument was decisive in showing that space must exist
independently of matter.
Kant :
In the eighteenth century the German philosopher Immanuel Kant developed a theory of knowledge in
which knowledge about space can be both a priori and synthetic.
[11]
According to Kant, knowledge about
space is synthetic, in that statements about space are not simply true by virtue of the meaning of the
words in the statement. In his work, Kant rejected the view that space must be either a substance or
relation. Instead he came to the conclusion that space and time are not discovered by humans to be
objective features of the world, but are part of an unavoidable systematic framework for organizing our
experiences.[12]
Non-Euclidean Geometry :
Euclid's Elements contained five postulates that form the basis for Euclidean geometry. One of these, the
parallel postulate has been the subject of debate among mathematicians for many centuries. It states that
on any plane on which there is a straight line L1 and a point P not on L1, there is only one straight line L2
on the plane that passes through the point P and is parallel to the straight line L1. Until the 19th century,
few doubted the truth of the postulate; instead debate centered over whether it was necessary as an
axiom, or whether it was a theory that could be derived from the other axioms.[13]
Around 1830 though,
the Hungarian János Bolyai and the Russian Nikolai Ivanovich Lobachevsky separately published
treatises on a type of geometry that does not include the parallel postulate, called hyperbolic geometry. In
this geometry, an infinite number of parallel lines pass through the point P. Consequently the sum of
angles in a triangle is less than 180° and the ratio of a circle's circumference to its diameter is greater
than pi. In the 1850s, Bernhard Riemann developed an equivalent theory of elliptical geometry, in which
no parallel lines pass through P. In this geometry, triangles have more than 180° and circles have a ratio
of circumference-to-diameter that is less than pi.

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Gauss and Poincaré
Although there was a prevailing Kantian consensus at the time, once non-Euclidean geometries had been
formalised, some began to wonder whether or not physical space is curved. Carl Friedrich Gauss, a
German mathematician, was the first to consider an empirical investigation of the geometrical structure of
space. He thought of making a test of the sum of the angles of an enormous stellar triangle and there are
reports he actually carried out a test, on a small scale, by triangulating mountain tops in Germany.[14]
Henri Poincaré, a French mathematician and physicist of the late 19th century introduced an important
insight in which he attempted to demonstrate the futility of any attempt to discover which geometry
applies to space by experiment.
[15]
He considered the predicament that would face scientists if they were
confined to the surface of an imaginary large sphere with particular properties, known as a sphere-world.
In this world, the temperature is taken to vary in such a way that all objects expand and contract in similar
proportions in different places on the sphere. With a suitable falloff in temperature, if the scientists try to
use measuring rods to determine the sum of the angles in a triangle, they can be deceived into thinking
that they inhabit a plane, rather than a spherical surface.
[16]
In fact, the scientists cannot in principle
determine whether they inhabit a plane or sphere and, Poincaré argued, the same is true for the debate
over whether real space is Euclidean or not. For him, which geometry was used to describe space, was a
matter of convention.
[17]
Since Euclidean geometry is simpler than non-Euclidean geometry, he assumed
the former would always be used to describe the 'true' geometry of the world.[18]
Einstein
In 1905, Albert Einstein published a paper on a special theory of relativity, in which he proposed that
space and time be combined into a single construct known as spacetime. In this theory, the speed of light
in a vacuum is the same for all observers—which has the result that two events that appear simultaneous
to one particular observer will not be simultaneous to another observer if the observers are moving with
respect to one another. Moreover, an observer will measure a moving clock to tick more slowly than one
that is stationary with respect to them; and objects are measured to be shortened in the direction that they
are moving with respect to the observer.
Over the following ten years Einstein worked on a general theory of relativity, which is a theory of how
gravity interacts with spacetime. Instead of viewing gravity as a force field acting in spacetime, Einstein
suggested that it modifies the geometric structure of spacetime itself.[19]
According to the general theory,
time goes more slowly at places with lower gravitational potentials and rays of light bend in the presence
of a gravitational field. Scientists have studied the behaviour of binary pulsars, confirming the predictions
of Einstein's theories and non-Euclidean geometry is usually used to describe spacetime.
Mathematics
In modern mathematics spaces are defined as sets with some added structure. They are frequently
described as different types of manifolds, which are spaces that locally approximate to Euclidean space,
and where the properties are defined largely on local connectedness of points that lie on the manifold.
There are however, many diverse mathematical objects that are called spaces. For example, vector
spaces such as function spaces may have infinite numbers of independent dimensions and a notion of
distance very different from Euclidean space, and topological spaces replace the concept of distance with
a more abstract idea of nearness.

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Classifications of Space with respect to Forty Two (42) Branches of
Modern Geometry :
1.Euclidean Space /Plane
2.Non-Euclidean Space
3.Hilbert Space
4.Banach Space
5.Fermi-Dirac Space
6.Cartesian/Coordinate Space
7.Hyperbolic Space
8.Parabolic Space
9.Hermitian Space
10.Spherical Space
11.Riemannian Space
12.Projective Space
13.Vector Space
14.Gaussian Space
15.Curvillinear Space
16.Einstein’s Space and Time ( The Theory of Relativity )
Remarks :
Would be included : Definition , Descriptions , Examples , Figures , Sub-classifications
etc of above mentioned spaces .