Download On Euclidean and Non-Euclidean Geometry by Hukum Singh DESM

On Euclidean and Non-Euclidean Geometry
by
Hukum Singh
DESM, NCERT
New Delhi
Abstract : Geometry is one of the most ancient branch of mathematics. Its importance
is now felt in every day life, different branches of Science, Engineering, Navigation,
Technology and other branches of knowledge. There are mainly two branches of geometry
namely Euclidean and non-Euclidean geometry. Study of Euclidean geometry is based on
Euclid axioms and postulates whereas non-Euclidean geometry have further been divided in
two parts namely hyperbolic geometry and elliptical or spherical geometry. Study of the
later parts of geometry is not only based on axioms but also on certain metrics and other
related concepts. The credit for discovering these geometries goes to Baudhayana,
Pythagoras, Euclid, Aryabhata, Brahmagupta, Gauss, Lobachevsky, Bolyai, Riemann,
Beltrami etc. The purpose of this article is to highlight about these geometries in a
systematic and lucid manner.
Introduction: The notions of geometry arose in ancient time from necessity. About four
thousand years ago Egyptians and Babylonians compiled some rules for finding areas and
volumes of some geometric objects. These geometric findings were taken over by Greeks
about two thousand five hundred years ago. It was Greeks who gave the name geometry
which means the measurement of the earth.
In India the knowledge of geometry was
important features in sulva period during 1000 B.C. to 500 B.C. In ancient time Aryans
believed that yajna is the only means for the attainment of supreme life. Different
geometrical altars were used for performing yajna. Sulva was used as the rule of cord in
the construction of altars. The knowledge of Pythagoras theorem was prevalent by
Baudhayana (1000 B.C.) before the Pythagoras (600 B.C.). Aryabhata (476 A.D.) gave the
value of π as
62832
= 3.1416 which is correct upto fourth decimal places. He stated that
20,000
the earth is spherical and is moving around the sun. This was described in Aryabhatiya
Goal-pad written in 499 A.D. Mahavira (900 A.D.) gave the formula for finding the area of
the quadrilateral as s(s - a)(s - b)(s - c)(s - d) , where a, b, c and d are the sides and s is the
semi-perimeter of the quadrilateral. Mahavira also gave the formula for calculating the area
of an ellipse. Bhaskara (1114 A.D.) gave the formulae for fnding the area and volume of
sphere. The elegant formulae given by Srinivasa Ramanujan (1887-1920) was used for
calculating the value of
upto millions of decimal places.
-2-
Euclidean Geometry: Euclid geometry is based on Euclid axioms which were given in the
book
namely “The Elements”. This book was written by Euclid in about 300 B.C. The
book consisted 13 volumes. The first six volumes consisted study of geometry, seven to ten
consisted number theory and last three consisted three dimensional solid geometry. The
Euclid axioms are [1], [3]
(a)There lie infinite number of points on a line (b) Infinite number of lines passes
through a point (c) One and only one line passes through two given points (d) There
is one and only one line passing through a given point and parallel to a given line
which is not containing the given point (e) Two straight lines in a plane are either
parallel or intersecting (f) The sum of the angles of a triangle is 180◦
The five Euclid’s Postulates are [1], [3]
(a) A straight line can be drawn from any point to any other point (b)A finite straight
line can be produced continuously in a straight line(c) A circle can be described
with any centre and any radius (d)All right angles are equal to one another (e) If
a straight line falling on two straight lines make the interior angle on the same
side less than two right angles, the two straight lines, if produced indefinitely,
meet on that side on which are the angles less than two right angles.
The geometry which we teach today in schools is Euclidean geometry. Plato was
a great geometrician lived a couple of generations before Euclid. It is said that
Plato put a sign above the door to his academy saying “Let no one enter who is
unfamiliar with geometry”. Plato probably learnt geometry from Pythagoras
(600 B.C.) whereas Pythagoras learnt from Thales (600 B.C.).
The introduction of coordinates in plane was made by French mathematician
Rene Descartes in 1637. Descartes explained that figures and numbers are
closely related [3]. Later on coordinate geometry was developed by Eucler
(1707-1783 A.D.). In 1748, Euler studied on theory of curves of the second
order and used coordinates for the curves in space. In coordinate geometry,
two dimensional and three dimensional Euclidean spaces R2 and R3 are
represented by planes and physical spaces, respectively.
-3-
Non-Euclidean Geometry: The discovery of non-Euclidean geometry was made
independently by Russian mathematician N.I. Labachevsky (1792-1856 A.D.) in 1826 and
the Hungarian mathematician J.Bolyai (1802-1860 A.D.) in 1832. German mathematician
C.F. Gauss (1777-1855 A.D.) had also investigated the idea of non-Euclidean geometry but
his work could not be published. The idea of non-Euclidean geometry differ in axioms from
those of Euclid. Lobachevsky replaced Euclid’s fifth postulate by “There exist two lines
parallel to a given line through a given point not on the line”. In non-Euclidean geometry
the sum of the angles of a triangle less than 180◦ in the hyperbolic space and is greater
than 180◦ in the elliptical or spherical space. In todays study non-Euclidean geometry is
much more applicable than Euclidean geometry. In 1868, the Italian mathematician
E.Beltrami (1835-1900) produced a model for 2-dimensional non-Euclidean geometry in
3-dimensional Euclidean geometry. In general, any n-dimensional non-Euclidean space can
be embedded in
n(n+1)
2
dimensional Euclidean space.
The geometry of surface in
3-dimensional Euclidean space was studied by Gauss. German mathematician B.Riemann
(1826-1866) discovered n-dimensional geometry which is now known as Riemannian
geometry. Riemann also studied on spherical geometry and showed that every line passing
through a point R not on the line PQ meets the line PQ. The generalisations of Riemannian
geometry is Finsler geometry whose metric depends on position as well as direction where
as Riemann metric depends on position only.
Applications: Euclidean geometry has its maximum application in daily life in one form or
the other. Its results are mainly used in engineering. In case of non-euclidean geometry
i.e. if L is a surface in 3-dimensional Euclidean Space, then there are two principal directions
at any point of the surface and hence the two lines of curvatures. The point of intersection
of consecutive normals along a line of curvature is a centre of curvature for all points of L
and form in general a surface which is called surface of centres or centro surface of L. In
general centro surface consists of two sheets, corresponding to the two families of lines of
curvatures. The centro surface plays a basic role in the theory of shell. Secondly, if R3 is
the Euclidean 3-space and L is a circle
x2+y2 = 1
,
then the focal point [5] of L in R3 is precisely the z-axis.
-4Thirdly, let M be a curved space-time and Rij and R be Ricci tensor and scalar curvature of
M. Then Einstein Tensor Gij is given by [4]
Gij = Rij − 1 R gij,
2
(1)
where gij is metric tensor of M.
Using this, the Einstein field equation is given by [ 4 ]
Gij = 8
Tij,
(2)
where Tij is the Energy momentum tensor.
REFERENCES
[1] Birkhof, G. and Bennett, M.K., Felex Klein and his Erlanger Programme, History and
Philosophy of modern mathematics, University of Minn. Press, 1988.
[2] Beem J.K. Ehrlich, P.Global Lorentzian Geometry, Marcel dekker, New York, 1981
[3] Descartes, R. Geometria, Paris, 1637
[4] Hawking, S.and Elis, G. the larges scale structure of space-time, Cambridge University
Press, Cambridge, 1973
[5] Hukum Singh, on the cut locus and the focal locus of submanifolds in a Riemannian
manifold, Annales University, Science Budapest, Secti. Mathematic (a) 30 (1987), 139-144