Transcript History and Philosophy of Mathematics MA0010

History and Philosophy of Mathematics
MA0010
PLANE, SOLID AND COORDINATE
GEOMETRY
CONDUCTED BY
D E PA RT M E N T O F M AT H E M AT I C S
U N I V E R S I T Y O F M O R AT U WA
MS SHANIKA FERDINANDIS
MR. KEVIN RAJAMOHAN
Plane Geometry
Department of Mathematics,UOM
2 April 2016
Euclid ( Father of Geometry)
Euclidean Geometry
 Euclidean geometry is a mathematical system attributed to
the Greek mathematician Euclid of Alexandria. Euclid's
Elements is the earliest known systematic discussion of
geometry.
 The method consists of assuming a small set of intuitively
appealing axioms, and then proving many other
propositions (theorems) from those axioms.
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2 April 2016
Some basic results in Euclidean Geometry
 The sum of angles A, B, and C is equal to 180 degrees.
 The Pythagorean theorem: The sum of the areas of the two squares
on the legs (a and b) equals the area of the square on the
hypotenuse (c).
 Thales' theorem: if AC is a diameter then the angle at B is a right
angle
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2 April 2016
Axioms of Euclid’s Geometry
 Euclid gives five postulates for plane geometry, stated in terms
of constructions:
Let the following be postulated:
1.
2.
3.
4.
5.
[It is possible] to draw a straight line from any point to any point.
[It is possible] To produce [extend] a finite straight line continuously in
a straight line.
[It is possible] To describe a circle with any center and distance [radius].
That all right angles are equal to one another.
The parallel postulate: That, if a straight line falling on two straight lines
make the interior angles 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 the two right angles.
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2 April 2016
Common Notions (Axioms)
1.
2.
3.
4.
5.
Things that equal the same thing also equal one another.
If equals are added to equals, then the wholes are equal.
If equals are subtracted from equals, then the remainders
are equal.
Things that coincide with one another equal one another.
The whole is greater than the part.
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2 April 2016
Nine point circle
 The nine-point circle is a circle that can be constructed for any given
triangle. It is so named because it passes through nine significant
points, six lying on the triangle itself (unless the triangle is obtuse).
They include:
 The midpoint of each side of the triangle.
 The foot of each altitude.
 The midpoint of the segment of each altitude from its vertex to the
orthocenter (where the three altitudes meet).
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2 April 2016
Centroid
 The centroid (G) of a triangle is the common intersection of
the three medians of a triangle. A median of a triangle is
the segment from a vertex to the midpoint of the opposite
side.
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2 April 2016
Orthocenter
 The orthocenter (H) of a triangle is the common
intersection of the three lines containing the altitudes. An
altitude is a perpendicular segment from a vertex to the
line of the opposite side.
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2 April 2016
Circumcenter
 The circumcenter (C) of a triangle is the point in the plane
equidistant from the three vertices of the triangle. Since a
point equidistant from two points lies on the perpendicular
bisector of the segment determined by two points, (C) is on
the perpendicular bisector of each side of the triangle. Note
(C) may be outside the triangle.
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2 April 2016
Euler Line
 In
geometry, the Euler
line, named after Leonhard
Euler, is a line determined
from any triangle that is
not equilateral; it passes
through several important
points determined from
the triangle. In the image,
the Euler line is shown in
red. It passes through the
orthocenter (blue), the
circumcenter (green), the
centroid (orange), and the
center of the nine-point
circle (red) of the triangle..
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2 April 2016
Pythagorean Theorem: Different Proofs
 This is a theorem that may have more known proofs than any
other; the book Pythagorean Proposition, by Elisha Scott Loomis,
contains 367 proofs.
 Proof using similar triangles
 Let ABC represent a right angle triangle.
 Draw an altitude from point C and call H its intersection with the side AB.
 The new triangle ACH is similar to ABC. (by definition of the altitude, they
both have a right angle)
 Similarly, triangle CBH is similar to ABC.
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Proof using similar triangles cont…
 The similarities lead to the two ratios:
 These can be written as
 Summing these two equalities, we obtain
 In other words, The Pythagorean theorem:
 Exercise: Prove the Pythagorean theorem in one other way.
Department of Mathematics,UOM
2 April 2016
The Pythagorean Theorem in 3D
 The Pythagorean Theorem, which allows you to find the
hypotenuse of a right triangle, can also be used in three
dimensions to find the diagonal length of a rectangular prism.
This is the distance d from one corner of the box to the
furthest opposite corner, as shown in the diagram at the right.
 The distance can be calculated using:
Department of Mathematics,UOM
2 April 2016
Polygons
 In geometry a polygon is traditionally a plane figure that is
bounded by a closed path or circuit, composed of a finite
sequence of straight line segments (i.e., by a closed
polygonal chain). These segments are called its edges or
sides, and the points where two edges meet are the
polygon's vertices or corners.
 The following are examples of polygons:
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2 April 2016
Question:
 State whether the figure’s below are polygons or not ?
a.
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b.
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Vertex
 The vertex of an angle is the point where the two rays that
form the angle intersect.
 The vertices of a polygon are the points where its sides
intersect.
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2 April 2016
Regular Polygon
 A regular polygon is a polygon whose sides are all the same
length, and whose angles are all the same. The sum of the
angles of a polygon with n sides, where n is 3 or more, is
180° × (n - 2) degrees.
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2 April 2016
Triangle- Three sided polygon
 Equilateral Triangle or Equiangular Triangle
A triangle having all three sides of equal length. The angles
of an equilateral triangle all measure 60 degrees.
 Isosceles Triangle
A triangle having two sides of equal length.
 Right Triangle
A triangle having a right angle. One of the angles of the
triangle measures 90 degrees. The side opposite the
right angle is called the hypotenuse.
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2 April 2016
Four sided Polygons
 Parallelogram
A four-sided polygon with two pairs of parallel sides.
 Rhombus
A four-sided polygon having all four sides of equal length.
 Trapezoid
A four-sided polygon having exactly one pair of parallel sides. The two
sides that are parallel are called the bases of the trapezoid.
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2 April 2016
Tessellation
A Tessellation is created when a shape is repeated
over and over again covering a plane without any
gaps or overlaps.
Only three regular polygons tessellate in the Euclidean Plane:
Triangles, Squares or Hexagons.
A tessellation of triangles
A tessellation of squares
A tessellation of hexagons
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2 April 2016
Compass and straightedge
 Compass-and-straightedge or ruler-and-compass
construction is the construction of lengths, angles, and other
geometric figures using only an idealized ruler and compass.
 Every point constructible using straightedge and compass
may be constructed using compass alone. A number of
ancient problems in plane geometry impose this restriction.
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2 April 2016
Trisecting an angle
 Angle trisection is the division of an arbitrary angle into three equal
angles. It was one of the three geometric problems of antiquity for which
solutions using only compass and straightedge were sought. The problem
was algebraically proved impossible by Wantzel (1836) French
mathematician.
Angles may not in general be trisected
 The geometric problem of angle trisection can be related to algebra –
specifically, the roots of a cubic polynomial – since by the triple-angle
formula,
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2 April 2016
Gauss
Johann Carl Friedrich Gauss was a German mathematician and
scientist who contributed significantly to many fields, including
number theory, statistics, analysis, differential geometry, geodesy,
electrostatics, astronomy and optics. Sometimes known as the “the
Prince of Mathematicians" or "the foremost of mathematicians") and
"greatest mathematician since antiquity", Gauss had a remarkable
influence in many fields of mathematics and science and is ranked as
one of history's most influential mathematicians. He referred to
mathematics as "the queen of sciences."
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2 April 2016
Coordinate
Geometry
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2 April 2016
Coordinate Geometry
 Cartesian Coordinates
In the two-dimensional Cartesian coordinate system, a point P in the xyplane is represented by a pair of numbers (x,y).
•x is the signed distance from the y-axis to the point P, and
•y is the signed distance from the x-axis to the point P.
In the three-dimensional Cartesian coordinate system, a point P in the
xyz-space is represented by a triple of numbers (x,y,z).
•x is the signed distance from the yz-plane to the point P,
•y is the signed distance from the xz-plane to the point P, and
•z is the signed distance from the xy-plane to the point P.
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2 April 2016
Coordinate Geometry
 Polar Coordinates
 The polar coordinate systems are coordinate systems in which a point is
identified by a distance from some fixed feature in space and one or more
subtended angles. They are the most common systems of curvilinear
coordinates.
 The term polar coordinates often refers to circular coordinates (two-
dimensional). Other commonly used polar coordinates are cylindrical
coordinates and spherical coordinates (both three-dimensional).
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2 April 2016
Converting Polar and Cartesian coordinates
 To convert from Cartesian Coordinates (x,y) to Polar
Coordinates (r,θ):
 To convert from Polar coordinates (r, θ) to Cartesian
coordinates
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2 April 2016
Circle
 A circle is the set of points in a plane that are equidistant
from a given point . The distance from the center r is called
the radius, and the point o is called the center. Twice the
radius is known as the diameter .
 In Cartesian coordinates, the equation of a circle of radius r
centered on (h,k) is
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2 April 2016
Area of a Circle
 This derivation was first recorded by Archimedes in
Measurement of a Circle (ca. 225 BC).
 If the circle is instead cut into wedges, as the number of
wedges increases to infinity, a rectangle results, so
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2 April 2016
Further Terminology
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Ellipse
 The ellipse is defined as the locus ( A the set of all points
satisfying some condition) of a point (x,y) which moves so
that the sum of its distances from two fixed points (called
foci, or focuses ) is constant.
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2 April 2016
Ellipse cont…
 Ellipses with Horizontal Major Axis
 Ellipses with Vertical Major Axis
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2 April 2016
Hyperbola
 The word "hyperbola" derives from the Greek meaning
"over-thrown" or "excessive", from which the English term
hyperbole derives. In mathematics a hyperbola is a
smooth planar curve having two connected components or
branches, each a mirror image of the other and resembling
two infinite bows aimed at each other.
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2 April 2016
Hyperbola cont..
 Horizontal transverse axis
 Vertical transverse axis
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2 April 2016
Parabola
A parabola is the set of all points in the plane equidistant from
a given line (the conic section directrix) and a given point not
on the line (the focus). The focal parameter (i.e., the distance
between the directrix and focus) is therefore given by P=2a,
where a is the distance from the vertex to the directrix or focus.
The surface of revolution obtained by rotating a parabola about
its axis of symmetry is called a parabolid.
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2 April 2016
Spiral
 A spiral is typically a planar curve (that is, flat), like the
groove on a record or the arms of a spiral galaxy.
 A spiral emanates from a central point, getting
progressively farther away as it revolves around the point.
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2 April 2016
Two-dimensional spirals
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2 April 2016
Cycloid
 A cycloid is the locus of a point on the rim of a circle of
radius a rolling along a straight line. The cycloid was first
studied by Cusa when he was attempting to find the area of
a circle by integration. It was studied and named by Galileo
in 1599.
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2 April 2016
Hypocycloid
 The path traced out by a point on the edge of a circle of
radius b rolling on the outside of a circle of radius a.
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Solid Geometry
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Sphere
 Spherical surface has been defined as the locus of points in
three-dimensional space, at a given distance from a given
point.
 The given point is called the center. The given distance is
called a radius.
 Sphere is a solid bounded by a spherical surface.
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 In analytic geometry, a sphere with center (a, b, c) and
radius r is the locus of all points (x, y, z) such that
 Refer on the Properties of the sphere.
 The points on the sphere with radius r can be
parameterized by
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2 April 2016
Ellipsoid
 An ellipsoid is a type of quadric surface that is a higher
dimensional analogue of an ellipse. The equation of a
standard axis-aligned ellipsoid body in an xyz-Cartesian
coordinate system is
 Where a and b are the equatorial radii (along the x and y
axes) and c is the polar radius (along the z-axis).
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2 April 2016
Hyperboloid
 A hyperboloid is a type of surface in three dimensions,
described by the equation
 Refer the importance of Hyperboloid
structures in Construction engineering.
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2 April 2016
Plot 3d Figures in Matlab
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2 April 2016
Platonic solids
 Tetrahedron, Cube, Octahedron, Dodecahedron & Icosahedron –
These 5 solids are called Perfect solids or Platonic solids (in
which a constant number of identical regular faces meet at
each vertex)
 They are known as Perfect, because of their unique
construction-They are the only forms we know of, that have
multiple sides which all have the same shapes & size.
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2 April 2016
Archimedean Polyhedra
 They are formed from Platonic Solids by cutting off the
corners ( Truncated Polyhedra).
 It is a solid made out of, more than one polygon.
 All the vertices are identical.
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2 April 2016
The 13 Archimedean Solids
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2 April 2016
Further Topics in
Geometry
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2 April 2016
Euclid’s 5th Postulate
 “That if a straight line falling on two straight lines makes
the interior angles less that two right angles, the two
straight lines, if produced indefinitely , meet on that side on
which the angles are less than tow right angles.
 In other words: “Through an exterior point of a straight
line ( a line not on the straight line) one can construct one
and only parallel to the given straight line.
 The 5th Postulate is logically consistent in itself and forms
an axiomatic system with the other 4 postulates.
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2 April 2016
 But while forming an axiomatic system, the 5th postulate
was thought to be dependant on the first 4.
 Therefore mathematicians through out the past, redefined
the 5th postulate with new theories and gave way to nonEuclidian geometry. E.g. Hyperbolic geometry, Elliptic
geometry.
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2 April 2016
Non-Euclidian Geometry
 The axioms of Geometry were formerly regarded as laws of
thought which an intelligent mind could neither deny nor
investigate.
 However, that it is possible to take a set of axioms, wholly
or in part contradicting those of Euclid, and build up a
Geometry as consistent as his.
 Examples of non-Euclidean geometries include the
hyperbolic and elliptic geometry, which are contrasted with
a Euclidean geometry.
 The essential difference between Euclidean and nonEuclidean geometry is the nature of parallel lines.
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2 April 2016
 Another way to describe the differences between these
geometries is to consider two straight lines indefinitely
extended in a two-dimensional plane that are both
perpendicular to a third line:
1. In Euclidean geometry the lines remain at a constant
distance from each other, and are known as parallels.
2. In hyperbolic geometry they "curve away" from each
other, increasing in distance as one moves further from
the points of intersection with the common
perpendicular; these lines are often called ultra parallels.
3. In elliptic geometry the lines "curve toward" each other
and eventually intersect.
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2 April 2016
Triangles in different spaces
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2 April 2016
Hyperbolic Geometry
 Hyperbolic geometry (also called Lobachevskian geometry)
was created in the first half of the nineteenth century in the
midst of attempts to understand Euclid's axiomatic basis
for geometry.
 It is one type of non-Euclidean geometry that discards
Euclid's 5th postulate.
 It is replaced by the postulate which states that “Given a
line and a point not on it, there is more than one line
(infinitely many lines) going through the given point that is
parallel to the given line”.
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2 April 2016
 The parallel postulate in Euclidean geometry is equivalent
to the statement that, in two dimensional space, for any
given line l and point P not on l, there is exactly one line
through P that does not intersect l; i.e., that is parallel to l.
In hyperbolic geometry there are at least two distinct lines
through P which do not intersect l, so the parallel postulate
is false.
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2 April 2016
 An example of such a case in hyperbolic geometry , is the
hyperbola. Where the hyperbola, though it approaches the
asymptote it never meets it.(This violates Euclid’s parallel
postulate)
 Applications of hyperbolic geometry includes topics such
as Toplogy, Group Theory and Complex variables &
conformal mapping.
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2 April 2016
Problems unsolved in geometry.
 The Hadwiger problem.
 The Polygonal illumination problem.
 The Chromatic Number of the plane.
 Kissing Numbers.??
 Perfect cuboids.
 The Kabon Triangle Problem??
 There are many more.. Google and explore!!!
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2 April 2016
Kissing numbers
 In d dimensions, the kissing number K(d) is the maximum
number of disjoint unit spheres that can touch a given
sphere.
 What could be K(2) and K(3)??
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2 April 2016
Kabon triangle problem
 The problem is to find how many disjoint triangles can
be created with n lines in the plane (K(n))
 What could be the sequence of K(n) ??
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2 April 2016
At the end of this lecture…
 We hope you would have been enlightened about the
broader perspective of geometry. Namely plane, solid
and coordinate geometry.
 You would have realized the need for Mathematical
thinking and reasoning…!!
 We also hope you would take the formulae of different
curves in your minds and apply them when you come
across mathematical problems.
 Please go through any new words you came across..!!
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2 April 2016
The End
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2 April 2016