This document discusses three-dimensional space and geometry. It begins by defining dimension and explaining that a point in 3D space is defined by three coordinates: x, y, and z. Different coordinate systems for 3D space are presented, including Cartesian and cylindrical/spherical coordinates. Common 3D shapes are described such as polyhedrons, prisms, cylinders, cones, and spheres. Higher dimensions beyond 3D are briefly touched on. The document also discusses visualizing 3D space through graphs of functions with multiple variables.

1. Three Dimensional
Space
Dr. Farhana Shaheen
Assistant Professor
YUC-SA

6. 3-D Picture

8. 3-D

9. 3-D

10. 3-D Illusions

11. Dimension
 In Mathematics and Physics, the dimension of a space or
object is informally defined as the minimum number of
coordinates needed to specify each point within it. Thus a
line has a dimension of one because only one coordinate is
needed to specify a point on it. A surface such as a plane
or the surface of a cylinder or sphere has a dimension of
two because two coordinates are needed to specify a point
on it (for example, to locate a point on the surface of a
sphere you need both its latitude and its longitude). The
inside of a cube, a cylinder or a sphere is three-
dimensional because three co-ordinates are needed to
locate a point within these spaces.

12. A drawing of the first four dimensions
 On the left is zero dimensions (a point)
and on the right is four dimensions (a
tesseract). There is an axis and labels on
the right and which level of dimensions it
is on the bottom. The arrows alongside
the shapes indicate the direction of
extrusion.

13. A diagram showing the first four
spatial dimensions

14.  Below from left to right, is a square, a cube, and
a tesseract.
 The square is bounded by 1-dimensional lines,
the cube by 2-dimensional areas, and the
tesseract by 3-dimensional volumes.
 A projection of the cube is given since it is
viewed on a two-dimensional screen. The same
applies to the tesseract, which additionally can
only be shown as a projection even in three-
dimensional space.

15. Metatron's Cube

16. Three-dimensional space
 Three-dimensional space is a
geometric model of the physical universe
in which we live. The three dimensions
are commonly called length, width, and
depth (or height), although any three
mutually perpendicular directions can
serve as the three dimensions.

17.  In mathematics, analytic geometry (also called Cartesian
geometry) describes every point in three-dimensional
space by means of three coordinates. Three coordinate
axes are given, each perpendicular to the other two at the
origin, the point at which they cross. They are usually
labeled x, y, and z. Relative to these axes, the position of
any point in three-dimensional space is given by an
ordered triple (x, y, z) of real numbers x, y, z,
each number giving the distance of that
point from the origin measured along the
given axis, which is equal to the distance
of that point from the plane determined
by the other two axes.
Analytic geometry

18.  Three dimensional Cartesian coordinate
system with the x-axis pointing towards
the observer.

19. Three Dimensional Plane

20. Shapes in 2-D
and 3-D

21. Polyhedron
Cylinder
Cone
Sphere
 Space figures are figures whose points do
not all lie in the same plane.
For examples:

22. Space figures
 Polyhedrons are space figures with flat surfaces, called
faces, which are made of polygons.
 Prisms and pyramids are examples of polyhedrons.
 A cylinder has two parallel, congruent bases that are
circles.
 A cone has one circular base and a vertex that is
not on the base.
 A sphere is a space figure having all its points an equal
distance from the center point.
Note that cylinders, cones, and spheres are not
polyhedrons, because they have curved, not flat, surfaces.

23. Polygons: Triangles,
Squares, Pentagons
 Three-dimensional geometry, or space geometry, is used to
describe the buildings we live and work in, the tools we
work with, and the objects we create. First, we'll look at
some types of polyhedrons.
 A polyhedron is a three-dimensional figure that has
polygons as its faces. Its name comes from the Greek
"poly" meaning "many," and "hedra," meaning "faces." The
ancient Greeks in the 4th century B.C. were brilliant
geometers. They made important discoveries and
consequently they got to name the objects they
discovered. That's why geometric figures usually have
Greek names!

24. Examples of polyhedrons

25. PRISM
 We can relate some polyhedrons--and
other space figures as well--to the two-
dimensional figures that we're already
familiar with. For example, if you move a
vertical rectangle horizontally through
space, you will create a rectangular or
square prism.

26. TRIANGULAR PRISM
 If you move a vertical triangle
horizontally, you generate a triangular
prism. When made out of glass, this type
of prism splits sunlight into the colors of
the rainbow.

27. Prism splitting sunlight into
rainbow colours

28. Platonic Solids
 Platonic Solids are polyhedrons where all
the faces are regular polygons, and all the
corners have same number of faces
joining them, and all the faces are
exactly the same size and shape.

30. Crystal Platonic Solids

31. Tetrahedron
 In geometry, a tetrahedron (plural:
tetrahedra) is a polyhedron composed of
four triangular faces, three of which meet
at each vertex. A regular tetrahedron is
one in which the four triangles are
regular, or "equilateral", and is one of the
Platonic solids. The tetrahedron is the only
convex polyhedron that has four faces. A
tetrahedron is also known as a triangular
pyramid.

32. Tetrahedron

33. Hexahedron

34. Octahedron and Icosahedron

35. Icosahedron models

36. Dodecahedron

37. MerKaBa

38. Merkaba
 The Merkaba is an extremely powerful symbol.
It is a combination of two star tetrahedrons - one
pointing up to the heavens,
channelling energy down to the earth plain, and
one pointing downwards, drawing up energy
from the earth beneath. The top,
or upward pointing tetrahedron is male and
rotates clockwise, with the bottom or downwards
pointing one being female, which
rotates counter-clockwise

39. CONE
 A Cone is another familiar space figure
with many applications in the real world.
 A cone can be generated by twirling a
right triangle around one of its legs.
 If you like ice cream, you're no doubt
familiar with at least one of them!

40. Cone Shells and Cone Ice Cream

41. CYLINDER
 Now let's look at some space figures that
are not polyhedrons, but that are also
related to familiar two-dimensional
figures. What can we make from a circle?
If you move the center of a circle on a
straight line perpendicular to the circle,
you will generate a cylinder. You know this
shape--cylinders are used as pipes,
columns, cans, musical instruments, and
in many other applications.

42. Cylindrical cans and flute

43. SPHERE
 A sphere is created when you twirl a circle
around one of its diameters. This is one of
our most common and familiar shapes--in
fact, the very planet we live on is an
almost perfect sphere! All of the points of
a sphere are at the same distance from its
center.


44. Spherical Shapes

45. Rhombicosidodecahedron"?
 There are many other space figures--an
endless number, in fact. Some have
names and some don't. Have you ever
heard of a "rhombicosidodecahedron"?
Some claim it's one of the most attractive
of the 3-D figures, having equilateral
triangles, squares, and regular pentagons
for its surfaces. Geometry is a world unto
itself, and we're just touching the surface
of that world.

46. Rhombicosidodecahedron
The 3-D figure, having
equilateral triangles,
squares, and regular
pentagons
for its surfaces.

47. Rhombicosidodecahedrons

48. Euclidean Space
 In mathematics, solid geometry was the
traditional name for the geometry of
three-dimensional Euclidean space — for
practical purposes the kind of space we
live in. It was developed following the
development of plane geometry.
Stereometry deals with the
measurements of volumes of various solid
figures: cylinder, circular cone, truncated
cone, sphere, prisms, blades, wine casks.

49. Cylindrical and Spherical
Coordinates
 Other popular methods of describing the
location of a point in three-dimensional
space include cylindrical coordinates and
spherical coordinates, though there are an
infinite number of possible methods.

50. Cylindrical coordinate system
(ρ, φ, z)
 A cylindrical coordinate system is a
three-dimensional coordinate system,
where each point is specified by the two
polar coordinates of its perpendicular
projection onto some fixed ρφ-plane
and by its (signed) distance z
from that plane.

51. Application of cylindrical
Coordinates
 Cylindrical coordinates are useful in
connection with objects and phenomena
that have some rotational symmetry
about the longitudinal axis, such as water
flow in a straight pipe with round cross-
section, heat distribution in a metal
cylinder, etc.

52. Example
 A cylindrical coordinate system with origin
O, polar axis A, and longitudinal axis L.
The dot is the point with radial distance
ρ = 4, angular coordinate φ = 130°, and
height z = 4.

53. Spherical Coordinates (r, θ, Φ)

55. Spherical Coordinates (r, θ, Φ)

57.  Spherical coordinates, also called
spherical polar coordinates, are a system
of curvilinear coordinates that are natural
for describing positions on a sphere or
spheroid. Define θ to be the azimuthal
angle in the xy-plane from the x-axis ,
Φ to be the polar angle (also known as
the zenith angle , and ρ to be distance
(radius) from a point to the origin. This is
the convention commonly used in
mathematics.

58. Set of points where ρ (rho) is
constant

59. Points where Φ (phi) is constant

60. Points where θ (theta) is constant

61.  What happens when ρ, θ, Φ are all
constant (one by one)
 (ρ, θ, Φ) = (Rrho, Pphi, Ttheta)

63. Viewing Three Dimensional Space
 Another mathematical way of viewing three-
dimensional space is found in linear algebra,
where the idea of independence is crucial. Space
has three dimensions because the length of a
box is independent of its width or breadth. In the
technical language of linear algebra, space is
three dimensional because every point in space
can be described by a linear combination of three
independent vectors. In this view, space-time is
four dimensional because the location of a point
in time is independent of its location in space.

64. Viewing Three Dimensional Space
 Three-dimensional space has a number of
properties that distinguish it from spaces of other
dimension numbers. For example, at least 3
dimensions are required to tie a knot in a piece
of string. Many of the laws of physics, such as
the various inverse square laws, depend on
dimension three.
 The understanding of three-dimensional space in
humans is thought to be learned during infancy
using unconscious inference, and is closely
related to hand-eye coordination. The visual
ability to perceive the world in three dimensions
is called depth perception.

65. Skew Lines
 In solid geometry, skew lines are two lines
that neither intersect nor are they parallel.
Equivalently, they are lines that are not both
in the same plane. A simple example of a
pair of skew lines is the pair of lines through
opposite edges of a regular tetrahedron (or
other non-degenerate tetrahedron). Lines
that are coplanar either intersect or are
parallel, so skew lines exist only in three or
more dimensions.

66. Example of skew lines

67. Four Skew Lines

68. Everyday example of Skew lines

69. Skew lines in Air Shows

70. Air displays

71. THANKYOU

72. Ur comments please….

73. Graphs in 2-D

74. 2D-Graphs

75. Surfaces in 3 - D

76. 3-D surface

82. Example of function of two
variable
 So far, we have dealt with functions of
single variables only. However, many
functions in mathematics involve 2 or more
variables. In this section we see how to find
derivatives of functions of more than 1
variable.
 Here is a function of 2 variables, x and y:
 F(x,y) = y + 6 sin x + 5y2
 To plot such a function we need to use a 3-
dimensional co-ordinate system.

83. F(x,y) = y + 6 sin x + 5y2

84. To find Partial Derivatives
 "Partial derivative with respect to x"
means "regard all other letters as
constants, and just differentiate the x
parts".
 In our example (and likewise for every 2-
variable function), this means that (in
effect) we should turn around our graph
and look at it from the far end of the y-
axis. So we are looking at the x-z plane
only.

85. Graph of F(x,y) = y + 6 sin x + 5y2
taking y constant

86. We see a sine curve at the bottom and this
comes from the 6 sin x part of our function
F(x,y) = y + 6 sin x + 5y2.
The y parts are regarded as constants.

88.  (The sine curve at the top of the graph is
just where the software is cutting off the
surface - it could have been made it
straight.)
 Now for the partial derivative of
 F(x,y) = y + 6 sin x + 5y2
 with respect to x:

89. Partial Differentiation with respect
to y
 "Partial derivative with respect to y"
means "regard all other letters as
constants, just differentiate the y parts".
 As we did above, we turn around our
graph and look at it from the far end of
the x-axis. So we see (and consider things
from) the y-z plane only.

90. Graph of F(x,y) = y + 6 sin x + 5y2
taking x constant

91. Parabola
We see a parabola. This comes
from the y2 and y terms in
F(x,y) = y + 6 sin x + 5y2.
The "6 sin x" part is now
regarded as a constant.

92.  Now for the partial derivative of
 F(x,y) = y + 6 sin x + 5y2
 with respect to y.
The derivative of the y-parts with respect to y is
1 + 10y. The derivative of the 6 sin x part is
zero since it is regarded as a constant when we
are differentiating with respect to y.