MATHS SYMBOLS - GEOMETRY - FIRST ELEMENTS - POINTS - LINE SEGMENTS - RAYS - STRAIGHT LINES - DIMENSIONS - SQUARES - CUBES - TESSERACT - PLANAR GEOMETRY - VERTICES - ANGLES - DEGREES - POLYGONAL CHAINS - POLYGONS - LIST of POLYGONS - THREE DIMENSIONS - POLYHEDRONS - PYRAMIDS - TRIANGULAR PYRAMIDS - TETRAHEDRONS - EDGES - LIST of POLYHEDRONS #angles #cubes #degrees #dimensions #edges #first_elements #geometry #line_segments #list_of_polygons #list_of_polyhedrons #planar_geometry #points #polygons #polygonal_chains #polyhedrons #pyramids #rays #squares #straight_lines #tesseract #tetrahedrons #three_dimensions #triangles #triangular_pyramids #vertex #vertices

1. ENZO EXPOSYTO
MATHS
SYMBOLS
GEOMETRY - FIRST ELEMENTS
Enzo Exposyto 1

2. GEOMETRY
-
FIRST ELEMENTS
Enzo Exposyto 2

3.
Enzo Exposyto 3

4. 1 - Basic Elements and First Four Dimensions 6
2 - Lines in Planar Geometry 21
3 - Vertices, Angles, Degrees in Planar Geometry 27
4 - Polygonal Chains and Polygons 41
5 - Vertices, Sides, Diagonals of Polygons 50
6 - A List of Polygons 60
Enzo Exposyto 4

5. 7 - Polyhedrons 64
8 - Vertices, Edges, Diagonals of Polyhedrons 69
9 - A List of Polyhedrons 79
10 - SitoGraphy 83
Enzo Exposyto 5

6. Basic Elements
and
First Four
Dimensions
Enzo Exposyto 6

7. Basic Elements - 1
Enzo Exposyto 7

8. Basic Elements - 2
. A geometric point don't have
any length, area, volume
or any other dimensional attribute.
It's,
it represents,
a unique location
in the dimensional space.
_____ A line segment is a part of a straight line
that is bounded by two distinct endpoints.
It contains every point on the straight line
between the 2 endpoints
and can include 0, 1 or both endpoints
Enzo Exposyto 8

9. Basic Elements - 3
ray
Let's consider a straight line
and a point A on it.
A divides this line into two parts:
for example, left part and right part.
Each part of the line is a ray.
Usually, the point A
is an element of the chosen ray
and is named
its initial point.
Enzo Exposyto 9
A
A

10. Basic Elements - 4
line or straight line (having no curvature)
In geometry, frequently,
the concept of line
is taken as a primitive.
It has only one dimension,
namely length,
without any width nor depth.
In analytic geometry,
a line, in the plane,
is often defined
as the set of infinite points
whose coordinates
satisfy a given linear equations
Enzo Exposyto 10

11. Dimension
dimension in Mathematics
and Physics,
the dimension
of a mathematical space
(or of an object)
is
informally
defined as
the minimum number
of coordinates
(x, y, z, w, …)
needed
to specify
ANY POINT
within it
Enzo Exposyto 11

12. Specify a point in 1 dimension: P(x)
Number Line
Enzo Exposyto 12
__|
0
(x)

13. Specify a point in 2 dimensions: P(x, y)
Cartesian Coordinate Plane
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(0, 0)
(x, y)

14. Enzo Exposyto 14
y
(0, 0, 0)
P(x, y, z)
x
z
z y
Specify a point in 3 dimensions: P(x, y, z)
3-d Cartesian Coordinate System
x

15. From 0 to 4 dimensions: how to form a tesseract from a point
1 - Two points can be connected to have a line segment
2 - Two parallel line segments can be connected to form a square
3 - Two parallel squares can be connected to form a cube
4 - Two parallel cubes can be connected to form a tesseract
0 - D 1 - D 2 - D 3 - D 4 - D
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16. 0 dimension (1 point) 1 dimension (1 line segment)
Enzo Exposyto 16
x

17. 2 dimensions (a square)
Enzo Exposyto 17
x
y

18. 3 dimensions (a cube)
Enzo Exposyto 18
x
y z

19. 4 dimensions (a tesseract)
Enzo Exposyto 19
x
zy
w

20. From 0 to 4 dimensions:
how to form a tesseract from a point
0) Let’s start from
a 0-dimensional point
1) If we translate this point of one unit length, along x-axis, we get
an 1-dimensional line segment
2) If we move the line segment of one unit length, along y-axis,
in direction perpendicular to x-axis, we form
a2-dimensional square
3) If we translate this square of one unit length in the direction
normal to the plane which it lies on, along z-axis, we have
a 3-dimensional cube
4) If we move the cube of one unit length into the 4th dimension,
along w-axis, we obtain
a 4-dimensional tesseract
n) This can be generalised to any number of dimensions.
Enzo Exposyto 20

21. Lines
in
Planar
Geometry
Enzo Exposyto 21

22. Vertical Line and Horizontal Line
| vertical line
__ horizontal line
Enzo Exposyto 22

23. Diagonal
/ or diagonal;
informally,
any sloping line
is called diagonal;
inclined obliquely
from a reference line
(as the vertical);
synonyms of diagonal:
inclined,
oblique,
slanted,
sloped,
sloping,
...
Enzo Exposyto 23

24. Intersecting Lines
l and m
are two intersecting lines
which share exactly
one point,
named P.
This shared point
is called
the point
of intersection.
Enzo Exposyto 24

25. Parallel Lines
// parallel lines;
they are lines
which don't meet.
Two lines in a plane
that
don't intersect
or don't touch
each other
at any point
are said
to be parallel l and m are parallel lines
not parallel lines
|| parallel vertical lines
Enzo Exposyto 25
l m

26. Perpendicularity
perpendicularity sign
... is perpendicular to ...
... is normal to ...
For example:
a vertical line
is perpendicular to
or
is normal to
a horizontal line
Enzo Exposyto 26

27. Vertices,
Angles,
Degrees
in
Planar
Geometry
Enzo Exposyto 27

28. Vertex
vertex (plural vertices or vertexes)
Generally speaking,
it's a special kind
of POINT
that describes
the corner
or
the intersection
of geometric shapes
Enzo Exposyto 28

29. Vertex of an angle - 1
vertex (of an angle)
The vertex of an angle
is the end point
where
two rays
or
two line segments
come together.
Enzo Exposyto 29

30. Vertex of an angle - 2a
vertex (of an angle)
More precisely,
the vertex of an angle
is the point
where two rays
begin or meet
or
where two line segments
join or meet
or
where two lines
intersect
(cross each other)
Enzo Exposyto 30

31. Vertex of an angle - 2b
l and m
are two intersecting lines
which share exactly
one point,
named P.
P is the vertex
of 4 angles
Enzo Exposyto 31
l
P
m
.

32. Angle
angle
In PLANAR GEOMETRY,
an angle
is the figure formed,
for example,
by TWO RAYS
or by TWO LINE SEGMENT.
The rays or the line segment,
named the SIDES of the angle,
share a common endpoint,
called the VERTEX of the angle.
Angle
is also used
to indicate
the MEASURE
of an ANGLE
or of a ROTATION
Enzo Exposyto 32

33. Angle
Enzo Exposyto 33

34. Degrees
° degrees (angles);
the degree symbol;
we use a little circle °
following the number
to mean degrees;
we can measure angles
in degrees;
there are 360 degrees (360°)
in one full rotation
(one complete circle around);
half a circle is 180°
(called a straight angle);
quarter of a circle is 90°
(called a right angle);
Enzo Exposyto 34

35. Types of Angles and Degrees
Right Angle (90°)
Enzo Exposyto 35

36. Types of Angles and Degrees
Right Angle (90°) and Perpendicularity
perpendicularity sign
... is at 90° (90 degrees) to ...
... forms a right angle with ...
AB CD
a line segment (AB) drawn so that
it forms 2 right angles (90°)
with a line (CD)
Enzo Exposyto 36

37. Types of Angles and Degrees
a - Acute Angle (less than 90°)
b - Obtuse Angle (greater than 90° and less than 180°)
c - Straight Angle (180°)
Enzo Exposyto 37

38. Types of Angles and Degrees
Reflex Angle
(greater than 180° and less than 360°)
Enzo Exposyto 38

39. Types of Angles and Degrees
a Full Circle is 360°
Enzo Exposyto 39

40. Types of Angles and Degrees
a Synthesis
Enzo Exposyto 40

41. Polygonal
Chains
and
Polygons
Enzo Exposyto 41

42. Polygonal Chains - Examples
a simple open polygonal chain
a self-intersecting polygonal chain
a simple closed polygonal chain
Enzo Exposyto 42

43. Polygonal Chains - 2
polygonal it is a finite sequence
chain of connected line-segments,
called sides.
This sides are connected
by consecutive points,
called vertices.
For example:
an angle
has
a simple open polygonal chain;
a triangle,
a square, …
have
a simple closed polygonal chain
Enzo Exposyto 43

44. Polygonal Chains - 3
polygonal More precisely,
chain a simple polygonal chain
is one in which
only consecutive
line-segments
intersect
and only
at their endpoints.
Enzo Exposyto 44

45. Polygonal Chains - 4
polygonal Besides,
chain a closed polygonal chain
is one in which
the first vertex
coincides
with the last one,
or, alternatively,
the first and the last vertices
are also connected
by a line segment.
A simple closed polygonal chain
in the plane
is the boundary
of a simple polygon.
Enzo Exposyto 45

46. Polygons - 1
polygon is a plane figure
that is bounded
by a finite sequence
of straight line-segments
which form
a closed polygonal chain.
Often the term "polygon"
is used in the meaning
of "closed polygonal chain",
but, in some cases,
it's important
to draw a distinction
between
a polygonal area
and a polygonal chain
Enzo Exposyto 46

47. Polygons - 2
polygon Since two line-segments
(triangle) of an angle
always form
a simple open polygonal chain,
are needed, at least,
three line-segments
to have a closed polygonal chain
and, then, a polygon.
Moreover,
we can form a triangle
if, and only if,
every sum
of the measures of two sides
is greater than
the measure of the third side.
Enzo Exposyto 47

48. Polygons - 3
polygon In other words,
(triangle) for every
three line-segments
which form a triangle
and
whose measures are a, b, c,
it's
a + b > c
a + c > b
b + c > a
Enzo Exposyto 48

49. Polygons - Example
polygon Since a = 4.55
(triangle) b = 4.31
c = 10.15
and a + b = 4.55 + 4.31 = 8.86
then a + b < c because 8.86 < 10.15
It's impossible
to form a triangle
by line-segments
with a, b, c lengths
Enzo Exposyto 49
a
c
b

50. Vertices,
Sides,
Diagonals
of
Polygons
Enzo Exposyto 50

51. Vertex
vertex (of a polygon)
A vertex
is a corner point
of a polygon,
formed by
the intersection
of two
consecutive sides.
A side is called
an edge as well
(see polyhedrons).
Enzo Exposyto 51

52. Vertices and sides
A polygon (a pentagon)
with 5 vertices (points)
and 5 sides (line-segments)
Enzo Exposyto 52
side
sideside
side
side
vertex
vertex
vertex
vertex
vertex

53. Side and Vertices
In a POLYGON
(2 dimensions)
a SIDE
is
a particular type
of LINE SEGMENT
JOINING
TWO
CONSECUTIVE
VERTICES
of the shape
Enzo Exposyto 53

54. Side
In a POLYGON,
a SIDE
is
a LINE-SEGMENT
ON THE BOUNDARY.
It's often
called
an EDGE
(see polyhedrons).
Enzo Exposyto 54

55. Side (Edge)
A regular polygon
(a square)
with 4 sides
(4 edges),
each between
two vertices
Enzo Exposyto 55
side
. vertex
.
.
. vertexvertex
vertex
side
side
side

56. Side and Diagonal
Remember that
a LINE-SEGMENT
which joins
two vertices
and that
PASSES THROUGH
the INTERIOR
or
the EXTERIOR
of the polygon
is NOT an SIDE
but INSTEAD
is called
a DIAGONAL
(see next page)
Enzo Exposyto 56

57. Diagonal
/ or diagonal;
a diagonal
of a polygon
is
a line-segment
joining
two
non-consecutive
vertices
of the shape
Enzo Exposyto 57

58. Diagonal and Vertices
A polygon (an hexagon)
with 6 vertices
There are some diagonals:
they join non-consecutive vertices
Enzo Exposyto 58

59. Diagonal and Vertices
A star
The diagonal
that passes through
the exterior of the polygon
joins two
non-consecutive vertices
Enzo Exposyto 59

60. a List
of
Polygons
Enzo Exposyto 60

61. In the next pages,
a LIST
of
GEOMETRIC
SHAPES
2-D
In the 1st column, types of triangles
in the 2nd, types of quadrilaterals
in the 3rd, types of regular polygons
(from math-salamanders.com)
Enzo Exposyto 61

62. Geometric Shapes 2D - 1
Enzo Exposyto 62

63. Geometric Shapes 2D - 2
Enzo Exposyto 63

64. Polyhedrons
Enzo Exposyto 64

65. Polyhedrons - 1
polyhedron (plural polyhedra or polyhedrons)
is a solid
in three dimensions
with flat polygonal faces,
straight edges
and sharp corners called vertices.
Pyramids,
truncated cones,
prisms,
cones,
cylinders,
spheres,
are polyhedrons
(three-dimensional shapes)
or solid figures
Enzo Exposyto 65

66. Polyhedrons - 2
polyhedron Since any polygon (2-dimensional shape)
(pyramid) lies in a plane,
is needed, at least,
a point located above or below
the plane of the polygon
to form a polyhedron (3-dimensional figure).
Therefore,
we can form
a triangle-based pyramid
if, and only if,
we connect
the vertices of the triangle
with a point, called apex,
located above or below
the plane of the triangle
Enzo Exposyto 66

67. Polyhedrons - 3
polyhedron Note that
(pyramid) a triangle-based pyramid
has, in total, 4 triangular faces:
the base
and the 3 faces
which we get by connecting
the 3 vertices of the base
with the apex.
So, this solid
is, more often, called
a tetrahedron.
Enzo Exposyto 67

68. Polyhedrons - 4
polyhedron
(tetrahedron)
Enzo Exposyto 68

69. Vertices,
Edges,
Diagonals
of
Polyhedrons
Enzo Exposyto 69

70. Vertex
vertex (of a polyhedron)
A vertex
is a corner point
of a polyhedron
formed
by the intersection of edges
or
by the intersection of faces
(a face is
a polygon
on the boundary
of a polyhedron)
Enzo Exposyto 70

71. Vertices, edges and faces
A regular polyhedron
(a cube)
with vertices,
edges
and faces
Enzo Exposyto 71

72. Edge and Vertices
In a polyhedron
(3 dimensions),
an EDGE
is
a particular type
of line segment
JOINING
TWO
CONSECUTIVE
VERTICES
of the shape
Enzo Exposyto 72

73. Edge and Faces
In a polyhedron,
an EDGE
is
a line segment
where
two
2-dimensional faces
meet
Enzo Exposyto 73

74. Edge and Faces
In a polyhedron,
like this cube,
every EDGE
is shared
by two FACES
Enzo Exposyto 74

75. Edge and Diagonal
Remember that
a SEGMENT
which joins
two vertices
and that
PASSES THROUGH
the INTERIOR
or
the EXTERIOR
of the polyhedron
is NOT an EDGE
but INSTEAD
is called
a DIAGONAL
(see next page)
Enzo Exposyto 75

76. Diagonal
/ or diagonal;
a diagonal
of a polyhedron
is
a line segment
joining
two
non-consecutive vertices
of the shape
Enzo Exposyto 76

77. Diagonal and Vertices
A regular polyhedron
(a cube)
Note the vertices A, A’, C, C’
and the diagonals AC and AC’
which join
non-consecutive vertices
Enzo Exposyto 77

78. Diagonal and Vertices
A stellated polyhedron
The diagonal
that passes through
the exterior of the shape
joins two
non-consecutive vertices
Enzo Exposyto 78

79. a List
of
Polyhedrons
Enzo Exposyto 79

80. In the next pages,
a LIST
of
GEOMETRIC
SHAPES
3-D
In the 1st column,
the 5 Platonic Solids
(from math-salamanders.com)
Enzo Exposyto 80

81. Geometric Shapes 3D - 1
Enzo Exposyto 81

82. Geometric Shapes 3D - 2
Enzo Exposyto 82

83. SitoGraphy
Enzo Exposyto 83

84. http://en.wikipedia.org/wiki/Dimension
https://en.m.wikipedia.org/wiki/Hypercube
http://www.mathopenref.com/
https://en.m.wikipedia.org/wiki/Angle
http://geometry157.blogspot.it/
https://en.m.wikipedia.org/wiki/Polygon
https://en.m.wikipedia.org/wiki/Polyhedron
http://www.math-salamanders.com/list-of-geometric-shapes.html
…
Enzo Exposyto 84