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198 views18 pagesGeometric Design
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» CHAPTER2 GEOMETRIC DESIGNS
We see patterns and diagrams every day and one example of these patterns is
tessellations. Tessellations are planes that are covered (or tiled) with shapes without gaps
or overlaps. We can find them all around, it can be in the classrooms, offices, soccer field,
bathroom, among others. Geometric patterns and diagrams commonly use different shapes
such us common shapes (circle, triangle, square, among others.). Other geometric patterns
use more complicated shapes, with some irregular shapes that we don't have names for.
This chapter focuses on the mathematical concepts involved in the design of what
appear in paintings, sculptures, and tilling, and textile imprints. Moreover, it also focuses
on the context of the role of symmetry in nature, and the geometric concepts, particularly
the design principles. Lastly, this chapter emphasizes the appreciation and the importance
of the geometric design patterns that have artistic and cultural significance.
Basic Concepts of Euclidean Geometry
Euclidean geometry, sometimes called parabolic geometry, is a geometry that follows a
set of proportions that are based on Euclid’s five postulates.
Types of Euclidean Geometry
1. Plane geometry. Itis the study of figures on a two-dimensional surface, that is, on a
plane. Particularly, it deals in objects that are flat, such as triangles and lines, which
can be drawn on a flat piece of paper.
2. Solid geometry. Itis the geometry of three-dimensional Euclidean space. It includes
the measurements of volumes of various solid figures (three-dimensional figures).
These include pyramids, cylinders, cones, spheres, and prisms.
Basic Geometric Terms
The simplest figure in geometry are points, lines, and angles. The term point and line
will remain undefined (later on it will be defined base in its characteristics), and an angle
will be defined in terms of rays, which are themselves defined in terms of segments. In
order to understand Euclidean Geometry, it is critical to have a solid set of definitions and
illustrations for these basic terms.
Table 2.1. Euan Seomety 2 Definitions
itis a location in space. It has
no magnitude.
+ A point has no size.
+ A point is named with a capital
letter.
The name of the points are
Point A point L and pot F
poe -+--Continued..
103 Line
“Extends infinitely in opposite
directions. Ithas length without
breadth. :
Two points define a unique
line.
‘A line can be named using 4
lowercase letter (perhaps with
Line segment
a subscript).
Tt is part of a line: two points
(the endpoints) together with
that part of the line between the
endpoints.
A line segment has a measure:
its length.
It can be named by naming its
endpoints.
A
It can be named as AB or BA
Its start with a point and extends
indefinitely in one direction.
It has also measure: itslength.
It can be named by naming its
one endpoint and any point in a
line.
Plane
It is a flat surface with no
thickness and no boundaries.
A plane can be represented
by three points which do not
lie on the same line or it can be
named by a capital letter.
This can be labelled as plane
ABC or plane M.
Lines in a plane
can be:
+ Intersecting
lines
+ Parallel lines
+ Perpendicular
lines
+ Concurrent
lines
Contain two or more lines cross
at one point in the plane.
Two or more lines in a plane
that ‘never intersects. ‘The
distance between the two lines
are the same.
Two or more lines that
intersects at 90 degrees angle,
Three or more lines passing
through the same point.
Line 1 is Parallel to line m.
Line n intersects line | and line
m. |
| .
| Recognizing and Analyzing Geometric Shapes
| Geometric shape is generally defined by the set of points and lines that are connected to
| form a closed figure. Particularly, it can be defined as figure or area closed by a boundary
which is created by combining the specific amount of curves, points, and lines. There are two
types of geometric shapes: the two-dimehsional and three-dimensional geometric shapes.
Generally, these basic two-dimensional and three-dimensional geometric shapes are
seen in the real world, They can be used as basic pattern for an inevitably beautiful wall
paper design, windows and doors in the house, painting, donuts, glass, towers, pyramids,
among others. Really it is true that shapes are all around us. Moreover, shapes are also used
as symbols that emulate different cultures and beliefs of different people which will be
discussed in the next lesson.
Two-dimensional Geometric Shapes
These, are closed figures made ‘up of vertices joined together by lines. Common
examples are triangles, quadrilaterals, and other polygons. These also include closed figure
conies such as circles and ellipses. Table 2.2 lists and describes these shapes and figures.
Table 2.2. Common Two-Dimensional Geometric Shapes (2-D)
Name of Stapes | gure Description aac ropenies —T |
+ The sum of the angles in a triangle
is 180% This is called the angle-sum
property.
+ The sum of the lengths of any two sides
ofa triangle is greater than the length of
the third side, Similarly, the difference
between the lengths of any two sides of
a triangle is less than the length of the
"third side.
Triangle
+ The side opposite to the largest angle is
the longest side of the triangle and the
All the properties of a rhombus apply
(the ones that matter here are parallel
sides, diagonals are perpendicular
bisectors of each other, and diagonals
bisect the angles)
+ All the properties of a rectangle apply
(Ge only one that matters here is
..fiagonals are congruent) |__ continued...
105 continuation.
a
Allangles are right angles by definition
Rectangle
, lel
+ Sides are paral
Opposite sides are congruent.
isects each other.,
« -Diagonals bisect 7 ae
‘i angles are right angles by definition,
«The diagonals are congruent.
Parallelogram
‘A quadrilateral that has two Pairs of
parallel sides
+ Two pairs of opposite angles are
congruent.
+. The consecutive angles are
supplementary.
Trapezoid
+ A quadrilateral with exactly one Pair
of parallel sides (the parallel sides are
called bases)
* A trapezoid in which the nonparallel
Sides (the legs) are congruent.
Rhombus
106
can’t be Used in
All the properties of a parallelogram
apply (the Ones. that matter here are
Parallel ‘sides, Opposite’ angles are
congruent, and consecutive angles are
Supplementary),
All sides are congruent by definition.
The diagonals bisect the angles,
The diagonals are perpendicular
bisectors of £ach othe:
* A quadrilateral in which two disjoint
Pairs of Consecut
airs of tive sides are congruent
Cisjoint pairg» means that one side
both pairs),
- continued. continuation.
Peotagon A polygon with five sides and five
angles.
|
Hexagon A polygon with six sides and six angles.
‘A polygon with seven sides and seven
H :
leptagon angles.
| A polygon with eight sides and eight
Octagoa angles.
|
‘A polygon with nine sides and nine
Nonagon shale:
Decca A polygon with ten sides and ten angles.
i continued.
107 Ellipse
‘oval shape, traced by a point
plane so that the sum of
its distances from two other points (the
foci) is constant, oF resulting when a
cone is cut by an oblique plane which
does not intersect the base.
A regular
moving in &
Three-dimensional Geometric Shapes
‘These are represented by lines joining set of | points and pl :
Polyhedrons are the usual three-dimensional shapes such us cube, prism,
Jane surfaces joining the line.
pyramid, cylinder,
cone, and the pyramid as well. Table 2.3 lists the common 3-D shapes.
Rectangular Prism/
Cuboid
It has six flat faces and all angles are
right angles.
And all of its faces are rectangles.
It is also a prism because it has the same
cross-section along a length.
Triangular prism
Just like cuboid, it has the same cross-
section along a length.
The cross section of this object is a
triangle,
Cube
108
Ithas six faces,
Each face has four edges (and is a
square).
Tthas 12 edges,
Ithas eight vertices (comer points) and
at each vertex three
~- Continued... continuation =~
Square Pyramid
ithas five faces.
‘The four side faces ar¢ triangles.
The base is a square.
It has five vertices (corner points).
Ithas eight edges.
Triangular Pyramid
It has four faces.
The three side fices are triangles.
The basé is also a triangle.
It has four vertices (corner points).
Ithas six edges.
It is also a tetrahedron.
Sphere
It is perfectly symmetrical.
‘All points on the surface are the same
distance “r” from the center.
Ithas no edges or vertices (corners).
It has one surface (not a “face” as it isn’t
flat)
Itis not a polyhedron.
Cylinder
Tthas a flat base and a flat top.
The base is the same as the top.
From base to top the shape stays the
same
It has one curved side
It is not a polyhedron as it has a curved
surface
Tt has a circle at one end, a point at the
other end and a curved side
It is not a polyhedron as it has a curved
surface
Torus
O>e0 Ole|>S
Image from
BricsCAD Help
Center
It can be made by revolving a small
circle (radius 1) along a line made by a
bigger circle (radius R).
It has no edges or vertices.
It is not a polyhedron.
109 Measurements in 2-D and 3-D Shapes
lume)
(Perimeter, Area, Surface Area, and Vol ne) ad 3D figures. Thee orn
There are common measurement made in both
for 3-D
i e area and volume
perimeter, circumference for circle and area for 2-D; § Sa dary ofa closed geometric
shapes. Perimeter is defined as the measure ofthe Hine on re. Each 2-D and 3-D geometric
figure. The area is the measure of the surface of the figure
° area, and volume.
shapes have formula for calculating the perimeter, area, eee the perimeter, area,
Table 2.3 and Figure 2.1 summarize the formula for deter :
surface area, and volume of 2-D and 3-D geometric shapes.
Table 2.3. Measurements for 2-D and 3-D Shapes
Area "Perimeter Circumaferenee
Figure
w
Triangle Ae beh P=MN+NP+PM
Name
Parallelogram A=bxh P=DE+EF+FG+GD
: Le P=b+b+b+b
Rhombus A=bxh ae
fe A
‘ cb
Ed
Rectangle
Ra P=Lt+w+Liw
A=L
oF P=2L+2w
Square
P=1414+14)
P=41
Trapezoid
P=MN+NP+PR+RM
Circle
Source: Image from http://www, ‘math-videos-online.g
3 ‘om/i
“imases/eommen geome
110 GENERAL CONE OR PYRAMID:
on —
siside 1
, volume: v= 3Ah
surface area: JP +8 +90?
RECTANGULAR RIGHT CIRCULAR CONE
I:length, w: width,
hrhelght
volume: v= lwh
surface area: s = 2lw + 2Ih
+ 2h
surface area: mrV7 +h? +n0*
FRUSTUM OF A CONE
SPHERE F: top radius
radius R=base radius
volume: v:
fiend TP
surface area: s = 4nr° si slant helght
volume: v= —(? +rR+R)h
surface area: ns (R +1) +207
aR?
RIGHT CIRCULAR
CYUNDER é SQUARE PYRAMID. i
aE s:side !
bi height jh hehelght i
wvearh volume: v= Z
surface area: s = 2xth +
xP surface area: s= s(s+ Vs¥+4nt)
nai REGULAR TETRAHEDRON
‘tube radius ie *
A torus radius volume: v= a5"
volume:
surface area: s = 3c
Figure 2.1. Three-Dimensional Geometry Formulas
Source: http://lugezl.com/images/
Transformation
In geometry, transformation refers to the movement of objects in the coordinate plane.
Particularly, it is an operation that moves, flips, or changes a shape to create a new shape in
‘a plane. Moreover, geometric transformations involve taking a pre-image and transforming
it in some way to produce an image. There are two different categories of transformati.
rigid and non-rigid transformations.
1. Rigid transformations (isometry) — does not change the shape or size of the pre-
11 Types of Transformation satobe syminetrical, and this symmetry resulted
Many objects or things around us are oe ks Within the rigid and non-rigid categories,
from the socalled geometric transformation fill in the rigid transformation
there are four main types of transformations. Thee asi types of transformation are as
category, and one is a non-rigid transformation. The "
follows: ; : a sib,
1. Rotation — rotating an object about a fixed point without cungins . hens - shape,
2. Translation — moving an object in space without changing » Shape or
3. Pilato. eee compression, enlargement or even expansion) expanding or
contracting an object without changing its shape or orientation, —
4. Reflection — flipping an object across a line without changing its size or shape.
5. Glide reflection—a special type of two-step isometry, uses combination of reflection
and translation.
Itis important to note that for the rigid transformation, after any of those transformations
(tur, flip, or slide), the shape still has"the same size, area, angles and line lengths.
Moreover, why is dilation the only non-rigid transformation? Remember that in a non-rigid
transformation, the shape will change its size, but it won’t change its shape. (See illustrations
page 112-113). .
Mlusttation of Transformation
Rotation is when we rotate a ae
degree around a oe seital.
Translation
is when
We slide a fi
di ide a figure in any
112 i ' Reflection is when we-flip a figure over aline. Dilation is when we enlarge or reduce a figure.
R¢6, 4)
Glide reflection is when we slide in any direction then we flip a figure over a line.
In addition, when one shape becomes another using only turns, flips and/or slides, then
the two shapes are congruent. Two shapes are similar when there is a need to resize one
shape to become another (by turns, flips and/or slides). So, when one shape becomes another
using transformation, the two shapes might be congruent or just similar.
Symmetry
An object is said to be symmetrical when one shape becomes exactly like another when
moved in some way. Ian Stewart (2013) defines the concept of symmetry as a transformation
of a mathematical structure of specific kind. Objects that are not symmetrical are called
asymmetrical. ; y
Principles of Symmetry , Glide
4 GGG
u Fees
tid 3
{
Mirror i 1
Goa
Glide Reflection
Source: www.admaf.org (ret: May 29, 2018)
Reflection
Reflection symmetry is also known as mirtor symmetry. This is because if’ you place a
mirror along the line of symmetry, it will give a reflection which is the same size and shape
as the original image. Glide reflection combines the principle of glide and reflection to
create alternating wall paper.
Tessellation
Tessellation is a pattern that uses different transformations
a of different geometric |
shapes. It creates repeating figures that do not overlap,
no empty spaces and fits together.
Types of Tessellation
114 2. Rotation Tessellation. Rotation tessellation is done by rotating the tile about 90
degrees to position the tile adjacent to the top. ‘This is done repeatedly until the
desired size is achieved.
https://www flickr.com/photos
3. Reflection Tessellation. In this type of tessellation, one has to cut, flip, and slide the
original tile. It can also be cut, slide, and flip. The flip is 180 degrees.
httpsi/web.techdlearning.com
115 nore the shape of the object is the same
4. Dilation Tessellation. Is a teal is er at flow show te concept of
; sgh
as the original but of different size.
dilation tessellation,
http://www.keywordbasket.com
Patterns and Diagrams
A geometric pattern is a kind of pattern formed by geometric shapes and typically
Tepeated like a wallpaper design. Any of the senses may directly observe patterns, Meanwhile,
geometric design is a branch.of computational geometry. It deals with the construction and
representation of free-form curves, surfaces, or volumes and is closely related to geometric
modeling. Core problems are curve and surface modelling and representation,
Some foreign and local samples of geometric patterns and diagrams,
ma
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