▪ Mathematics is integrated in the

symbology of culture and arts. In

arts, counting and patterns are the
needed elements to produce the
Mathematics design and style. They used to
as a Tool
projects aesthetics and logic of any
craft and create the real meaning of
structure especially of an artist.
 Objectives

At the end of this chapter, you must be

able to:

1. Apply geometric concepts especially

Geometric isometries in describing and creating designs;
Designs and
2. Appreciate the contribution of geometric
designs to the enrichment of the Filipino
Culture and Arts using concepts in geometry.
 ▪ Geometry can help enhance one’s artistic
prowess as well as enrich one’s own culture. According
to Rachel Bernstein and Eunice Lee (2011), Geometry
is everywhere. We can train ourselves to find the
Geometry in everyday objects and in works of art. Line,
shape, pattern, symmetry, scale, and proportion are the
Concepts building blocks of both art and math. Geometry offers
the most obvious connection between the two
disciplines. Both art and math involve drawing and the
use of shapes and forms, as well as understanding of
spatial concepts, two and three dimensions,
measurement, estimation, and pattern.
Line

Lines vary in width, length, curvature,

color, or direction.

Shape

Shape/plane figures are two-dimensional

figures in which all points lie in the same plane.
Shapes can be open or closed, free-form or
geometric.
▪ Form

▪ Forms/ space or solid figures are three-

dimensional (having length. Width, and depth),
enclose volume (or mass), and help us to
understand physical space. For example, a
triangle, which is two-dimensional, is a shape. But
a pyramid, which is three- dimensional, is a form.
Cubes, spheres, pyramids, cones, and cylinders
are examples of forms. They can be literally three-
dimensional or they can have the illusion of three –
dimensions.
 ▪ Repetition is the reccurence of elements of art
at regular intervals. When lines, shapes, and forms

▪ repeat in a predictable combination, they form a

pattern. The pattern of this quilt is the tessellation,
Pattern and a collection of shapes that fit together to cover a
Repetition surface without overlapping or leaving gaps.
Tessellations can be seen in almost every brick
wall, tiled floor or wall, quilt pattern, lace tablecloth,
fabric, and wallpaper pattern.

▪ Balance
▪ Balance is the arrangement of elements to create a sense of
equilibrium and harmony. There are three types of balance –
symmetry, asymmetry, or radial symmetry.

1. Symmetry is a type of balance in which the shapes and pattern

are identical on either side of a central boundary; the two
halves of a work mirror each other.

2. Asymmetry is an arrangement of parts in which the opposite

sides, divided by a central line, are not identical.

3. Radial symmetry is a form of symmetrical balance in which the

elements of a composition radiates from a central point in a
regular, repeating pattern. Radial symmetry can be found in
nature in flowers, starfish, jellyfish, crystals, and snowflakes.
▪ Proportion

▪ In art, proportion is the principle of

design concerned with the size of
relationships of parts of a composition
to each other and to the whole. In math,
proportion is the ratio or relation of one
part or another to the whole with
respect to size, quantity, and degree.
▪ Perspective

▪ Perspective is the system for representing three-

dimensional objects, viewed in spatial recession, on a two-
dimensional surface. The simplest form of perspective
drawing is near linear perspective, a system that allows
artists to trick the eye into seeing depth on a flat surface.
Linear perspective uses sets of implied lines called
converging or orthogonal lines that move closer together in
the apparent distance until they merge at an imaginary
vanishing point in the horizon. One point perspective uses
lines that lead to a single vanishing point in the horizon. One-
point perspective uses lines that lead to a single vanishing
point; two-point perspective uses lines that lead to two
different vanishing points.
▪ How to analyze geometric shapes
according to the level of geometric
thinking.

▪ Analyzing geometric shapes

(Vojkuvkoua, 2012) can be done using the Van
Heile Theory of Geometric Thinking. According
to Pierre van Heile, there are five levels of
thinking or understanding in geometry:
Level 0 – Visualization (Basic
visualization or Recognition)
Level 1 – Analysis (Description)
Level 2 – Abstraction (Informal
Deduction or Ordering or Relational)
Level 3 – Deduction (Formal
Deduction)
Level 4 – Rigor
 ▪ It is inherent in the Van Heile Theory that one considers
the ff. properties.

1. Fixed Sequence

▪ A student cannot be at Level N without having

Properties of gone through level (N-1). Therefore, the student
the Levels must go through the levels in order.

2. Adjacency

▪ At each level, what was intrinsic in the preceding

level becomes extrinsic in the current level.
1. Distinction

▪ Each level has its own linguistic symbols and its own network of
relationships connecting those symbols. The meaning of a
linguistic symbol is more than its explicit definition; it includes
experiences which the speaker associates with the given
symbol. What may be “correct” at one level s not necessarily
correct at another level.

2. Separation

▪ Two persons at a different levels cannot understand each other.

The teacher speaks a different “language” from the student who
is at a lower level. The Van Hieles thought says that this
property was one of the main reasons for failure in geometry.
1. Attainment

▪ The learning process leading to complete

understanding at the next level has five
phases – information, guided orientation,
explanation, free orientation, integration,
which are approximately not strictly
sequential.
▪ Transformation

▪ Accdg. To Jennifer Beddoe, (2003) in geometry,

transformation refers to the movement of objects in the
coordinate plane.

▪ Definitions of Transformations

▪ Geometric transformations involve taking a preimage

and transforming it in some way to produce an image. There
are two different categories of transformations:

1. The rigid transformation, which doesn’t change the shape

or size of the preimage.

2. The non-rigid transformation, which will change the size

but not the shape of the preimage.
▪ Types of Transformations

▪ Within the rigid and non-rigid categories, there are four types of
transformation. Three of them fall in the rigid transformation category,
and one is a non-rigid transformation.

1. Rotation is rotating an object about a fixed point without changing its

size or shape.

2. Translation is moving an object in one space without changing its size,

shape or orientation. A translation is performed by moving the preimage
to the requested number of spaces.

3. Dilation is expanding or contracting an object without changing its shape

or orientation. This is called resizing, contraction, compression,
enlargement or even expansion. The shape becomes bigger or smaller.

4. Reflection is flipping an object across a line without changing its size or

shape.
ROTATION
TRANSLATION
DILATION
REFLECTION
 What do you
think about the
picture? Is it
getting bigger? Or
smaller?
What do you think is
the type of
transformation
which is most
dominant from the
picture?
▪ The Psychology of Shapes
▪ From Siddanth Pillai (Kayla Darling,
2018), the most common shapes are the
rectangles, circles, triangles, rhombuses, and
hexagons. They can be broken down into
something like the ff:
1. Rectangle/squares – Balance and Tradition
2. Circles – Infinity, Protection, and Feminity
3. Triangles – Stability, Energy, and Aggression
4. Rhombus – Vibrant and Contemporary
5. Hexagon – Unity and Balance
 ▪ Accdg. To Kayla Darling (2018) shapes
are the bases of most images and
designs. Different shapes can evoke
different meanings. Combining these
shapes in different ways can create
entirely new meanings. For example,
creating something out of rectangles
and triangles could emphasize a call to
return to tradition.
▪ Geometric Patterns are a collection of
shapes, repeating or altered to create a
cohesive design.
 Euclid (fl. 300BC), sometimes given the name
Did you
Euclid of Know?
Alexandria to distinguish him from
Euclides of Megara was a Greek
mathematician, often referred to as the “
founder of geometry” or the “father of
geometry”.
Deduce the theorems of what is now called
Euclidean Geometry from a small set of
axioms.
Euclid is the anglicized version of the Greek
name which means “renowned, glorious”.
THANK YOU !