Mathematics in the Modern World
Section 3. Mathematics as a Tool
Core Idea
Geometry can help enhance one’s artistic
prowess as well as enrich one’s own culture.
Learning Outcomes
At the end of this lesson, you are expected to
apply geometric concepts in describing and
creating designs; and
contribute to the enrichment of the Filipino
culture and arts using the concepts in
geometry.
Expected Output
Integrating project
Museum Gallery
Production number
Portfolio of geometric indigenous
designs
etc…
Recognizing and Analyzing Geometric Shapes
Polygon
Simple and Complex
Convex and Concave
Regular and Irregular
Recognizing and Analyzing Geometric Shapes
Polygon
Simple and Complex
Convex and Concave
Regular and Irregular
Interior angles of a polygon:
In general, for a polygon with n sides,
sum of the internal angles is equal to
(n – 2) x 180°
Recognizing and Analyzing Geometric Shapes
Solids
Volume
Surface area
Polyhedron (Prism and Pyramid)
Non-polyhedron (Sphere, Cone, Torus and
Cylinder)
Platonic solids
Euler’s formula
F+V–E=2
Geometric Transformations
Translation is a transformation of an object where every point of it
moves a fixed distance and a given direction.
The translation of a point (x, y) by some vector (tx, ty) results in
the transformed point (x´, y´) = (x + tx, y + ty).
Geometric Transformations
Rotation is a transformation of an object by rotating about a given
point through a given angle.
When rotating an object around the point of origin by an angle θ,
the point (x, y) ends up at
(x´, y´) = (x·cosθ – y·sinθ, x·sinθ + y·cosθ).
Geometric Transformations
Reflection is a transformation of an object where every point of it
and its image are of the same distance from the line of symmetry.
Geometric Transformations
Glide Reflection is a composition of translation and reflection in a
line parallel to the direction of translation.
Geometric Transformations
Reduction Enlargement
Dilation is a transformation of an object by resizing to either reduce
it or enlarge it about a point with a given factor. The value of factor
(r) determines whether the dilation is enlargement or reduction.
Patterns and Designs
Patterns are one aspect in geometry which are usually found and
utilized. There are patterns around us; in our home, we see patterns
on wallpapers, on floor mats, on bed sheets, window pane and on
pieces of furniture. Patterns are also profusion in nature: on flowers,
in leaves, on animals and on other places.
Patterns and Designs
Symmetry
If a figure can be reflected over a line in such a way that the
resulting image coincides with the original, then the figure has
reflection symmetry. Reflection symmetry is also called bilateral
symmetry. The reflection line is called the line of symmetry.
You can test a figure for reflection symmetry by tracing and
folding it.
How many lines of symmetry do the leaves and chess board
have?
Patterns and Designs
Symmetry
If a figure can be rotated about a point in such a way that its
rotated image coincides with the original figure, then the figure has
rotational symmetry.
You can trace a figure and test it for rotational symmetry. Place
the copy over the original and rotate the copy about the suspected
symmetry point. Count the number of times the copy and the
original coincide with the copy until it is back in the position it
started in.
A figure has order n rotational symmetry
if 1/n of a complete revolution leaves the
figure unchanged.
Patterns and Designs
Rosette Pattern
A symmetry group is the collection of all symmetries of a plane
figure. The symmetry groups have all been one of two types:
Cyclic symmetry group has rotation symmetry only around a
center point. If the rotation has n order, the group is called Cn.
Dihedral symmetry group has rotation symmetry around a center
point with reflection lines through the center point. If there will be n
reflection lines and the group is called Dn.
Patterns and Designs
Rosette Pattern
Triskelion logo with Hexagon with
cyclic (C3) symmetry dihedral (D6) symmetry
The cyclic and dihedral symmetry groups are known as rosette
symmetry groups, and a pattern with rosette symmetry is known as a
rosette pattern.
Rosette patterns have been used as architectural and sculptural
decoration of the new century.
You can create your own rosette pattern using a pattern
generator. Visit the site http://math.hws.edu/eck/jsdemo/rosette.html
Patterns and Designs
Frieze Pattern
An infinite strip with a repeating pattern is called a frieze
pattern, or sometimes a border pattern or an infinite strip pattern.
The term "frieze" is from architecture, where a frieze refers to a
decorative carving or pattern that runs horizontally just below a
roofline or ceiling.
Patterns and Designs
Frieze Pattern
A frieze group is the set of symmetries of a frieze pattern, that is
geometric transformations built from rigid motions and reflections
that preserve the pattern. This group contains translations and may
contain glide reflections, reflections along the long axis of the strip,
reflections along the narrow axis of the strip, and 180° rotations.
Patterns and Designs
Frieze Pattern
Using the International Union of Crystallography (IUC)
notation, the names of symmetry groups are listed in the table next.
These names all begin with "p" followed by three characters. The
first is "m" if there is a vertical reflection and "1" if it has none. The
second is "m" if there is a horizontal reflection or "g" if there is a
glide reflection, otherwise use "1". The third is "2" if there is a 180°
rotation and "1" if there is none.
Mathematician John H. Conway also created nicknames for each
frieze group that relate to footsteps.
Patterns and Designs
Frieze Pattern
Patterns and Designs
Frieze Pattern
Patterns and Designs
Frieze Pattern
Patterns and Designs
Frieze Pattern
What is the symmetry group of the frieze pattern below?
Patterns and Designs
Tessellations
A tessellation is defined as a pattern of shapes that covers a
plane without any gaps or overlaps.
Tessellations can be found on pavements, patios and wallpapers.
In most cases, tessellations are formed by repeated pattern, however,
some utilize pictures or designs which in no way repeats.
Patterns and Designs
Tessellations
Geometric transformation of polygons such as translation,
reflection and rotation can be used to create patterns. Examples of
tessellation of regular polygons are shown below.
tessellation of squares tessellation of triangles
tessellation of hexagon
Patterns and Designs
Tessellations
The vertex point is the point where the shapes come together.
The sum of all the angles of each shape that come together at vertex
point is 360º. The shapes will overlap if the sum is greater than
360º; otherwise, there will be gaps if the sum is less than 360º.
Naming tessellation can be done by looking at one vertex
point. Looking around a vertex point, start with a shape with the
least number of sides, count the number of sides of each shape at
each vertex point. The name of tessellation then becomes these
numbers. Example in tessellation of triangles, the number of sides is
3 and there are 6 shapes, therefore it can be named as 3, 3, 3, 3, 3, 3.
For the squares, it can be labeled as 4, 4, 4, 4 and for the hexagon,
we can call it 6, 6, 6.
Patterns and Designs
Tessellations
Semi-regular tessellations can be formed using a variety of
regular polygons and the arrangement of these polygons at every
vertex is identical. Some examples of semi-regular tessellations are
follows.
3, 3, 3, 4, 4 3, 3, 3, 4, 4
3, 4, 4, 6
Patterns and Designs
Tessellations
Tessellations can be used to create art, puzzles, patterns and
designs. Some famous mathematicians and artists based their works
on the concept of tessellation. One of them was Maurits Cornelis
Escher who was a Dutch graphic artist who made mathematically
inspired woodcuts, lithographs, and mezzotints.
Hexagonal tessellation with animals:
Study of Regular Division of the Plane
with Reptiles (1939).
Escher reused the design in his 1943
lithograph Reptiles.
Mindanao Designs, Arts and Culture
Mindanao is home of eighteen tribal groups on which weaving is
their identity, culture and way of life. For these indigenous
communities, woven textile conveys their creativity, beliefs and
ideologies and there are some very interesting geometrical themes
imbedded in this art culture.
References
Aufmann et al. (2013). Mathematical excursions, 3rd ed. Belmont: Brooks/Cole, Cengage
Learning
COMAP, Inc. (2013). For all practical purposes: mathematical literacy in today’s world.
NY: W.H. Freeman & Co.
Johnson and Mowry (2012). Mathematics: a practical odyssey, 7th ed. Belmont: Brooks/
Cole, Cengage Learning
http://cnnphilippines.com/life/culture/2017/06/27/mindanao-weaves.html
https://www.wyzant.com
http://www.bbc.co,uk
http://mathforum.org/sum95/suzanne/whattess.html
http://hauteculturefashio.com/tboli-textile-tribe-philippines/
http://www,ethnicgroupsphilippines.com/2017/03/02/the=maranao-and-their-weaving-
tradition/
http://www.choosephilippines.com/specials/buy-local/3759/mindanao-woven-cloth-crafts
https://www.universalclass.com
https://mandayanblood.blogspot.com/2008/08/mandaya-dagmay-weaving.html
http://mathstat.slu.edu/escher/index.php/
https://en.wikipedia.org/
ACTIVITY
Create your own design inspired by some
geometric concepts.
Give meaning to the details of your design and
present it to the class.
References:
Aufmann et al (2013). Mathematical Excursions 3ed. Brooks/Cole ,Cengage
Learning.
Bluman, A. G. (2012). Elementary statistics: a step by step approach 8ed. New
York: McGraw-Hill.
COMAP, Inc. (2013). For all practical purposes: mathematical literacy in
today’s world. New York: W.H Freeman and Company.
Johnson & Mowry (2012). Mathematics: a practical odyssey. Brooks/Cole,
Cengage Learning
Lawsky et al (2014). CK-12 advanced probability and statistics, 2ed. CK-12
Foundation.
Nocon, R. & Nocon, E. (2016). Essential mathematics for the modern world..
QC: C & E Publishing, Inc.
Vistru-Yu, C. and Gozon, A. (2016). Statistics a review ppt. CHED’s GE First
Generation Training.
