Mathematics in the Modern World: Lecture 1 and 2 Flashcards
He mentioned that we live in a universe of patterns.
Ian Stewart
These are the things that are repetitive, which can be found in nature as color, shape, action, or some other sequences that are almost everywhere.
Patterns
It is a way to calculate or solve a problem.
Rule
It expresses patterns.
Mathematics
It is an exact correspondence of form and constituent configuration on opposite sides of a dividing line or plane or about a center or an axis.
Symmetry
It is also called mirror symmetry or line symmetry.
Reflection Symmetry
Reflection Symmetry is also called as, what?
Mirror Symmetry or Line Symmetry
It is also called radial symmetry.
Rotational Symmetry
Rotational Symmetry is also called as, what?
Radial Symmetry
In Biology, this kind of symmetry is exhibited by objects when their similar parts are regularly arranged around a central axis and the pattern looks the same after a certain amount of rotation. Note that if you rotate the given images below by several degrees, you can still
achieve the same appearance as the original position.
Rotation Symmetry
This kind of symmetry is exhibited by objects which do not change its size and shape even if it moved to another location. Note that the movement does not involve with reflection or rotation.
Translational Symmetry
What are the kinds of symmetry?
Reflection Symmetry; Rotational Symmetry; Translational Symmetry
These are never-ending patterns that are self-similar across different scales. The image just reappears over and over again no matter how
many times the object is magnified.
Fractals
Patterns are also exhibited in the external
appearances of animals.
Spots and Stripes
These are curved patterns made by series
Spirals
Flowers are easily considered as things of beauty.
Their vibrant colors and fragrant odors make them
very appealing as gifts or decorations.
Flower Petals
Are easily considered as things of beauty.
Flowers
What are the types of patterns in nature?
Symmetry; Spiral; Fractals; Spots and Stripes; Flower Petals
It is a series of numbers where a
number is found by adding up the two numbers before it.
Fibonacci Sequence
The sequence encountered in the rabbit problem 1,
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, … is
called the _________.
Fibonacci Sequence
The terms in the Fibonacci sequence is called _________.
Fibonacci Numbers
He is also known as Fibonacci
Leonardo of Pisa
It is the perfect rectangle.
Golden Rectangle
The golden ratio was first called as the ___________ in the early
1500s
Divine Proportion
It was first called as the Divine Proportion
Golden Ratio
This contains the drawings of the
five platonic solids and it was probably da Vinci.
De Divina Proportione
The drawings of five platonic solids is called, what in Latin?
Section aurea or Golden Secion
What is the formula of the golden ratio?
The number of petals in a flower is often one of
the following numbers: 3, 5, 8, 13, 21, 34 or 55.
For example, the lily has three petals, buttercups
have five of them, the chicory has 21 of them,
the daisy has often 34 or 55 petals, etc.
Flower Petals
In Fibonacci, written as a rule the expression is…
In both human and nonhuman,
abound with examples of the Golden Ratio. The
mouth and nose are each positioned at golden
sections of the distance between the eyes and the
bottom of the chin. Similar proportions can been
seen
Faces
These are produced at the center,
and then migrate towards the outside to fill all the
space. Sunflowers provide a great example of
these spiraling patterns.
Seed Heads
The Golden Section is manifested in the
structure of the human body. The human body is
based on Phi and the number 5.The number 5
appendages to the torso, in the arms, leg and
head. 5 appendages on each of these, in the
fingers and toes and 5 openings on the face.
Animal bodies exhibit similar tendencies.
Body Parts
Spiraling patterns can be found on
pineapples and cauliflower. Fibonacci
numbers are seen in the branching of trees
or the number of leaves on a floral stem;
numbers like 4 are not. 3’s and 5’s, however,
are abundant in nature.
Fruits, Vegetables, and Trees
These are the most common
galaxy shape. The Milky Way has several
spiral arms, each of them a logarithmic
spiral of about 12 degrees.
Spiral Galaxies
It follow the
logarithmic spiral, as does the cochlea of the inner
ear. It can also be seen in the horns of certain
goats, and the shape of certain spider’s webs.
Shells
It’s amazing how closely the powerful
swirls of ________ match the Fibonacci
sequence.
Hurricanes
The exterior dimension of the
___________ in Athens, Greece
embodies the golden ratio.
Pathernon
Here, Plato describes five
possible regular solids that relate
to the golden ratio which is now
known as Platonic Solids.
Timaeus
He was the first to give definition of the
golden ratio as “a dividing line in the extreme
and mean ratio” in his book the “Elements”.
Euclid
He was into many interests
such as invention, painting, sculpting, architecture,
science, music, mathematics, engineering,
literature, anatomy, geology, botany, writing,
history and cartography. He used the golden ratio
to define the fundamental portions in his works. He
incorporated the golden ratio in his own paintings
such as the Vitruvian Man, The Last Supper,
Monalisa and St. Jerome in the Wilderness.
Leonardo Da Vinci
It is “a dividing line in the extreme
and mean ratio” in his book the “Elements”.
Golden Ratio
He was considered the greatest living artists of his time.
He used golden ratio in his painting “The Creation of Adam” which can be seen on the ceiling of the
Sistine Chapel. His painting used the golden ratio showing how God’s finger and Adam’s finger
meet precisely at the golden ratio point of the weight and the height of the area that contains them.
Michaelangelo
di
Lodovico
Simon
More popularly known as Raphael was also a painter and
architect from the Rennaisance. In his painting “The School of Athens,”, the division between the
figures in the painting and their proportions are distributed using the golden ration. The golden
triangle and pentagram can also be found in Raphael’s painting “Crucifixion”.
Raffaello Sanzio da Urbino
In his work “The Sacrament of the Last Supper”, golden ratio can be found.
Salvador Dali
In his works(“Bathers at Assinieres”, “Bridge of Courbevoie” and “A Sunday on
La Grande Jette”, golden ratio can be found.
George-Pierre Surat
In his work “Birth of Venus”, golden ratio can be found.
Sandro Botticelli
Built 4700 BC in Ahmes Papyrus of Egypt is with proportion according
to a “Golden Ratio”. The length of each side of the base is 756 feet with a
height of 481 feet. The ratio of the base to the height is roughly 1.5717,
which is close to the Golden ratio.
Great Pyramid of Giza
Is a Gothic Cathedral in Paris, which was built in
between 1163 and 1250. It appears to have a golden ratio in a number
of its key proportions of designs.
Notre dame
The _________ in India used the golden ratio in its construction
and was completed in 1648. The order and proportion of the arches of
the _______ on the main structure keep reducing proportionately
following the golden ratio.
Taj Mahal
The _______ in Paris, France also
exhibits the Golden ratio.
Cathedral of Our Lady of Chartres
In the ________, the window configuration reveal
golden proportion
United Nation Building
The __________ in Paris, France, erected in 1889 is an iron lattice.
The base is broader while it narrows down the top, perfectly following
the golden ratio.
Eiffel Tower
The _______ in Toronto, the tallest tower and freestanding
structure in the world, contains the golden ratio in its design. The ratio
of observation deck at 342 meters to the total height of 553.33 is
0.618 or phi, the reciprocal of phi.
CN Tower
The five possible regular solids is now called what?
Platonic Solids
