Mathematics

as Art

TOPICS TO BE COVERED:

•The Platonic solids and polyhedral

•The golden ratio and its applications:

a.Architecture

b.Painting

c. Book Design

• Applications of geometry like :

a. Kaleidoscopes

b. Mazes and labyrinths

c. The fourth dimension and

d. Optical illusions

• Music

The Platonic solids and Polyhedra

The Golden Ratio Then we have the golden ratio.

Looking at the rectangle we just drew,

you can see that there is a simple

formula for it. When one side is 1, the

other side will be:

The square root of 5 is approximately

2.236068, so The Golden Ratio is

approximately (1+2.236068)/2 =

3.236068/2 = 1.618034. This is an

easy way to calculate it when you

need it.

a. Architecture

• The Parthenon's façade as well as elements of its façade and elsewhere are said by some to be circumscribed by golden rectangles. Other scholars deny that the Greeks had any aesthetic association with golden ratio.

• Near-contemporary sources like Vitruvius exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

b.Painting

•The drawing of a man's body in a pentagram suggests relationships to the golden ratio.

•The 16th-century philosopher Heinrich Agrippa drew a man over a pentagram inside a circle, implying a relationship to the golden ratio.[2]

Leonardo da Vinci's illustrations of polyhedra in De divinaproportione (On the Divine Proportion) and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by anything in Leonardo's own writings. Similarly, although the Vitruvian Man is oftenshown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.

Salvador Dalí, influenced by the works of MatilaGhyka, explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.

c. Book design

•Depiction of the proportions in a medievalmanuscript. According to Jan Tschichold:"Page proportion 2:3. Margin proportions1:1:2:3. Text area proportioned in the GoldenSection."

•According to Jan Tschichold, there was a timewhen deviations from the truly beautiful pageproportions 2:3, 1:√3, and the Golden Sectionwere rare. Many books produced between1550 and 1770 show these proportionsexactly, to within half a millimeter.

Mazes and Labyrinths• Many ornamental patterns

are related to topology, forexample mazes. Is there adifference between a maze and alabyrinth? Traditionally, theterms have been considered to besynonymous, but around 1990people interested in the spiritualaspects of labyrinths devised aterminology where a labyrinth isunicursal and a mazemulticursal. This means that alabyrinth has only one path withno branches and no dead ends, inother words, no choice, while amaze is a logical puzzle withbranches and possibly dead ends.

This maze appears in several medieval Hebrew manuscripts.Although this maze has a superficial resemblance to the Cretanmaze, a close comparison shows they are quite different. TheJericho maze has 7 levels, whereas the Cretan maze has 8, andthe sequence in which the levels are reached differs from onemaze to the other.

The best known of these mazes is the Cretan maze:

Jericho Maze:

OPTICAL ILLUSIONS

Geometrical-optical illusions are visual illusions, also optical illusions, in which the geometrical properties of what is seen differ from those of the corresponding objects in the visual field.

Explanations of geometrical-optical illusion are based on one of two modes of attack:

• the physiological or bottom-up, seeking the cause of the deformation in the eye's optical imaging or in signal misrouting during neural processing in the retina or the first stages of the brain, the primary visual cortex, or

• the cognitive or perceptual, which regards the deviation from true size, shape or position as caused by the assignment of a percept to a meaningful but false or inappropriate object class.

KALEIDOSCOPES

Three mirrors, generally from four or five to ten or twelveinches long, and with a width of about an inch when thelength is 6 inches, and increasing in proportion as thelength increases, are put together at an angle of 60degrees.

With one mirror, an object is reflected such that the angle ofincidence equals the angle of reflection.

Combining this with themirror configuration above, acomplex pattern formsbetween the mirrors due tothe perspective of the eyelooking down this long mirrorshaft. The pattern forms witha real piece of glass seen inthe actual opening at theend of the mirror. It is thenreflected as seen below, andthis complex pattern reflectsoutward yielding an illusionof an endless landscape ofthe glass piece. Below is thecenter most subset of thepattern.

FOUR-DIMENSIONAL SPACE

• In mathematics, four-dimensional space ("4D") is a geometric space with four dimensions. It is typically meant to mean four-dimensional Euclidean space, generalizing the rules of three-dimensional Euclidean space. It has been studied by mathematicians and philosophers for over two centuries, both for its own interest and for the insights it offered into mathematics and related fields.

• Comparatively, 4-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled w. To describe the two additional cardinal directions, Charles coined the terms ana and kata, from the Greek words meaning "up toward" and "down from", respectively. A length measured along the w axis can be called spissitude, as coined by Henry More.

PROJECTIONS

• A useful application of dimensional analogy in visualizingthe fourth dimension is in projection. A projection is a way forrepresenting an n-dimensional object in n − 1 dimensions. Forinstance, computer screens are two-dimensional, and all thephotographs of three-dimensional people, places and things arerepresented in two dimensions by projecting the objects onto aflat surface. When this is done, depth is removed and replacedwith indirect information. The retina of the eye is also a two-dimensional array of receptors but the brain is able to perceivethe nature of three-dimensional objects by inference fromindirect information (such as shading, foreshortening, binocularvision, etc.). Artists often use perspective to give an illusion ofthree-dimensional depth to two-dimensional pictures.

Music

First octave

In music theory, the first octave, also called the contra octave, ranges from C1, or about 32.7 Hz, to C2, about 65.4 Hz, in equal temperamentusing A440 tuning. This is the lowest complete octave of most pianos (excepting the Bösendorfer Imperial Grand). The lowest notes of instruments such as double bass, electric bass, extended-range bass clarinet, contrabass clarinet, bassoon, contrabassoon, tuba and sousaphone are part of the first octave.

Ex: Siberian Throat Singers

The ability of vocalists to sing competently in the first octave is rare, even for males. A singer who can reach notes in this range is known as a basso profondo, Italian for "deep bass". A Russian bass can also sing in this range, and the fundamental pitches sung by Tibetan monks and the throat singers of Siberia and Mongolia are in this range.

Mathematics in History

Mathematics in Ancient Egypt

Counting in Early Egypt

In ancient Egypt, texts would be written in Hieroglyphs. Their number system always starts by 10. Their number one came from a simple stroke. The number with two strokes and so on. The numbers 10, 100, 1,000, 10,000 and 1,000,000 have their own hieroglyphs. . Number 10 is a hobble for cattle, number 100 is represented by a coiled rope, the number 1000 is represented by a lotus flower, the number 10,000 is represented by a finger, the number 100,000 is represented by a frog and a million was represented by a god with his hands raised in adoration.

Addition in early Egypt In adding two numbers, just add the

numbers then collect all symbols of

similar type and replace a ten of one

type by one of the next higher symbol.

Example: Add 35 and 17.

add

Subtraction in Early EgyptSubtraction goes with the same but opposite process , For example:

Subtract 17 from 35.

Minus

Multiplication

in Early EgyptIn multiplying two numbers, all you needed to understand was the

double of the number.

For example:

multiply 35 by 11.

1 35

2 70

4 140

8 280

1 +2+8=11 35+70+280=385

Multiplication

in Early Egypt

Division in

Early EgyptIn Division, just do the reversal process of multiplication.

For example: divide 1075 by 25

1 25

2 50

4 100

8 200

16 400

32 800

1+2+8+32= 43 25+50+200+800= 1075

Division in

Early Egypt

SUMMARY•Mathematics in Ancient Egypt is composed of four main operation. Addition, subtraction, multiplication and division which is also used nowadays. The only difference is instead of numbers they use symbols called hieroglyphics/counting glyphs.

Patterns in Real Life

Objectives:

To cite examples of some applications of mathematics in our everyday lives.

To define the meanings of the different mathematical patterns applied to our daily lives.

To prove that mathematics has importance not only in science but in our surroundings as well.

Topics to be Covered: Number Patterns

o Arithmetic Sequence

o Geometric Sequence

oFibonacci Sequence

Fibonacci in Nature

o Fibonacci in Plants

oFibonacci in Animals

oFibonacci in Humans

Number Patterns

o Arithmetic Sequence

An arithmetic sequence has a constant difference between terms. The first term is a1, the common difference is d, and the number of terms is n.

Explicit Formula: an = a1 + (n – 1)d

o Geometric Sequence

A geometric sequence has a constant ratio between terms. The first term is a1, the common ratio is r,

and the number of terms is n.

Explicit Formula:

an = a1rn-1

o Fibonacci Sequence

The sequence of numbers which the next term is found by adding the previous two

terms.

Fibonacci in Nature

The Fibonacci numbers are nature's numbering system. They appear everywhere in nature, from the leaf arrangement in plants, to the

pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The

Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees,

and even all of mankind.

Why do these arrangements occur?

In the case of leaf arrangement, or phyllotaxis, some of the cases may be related to maximizing the space for each leaf, or the average amount of light falling on

each one. Even a tiny advantage would come to dominate, over many generations. In the case of

close-packed leaves in cabbages and succulents the correct arrangement may be crucial for availability of

space.

In the seeming randomness of the natural world, we can find many instances of mathematical order

involving the Fibonacci numbers themselves and the closely related “golden" elements.

Fibonacci in Plants

Phyllotaxis is the study of the ordered position of leaves on a stem. The

leaves on this plant are staggered in a spiral

pattern to permit optimum exposure to

sunlight. If we apply the Golden Ratio to a circle we can see how it is that

this plant exhibits Fibonacci qualities.

By dividing a circle into golden proportions, where the ratio of the arc length are equal to the Golden

Ratio, we find the angle of the arcs to be 137.5 degrees. In fact, this is the angle at which adjacent leaves are

positioned around the stem. This phenomenon is observed in many types of plants.

Inside the fruit of many plants we can observe the presence of Fibonacci order.

The banana has three (3) sections

The apple has five (5) sections

Fibonacci in Animals

The shell of the chambered Nautilus has Golden proportions. It is a

logarithmic spiral.

The eyes, fins and tail of the dolphin fall at golden sections along the

body.

A starfish has 5 arms.

Fibonacci in Humans

It is also worthwhile to mention that we have eight (8) fingers in total, five (5) digits on each hand,

three (3) bones in each finger, two (2) bones in one (1) thumb, and one (1) thumb on each hand.

Fallacies in Logic

What are Fallacies?A fallacy is an argument that uses

poor reasoning. An argument can

be fallacious whether or not its

conclusion is true.

• Semantic Fallacies – are those which

have to do with errors due to ambiguity in

the meaning of words used or errors due

to confusion resulting from incorrect

grammatical construction, or judgment or

reasoning.

Types of Semantic Fallacies

Fallacy of Equivocation –committed when one assumes that

words that have the same spelling or

sound are used with the same

meaning in an argument.

Examples:

• Star is a heavenly body,

but Elizabeth is a star,

therefore, Elizabeth is a heavenly

body

• Light comes from the sun,

but feathers are light,

hence, feathers come from the sun

Fallacy of Composition –

taking ideas or things together

when they should be taken

separately or individually.

Example:

Aliens and those below 18 years are

not eligible to vote,

but I should be allowed to vote

because while I am an alien I am of

voting age.

Fallacy of Division – is the opposite

of fallacy of composition and results

from taking ideas or things

separately when they should be

taken collectively.

Example:

All his merchandise cost P500,

but his ballpoint pen is part of his merchandise

therefore, his ballpoint pen costs P500

•Material Fallacies – have to

do with errors which have to

do with errors which spring

from inattention or abuse of

the subject matter or

content of an argument.

Types of Material Fallacies

Fallacy of Accident – consists in

arguing that what is affirmed or denied

of a thing under accidental condition

can also be affirmed or denied of its

essential nature.

Example:

A person who is drunk is irrational,

but this person is drunk,

therefore this person is irrational.

Fallacy of Confusing the Absolute and

Qualified Statement – is the result of

concluding that a qualified statement is

true because the absolute statement is

true, or that the absolute statement is true

because the qualified statement is true.

Example:

Filipinos are hospitable,

this person is a Filipino,

hence, this person is hospitable