Dpt. de Matemáticas

EL NÚMERO ÁUREO Y
LA COLEGIATA DE SALAS

El número áureo y la colegiata de Salas from ProyectoComenius

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LET’S GO TO LA COLEGIATA!

OUR AIMS: to look for and find the Golden Number in La Colegiata in Salas and collaborate in the Comenius Project.

JUSTIFICATION:  possible similarities in the architectural elements used in the Gothic-Renaissance styles, Mannerism and Neoclassical style which predominated sometime in the XVIII century in agreement with rationalist ideas and the principles of reason.

LIMITATIONS: we have only used tape-measures, pencils and calculators where we had an easy access, always adapting ourselves to the curricular level of 1^stESO.

WHAT WE DO: we want to find the Golden Number en La Colegiata, and so:

-we select among the figures which contain rectangles, those which we think are aesthetically more proportionate.

-we take photographs of the elements we are going to study.

-we number and identify the figures we have chosen.

-we start to measure and take notes.

-we check the existence of the Golden Mean

CONCLUSIONS

-Our investigation has not been very exhaustive but yet, representative.

-The Golden number has mainly been observed in the main altarpiece and in some parts of the Mausoleum.

-There are many other parts which can be studied applying other principles like that of similarity (Thales Theorem), Pythagoras Theorem …

-We can also use other instruments of measure like the laser to obtain greater accuracy.

-The students have become familiar with the Golden Number and have also become aware of the artistic heritage of their village.

-but there is still a lot of work to do …

And our intention is to continue with the study of the Golden Number from both an arithmetic and a geometric point of view, look for it in the buildings around us (The Monastery in Cornellana, the Tower of the Castle, San Martin’s Church…) and use it in our every day life, and it is for sure that we will improve a lot!!!

This project has been carried out by the students of 1^st ESO  (2012 -2013) with the help of their teachers of Mathematics and English (Enedina García and Carolina Suárez)

THE GOLDEN RATIO IN LA COLEGIATA IN SALAS

A presentation on a number which is as widely used as unknown.

WHAT IS THE GOLDEN NUMBER?

It is an algebraic number (since it is the solution to a polynomial equation) and also irrational (because it has got infinite different decimal figures)

WHAT DOES IT MEAN GEOMETRICALLY?

Out of this proportion we get the equation whose solution provides the Golden Ratio.

A LITTLE BIT OF MATHS HISTORY

The Golden Ratio is nowadays represented with the Greek letter “PHI” to acknowledge Phidias, a 5th century B.C. sculptor , painter and architect of ancient Greece (490 - 430 B.C.) who used the Golden Ratio in his works.

The association between this letter and the Golden Ratio did not take place until the 20^th century.

In ancient Greece, it was already considered that that way of dividing a segment was logical and, taking it into account, figures which provided proportion and beauty could be created.

The first person who formally studied the Golden Segment was Euclid (325 – 265 B.C.), a Greek mathematician often referred to as the “Father of Geometry”

Pythagoras (582 – 500 B.C.) and the Pythagorean School had the star pentagon or pentagram as its symbol, although it seems they did not know about the existence of the Golden Ratio, at least in a conscious way, because they only worked with fractions.

FIBONACCI

Leonardo of Pisa most commonly known as Fibonacci (1170 – 1250 A.D), spread the Hindu-Arabic numeral system in Europe.

In 1202, he wrote the Liber Abaci. The book advocated numeration with the digits 0 – 9, and showed the practical importance of the new numeral system by applying it to commercial bookkeeping, conversion of weights and measures, the calculation of interest, money-changing and other applications.

THE ENLIGHTENMENT (end of XVII century – end of XVIII c. or beginning of XIX c.)

Fibonacci’s book was widely used in Europe during the Enlightenment and greatly influenced on the European mathematical thinking since it was based on a logical reasoning.

FIBONACCI’S SEQUENCE

Fibonacci described his sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 …as the solution to an arithmetical problem on the growth of rabbits:

“Suppose a newborn pair of rabbits, one male and one female, is put in the wild. The rabbits mate at the age of one month and at the end of its second month a female can produce another pair of rabbits. Suppose that the rabbits never die and that each female always produces one new pair, with one male and one female, every month from the second month on.  How many pairs will there be in one year?”

ROBERT SIMPSON, SCOTTISH BOTANIST AND MATHEMATICIAN

In 1753, he discovered that the ratios of successive numbers in the Fibonacci sequence, when these tended to infinity, moved towards the Golden Number.

-check doing without the 1^st

-1/1; 2/1; 3/2; 5/3; … 1,618033…

THE GOLDEN NUMBER - APPLICATIONS

-The Fibonacci sequence has many applications in Computer Studies, Mathematics, The Game Theory, Biology …

-The Golden Ratio is found in multiple forms of architecture, painting … and in objects that we very often use as we will see below

THE GOLDEN NUMBER AND THE WORLD OF ARTS

The proportions which follow the Golden Ratio are considered aesthetically pleasing.

Many architects, painters, sculptors … since ancient times have created their works basing them on the golden proportions.

THE PARTHENON in GREECE

THE VITRUVIAN MAN by LEONARDO DA VINCI

THE MONA LISA by LEONARDO DA VINCI

PYTHAGORIAN LEDA by DALI

THE PYRAMIDS in EGYPT

RUPESTRE DE MIRA IN ASIA (IV c. B.C.)

NOTRE DAME in PARIS AND THE GOLDEN RECTANGLES IN THE UNIVERSITY OF SALAMANCA

THE FAMILY of CHARLES IV by GOYA

THE GOLDEN ROSE IN GERMANY

NATURE

Nature hides the Golden Ratio in many of its forms: in sunflower seeds,   in galaxies and the nautilus shell, flowers and fruit.

In the case of human beings since we are a foetus. UNTIL we become a man or a woman aesthetically “perfect”: distance between fingertips and shoulders and elbows, length and width of the head, height and distance between our navel and feet. Beauty standards remain …And even another Simpson (Homer in this case), tries to follow these standards although we must make a special visual effort.

LA COLEGIATA SANTA Mª LA MAYOR IN SALAS

The main building belongs to the first part of the XVI century. The two chapels and the bell tower were added afterwards.

-The main building is from the first half of the XVI century

-The altarpieces are from the XVII century

-The mausoleum is from the late XVI century

THE PROMOTERS

-The Valdés-Salas family as a funerary monument

-D. Fernando Valdés-Salas, a politician and religious man who promoted and left as his last will, the foundation of the University of Oviedo. His name is observed in the name of our secondary school.

-The emblem of the University of Oviedo corresponds to the coat of arms of the Valdés family

MAIN PARTS AND STYLES

-The main part: Gothic and Renaissance styles

-The two side chapels and the bell tower were added afterwards

-The main altarpiece (beginning of the XVII century) belongs to the Mannerism style

-The altarpiece in The Chapel of Malleza (middle XVII century) was created by Luis Fdez de la Vega

-The Mausoleum

THE MAUSOLEUM

-D. Fernando Valdés-Salas funerary monument

-Piece created by Pompeyo Leoni, a reinassance italian artist who worked for Philip II in El Escorial and also sculpted the Mausoleum in Guadalajarabetween 1576 and 1582