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Inovácia obsahu a metód vzdelávania prispôsobená potrebám vedomostnej spoločnosti Golden cut in mathematics, nature Inovácia obsahu a metód vzdelávania prispôsobená potrebám vedomostnej spoločnosti Golden cut in mathematics, nature and visual arts VYPRACOVAL: Mgr. Lucia Findoráková máj 2015 „Moderné vzdelávanie pre vedomostnú spoločnosť/Projekt je spolufinancovaný zo zdrojov EÚ“

TABLE OF CONTENTS • • 1. MATHEMATICS 2. NATURE 3. ARCHITECTURE Ancient Egypt 4. TABLE OF CONTENTS • • 1. MATHEMATICS 2. NATURE 3. ARCHITECTURE Ancient Egypt 4. ARCHITECTURE: gothic period in France ARCHITECTURE of the 20 th century 6. VISUAL ARTS: DRAWING AND PAINTING 7. PHOTOGRAPHY 8. CORPORATE IDENTITY

1. MATHEMATICS • Golden ratio or golden section or golden cut is number approximately 1. MATHEMATICS • Golden ratio or golden section or golden cut is number approximately equal to 1, 618. . . • The ratio of the golden cut can be seen at triangles, rectangles and other geometric forms. • Golden ratio can be counted according to length of sides

 • Golden ratio or golden section or golden cut is number approximately equal • Golden ratio or golden section or golden cut is number approximately equal to 1, 618. . . • The ratio of the golden cut can be seen at triangles, rectangles and other geometric forms. • Golden ratio can be counted according to length of sides • a+b is to a as a is to b

Famous mathematicians • Golden section was firstly discussed by ancient mathematitians, such as Euclid. Famous mathematicians • Golden section was firstly discussed by ancient mathematitians, such as Euclid. • Luca Paioli defined golden cut as a divine proportion in his Divina proportione. • The first known decimal approximation of golden cut as golden fraction was published by Michael Maestlin in 1597.

2. NATURE Golden cut observable in nature • Nature amazes us everyday. It offers 2. NATURE Golden cut observable in nature • Nature amazes us everyday. It offers wide range of golden cut samples to see. Some of them can be observed only with microscope but there are many beautiful structures visible tu human eyes. Take a look at sunflower seeds.

Fibonacci spirals in nature The first photograph of vivid colours depicts an ovary. It Fibonacci spirals in nature The first photograph of vivid colours depicts an ovary. It seems unbelievable , but golden cut in spiral form can be observed in nature, too (picture below - hurricane Irene).

3. ARCHITECTURE Ancient Egypt The Great pyramid, also known as the Cheops pyramid, was 3. ARCHITECTURE Ancient Egypt The Great pyramid, also known as the Cheops pyramid, was built in 2580 – 2560 BC according to golden cut. It had 146, 5 m height (nowadays it is just 138, 8 m) and its base was 230, 4 m wide. It was the tallest man-bade structure in the world for very long time – over 3800 years. It is not considered a bulding, because it was not -habitable.

4. ARCHITECTURE: gothic period in France On the right: Notre-dame of Laon Cathedral is 4. ARCHITECTURE: gothic period in France On the right: Notre-dame of Laon Cathedral is an example of beauty that was counted mathematically according to the divine proporction. Chartes Cathedral or Notre Dame in Paris are other cathedrals proportionally build according to golden cut.

5. ARCHITECTURE of the 20 th century Plan of Farnsworth house, in Plano, Illinois 5. ARCHITECTURE of the 20 th century Plan of Farnsworth house, in Plano, Illinois (USA) designed by Ludwig Mies van der Rohe followes golden ratio. Among other architects inspired to build in divine proportions was for example Le Corbusier. Golden cut influences interiors, too. Furniture proportions are adjusted to human proportions.

Farnsworth house, in Plano, Illinois (USA) Ludwig Mies van der Rohe: Farnsworth house This Farnsworth house, in Plano, Illinois (USA) Ludwig Mies van der Rohe: Farnsworth house This summer house was designed after van der Rohe met wealthy Dr. Edith Farnsworth, at the dinner party in 1945. Dr. Edith Farnsworth was interested in contemporary art and supported the idea of simple concept.

6. VISUAL ARTS: DRAWINGS AND PAINTINGS Vitruvian man, famous 15 th century drawing by 6. VISUAL ARTS: DRAWINGS AND PAINTINGS Vitruvian man, famous 15 th century drawing by Leonardo da Vinci, is a drwaing displayed to public only occassionally. This precious drawing following text, called also Canon of proportions, is baseed on Book III, part of De architectura is a book written by Roman architect Vitruvius. He observed human body proportions and counted ideal proportions of human body.

A philosoprer Heinrich Agrippa experimented with depiction of human proportions in pentagram. In 16 A philosoprer Heinrich Agrippa experimented with depiction of human proportions in pentagram. In 16 th century.

Salvador Dalí: The Sacrament of the Last Supper Golden cut influenced the placement of Salvador Dalí: The Sacrament of the Last Supper Golden cut influenced the placement of main protagonists. Also the polyhedron placed in the background consists of 12 regular pentagons and has fundamental golden connections. 2, 67 m x 1, 67 m, 1955, oil

7. PHOTOGRAPHY There are many compositional rules to help you select and adjust a 7. PHOTOGRAPHY There are many compositional rules to help you select and adjust a composition. Golden cut enables wide range of possibilities. Placing the main subject, focal point, in the golden cut intersections, turns a photographic composition to interesting image aesthetically pleasing to its viewer.

8. CORPORATE IDENTITY As the divine proportion is pleasing to human eyes, many organizations, 8. CORPORATE IDENTITY As the divine proportion is pleasing to human eyes, many organizations, institutions, corporations , brands or products use corporate logos constructed upon phi φ.

SOURCES • • • http: //britton. disted. camosun. bc. ca/goldslide/gold 08. jpg http: //constellationcontemplation. SOURCES • • • http: //britton. disted. camosun. bc. ca/goldslide/gold 08. jpg http: //constellationcontemplation. blogspot. sk/2011/04/yod-eling-to-firey-breakthrough. html http: //en. wikipedia. org/wiki/List_of_works_designed_with_the_golden_ratio http: //en. wikipedia. org/wiki/Vitruvian_Man http: //io 9. com/5985588/15 -uncanny-examples-of-the-golden-ratio-in-nature http: //www. patheos. com/blogs/billykangas/2014/11/exploring-dalis-sacrament-of-the-lastsupper. html http: //wp. production. patheos. com/blogs/billykangas/files/2014/11/dali-last-supper. jpg http: //tinyrottenpeanuts. com/golden-ratio-for-kids/ http: //www. livescience. com/37704 -phi-golden-ratio. html http: //www. webdesignstuff. co. uk/nl 004/golden-ratio/application-examples-in-moderndesign/