Form and Function in Art

© Tassili-n-Ajjer, Nigeria, Section of rock-wall painting, ca. 5000-2000 BCE

“…things are beautiful when they work. Art is function.”

Giannina Braschi

© Nebamun Hunting in the Marshes, c. 1350 BCE

“One of the functions of art is to offer a more desirable reality – a model, as it were, of another style of existence with its own pace and its own cultural reference.” 

Peter Schmidt

© The Aldobrandini Wedding, c. 27 BCE – 14 BCE

Artists are concerned primarily with three functions.

1. Complete visual perception, internal, external, spatial, front, back and sides of an object.

© Masaccio, The Tribute money, c. 1425-28

2. Movement and the capturing of movement. From the the earliest cave paintings to the present day, artists have at various times studied the problem of showing movement on a two dimensional surface. Movement has always been there and it has been analysed in various fashions without really solving it.

Antoine Watteau, The Embarkation for Cythera, 1717 CE

3. An analysis of the whole, and the inter-relationships of the parts.

© Hokusai,The Underwave off Kanagawa, 1829-1833

“The painter’s function, generally speaking, is to explore and demonstrate in their work the interdependency of forms.” 

Robert McBryde

© Giacomo Balla, Dynamism Of A Dog On Leash, 1912

“Trying to capture the beauty of movement is a lofty and elusive endeavour…” 

Nisla

Roy Lichtenstein, Whaam!, 1963

Click here to read and view more.

https://goffjamesart.wordpress.com

Image Credit © Tassili-n-Ajjer, Nigeria, Section of rock-wall painting, ca. 5000-2000 B.C.E.

Source Credit https://www.artsy.net/artwork/tassili-n-ajjer-nigeria-section-of-rock-wall-painting

Image Credit © Nebamun Hunting in the Marshes, c. 1350 BCE

Source Credit https://www.britishmuseum.org/visiting/galleries/ancient_egypt/room_61_tomb-chapel_nebamun.aspx

Image Credit © The Aldobrandini Wedding, c. 27 BCE – 14 BCE

Source Credit https://en.wikipedia.org/wiki/Aldobrandini_Wedding#/media/File:Aldobrandini_wedding.JPG

Image Credit © Masaccio, The Tribute money, c. 1425-28

Source Credit https://theculturetrip.com/europe/italy/articles/10-painters-of-the-italian-renaissance-you-should-know/

Image Credit © Antoine Watteau, The Embarkation for Cythera, 1717 CE

Source Credit https://www.wikiart.org/en/antoine-watteau/not_detected_229435-1717

Image Credit © Hokusai,The Underwave off Kanagawa, 1829-1833

Source Credit https://artsandculture.google.com/asset/de-grote-golf-bij-kanagawa/JgEdIdn4pME3Mw?hl=en-GB

Image Credit © Giacomo Balla, Dynamism Of A Dog On Leash, 1912

Source Credit https://www.artsology.com/motion_in_art.php

Out of Egypt and Babylon, ‘The Prisoner’s Tale’ – Mathematics in Art – Part 4

© Fig. 1, Architecture Babylon, (Artist and Date Unstated)

“A thought is an idea in transit.” 

Pythagoras

© The Pyramids, Egypt, c.2630 BCE

Previously I have offered brief discussions with regard to the role that Phi (Φ) supposedly plays in relation to form and the understanding of artistic creativity.

© Fig. 2, Artist Unstated, (Title Unstated) The Parthenon, (Date Unstated)

From earliest times both artists and architects have used the formula (π / Phi)f or their paintings and buildings.

Fig.3, Statue of Zeus at Olympia, Entwurff einer historischen Architectur by Johann Bernhard Fischer von Erlach; engravings by Johann Adam Delsenbach (Leipzig, 1725)

“Geometry is the right foundation of all painting.”

Albrecht Dürer

© Raphael, The School of Athens, 1509 -11

Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. The division of a line into “extreme and mean ratio” (the golden section) is important in the geometry of regular pentagrams and pentagons. 

Fig.4 Golden Ratio of the Pentagram

The Greeks usually attributed discovery of this concept to Pythagoras (560-480 BC) or his followers. One of the Pythagoreans’ main symbols was the Pythagorean Pentacle (above), a pentacle inscribed within a pentagon. 

Fig 5, Heinrich Agrippa, Pentagram overlaid over human body, (Date Unstated)

Pythagoras espoused the Pythagorean Theorem (a squared + b squared = c squared) and venerated the pentacle and the golden ratio. He viewed the pentacle as a symbol of mathematical and natural perfection. 

Heinrich Agrippa, Children of the Sun, Italian manuscript “De Sphaera”, c. 1460 CE

“There is geometry in the humming of the strings, there is music in the spacing of the spheres.”

 Pythagoras

© Leonardo da Vinci possibly/Andrea del Verrocchio, c. 1472–1475

“Mighty is geometry; joined with art, resistless.”

Euripides

To read previous articles click below –

Mathematics in Art – Lines and shapes or Phi (in the sky)- Part 1.

Reaching for the Sky – Mathematics in Art – Part 2

Marble the Medium of Dreamers and Idealists – Mathematics in Art – Part 3

Click here to read more articles.

https://goffjamesart.wordpress.com

Image Credit © Fig.1, Architecture Babylon, (Artist and Date Unstated)

Source Credit https://prezi.com/bf-zgr4spvq-/babylon-architecture/

Image Credit © The Pyramids, Egypt, c.2630 BCE

Source Credit https://www.readersdigest.ca/travel/world/10-architectural-wonders-ancient-world/

Image Credit © Fig. 2, Artist Unstated, (Title Unstated) The Parthenon, (Date Unstated)

Source Credit https://seetheworld.travelforkids.com/wp-content/uploads/2016/02/leon-von-klenze-drawing-acropolis-715.jpg

Image Credit © Fig.3, Statue of Zeus at Olympia, Entwurff einer historischen Architectur by Johann Bernhard Fischer von Erlach; engravings by Johann Adam Delsenbach (Leipzig, 1725)

Source Credit https://www.britannica.com/topic/Statue-of-Zeus/media/1/656808/187144

Image Credit © Raphael, The School of Athens, 1509 -11

SourceCredithttps://en.wikipedia.org/wiki/The_School_of_Athens#/media/File:%22The_School_of_Athens%22_by_Raffaello_Sanzio_da_Urbino.jpg

Image Credit © Fig.4, Golden Ratio of the Pentagram, (Artist Unstated)

Source Credit https://www.pinterest.co.uk/pin/78883430952112310

Image Credit © Fig 5, Pentagram overlaid over human body, Heinrich Agrippa, (Date Unstated)

Source Credit https://willyoctora.wordpress.com/tag/pentagram/

Image Credit © Heinrich Agrippa, Children of the Sun, Italian manuscript “De Sphaera”, c. 1460 CE

Source Credit https://willyoctora.wordpress.com/tag/pentagram/

Image Credit © Leonardo da Vinci possibly/Andrea del Verrocchio, c. 1472–1475

Source Credit https://upload.wikimedia.org/wikipedia/commons/b/b0/Annunciation_%28Leonardo%29_%28cropped%29.jpg

Marble the Medium of Dreamers and Idealists – Mathematics in Art – Part 3

© Fig. 1 Artist Unstated, (Title Unstated)The Parthenon, (Date Unstated)

“It is fit that the Past should be dark; though the darkness is not so much a quality of the past as of tradition. It is not a distance of time, but a distance of relation, which makes thus dusky its memorials.

What is near to the heart of this generation is fair and bright still. Greece lies outspread fair and sunshiny in floods of light, for there is the sun and daylight in her literature and art. Homer does not allow us to forget that the sun shone,–nor Phidias, nor the Parthenon.”

Henry David Thoreau 

Fig 2 Parthenon Sculptures, (Details Unstated)

Previously I have offered brief discussions with regard to the role that Phi (Φ) supposedly plays in relation to form and the understanding of artistic creativity.

Fig. 3 Parthenon

The Golden Ratio (Φ) also known as the Golden Section (Latin: sectio aurea) or Golden Mean.

Fig. 4, Alexandros, Venus de Milo, ‘The statue adheres, intentionally or not, strictly to the Phi, Golden Ratio or Golden Proportion of 1.6180339887.’

Other names include extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, golden number, and mean of Phidias.

Fig. 4, Phidias, Heracles, c. 500 BCE

Phidias, or The Great Pheidias (in Ancient Greece Φειδίας; c. 480 – 430 BC), regarded as one of the greatest of all sculptors of Classical Greece, painter and architect, who lived in the 5th century BCE..  His Statue of Zeus at Olympia was one of the Seven Wonders of the Ancient World. 

Fig.5, Statue of Zeus at Olympia, Entwurff einer historischen Architectur by Johann Bernhard Fischer von Erlach; engravings by Johann Adam Delsenbach (Leipzig, 1725)

To view video click https://www.britannica.com/topic/Statue-of-Zeus/media/1/656808/191975

Although it appears that no original works in existence can be attributed to him with certainty, numerous Roman copies in varying degrees of supposed fidelity are known to exist.

Fig. 6, The Varvakeion Athena reflects the type of the restored Athena Parthenos: Roman period, 2nd century CE

Phidias designed the statues of the goddess Athena on the Athenian Acropolis, namely the Athena Parthenos inside the Parthenon and the Athena Promachos a colossal bronze statue of Athena which stood between it and the Propylaea,a monumental gateway that served as the entrance to the Acropolis in Athens. 

Fig.7, Horsemen, detail of a frieze from the Parthenon at Athens, British Museum, London

These marbles, the works of the dreamers and idealists of old, live on, leading and pointing to good. They are the works of visionaries and dreamers, but they are realizations of soul, the representations of the ideal.

They are grand, beautiful, and true, and they speak with a voice that echoes through the ages. Governments have changed; empires have fallen; nations have passed away; but these mute marbles remain–the oracles of time, the perfection of art.

Herman Melville

Click here to read more.

https://goffjamesart.wordpress.com

Image Credit © Fig. 1 Artist Unstated, (Title Unstated)The Parthenon, (Date Unstated)

Source Credithttps://seetheworld.travelforkids.com/the-parthenon-athens-london-ancient-greece/

Image Credit © Fig 2 Parthenon Sculptures, (Details Unstated)

Source Credithttp://www.aboutbritain.com/articles/elgin-marbles.asp

Image Credit © Fig. 3 Parthenon

Source Credithttps://idfive.com/ideas/instantly-improve-brand-designs-golden-ratio/

Image Credit © Fig. 4, Alexandros, Venus de Milo,

Source Credithttps://canukeepup.wordpress.com/2009/07/17/the-golden-ratio-in-greek-art-architecture/

Image Credit © Fig. 4, Phidias, Heracles, c. 500 BCE

Source Credithttps://www.britannica.com/biography/Phidias

Image Credit © Fig.5, Statue of Zeus at Olympia, Entwurff einer historischen Architectur by Johann Bernhard Fischer von Erlach; engravings by Johann Adam Delsenbach (Leipzig, 1725)

Source Credithttps://www.britannica.com/topic/Statue-of-Zeus/media/1/656808/187144

Image Credit © Fig. 6, The Varvakeion Athena reflects the type of the restored Athena Parthenos: Roman period, 2nd century CE

SourceCredithttps://en.wikipedia.org/wiki/Athena_Parthenos#/media/File:NAMA_Ath%C3%A9na_Varvakeion.jpg

Image Credit © Fig.7, Horsemen, detail of a frieze from the Parthenon at Athens, British Museum, London

Source Credithttps://www.britannica.com/biography/Phidias/media/1/455782/99887

Reaching for the Sky – Mathematics in Art – Part 2

Megalithic Temple, Malta, c. 3000 -700 BCE

“Since ancient times artists and architects have seen in the golden mean the most aesthetically satisfying geometric ratio.” 

 Stephen M. Barr

The Pyramids, Egypt, c.2630 BCE

“Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.”

Johannes Kepler

The Parthenon, Greece, c. 438 BCE, Image © THINKSTOCK

Previously I offered a brief overview of the role that Phi (Φ) supposedly plays in relation to form and the understanding of creating images. 

Al Khazneh, Petra, Jordan, c. 100 ACE

Frequently some commentators imply that following such a set of mathematical rules is all that is necessary to imbibe the creative process with harmonious elements and can be applied in all circumstances and disciplines.

Leshan Giant Buddha,China, c.713-803 ACE, Image © THINKSTOCK

“The good is always beautiful, and the beautiful never lacks proportion.”

Plato

Sacsaywaman, Peru, c.900-1200 ACE

Many, whether they are artists, architects, musicians or in other creative fields have endeavoured to apply such a concept to their philosophy and thinking as an integral element in their attempts to create a form that is both “beautiful” and harmonious. 

Borobudur, Indonesia, c. 825 ACE, Image © THINKSTOCK

If the ancient architects deliberately included specific mathematical ratios and shapes in their structures it may have had nothing to do with aesthetics, but with a desire to achieve some mystical harmony with nature rather than an aesthetic goal. 

Machu Picchu, Peru, c. 1450 ACE

“[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written.”

Galileo Galilei

Taj Mahal, India, 1632

“The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.”

Albert Einstein

Sagrada Família, Barcelona, Spain, 1882 – to date

Click here to read and view more.

https://goffjamesart.wordpress.com

Image Credit © Megalithic Temple, Malta, c. 3000 -700 BCE

Source Credithttps://www.readersdigest.ca/travel/world/10-architectural-wonders-ancient-world/

Image Credit © The Pyramids, Egypt, c.2630 BCE

Source Credithttps://www.readersdigest.ca/travel/world/10-architectural-wonders-ancient-world/

Image Credit The Parthenon, Greece, c. 438 BCE, Image © THINKSTOCK

Source Credithttps://www.readersdigest.ca/travel/world/10-architectural-wonders-ancient-world/

Image Credit © Al Khazneh, Petra, Jordan, c. 100 ACE

Source Credithttps://www.readersdigest.ca/travel/world/10-architectural-wonders-ancient-world/

Image Credit © Leshan Giant Buddha,China, c.713-803 ACE, Image © THINKSTOCK

Source Credithttps://www.readersdigest.ca/travel/world/10-architectural-wonders-ancient-world/

Image Credit © Sacsaywaman, Peru, c.900-1200 ACE

Source Credithttps://www.readersdigest.ca/travel/world/10-architectural-wonders-ancient-world/

Image Credit © Machu Picchu, Peru, c. 1450 ACE

Source Credithttp://www.arthistoryarchive.com/arthistory/architecture/images/Machu-Picchu.jpg

Image Credit © Taj Mahal, India, 1632

Source Credithttps://ancientworldwonders.com/uploads/posts/Taj_Mahal/3_Taj_Mahal.jpg

Image Credit © Sagrada Família, Barcelona, Spain, 1882 – to date

Source Credithttps://en.wikipedia.org/wiki/Sagrada_Fam%C3%ADlia

Mathematics in Art – Lines and shapes or Phi (in the sky)- Part 1.

© Leonardo Da Vinci, The Last Supper, 1494-99

“Without mathematics there is no art,”

Luca Pacioli

© Leonardo da Vinci possibly/Andrea del Verrocchio, c. 1472–1475

Humans observe and experience all the various shapes within our environment by recognizing all of their constructive elements and qualities, engrafting additional impressions and information until an entire concept of any object’s character is fairly complete.

© Sandro Botticelli, The Birth of Venus, 1483-85,

As humans survey the environment which surround them there is the capability to frequently compare the ratio of one shape with ratio of another. For example, it can be argued that if one compared the two parts of a divided line or two sides of a satisfying rectangle one would discover that the longer is approximately 1.618 times as long as the shorter. In other words the ratio between is 1.618:1. 

© Michelangelo, The Creation of Adam, 1508-12

Whether one is observing the latest technological invention or the familiarity of pet creature, a comparison of height against width informs one about a part of the the object’s character. It can be expressed in terms of ratio and geometrical lines and shapes.

© Raphael, The School of Athens, 1509 -11

“The negative root of the quadratic equation for φ (the “conjugate root”) is The absolute value of this quantity (≈ 0.618) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, b/a), and is sometimes referred to as the golden ratio conjugate. It is denoted here by the capital Phi (Φ):

Alternatively, Φ can be expressed as This illustrates the unique property of the golden ratio among positive numbers, that or its inverse:

This means 0.61803…:1 = 1:1.61803….”

Eric W Weisstein,-“Golden Ratio Conjuagate”

“Since ancient times artists and architects have seen in the golden mean the most aesthetically satisfying geometric ratio.”


Stephen M. Barr

© Leonardo da Vinci, Mona Lisa, c. 1503– 07

Click here to read and view more.

https://goffjamesart.wordpress.com

Image Credit © Leonardo Da Vinci, The Last Supper, 1494-99

Source Credithttps://news.artnet.com/art-world/golden-ratio-in-art-328435

Image Credit © Leonardo da Vinci possibly/Andrea del Verrocchio, c. 1472–1475

Source Credithttps://en.wikipedia.org/wiki/Annunciation_(Leonardo)#/media/File:Annunciation_(Leonardo)_(cropped).jpg

Image Credit © Sandro Botticelli, The Birth of Venus (1483-85)

Source Credithttps://news.artnet.com/art-world/golden-ratio-in-art-328435

Image Credit © Michelangelo, The Creation of Adam, 1508-12

Source Credithttps://news.artnet.com/art-world/golden-ratio-in-art-328435

Image Credit © © Raphael, The School of Athens, 1509 -11

SourceCredithttps://en.wikipedia.org/wiki/The_School_of_Athens#/media/File:%22The_School_of_Athens%22_by_Raffaello_Sanzio_da_Urbino.jpg

Image Credit © Leonardo da Vinci, Mona Lisa, c. 1503– 07

SourceCredithttps://en.wikipedia.org/wiki/Mona_Lisa#/media/File:Mona_Lisa,_by_Leonardo_da_Vinci,_from_C2RMF_retouched.jpg

Form as an extension of creation and content in Art; and, that which lies behind it. What the hell is that then?

“Form is a revelation of essence.” 

Meister Johann Eckhart

Man from Brno,
CzechRepublic,
c.30,000 – 25,000 BCE

“Creation is only the projection into form of that which already exists.” 

The Bhagavad Gita

Stag Hunt, floor mosaic, Pella, Greece, c. 300 BCE

Form within Art, in essence, is delicate and sensitive; and, in so being provides vitality to the mode, context, content and particular needs of the artist. If form remains so and critically obeys the criteria undertaken an astonishing realism and expressive freedom can be captured. 

Bodhisattva Padmapani (detail), Cave 1, Ajanta, India, c.700 ACE

Form can be mathematically precise if its primary goal and virtues are to encapsulate simplicity, balanced design, outline clarity and clear organized compositional arrangement.

 Dish, East Persia, c. 1000 ACE

Imagine for a moment that one is trekking over a vast expanse of desert terrain which expands outward and away from one and stretching as far as one can see. In the far distance an object emerges and appears at the edge of the horizon. 

It is dusk and in the enveloping evening darkness one is unable to identify what the image is, observe its true outline shape, colour, its textured surface or from which material it might be constructed from.

Since there are no other objects within this particular sparse environment, such as trees or people and one doesn’t know how far away it is; one is unable to make judgements with regard to its size. 

Henry Moore, Animal Head, 1951

However, in spite of this, a part of the object’s elemental character will have impressed itself into the psyche and one will have been enabled to make comparisons with regard to its height relative to its width.

As one approaches the the object and increasingly recognizes all of its constructive elements and qualities, one can engraft additional impressions and information until the entire concept of the object’s character is fairly complete. 

However one’s original impression – comparison of height with width – will have conveyed an important part of the object’s character.

Pieter Brueghel The Elder,The Harvesters, 1565 ACE

Artists whether they be Realists or Abstractionists create artistic forms that embrace or challenge individual viewer’s experiences and expectations.

Jackson Pollock, autumn Rhythm (Number 30), 1950 ACE

“Painting around an object will help one look at ones subjects in a new way – as shapes and spaces…”

Mary Todd Beam

Paul Klee, Sunset, 1930

“When the whole and the parts are seen at once, as mutually producing and explaining each other, as unity in multeity, there results shapeliness.” 

Samuel Taylor Coleridge

Arshile Gorky, Betrothal 1, 1947

Click here to read and view more.

https://goffjamesart.wordpress.com

Image Credit © Man from Brno, Czech Republic, c.30,000 – 25,000 BCE

Source Credithttps://linipin1486.files.wordpress.com/2015/07/screen-shot-2015-07-07-at-20-11-58.png

Image Credit © Stag Hunt, floor mosaic, Pella, Greece, c. 300 BCE

Source Credithttps://en.wikipedia.org/wiki/Stag_Hunt_Mosaic#/media/File:Deer_hunt_mosaic_from_Pella.jpg

Image Credit © Bodhisattva Padmapani (detail), Cave 1, Ajanta, India, c.700 ACE

Source Credithttps://www.khanacademy.org/humanities/art-asia/south-asia/buddhist-art2/a/the-caves-of-ajanta

Image Credit ©  Dish, East Persia, c. 1000 ACE

SourceCredithttps://en.wikipedia.org/wiki/Islamic_art#/media/File:Iran_orientale_o_asia_centrale,_coppa,_x_sec_02.JPG

Image Credit © Henry Moore, Animal Head, 1951

Source Credithttps://www.tate.org.uk/art/artworks/moore-animal-head-t02271

Image Credit © Pieter Brueghel The Elder,The Harvesters, 1565 ACE

SourceCredithttps://en.wikipedia.org/wiki/The_Harvesters_(painting)#/media/File:Pieter_Bruegel_the_Elder-_The_Harvesters_-_Google_Art_Project.jpg

Image Credit © Paul Klee, Sunset, 1930

Source Credithttps://www.artic.edu/artworks/61608/sunset

Image Credit © Arshile Gorky, Betrothal 1, 1947

Source Credithttps://www.moca.org/collection/work/betrothal-i-2