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

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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