Here we start to learn about the intersection of physics, geometry and art.
This intro has been excerpted and adapted from Mathematics and art, Wikipedia.
Artists have used mathematics since the 5th century BCE when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions based on the ratio 1:√2 for the ideal human form.
In the Italian Renaissance, Luca Pacioli wrote the influential treatise De Divina Proportione (1509), illustrated with woodcuts by Leonardo da Vinci, on the use of the golden ratio in art.
Not long after the physics of geometrical optics became a topic of study and use in European art. The Italian painter Piero della Francesca developed Euclid’s ideas on perspective in treatises such as De Prospectiva Pingendi, and in his paintings. The engraver Albrecht Dürer made many references to mathematics in his work Melencolia I.
In modern times, the graphic artist M. C. Escher made intensive use of tessellation and hyperbolic geometry, with the help of the mathematician H. S. M. Coxeter, while the De Stijl movement led by Theo van Doesberg and Piet Mondrian embraced geometrical forms.
The artist David Hockney has argued that artists from the Renaissance onwards made use of the camera lucida to draw precise representations of scenes; the architect Philip Steadman similarly argued that Vermeer used the camera obscura in his distinctively observed paintings. These would be practical uses of geometric optics.
The following is from the course “Geometry in Art and Architecture”, Paul Calter, 1998, Dartmouth College.
Like most discoveries, perspective theory did not emerge out of a vacuum. The underlying ideas had been accumulating for centuries. While the main application of perspective is in art, it is an optical phenomenon and thus has its principal root not in art but in geometrical optics.
Euclid’s Optica, C. 300 B.C., was the first text on geometrical optics, in which are defined the terms visual ray and visual cone.
Vitruvius’ Ten Books on Architecture which appeared about 25 B.C., was the only book on architecture to survive from antiquity. It profoundly influenced Renaissance architecture and thinking, including that of Alberti, who quoted Vitruvius in his Della pittura. Vitruvius wrote,
Perspective is the method of sketching a front with the sides withdrawing into the background, the lines all meeting in the center of a circle.
Ptolemy’s Optica, c. 140 CE, was another early text on geometrical optics, and included theories on refraction. The centric ray is defined by Ptolemy as the ray that does not get refracted.
In his Geographia, c. 140 A.D., Ptolemy applies the principles of geometric optics to the projection of the spherical surface of the earth onto a flat surface, to produce a map. He is said to have made the first known linear perspective construction for drawing a map of the world.
Galen’s De usu partium, c. 175 A.D., contains an early but erroneous description of how the eye creates images. The book was still important, however, as a stepping stone in the development of the theory of perspective.
From Islam, Alhazen’s Perspectiva, c. 1000 A.D., was an important compendium on optics. It integrated the works of Euclid, Ptolemy, and Galen.
Roger Bacon’s Opus Majus, c. 1260 A.D., included a section on optics, whose geometric laws, he maintained, reflected God’s manner of spreading His grace throughout the universe.
John Pecham’s Perspectiva communis, c. 1270 A.D., was another treatise on optics that was widely available during the Renaissance. Finally, Blasius of Parma’s Quaestiones perspectivae, c. 1390 A.D., was a popular adaptation of the works of Bacon and Pecham.
The relationship of culture to rays of light from the sun
Apps & Exhibits
Hockney and Falco argue that Renaissance painters used optical devices
This web site discusses a startling new theory advanced by world-renowned artist David Hockney, working in collaboration with University-of-Arizona physicist Charles Falco, to the effect that, as far back as the 1420s, Master Painters in the European Tradition were employing optical devices to render lifelike images of people and their surroundings. This web site brings together Hockney, Falco, and their principal supporters and skeptics among art historians, art critics, scientists and painters for a full public airing of their views.
Most art historians believe that most European painters, since the Italian Renaissance, employed elaborate systems of mathematical perspective to achieve their effects. Over the past several years, however, Hockney and Falco have been arguing that, on the contrary, artists in the European Tradition, going all the way back to Bruges in the 1420s, were employing a variety of optical devices (including concave mirrors, lenses, the camera obscura and the camera lucida). In effect they suggest that painters (including Van Eyck, Caravaggio, Lotto, Velazquez, Vermeer, Chardin, Ingres, etc.) were using precursors of photographic cameras for centuries before the invention of camera film (chemical fixatives) in 1839; and that it was only with the spread of photography that European painters, suddenly disenchanted with the “optical look,” began to undertake the critique of photography implicit in impressionism, expressionism and cubism and the modernist tradition.
Needless to say, these claims are highly controversial: if true, they would have far-reaching implications for our understanding of art. Public awareness of this new interpretation became widespread with the publication of Hockney’s exposition of his thesis, Secret Knowledge: Rediscovering the Lost Techniques of the Old Masters, [1st edition, 2001; 2nd edition, 2006].
Also see Hockney–Falco thesis (Wikipedia)
We don’t always need perspective!
“Although every human being (of whatever ethnicity) experiences the natural visual illusion of parallel edges—like roadsides or railroad tracks—appearing to converge toward a point as they approach the horizon, it is not natural to reproduce this illusion in pictures. In other words, while everybody sees the same phenomenon in reality, no one, no matter how artistically talented, is innately predisposed to picture it (except, remarkably, certain autistic prodigies). Perspective is a technique that generally must be learned. Therefore there is no reason to believe that nature rather than nurture had anything to do with why artists in other ages and cultures did not pursue the “realism” preferred in the West.”
from Perspective, Other “Perspectives”: JRank Science & Philosophy website, taken from New Dictionary of the History of Ideas, 2005
Artist Scott McCloud explains it this way:
Richard Feynman, a nobel prize winning physicist, on ‘The Beauty of a Flower’:
I have a friend who’s an artist and he’s some times taken a view which I don’t agree with very well. He’ll hold up a flower and say, “look how beautiful it is,” and I’ll agree, I think. And he says, “you see, I as an artist can see how beautiful this is, but you as a scientist, oh, take this all apart and it becomes a dull thing.” And I think he’s kind of nutty.
First of all, the beauty that he sees is available to other people and to me, too, I believe, although I might not be quite as refined aesthetically as he is. But I can appreciate the beauty of a flower.
At the same time, I see much more about the flower that he sees. I could imagine the cells in there, the complicated actions inside which also have a beauty. I mean, it’s not just beauty at this dimension of one centimeter: there is also beauty at a smaller dimension, the inner structure…also the processes.
The fact that the colors in the flower are evolved in order to attract insects to pollinate it is interesting — it means that insects can see the color.
It adds a question — does this aesthetic sense also exist in the lower forms that are…why is it aesthetic, all kinds of interesting questions which a science knowledge only adds to the excitement and mystery and the awe of a flower. It only adds. I don’t understand how it subtracts.
Source: PBS NOVA episode: “Best Mind Since Einstein”
Paintings Too Perfect? The Great Optics Debate, By Sarah Boxer, 12/4/2001, The New York Times
Quotes on art
“Well, Art is Art, isn’t it? Still, on the other hand, water is water. And east is east, and west is west, and if you take cranberries and stew them like applesauce they taste much more like prunes than rhubarb does. Now, uh…now you tell me what you know. ”
– Groucho Marx
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
HS-PS4-1. Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling within various media. Recognize that electromagnetic waves can travel through empty space (without a medium) as compared to mechanical waves that require a medium
HS-PS4-3. Evaluate the claims, evidence, and reasoning behind the idea that electromagnetic radiation can be described by either a wave model or a particle model, and that for some situations involving resonance, interference, diffraction, refraction, or the photoelectric effect, one model is more useful than the other.
HS-PS4-5. Communicate technical information about how some technological devices use the principles of wave behavior and wave interactions with matter to transmit and capture information and energy. Emphasis is on qualitative information and descriptions. Examples of principles of wave behavior include resonance, photoelectric effect, and constructive and destructive interference.
SAT subject test in Physics: Waves and optics
• General wave properties, such as wave speed, frequency, wavelength, superposition, standing wave diffraction, and Doppler effect
• Reflection and refraction, such as Snell’s law and changes in wavelength and speed
• Ray optics, such as image formation using pinholes, mirrors, and lenses
• Physical optics, such as single-slit diffraction, double-slit interference, polarization, and color