Art and Mathematics?

[stlukesguild]

I know that we have a number of members with backgrounds in Mathematics and Science. Many would dismiss the notion of the relationship or connection between something as Subjective as Art and the more concrete Objective fields of Science and Mathematics. I know, for example, that I virtually hated Math when I was in grade school... and dreaded having to take a Math course as part of the requirements for my teaching license. "Mathematics for the Liberal Arts" was the name of the course I took which was taken by many students in fields of Education such as Visual Art, Music, Literature, History, etc... And yet... over the years I have increasingly employed Mathematics (Geometry, Linear Perspective, Tessellations, etc...) while still having an ambivalent relationship with Mathematics. Among my favorite Artistic cultures are the Italian Renaissance and the Art of Persia/Islam. Both of these embraced the relationship between Art and Mathematics. A pair of videos on YouTube by a poster who goes under the name, Amor Sciendi (roughly Love of Knowledge or Understanding) greatly intrigued me. I came upon the first of these videos a good number of years ago. Both explore Botticelli's Birth of Venus and his use of the Mathematic formula of Phi Φ or the "golden ratio".

I'll make no claims to use of Mathematics that in any way approaches what Amor Sciendi has posited can be found in the painting of Botticelli... but I have made a good deal of Mathematics in my paintings... especially for someone who admits an aversion to Mathematics.

My question then, is how many here employ a good degree of Mathematics in the creation of their Art? Do share this use with the rest of us.

 

[Hausamann]

Most of my paintings have been 1220 x750mm. 1220 / 750 = 1.626666

 

[E.J.H.]

When I think of art and mathematics I am immediately thinking of Escher.
(I was born near his home city btw)

 

[Hausamann]

My first book I studied on this subject was 'Sacred Geometry: Philosophy and Practice'. It had a great influence on me.

 

[stlukesguild]

My last 2 paintings have been 54" x 32.5" which comes to a ratio of 1.661538... which is not far from the "Golden Rectangle". My older full standing figure paintings all measure roughly 80" x 46-48" which comes to a ratio of 1.739... to 1.66666... Both are again close to the "Golden Ratio"... but I suspect that my choice of this ratio has more to do with the proportions of the human body than it does with any conscious effort to work within the "Golden Ratio". At least this was true of the earlier works. In the later works when I chose to work with the dimensions of 80"x48" I know I was more conscious of attempting to employ something closer to the "Golden Rectangle"

 

[stlukesguild]

I can't say I have consciously employed Phi φ often in my paintings, but I do use a good deal of Mathematics and measurements for the compositional structure, the patterns and tessellations, and even the placement of the figures.

 

[Hermes2020]

I guess it is surprising, given my degrees in the sciences, that I don't really use mathematics consciously in the stuff I make. My approach is to eyeball proportions and to use what whatever looks right to me. It would be interesting to analyze some of my better products to see whether there are hidden golden ratios to be found, but the activation energy hump to do that is too high. In other words, I am too lazy to do it!

 

[brianvds]

I guess it is surprising, given my degrees in the sciences, that I don't really use mathematics consciously in the stuff I make. My approach is to eyeball proportions and to use what whatever looks right to me. It would be interesting to analyze some of my better products to see whether there are hidden golden ratios to be found, but the activation energy hump to do that is too high. In other words, I am too lazy to do it!

Same with me, but my degree is in life sciences, so not very mathematical anyway. I studied botany for years before I finally realized that what attracted me was the beauty of plants rather than the minutiae of photosynthesis (though that is actually also all very interesting!)

I think most artists use math like a calculator, i.e. we apply rules of perspective or tessellation without necessarily knowing how they were developed or why they work. But even so my interest in it is limited. E.g. I have seen YouTube videos explain how to draw a cube in correct perspective, or how to draw shadows correctly, and much of it strikes me as so convoluted and complicated that it is easier to just eyeball it. That's after all what the Netherlandish masters did, and it worked perfectly well for them.

I have thought of going into Arabic design, simply so I can use it in art classes, but from what I have seen, it won't work for that purpose: the kids I work with is, erm, a bit limited. Can't draw a straight line with a ruler, can't focus for more than thirty seconds at a time - they'll never make it with designs that can take hours to work out.

 

[Artyczar]

I can hardly do division.

 

[Hausamann]

I can hardly do division.

Thank AI for calculators

 

[Hausamann]

When I make a large canvas frame I often use the 3, 4, 5, triangle principle for creating square corners.
For those unfamiliar to this...

Let's say I want to make a frame that is 75cm x 122cm. Assuming that (3) represents 75cm we can work out what (4) is.
73 / 3 = 25
25 x 4 = 100
So (4) represents 100cm. I mark the 12cm length at 100cm.
Then I need to now what (5) is.
25 x 5 = 125
So now I arrange the 75 and 122cm length to make a rough right-angle corner. Then I move and measure between the 75 and 100cm marks (the diagonal) to read 125cm. Then I secure the two lengths at the corner knowing that it is a perfect 90 degree square angle.
I follow this method for all corners.

 

[JStarr]

My family is made up of eyeballers. Measuring is... an extra step which only seems useful if there's some absolute reason for it: Cutting the mat, how deep a rabbet, inlaying the walnut-- my Mother cut patterns out of newspaper by eye and then used them to make clothing; my Father could look at *any* piece of furniture and build it- he was a Master carpenter**. We learned to cook by eyeballing amounts- the only time you really have to measure is when you're making candy- that's far more physics than soup.

At first, I'd whack off a piece of paper howsoever big I thought it needed to be- which meant DH had to make odd-sized frames. I figured out, though, when I found a stash of empty frames at an old barn sale, to measure to the usual suspect size of frames- made it easier on everyone.

But really, that's as far as I go with Fibonacci and Golden Mean and all that; I may eyeball the tic-tac-toe lines to see if I can hit the mark, but I'm not going to let it faze me if I can't. I agree with what @brianvds said, "...and much of it strikes me as so convoluted and complicated that it is easier to just eyeball it. That's after all what the Netherlandish masters did, and it worked perfectly well for them."

Could do worse, I suppose, than to emulate Rembrandt....

**For the older folks here, back in 1984, Reagan gave the Commencement Address at the US Air Force Academy where my Father was Head of the Carpentry Shop; he built the desk Reagan used while at the Academy that day, and he stole two jelly beans out of the bowl on the desk, went back to the shop and made a tiny little display case for them out of walnut and glass. I have it somewhere- likely with his extra carpentry pencils....

 

[Hermes2020]

On rereading my post of the 14th, I think it would be good to clarify that I was only referring to the aesthetic or artistic side of my paintings and sculptures. I rely on eyeballing to decide on where to place elements in paintings and sculptures, but not when it comes to the measurements and technical details in my sculptures. I rely heavily on mathematics to work out the engineering components. In the software I use, mainly Rhino 3D, it is possible to work to fractions of a millimetre; the default tolerance is 0.001 mm. Such high precision is not necessary in my concrete sculptures, so I work more coarsely. I find it is quite good enough to work within 1 mm tolerances.

 

[stlukesguild]

JStarr- ...that's as far as I go with Fibonacci and Golden Mean and all that; I may eyeball the tic-tac-toe lines to see if I can hit the mark, but I'm not going to let it faze me if I can't. I agree with what @brianvds said, "...and much of it strikes me as so convoluted and complicated that it is easier to just eyeball it. That's after all what the Netherlandish masters did, and it worked perfectly well for them."

Could do worse, I suppose, than to emulate Rembrandt....

Arguably, the two biggest innovations of the Renaissance in the visual arts were the development of the Mathematic-based formulae of linear perspective, proportion, and anatomy in Italian Art and that of oil paint in Northern European Art. Combined, they led to the mastery of the illusion of form and space on a flat surface.

I'll assume that Brian was speaking of the early (1400s) Northern European artists when he spoke of those who "eyeballed" linear perspective. Van Eyck might be the best example of this. His famous Arnolfini Wedding...

... appears correct according to the rules of perspective. Van Eyck recognized that the lines of the floor boards, the edge of the bed, the cabinets, the window frame, etc... receded at a diagonal away from him (the viewer). He also picked up solely from observation how objects further away receded in scale. But none of receding lines converge upon a single vanishing point. I forget which Netherlandish artist first employed the mathematic-based linear perspective. Perhaps Gerard David? Certainly, linear perspective was mastered in Northern Europe well before Rembrandt. Dürer, who traveled and studied in Italy (especially under Giovanni Bellini) was quite taken by perspective and proportion. Rembrandt famously exhibited his mastery of perspective with the spiral staircase in his painting of the Philosopher in Meditation.

A spiral staircase in perspective was one of the most torturous assignments I had to complete in Art School.

I would never suggest linera perspectic, Phi, or any other mathematic-based system was the end-all be-all of art, but as the short videos suggested, we are hardwired to recognize patterns and these are yet one more way to organize works of art.

 

[Terri]

View attachment 32207

I know that we have a number of members with backgrounds in Mathematics and Science. Many would dismiss the notion of the relationship or connection between something as Subjective as Art and the more concrete Objective fields of Science and Mathematics. I know, for example, that I virtually hated Math when I was in grade school... and dreaded having to take a Math course as part of the requirements for my teaching license. "Mathematics for the Liberal Arts" was the name of the course I took which was taken by many students in fields of Education such as Visual Art, Music, Literature, History, etc... And yet... over the years I have increasingly employed Mathematics (Geometry, Linear Perspective, Tessellations, etc...) while still having an ambivalent relationship with Mathematics. Among my favorite Artistic cultures are the Italian Renaissance and the Art of Persia/Islam. Both of these embraced the relationship between Art and Mathematics. A pair of videos on YouTube by a poster who goes under the name, Amor Sciendi (roughly Love of Knowledge or Understanding) greatly intrigued me. I came upon the first of these videos a good number of years ago. Both explore Botticelli's Birth of Venus and his use of the Mathematic formula of Phi Φ or the "golden ratio".

I'll make no claims to use of Mathematics that in any way approaches what Amor Sciendi has posited can be found in the painting of Botticelli... but I have made a good deal of Mathematics in my paintings... especially for someone who admits an aversion to Mathematics.

My question then, is how many here employ a good degree of Mathematics in the creation of their Art? Do share this use with the rest of us.

I've watched this video a couple of times before, and have suggested to others to watch it, as well - if only to point out the exquisiteness of approach in certain artworks and, thus, help gain understanding in what we're looking at.

It's interesting to come across this thread, as I recently dug out my copy of Interface: The Painter and the Mask, by Francoise Gilot, due to her recent passing and wanting to refresh my memory. There's a later chapter where she talks about this very subject, has mathematical equations in the text, and talks about how hard she worked in art school on these principles. I was taken aback and felt rather lost by that chapter - looking at it again now, her actual point is that once she eased off a bit, she felt she did better.

I personally don't do much beyond having used a loose grid when going for perspectives or arrange composition. I don't paint on canvas so immediately create small space restrictions by taping the edges of oil paper all the way around.

 

[Bongo]

When I make a large canvas frame I often use the 3, 4, 5, triangle principle for creating square corners.
For those unfamiliar to this...
View attachment 32255
Let's say I want to make a frame that is 75cm x 122cm. Assuming that (3) represents 75cm we can work out what (4) is.
73 / 3 = 25
25 x 4 = 100
So (4) represents 100cm. I mark the 12cm length at 100cm.
Then I need to now what (5) is.
25 x 5 = 125
So now I arrange the 75 and 122cm length to make a rough right-angle corner. Then I move and measure between the 75 and 100cm marks (the diagonal) to read 125cm. Then I secure the two lengths at the corner knowing that it is a perfect 90 degree square angle.
I follow this method for all corners.

why not just use a carpenter's square?

 

[Hausamann]

While waiting for college to start a new year (1974) I spent three months working for an architectural rendering firm. My job was to convert architectual plans of large buildings into a perspective drawing outlays. They had to be more than precise, they also had to look realistic. At the firm I had a great teacher. I learnt so much more about perspective drawing than I ever had when learning Technical drawing and drafting at school.

The objective was to illustrate what the large building would look like when built. It was an important part of getting approval to build.
Everything had to look solid and right, not looking as if the tall building was leaning, especially backwards or forward.

For tall buildings the vanishing points, on either side of the horizon, had to be many feet away. Otherwise the building would look peaky and/or out of proportion. The firm created wooden curves that got clamped along the edge of the drafting table. These curves were a portion of a large cirfumference, say with a 12 foot radius. The T-Square would traverse the curve, simulating a vanishing point 12 feet away.

Even so, a tall building may end up looking as if leaning forward or backward just a tad. This is were false perspectives are needed. The reason being is that the Earth is curved. One may need to raise/lower the vanishing point as one draws the top or bottom of the building, depending from were the observer is. If the building had a helicopter landing port, as one of its important features, the observation point would be above the building instead from the ground. Often the vanishing point got slightly raised/lowered per floor drawn.

When the drawing was finished with the Renderers approval, the Renderer took over to paint it in watercolours. These renderers were a marvelous sight to watch them paint.

So, a large part of perspective is to 'know' how to shift vanishing points and horizons to get the right look. I can honestly tell you that if my technical-drawing teacher saw me shifting the vanishing point, he would have a fit.

 

[brianvds]

I'll assume that Brian was speaking of the early (1400s) Northern European artists when he spoke of those who "eyeballed" linear perspective. Van Eyck might be the best example of this. His famous Arnolfini Wedding...

They were indeed whom I had in mind. I think even Bruegel's perspective is not quite right yet? It took a while for the Italian discoveries to penetrate northwards.

A spiral staircase in perspective was one of the most torturous assignments I had to complete in Art School.

I can imagine - I would frankly have no idea how to do it. Now that everyone uses reference photos for everything, we have become lazy; we simply work from the reference. Problem is of course that the camera can introduce severe perspective distortions, and I have never been able to copy anything correctly anyway.

I think widespread adoption of perspective technique to some extent required availability of relative cheap but large pieces of paper. Can't really have a million constructions lines on the back of an envelope.

 

[stlukesguild]

Hausamann- For tall buildings the vanishing points, on either side of the horizon, had to be many feet away. Otherwise, the building would look peaky and/or out of proportion...

Even so, a tall building may end up looking as if leaning forward or backward just a tad. This is where false perspectives are needed...

a large part of perspective is to 'know' how to shift vanishing points and horizons to get the right look...

The Renaissance and later masters definitely made use of distortions of perspective. Paolo Uccello famously became obsessed with linear one-point perspective to such an extent that Vasari commented on it to a comic effect in his Lives of the Artists. His series of paintings of The Battle of San Romano employ one-point perspective in a manner that certainly looks "wrong" and even silly... as all the soldiers and horses killed in battle and their lances fall horizontal to the picture plane or follow the receding lines toward the vanishing point.

Later artists such as Donatello and Michelangelo used distortions in perspective (for example, in the painted architecture of the Sistine Ceiling) in order to create the illusion of a greater degree of depth and scale and/or to account for the point of view of the audience. The slightly oversized head of Michelangelo's David as well as the deep-cut eyes were intentionally employed to deal with the fact that the sculpture was intended to be seen high up on a building in Florence. Leon Battista Alberti, who codified the "rules" of perspective in the treatise De Pictura, employed an optical illusion in his design for the Palazzo Ruccelai. The scale and roughness of the blocks of the facade of the Palazzo decrease with each floor of the building mirroring and amplifying the optical effect of atmospheric perspective.

Having said all of this, as I noted elsewhere, my own approach to drawing/painting makes a good deal more use of the instinctive or intuitive as opposed to conscious measurement. I will alter something (a color, shape, line, etc...) that I feel is wrong and then sit back and look at it critically and ask myself whether it is now better or worse. There are times, however, when my criticism will be based on measurements of some form... checking patterns and tessellations, or the proportions. I imagine that the Renaissance artists recognized that mathematic structures were a method of lending a logical means of organization to their art. It was also a means of leading the eye to important elements.

 

[stlukesguild]

Brian- Now that everyone uses reference photos for everything, we have become lazy; we simply work from the reference. Problem is of course that the camera can introduce severe perspective distortions, and I have never been able to copy anything correctly anyway.
Photographs also result in distortions of proportion and flatten forms. Degas had a famous quote to the effect of "Drawing is not rendering what you see but what you can make others see." This point was hammered home to me, to a great extent, through copying the drawings of the "old masters". One of my favorite drawings is this study for The Libyan Sibyl by Michelangelo, It made me conscious of the impact of the "weight" of lines and how these "read" visually. The right arm in this drawing gets progressively lighter drawn with lines of a decreasing visual weight. This exaggerates the optical effect of atmospheric perspective.

I think widespread adoption of perspective technique to some extent required availability of relative cheap but large pieces of paper.

Yes, access to quality paper (as well as linen canvas) were major influences on the Renaissance. I think even more than perspective, affordable paper allowed for rapid sketching... gestural drawings that suggested movement. Antonio del Pollaiuolo's engraving The Battle of the Naked Men, was one of the largest and finest prints of the early Renaissnce. The anatomy is masterful... but looks stilted...

... because Pollaiuolo failed to understand physiology... or how the body moved. Leonardo was constantly making rapid sketches in his notebooks, and as a result, his drawings are far more life-like.

We see this in Raphael's and Michelangelo's drawings as well.