These are pages I worked on last year from old books of W.H.J. Boot, Eugène Chevreul, Faber Birren, Arthur Guptill, John Ruskin and one called ‘the sketcher’s manual’ by Frank Howard which gives a few pointers on how to arrange lights and darks in a composition. You can read them for free at Hathitrust.org or Archive.org
I realize I should have grouped the pages by subject but….. for what its worth…:
These two pages are following Steve Huston’s lectures, you can find them on youtube here: Guptill: Ruskin: Random page:
During the pythagorean time, a mysterious new type of number, now called the irrational number was concluded to exist from the famous Pythagorean theorem which states that
The sum of the squares of the sides of a right-angle triangle is equal to the square of the hypotenuse (the side opposite the right angle).
Now, if we take the two smaller sides in a right-angled triangle to be 3 and 4 units in length, then the remaining side would be 5 units in length because 5*5 = 4*4 + 3*3. So far, so good; but what if the shorter sides were 1 unit each in length? The larger side would have to be x where x can be written as x*x = 1+1 = 2. But what can be the value of x which when multiplied by itself would give 2? It must lie somewhere between 1 and 2, so that means it should be a fraction. But it turns out that there is no fraction that when multiplied by itself gives 2. It was finally calculated to be a value running forever without any regular pattern: 1.41421356…….. and written as squareroot of 2 and is an example of an irrational number. A rational number can also run forever but it has a repeating pattern to it (for e.g 1/3 = 0.3333…. ) That these type of numbers could exist which are not fractions of whole numbers was so shocking that the truth was suppressed at the beginning. Now we know that irrational numbers are the norm rather than the exception. Between every two consecutive integers and two fractions there can exist an infinite number of irrational numbers.
The root of 2 may not be a very special irrational number but there is another irrational number that is, and which is considered almost magical – phi (1.616033988749894…………) and has become the most popular number to be used and studied in biology, art, architecture and music since its discovery. The golden ratio or golden or divine proportion as it is called is formed when a point B is chosen on a line AC such that the distances AC:AB = AB:AC (fig a). Phi can be calculated from the arithmetic mean of 1 and the square root of 5. It can also be obtained from the Fibonacci series (the integer obtained from adding the two previous integers before it with the first two integers being 1 and 1) The Fibonacci sequence produces numbers that, when divided, forever keep getting closer and closer to the value of phi.
A golden rectangle is a rectangle which has a length over width ratio equal to the golden ratio. A golden spiral can be drawn inside this retangle by dividing it into squares and drawing a quarter circle with radius equal to the side of each square. (An approximate golden spiral can also be made from the numbers in the Fibonacci series) This spiral is a special type of what is called a logarithmic or equiangular spiral.
The logarithmic spiral has a property of self-similarity meaning that the shape of the spiral remains the same for each successive curve. It keeps getting bigger and bigger but its shape remains unaltered. Logarithmic spirals are found very often in nature – in snail and sea shells, in the arms of galaxies, pinecones, fruits and vegetables like broccoli and pineapple, in the way a hawk approaches its prey etc. The cross-section of a nautilus sea shell forms one of the finest, most beautiful natural example of a logarithmic spiral. The inside of its shell has an iridescent mother-of-pearl coating. The animal living inside the shell needs the shell to grow along with it in order to keep accomodating it. As it grows it moves into a larger area and seals off the vacated chamber. As it always places new shell material at the open edge and faster at one side than the other, it makes the shell grow in a spiral manner. (If the shell was circular in shape, the poor creature would not be able to grow) Each type of animal has a different rate of deposition and just slight changes in these rates result in a stunning array of beautiful shell shapes. Horns of animals make a two-dimensional version of the same concept. They would grow straight if the rate of material which is added at the base is equal on both sides (like in nails) but if one side grows faster than the other, the horn starts to coil to the side which has less material added. Below are some varieties of sea shells:
In the heads of flowers, the most beautiful and well known example being that of the sunflower, phi is used in seed organisation. The heads consist of small florets which later mature into seeds. The seeds are produced at the centre and migrate outward to eventually fill out all the space. Each seed is produced at an angle to the previous one. Let us suppose that the seeds are produced for every half turn, that is 180 degrees. (fig a) This will cause the pattern of seeds to be in a straight line which is clearly not a very efficient way of filling the space since there are a lot of gaps. The same is the case for one quarter of a turn (fig b) or for any decimal fraction(fig c) for that matter. Fig d shows the seeds being produced at an angle of pi (3.14..) which is an irrational number. Now there appears a spiral pattern but it is still not an optimal filling. For that to happen, the rotation has to be the most irrational number possible and that is phi. The corresponding angle, that is the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden section. This is approximately equal to 137.51 degrees. You can see the resulting pattern corresponding to this golden angle in the daisy head photo. It appears as two sets of crisscrossing spirals one going left and the other going right and the number of these spirals in each direction will be two consecutive numbers in the Fibonocci series. i.e. they can be 5 and 8 or 8 and 13 or 34 and 55 etc depending on the size of the flower head. Also, since the petals are formed at the extremities of the spirals the number of petals in a large number of flowers are also Fibonocci numbers. What is simply amazing is that a flower can make such a precise measurement of angle, even a very slight change of 1/10th of an angle can disrupt the optimal filling! In the same way, a plant uses this angle in leaf growth to ensure that all leaves receive the maximum amount of sunlight and dew (fig e). Leonardo da Vinci was the first to describe this type of leaf arrangement in plants.
Phi is also found in the human body, like in the ratio of the forearm to the hand, in the spiral of the ear, in the DNA molecule… In fact, the occurence of phi is so common in nature that any form that is constructed based on these proportions is generally believed to be aesthetically pleasing too. For e.g in art, a canvas in the proportions of the golden ratio, can be sectioned again in this ratio lengthwise to get the horizon line. I`ve used them in this drawing of the human face; the length of the face:width of the face, the distance from the eyes to the chin:distance of the eyes to mouth, distance of eyes to mouth: distance of eyes to nose.
The Vitruvian man, a drawing made by Leonardo da Vinci to illustrate a book entitled `De divinia proportione` (The divine proportion) by the mathematical innovator Luca Pacioli has become a universal symbol of human perfection and the integration of mind,body and spirit. The name Vitruvian comes from the Roman architect Marcus Vitruvius(ca. 70-25 BC) who was the source of many of Pacioli`s ideas. Leonardo di Ser Piero da Vinci was a multitalented genius from Italy who lived in the 15th century. He was a painter, sculptor, architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist and writer and had an unquenchable curiosity to understand the universe. He described his anatomical works as a cosmografica del minor mondo, a cosmography of the microcosm and believed that an anology could be drawn from the workings of the universe to the workings of the human body. The Vitruvian man is the first of its kind to bring together artistic and scientific objectives in the proportions of the (male) human body. Each part is a simple fraction of the whole. The body is inscribed in both a circle (symbolising the spiritual existence) and a square (symbolising the material existence) with the centres of the two being different. The combination of the two different arm positions and two different leg positions creates a very dynamic composition of sixteen different poses.
The proportions as written by him in the accompanying text in mirror writing are as follows:
a palm is the width of four fingers
a foot is the width of four palms (i.e., 12 inches)
a cubit is the width of six palms
a pace is four cubits
a man’s height is four cubits (and thus 24 palms)
the length of a man’s outspread arms (arm span) is equal to his height
the distance from the hairline to the bottom of the chin is one-tenth of a man’s height
the distance from the top of the head to the bottom of the chin is one-eighth of a man’s height
the distance from the bottom of the neck to the hairline is one-sixth of a man’s height
the maximum width of the shoulders is a quarter of a man’s height
the distance from the middle of the chest to the top of the head is a quarter of a man’s height
the distance from the elbow to the tip of the hand is a quarter of a man’s height
the distance from the elbow to the armpit is one-eighth of a man’s height
the length of the hand is one-tenth of a man’s height
the distance from the bottom of the chin to the nose is one-third of the length of the head
the distance from the hairline to the eyebrows is one-third of the length of the face
the length of the ear is one-third of the length of the face
the length of a man’s foot is one-sixth of his height
The original drawing of da Vinci`s can be seen here:
After a long weekend, I was on the way to pick up my son from his kindergarten and I had a fragrant surprise waiting for me……. the tulips, daffodils, snowdrops and primroses were all gone and instead were sweet smelling lilacs, pansies, peonies……everywhere. Its amazingly beautiful and utterly fascinating how the landscape changes so suddenly. It can be ahem.. dangerous too if your`re anything like me in memorizing directions – left at the pink rhododendron bush, right at the yellow sunflowers …. Once we had shifted to a new house and summer was nearing its end. I went out for groceries not noticing while going that the yellow rose shrub at the corner had completely stopped flowering and while coming back I had lost my way! No worries, now I look to the buildings for directions not plants which I know will cheat on me! Well, anyway, coming back to the lilacs, I took this photo while it was still raining. The peonies are so huge they were drooping very heavily to the ground in the rain so that snap is from when they had picked up a little but not completely dry. I worked out a composition from these two and another one of sparrows. These cute little birds are very common here hopping their way about searching for food. It doesn`t matter if your sketch done on location or photo doesn`t have all the details. You can fill these out from other sources like the net or field books.
I`m still working on the drawing but some tips for a drawing like this – the stems are the directional elements so place these accurately first, you can shift them about if you feel it helps the composition, follow with roughly placing the flowers and foreground leaves. You can use a measuring scale or a grid. Now that you know these are accurately placed, you can take your time at filling in the details, maybe some bg elements (these can also be done later at the painting stage so that the drawing doesn`t look too confusing and you don`t have to find your place all the time) Also in a maze of leaves like these, it would help to sit for a longer period of time rather than short sessions because again you will end up having to keep finding your place in the pattern. Some of the lilacs are blurred and this is good, it adds interest to the painting if there is a combination of detailed ones and blurred ones; hard edged and soft edged. You can leave out some leaves and stems, choose the ones which add to the realistic effect and leave out ones which complicate the pattern unnecessarily. This is easier said than done for me. I think to myself that the highlight on this leaf is so pretty, the fall of this leaf and this stem is so graceful and so on and so forth that I usually end up having most everything…. But you can choose to have it as loose as you want to. For me, it is a pleasure to lose myself in this world of nature and forget for the time being the daily battles in life; and the longer that time can be the better. I have transferred the main elements from another sheet of paper to the watercolor paper and am drawing out the rest freehand.
For the peonies, the color choice was simple, the orange red is almost the exact hue of M Graham`s quin red (PR209). Another red which I chose for them, one which is warmer is PV 19, Schminke`s ruby red and Maimeri blu`s rose lake are almost the same hue (is the ruby red a tad cooler?) and though one is listed as semi transparent and the other semi opaque, they have the same level of transparency. For the lilacs, I`m going to change the color from this purple in the photo to a lovely burgundy shade that I had seen some time back (the first I had seen on lilacs). Burgundy is the tertiary shade (shade is a color mixed with black, tint is a color mixed with white and tone is a color mixed with grey) of violet-red. (tertiary color is a color made by mixing a primary with a secondary).Using this color ensures that there is harmony when the eye moves from the orange-red, red of the peonies to the red-violet of the lilacs instead of the jump there would be from the orange-red of the peonies to the violet of the original lilacs. (that said, I also feel that any riot of colors would work for flowers). Winsor and Newton perylene maroon (PR179) mixed with Winsor and Newton quin magenta (PR122) is the color I was looking for. Perylene maroon can also be used as the shadows for the peonies which would tie the two flowers together. Coming to the blue, I had my heart set on Winsor and Newton`s dumont`s blue for the sky which leans towards violet, the next spoke in the color wheel instead of a pthalo blue which leans towards green and which would have been a jump. But Dumont`s blue which has some red in it does not make clear greens, so I have opted for Daniel Smith`s ultramarine blue and new gamboge.