painting in layers according to a predefined and rigorous model

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the rule of three

Paint is a mixture that is made up of three types of ingredients: pigments that give the paint its color, binders that keep the pigments in place after drying and solvents that thin the paint, lower its strength and make it flow. Other types of ingredients may exist, such as siccatives or retardants but they are not relevant to the concepts explained here. Working by example, the first thing we will do is measure these three types of ingredients on a weighing scale before we mix them to make a paint.

Say we put 4 mg of mars black, 30 mg of egg tempera and 480 mg of water together, it gives us 514 mg of paint in total.

Feo Mars noir	Egy Egg yolk	H2O Distilled water
4 mg	30 mg	480 mg

In order to find the percentile quantities of this mixture we must do no more than apply the rule of three to each ingredient, where B is the ingredient quantity, A is the total quantity 514 and C is 100. x = (B/A)*C.

This will give us approximately:

• mars black: (4/514)*100 ≈ 0.77%
  
• tempera: (30/514)*100 ≈ 13.33%
  
• water: (480/514)*100 ≈ 85,9%
  

Feo Mars noir	Egy Egg yolk	H2O Distilled water
0.77%	13.33%	85.9%

The rule of three extracts a ratio of B to A, and applies this ratio to a pair of which only one variable is known, C, in order to get x.

properties of interest

So far, this gives us some properties of the paint that we can already work with. We can store the recipe in percentile form and use the rule of three to recreate this exact recipe at a later time in any quantity.

In order to make 650 mg of our recipe, we mix approximately:

• mars black: (0.77/100)*650 ≈ 5 mg
• tempera: (13.33/100)*650 ≈ 87 mg
• water: (85.9/100)*650 ≈ 557 mg

A similar method is applied in the kitchen when making pancakes for 10 people from a base recipe for 5. However, percentile quantities do not tell us all we want to know about a certain mixture when this mixture is to be used underneath or on top of other paint layers. We will now proceed to define a set of properties that tell us all we want to know about a mixture when it is used as an element in an overall model.

power and temper

The most important properties of paint are power and temper. With only these two we would already be able to say a great deal about any mixture we choose to make:

• Power (pow) is the sum of all pigment percentiles
• Temper (tem) is the ratio of the amount of binder to the the sum of all pigments

The notation we will use for this is pow|tem.

To calculate the power property of our previous example, we can simply take the percentile amount of mars black because it's the only pigment in the mixture, so pow = 0.77. To calculate the temper property we must divide the percentile amount of binder by the percentile amount of pigment,so 13.33/0.77 ≈ 17.31.

In the notation we will quote the pigment code before the numbers, so the final notation for our previous example will be Feo: .77|17. With this notation we lose some accuracy, but it will still be accurate enough. We will calculate quantities starting from the values of the properties, and not the other way around as we have done above.

In order to make 650 mg of our so called Feo: .77|17, we mix approximately:

• mars black: (0.77/100)*650 ≈ 5 mg
• tempera: ((0.77*17)/100)*650 ≈ 85 mg
• water: 650-5-85 = 560 mg

When multiple pigments and binders are used in a mixture, we will have to extend on the notation and the calculations quite a bit, but before we get more geeky about paint let's explore some basic concepts we wish to monitor when making all these calculations. Hopefully it will make more sense as to why anyone would even want to go through this trouble in the first place.

rock paper scissors

After defining power and temper I might as well introduce a third property called flow:

• Flow (flow) is the solvent percentile

To return once more to our previous example, the flow of Feo: .77|17 would be 86. Flow obviously gives us an indication of how diluted, watery or non-viscous a mixture is. But notice that we were able to calculate all ingredients that go into the recipe without making use of this property. That is because flow is not a value of choice when the other two properties are already defined. The three properties affect each other in a circular manner:

• When pow increases, tem goes down
• When tem increases, flow goes down
• When flow increases, pow goes down

When we started off using absolute quantities or even percentiles there really was no guideline for what makes a 'good' paint and what not. By defining properties of interest we have turned the mixing of paint into a game of rock paper scissors where any choice you make restricts other choices that can be made. This makes it possible to judge mixtures as bad or good, and to prefer one mixture over another.

fat over lean

Most artists will at some point in their education have heard of the 'fat over lean' rule. But there is often some confusion about what it implies exactly.

For a long time I thought I was painting fat over lean when I would dilute the first layers with a solvent and then use more and more paint directly from the tube as I painted on. This is off course not the case, because the fatness or amount of binder in this process remains proportionally the same. Solvents evaporate so to use more or less of them does not influence the composition of what is left behind.

Fat over lean means that the proportion of binder to pigment goes up as you go up the layers. Cracks in a surface will appear when a more brittle layer is placed on top of a more flexible layer. So to state the fat over lean rule in the reverse way: in order to create stable layers the brittleness of the mixtures must keep on decreasing as you add new layers. How would one define brittleness? Imagine a block of concrete as a mixture of cement and pebbles. If it would have too many pebbles, it would be easier to chip off a piece with a hammer. So the proportion of cement to pebbles would define the brittleness. In our case, the ratio of binder to pigment would define the brittleness. So to conclude, the "tem" property as defined above is not so much a matter of taste as it is a technical restriction, it must have a certain value in order to guarantee stability.

scalability

A system can be considered scalable when there is no need to change the way it is set up when certain key parameters are tweaked. In our case, scalability refers to the "slicing" of the layers in the model. Let us work from another example using vermillion imitation(Ver).

To make 800 mg of Ver: 2.0|8.0, we will mix approximately:

• vermillion: (2.0/100)*800 = 16 mg
• tempera: ((2.0*8.0)/100)*800 = 128 mg
• water: 800-16-128 = 656 mg

Of this, we will apply three layers.

Ver Vermillion imitation 2	Egy Egg yolk	H2O Distilled water
2%	16%	82%

layers: 3

The amount of color left behind by this process might be expressed by multiplying power by the number of layers 2*3 = 6. Which brings us right away to the definition of another property: result.

• Result (res) is power multiplied by the number of layers

In order to scale this process up, we will choose to apply the color in more consecutive layers while keeping the original result value of 6. This will yield, at least in theory, a more refined look without changing how much paint is deposited. So to upscale our process to six layers we will choose a power of 1, while keeping a result of 6, res/n = pow, 6/6 = 1.

To make 800 mg of Ver: 1.0|8.0, we will mix approximately:

• vermillion: (1.0/100)*800 = 8 mg
• tempera: ((1.0*8.0)/100)*800 = 64 mg
• water: 800-8-64 = 728 mg

Of this, we will apply six layers.

Ver Vermillion imitation 2	Egy Egg yolk	H2O Distilled water
1%	8%	91%

layers: 6

Notice that the tem property remains the same, because it is a ratio, only power needs to be recalculated when re-scaling. Similarly, we can choose to downscale the process, yielding a more brutal and direct look. If we choose to apply the result in two layers, the power of our recipe will be 6/2 = 3.

To make 800 mg of Ver: 3.0|8.0, we will mix approximately:

• vermillion: (3.0/100)*800 = 24 mg
• tempera: ((3.0*8.0)/100)*800 = 192 mg
• water: 800-24-192 = 584 mg

Of this, we will apply two layers.

Ver Vermillion imitation 2	Egy Egg yolk	H2O Distilled water
3%	24%	73%

layers: 2

Each of these three processes will in the end have deposited the same amount of colour and binder. The concept of scalability is especially important in projects that take a long time to complete. Often when making a drawing, after only a few hours you will notice that there is nothing more you can add to make it better, only worse. The point of saturation is reached quite quickly, no amount of labour on your part can then add more value. I would call this the Francis-point after my fantastic drawing teacher Filip Francis who stated "When is a drawing finished? Just before you finished it, it was finished -but you didn't see it!".

Painting is extremely well suited to push the Francis-point back endlessly, for months and even years, if only the quantities are carefully monitored throughout. In theory it would be possible to work on a painting for hundreds of years passing it on for generations like a bonsai tree or a dog breed. Scaling-up therefore is a tactic that allows labour to keep being converted into value. That being said, there are restrictions as to how much you can actually dilute a paint before the absolute amount of binder in the mixture becomes too low to perform its job.

b:a and c:a

Now that we have explored some basic concepts we can start working with multiple pigments in a mixture. Apart from whites, we will never mix more than 3 pigments into a recipe. This keeps the colors clear and crisp. We will call these three pigments a, b and c where a is the main pigment to which the others will be defined relatively.

To give the three non-white pigments a place in the system, we will introduce 2 more properties.

• b ratio (b:a) is the ratio of the amount of b pigment to the amount of a pigment
• c ratio (c:a) is the ratio of the amount of c pigment to the amount of a pigment

As in our previous examples we will first make a mixture to see what the value of these properties would be and later work backwards, starting from the properties and then mix by a formula.

Let us start then by putting lion green, sienna natural and burnt umber together:

• 15 mg of lion green: (15/1000)*100 = 1.5%
• 5 mg of sienna natural: (5/1000)*100 = 0.5%
• 20 mg of burnt umber: (20/1000)*100 = 2%
• 200 mg of tempera: (200)/1000)*100 = 20%
• 760 mg of water: (760)/1000)*100 = 76%

Grl Vert lion	Sina Sienna nat	Bu Burnt umber (lefebvre)	Egy Egg yolk	H2O Distilled water
1.5%	0.5%	2%	20%	76%

The power of this mixture is the sum of all pigment percentiles 1.5+0.5+2 = 4, the temper is 20/4 = 5. To capture the relative distribution of pigments within that 4% we must calculate b:a which is 0.5/1.5 ≈ 0.33 and c:a which is 2/1.5 ≈ 1.33. Before we reverse engineer this recipe starting from the properties, let's look at the notation, which should speak for itself Grl-Sina-Bu: 4.0|5.0 b.33 c1.3.

Now let us go ahead and make 2000 mg of Grl-Sina-Bu: 4.0|5.0 b.33 c1.3:

• lion green: ((1/(1+0.33+1.3)*4.0)/100)*2000 ≈ 30 mg
• sienna natural: ((0.33/(1+0.33+1.3)*4.0)/100)*2000 ≈ 10 mg
• burnt umber: ((1.3/(1+0.33+1.3)*4.0)/100)*2000 ≈ 40 mg
• tempera: ((4.0*5.0)/100)*2000 ≈ 400 mg
• water: 2000-30-10-40-400 ≈ 1520 mg

As before I have marked each value in the notation with a color so that you can track where it is used in the formula. To calculate the final quantities we must first get the percentile of each pigment. To do that, we use another rule of three: the sum a:a(which is always 1)+b:a+c:a seen as (1+0.33+1.3) is to each separate pigment as pow (the percentile total) is to each separate pigment's percentile.

luminosity

Luminosity is the final property we need to define in order to have the full notation and calculation. We will first proceed to demonstrate this property with a single base pigment before we introduce b:a and c:a. As we mentioned before, we will never mix more than 3 pigments into a recipe, -apart from whites. White pigments such as titanium white, zinc white or lead white are a class apart because they bring special qualities to the paint. That is why they have their own property which is a ratio that sits on top of a,b and c pigments instead of alongside them.

So let's define this property exactly:

• Luminosity (lum) is the ratio of all white pigments to all other pigments

Again as in our previous examples we will first make a mixture to see what the value of the property would be and later work backwards, starting from the properties and then mix by a formula.

Let's get on and mix some paint containing permanent blue dark and titanium white.

• 40 mg of permanent blue dark: (40/2000)*100 = 2%
• 30 mg of titanium white: (30/2000)*100 = 1.5%
• 560 mg of tempera: (560)/2000)*100 = 28%
• 1370 mg of water: (1370)/2000)*100 = 68.5%

The power is 2+1.5 = 3.5, the temper is 20/2.5 = 8. The luminosity then would be 1.5/2 = 0.75. The notation is Bpf: 3.5|8.0 ∝.75.

Let us go ahead and make 3000 mg of Bpf: 3.5|8.0 ∝.75:

• permanent blue dark: ((1/(1+0.75)*3.5)/100)*3000 = 60 mg
• titanium white: ((0.75/(1+0.75)*3.5)/100)*3000 = 45 mg
• tempera: ((3.5*8.0)/100)*3000 = 840 mg
• water: 3000-60-45-840 = 2055 mg

Bpf Bleu perm fonce	Tit Titanium white	Egy Egg yolk	H2O Distilled water
2%	1.5%	28%	68.5%

Now let's reintroduce b:a and c:a to see the full calculation of all properties in one example.

Let's make 5000 mg Grl-Sina-Lem: 4.0|8.0 ∝.75 b0.5 c1.5:

• lion green: ((1/(1+0.75)*(1/(1+0.5+1.5))*4.0)/100)*5000 ≈ 38 mg
• sienna natural: ((1/(1+0.75)*(0.5/(1+0.5+1.5))*4.0)/100)*5000 ≈ 19 mg
• cadmium yellow lemon: ((1/(1+0.75)*(1.5/(1+0.5+1.5))*4.0)/100)*5000 ≈ 57 mg
• titanium white: ((0.75/(1+0.75)*4.0)/100)*5000 ≈ 86 mg
• tempera: ((4.0*8.0)/100)*5000 = 1600 mg
• water: 5000-38-19-57-86-1600 = 3200 mg

Grl Vert lion	Sina Sienna nat	Lem Cadmium lemon dark	Tit Titanium white	Egy Egg yolk	H2O Distilled water
0.76%	0.38%	1.14%	1.71%	32%	64%

So here we use three separate, nested rules of three to get a,b and two nested rules of three to get the white part. As far as recipe calculation goes, this is as complicated as it will get.

The important point is that by making the lum property dominant to the b:a c:a properties, a color transition towards white can be achieved by tweaking only the lum value and leaving the other properties unaffected. To summarize, let's make a general formula that allows us to make any amount of paint with any the properties that we have defined:

When making a total of pow | tem ∝ lum b b:a c c:a , we mix:

• a = ((1/(1+lum)*(1/(1+b:a+c:a))*pow)/100)*total
• b = ((1/(1+lum)*(b:a/(1+b:a+c:a))*pow)/100)*total
• c = ((1/(1+lum)*(c:a/(1+b:a+c:a))*pow)/100)*total
• white = ((lum/(1+lum)*pow)/100)*total
• binder = ((pow*tem)/100)*total
• solvent = total-a-b-c-white-binder

continuous progressions

Properties like lum and b:a are especially helpful when creating color transitions as you progress through the layers. For example, when transitioning from a darker to a brighter blue, four mixtures might be laid in consecutively where every recipe has a lum value that is higher than the last. Let's take a look:

Bpf Bleu perm fonce	Tit Titanium white	Egy Egg yolk	H2O Distilled water
0.05%	3.6%	34%	62.35%

fourth layer   - Bpf: 3.7|9.3 ∝72


Bpf Bleu perm fonce	Tit Titanium white	Egy Egg yolk	H2O Distilled water
0.11%	3.73%	33.4%	62.79%

third layer    - Bpf: 3.8|8.7 ∝34


Bpf Bleu perm fonce	Tit Titanium white	Egy Egg yolk	H2O Distilled water
0.36%	3.5%	32.6%	63.54%

second layer   - Bpf: 3.8|8.4 ∝9.7


Bpf Bleu perm fonce	Tit Titanium white	Egy Egg yolk	H2O Distilled water
1.3%	2.73%	32%	63.97%

first layer    - Bpf: 4.0|7.9 ∝2.1

The layers are always presented chronologically from bottom to top. The first thing to notice is that the power remains roughly the same. The temper will increase slightly, it climbs from 7.9 to around 9.3. The luminosity however climbs exponentially:

• first layer: 2.73/1.3 ≈ 2
• second layer: 3.49/0.36 ≈ 10
• third layer: 3.73/0.11 ≈ 34
• fourth layer: 3.59/0.05 ≈ 72

When the luminosity value transitions are mapped on a graph, we get this curve:

On this graph we can see that in order to transition gradually to a whiter value, the proportional amount of white pigment must rise slowly in a bent curve. If the values were to rise linearly, the ratios in the middle would be too white to make for a smooth transition:

What makes a good transition is partially a matter of preference but most of the time, useful transitions will not be linear. Now let's take it one step further: it is possible to map out a transition beforehand by defining an algebraic function ƒ(x) = xⁿ and then use a snippet of that function from x = 0 to x = 100 to define a value transition.

This has a number of advantages. Firstly, while working on a transition you can stop at any point and pick up right where you left off weeks or months later. Tempera paint goes bad in just a few days, and having a stored transition makes it easy to work at your own pace instead of needing to worry about this. What is especially interesting about this approach is that when using a function there is an infinite amount of transitions at your disposal, depending on how many points on your curve you wish to call in.

All points on a curve defined by a function follow each other continuously and there is in theory always a new recipe to be found in between any two recipes. This makes the continuous approach highly scalable. In order to tune your function to have a certain curvature, parameters around x have to be added to change and bend the shape. Let's take the simplest quadratic function ƒ(x) = x². By adding a parameter we can push the curve up or down: ƒ(x) = (x²)+3500.

We can make the curve flatter or steeper by adding another parameter: ƒ(x) = (1.5*x²)+3500.

The full set of parameters is:

• move up-dow (mud): ƒ(x) = (x²)+mud
• move left-right (mlr): ƒ(x) = (x+mlr)²
• steepness (steep): ƒ(x) = (steep*x)²
• zoom (zoom): ƒ(x) = (x²)*zoom
• raise (raise): ƒ(x) = x^raise

Which gives us in an overall formula:

• ƒ(x) = (((steep*(x+mlr))^raise)*zoom)+mud

A linear function can be made by setting the raise parameter to 1. If you want to see for yourself how this works, try the graph plotter. All properties of interest can be bound to a function and then scaled to preference, the formulas always remain the same. In this way it is possible to define an entire painting as a set of continuous recipes before even starting.

to conclude

It was my aim to give you some appreciation for how the work is made and to make you able to understand the information that is given in the toggle tree under each painting. The math applied here is very basic and the theory was not developed as a concept behind the work but as a solution to practical problems I encountered across the years in my many failures.

In practice there are many additional challenges that need to be handled. Mixing paint is an artform in and of itself. Keeping tiny amounts of dust and dirt out of your paints and paintings is also a constant challenge. When it comes to painting, there is plenty of literature out there that covers this topic in great detail. Daniel V Thompson's books 'The practice of tempera painting' and 'Materials and techniques of medieval painting' are two fantastic references to start out with. Another interesting book is 'Materials for a History of Oil Painting' by Charles Lock Eastlake.