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Human Proportion II.2

The geometry and algebra of the child: 2

A mathematical formula for child proportion

The one proportion which appears to be common to the adult and the child is the proportion of the trunk to the total height: in simple whole number fractions, about 3/8.

However, at birth the head is about a quarter of the total height, whereas in Leonardo's ideal man the head is an eighth of the total height.

child proportions

Anthropometry of the child: towards a canon of proportion (continued)

It follows that in Leonardo's adult male the head plus the trunk makes up 1/8 + 3/8 = 1/2 of the total height, leaving 1/2 for the lower limbs. In the baby, the head plus the trunk make up 1/4 + 3/8 = 5/8, leaving only 3/8 for the lower limbs.

In the example I have drawn here, the head is a little less than 1/4 and the lower limbs a little more than 3/8, corresponding to a typical putto of about 3 years old rather than a newborn baby.

Thus, by keeping the trunk at 3/8, making the head smaller, and correspondingly lengthening the lower limbs, we can make our child look older as much as we please. We may consider whether there is a formula allowing us to calculate the proportions of ages in between.

VitruviusWhy have such a formula? One may point out that real people's body proportions vary enormously, and that the representation of the ideal has fallen out of fashion. This is true. However, in learning a standard one has an instrument by which to measure the reality. For example, it is a common error in drawing a face to make the forehead not deep enough. If one has a formula that says that the ridge of the orbit just above the eyes lies about half way between the chin and the top of the head, one can measure ones drawing to see if it is way out on this account. One would also measure the person sitting in front of one, to see to what extent their proportions vary from the expected ones.

Some artists prefer to rely on measurement only, but this too can be inexact and misleading. Why not use an additional tool that may be of some service?

Assumptions:

  1. We take the proportions of an adult's head as 1/8 of the height (that is, the height is 8 heads), and of a newborn baby's head as 1/4 of the height (the height is 4 heads).
  2. For artistic purposes, we may say that the change in proportion of head size to total height is approximately linear with age (that is, the relation can be represented on a straight line graph).
  3. A child achieves their adult height at around the age of 18 years.

Skip notes and go straight to the picture and the algebra.

Notes to the assumptions:

  1. Traditionally a height of 7 heads was taken for an adult mortal and 8 was the proportion used by the Greeks to represent the gods. However I have used 8 because this produces a formula for children which fits with modern children's measurements. I have evidence when I get it together to organise it.
  2. The total actual height does not vary with age in a regular (linear) fashion, as a glance at paediatric growth charts will confirm. There is rapid growth from 0 to 2 years, and another growth spurt in adolescence (ages 11 to 17 approximately). We are concerned here, however, with the relative proportions of parts rather than with absolute sizes. If the rate of growth of all parts increased similarly during periods of rapid growth, the change in head size to total height would still be a linear relation. To understand this, make a mark one third the way along an elastic band, and then stretch the band any amount short of breaking. The mark will still be one third of the way along. However, most of the additional growth in the adolescent period is in the trunk and limbs, and the increase in head size is less dramatic. This means that there will be a deviation from a linear relation between head size and total height, and for the adolescent period this deviation will be maximum at around the age of 15 years. In effect, a 15 year old will have a smaller head in relation to total height (or to put it another way, more 'heads' in the total height) than a linear formula will predict. However the discrepancy will be small inrelation to natural variation, and as this is an artistic formula I shall keep it simple for the sake of keeping it useful.

An algrebraic formula for child proportion

child proportions By assumption ii, we may now feed these data into a standard linear equation, as follows:

Let H be the total height in number of heads, and
let a be the age in years.

The general formula for any linear equation is:

y = mx + c

where m is the gradient and c is a constant.

Our y is the height in heads H, and our x is the age of the child a. Thus our general equation becomes: H = m.a + c

To find c we set a=0, to obtain:

H = m.0 + c

that is:

c = H when a = 0

Now we already said that the height H of a baby aged 0 years is 4 heads (assumption i), so we get:

c = 4

(Note that c is a constant that applies whatever the values taken by the variables: for the technically minded, it represents the intercept on the y-axis.)

Now we substitute this value for c back into the equation, this time for the case of the 18-year-old. So we have a=18 and H=8 (by assumption iii above), thus:

8 = m.18 + 4

Using the usual Islamic methods (so who do you think invented algebra?) we rearrange to get:

m = (8 - 4) / 18

which simplifies to:

m = 2 / 9

Substituting this back into our original equation: H = m.a + c

We get:

H = 2a/9 + 4

So for example, if the age of a child a is 9, the child will be H= 2x9/9+4= 6 heads in height.

Once again applying the cunning of Muhammad ibn Musa al Khwarizmi, we may rearrange the equation so as to be able to estimate the age of a child in a painting. We count the number of heads H that would make up the total height, and substitute into the formula to get the age a:

a = 9(H - 4)/2

Quite neat.


to Human Proportion I
to Child Proportion 1 the constancy of the trunk measurement and its relation to the divine ratio.
to Child Proportion 3: application of the formula to calculating the age of the Infanta Margarita Teresa in one of Velasquez's portraits of her, and constructing a convincing nude study.
to Margarita Teresa: work in progress on a study for a copy of Velásquez's painting.

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