# Human Proportion II.1

## The geometry and algebra of the child: 1

### How does the geometry of the child vary from Leonardo's Vitruvian man?

Here is a diagram to indicate my discoveries about child proportions, which is a work in progress. Some key differences from the proportions of Leonardo's adult man are evident.

### Anthropometry of the child: towards a canon of proportion

We do not know the Greek Classical system of proportion: all we have is the Egyptian canons of proportion on the one hand, and the system of the Roman architect Vitruvius, as developed by Leonardo, on the other. What lies in between is currently conjecture (despite the fact that the evidence is waiting for us in many classical sculptures). I am also trying to generalise Leonardo's results into a formula that will include the proportions of children as well, and these pages show the first steps.

Unlike the case of Leonardo's Vitruvian man, the child's armspan will not be equal to the height. We notice that the head is comparatively large in relation to the total height, and the legs comparatively short. If the arms grow proportionately with the legs then the armspan will be less than the height.

I have developed a mathematical formula of great simplicity, revealed in the pages that follow, which relates the fundamental proportions of the child to age.

The discovery revealed on this page is that, roughly speaking, the trunk measurement as a proportion of total height remains constant with age. Following Leonardo, the measurement is taken from the chin to the pubic bone (Leonardo says to the root of the penis). In Leonardo's diagram this trunk measurement is 3/8 of the total height.

This trunk measurement also happens to be approximately what remains after the total height is divided in proportion to the divine ratio, and the greater part removed. The divine ratio, claimed to have been known to the ancient Egyptians and incorporated into the dimensions of the Great Pyramid, is usually designated, as in the diagram, by the Greek letter phi. It is the solution of the algebraic equation phi=1/(phi-1).

For those who like algebra, the derivation is as follows:

phi is defined as the proportion that results when a line is divided into two unequal parts, such that the ratio between the whole and the larger part is equal to the ratio between the larger and the smaller part. If we call the whole phi, and the part 1, then it follows that phi/1=1/(phi-1), i.e. phi=1/(phi-1) as above.

Applying the method of Al Khorezmi (c800AD) we can obtain the value of phi = (1+SQR[5])/2 = 1.61803.... Al Khorezmi's method is very beautiful.

If we take the height of a child or adult to be 1, then, if we divide this height into two unequal parts in the divine ratio, the larger part will be 1/phi = 0.61803..., and the smaller part will be 1/(phi)^2=0.38196.... This smaller part is the measurement of the trunk as defined above. The decimal expansion of 3/8 is 0.375 - close enough for the purposes of painting.

to Human Proportion I

to Child Proportion 2: derivation of a simple formula linking the height in head sizes to age.

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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