1:7 and a series of 8

1:7
Order of size: a series of 8 measures
Authentic and derived order of size
Make your own Abacus
Make your own Form-Bank

I 1:7

Through its presence, architecture communicates. Its massive walls are intensely connected to the space they enclose.

In order to achieve this, Van der Laan asks himself a second question: what is the maximum difference between two sizes so that we can still relate them?

The answer is 1:7.

The thickness of the wall is 1:7 to the space it encloses.

This is called NEARNESS.

II Order of size: a series of 8 measures

With the ratios 3:4 and 1:7, Van der Laan makes a series of eight measures that he calls the order of size or measure-system.
All the measures are related by ca. 3:4.
The smallest and the largest measure relate as 1:7.

Van der Laan’s proportional series is unique, as it combines whole numbers into a hierarchical series.

The key is in the rounding off. In order to give these measures countable values, Dom van der Laan translates them arithmetically into the whole numbers 3, 4 and 7. So the relationship defining the series 1, 4/3, 7/4, … is a practical approximation of 1,324718…

Like the keys of a piano, you can use the 8 measures to make compositions.

The sum of the first measure (the yardstick or unit of the series) and the second measure is the fourth measure in the series.

The sixth measure is the sum of the fifth and a unit.

III

The eighth measure is twice the fifth measure plus the unit.

The difference between the sixth and the fifth measure is the unit.

The sum of the fifth and the sixth measure is the eighth measure.

Thus, the Plastic Number reconciles counting and measuring.

Dom Van der Laan names the 8 measures as major and minor elements, pieces, parts and wholes.

III Authentic and derived order of size

In order to refine the possibilities when designing, Van der Laan interwove a second series with the first. As such, there is an authentic and a derived measure-system. Each authentic measure is coupled with a smaller derived measure that is referred to as its derivate.

Every derived measure is 6/7 of its authentic measure.

A derived measure is the double of an authentic measure.

An authentic measure exceeds its derivative with the same authentic measure of the underlying order of size. In a system of 8 measures the difference between the largest and its derivative is thus equal to the smallest.

A derived measure lies in the harmonic mean of two authentic measures.

a1:z1 = a2:z2 = 3:4

The harmonic mean is the middle term between two others that exceeds the one term and is exceeded by the other by the same proportion of each term.

An authentic measure lies in the arithmetic mean of two derived measures.

x1 = x2

The arithmetic mean is the middle term between two others that exceeds the one term by as much as it is itself exceeded by the other.

IV Make your own Abacus

Dom Hans van der Laan considered the act of making essential for gaining an understanding about how to create equilibrium through the placement of various parts. Studying the plastic number and the design method therefore went hand in hand with the making of an abacus and a morphotheque or form-bank. They made it possible to practice with and perfect different proportions. These concrete tools were used during the design process to test various options.

The first abacus dates from 1952.

The abacus consists of 38 bars 12 mm wide and 7 mm high. Two orders of size are represented: the order from 3.5 mm up to 24.5 mm and the order from 24.5 mm up to 175.5 mm. To make it all fit, 2.5 mm from the lower order is also added, as well as the derived measure of 4 mm.

All measures in a building are linked to each other based on measure-systems. Synchronising all the sizes of the parts of a building is what Van der Laan called symmetry. The term is therefore not used here in the modern sense of reflection symmetry, which refers to the opposite identity of two halves. He meant symmetry in the sense of the study of proportions, the ratios between similar measurements in the same direction, such as two lengths or two heights.

Superposition: the whole is compared to one of its parts.

2 authentic measures are compared.

Images from VDL AS IX, 13.

Superposition: the whole is compared to one of its parts.

An authentic measure and a derived measure are compared.

III

Superposition: symmetrical relations.

Juxtaposition: two parts are compared to each other; the whole has disappeared.

V Make your own Form-Bank

Eurhythmy is the study of the form, of the proportions between the three dimensions, in which the length is compared to the breadth, the breadth with height, and the height with length, within a single form.

In 1976, the abacus was supplemented by the minor form-bank. The latter consists of 36 authentic forms with the eight measures of the system:

– 10 blocks

– 10 bars

– 10 slabs

– 6 white forms

‘The forms making up the total gamut are therefore determined in three stages. First, three primary groups, blocks, bars and slabs, are separated out from a central group. Then each primary group is subdivided into a central form and three small groups qualified as short or long, narrow or broad, flat or thick. Lastly each of these small groups comprises three specimens. In the middle of these nine small groups lies the group of three outstanding white forms surrounded by the three forms transitional between the three primary groups.’ (VDL AS X, 9)

The white forms are the most important. They are a group of six forms that are neither block, nor bar, nor slab.

‘From this superior group they separate out into blocks, bars and slabs, as light divides into the colours of the rainbow. We will therefore call the forms of the core group, by analogy, “white forms”.’ (VDL AS X, 7)

The three white forms in the corners are essentially transitional forms between the forms of block and slab, between block and bar, and between bar and slab.

‘However, the three central white forms have something in common with all three forms. The three categories, block, bar and slab, meet together in an ideal form which does not exist as such but presents itself as blocklike in one form, barlike in another and slablike in the third. Because of their outstanding value these three forms play a leading role in architecture.’ (VDL AS X, 8)

Derived forms are also possible. They can be seen as a doubling of an authentic form.

The bars in the 1977 form-bank are 16 mm high. The smallest block is 16 mm x 16 mm x 16 mm, while the largest slab is 114 mm x 114 mm x 16 mm.