Between 1920 and 1991, the Dutch Benedictine monk and architect Dom Hans van der Laan (1904–91) developed his own proportional system based on the ratio 3:4, or the irrational number 1.3247…, which he called the plastic number. According to him, this ratio directly grew from discernment, the human ability to differentiate sizes, and as such would be an improvement over the golden ratio. To put his theories to the test, he developed an architectural language, which can best be described as elementary architecture. His oeuvre — four convents and a house — is published on an international scale. His buildings have become pilgrimage sites for practicing architects and institutions that want to study and experience his spaces. His 1977 book

To understand and evaluate Van der Laan’s application of the plastic number, this paper approaches it as a practical design tool. It analyses its genealogy and defines its key concepts. From that framework, Van der Laan’s architectonic space is interpreted as a design methodology that combines antique tectonic theories reminiscent of writers from Plato to Vitruvius with more recent atectonic approaches towards space through experience and movement.

I believe that the secret of the language of architecture does not lie in the being of space itself, but in the way in which we connect to it.

Dom Hans van der Laan (

Between 1920 and 1991, the Dutch Benedictine monk and architect Dom Hans van der Laan (1904–91) developed his own proportional system. Just as the Benedictine monk Dom Mocquereau (1849–1930), in the beginning of the twentieth century, defined a universal notational system to restore Gregorian chant, called ‘le nombre musical’, Van der Laan set out to develop a universal ordering system to restore architecture: ‘le nombre plastique’ (the plastic number) (

His oeuvre of only four convents and a house is published on an international scale (Fig.

Overview Dom Hans van der Laan’s elementary architecture, 1961 to 1995. Aerial photos are courtesy of the abbeys. Other photos by the author.

To critically evaluate the plastic number and Van der Laan’s theory of architectonic space, it is necessary to address the problem of the mythical image that Van der Laan created for himself. Van der Laan believed that architecture produced meaning through its affective qualities, and he saw his proportional system as an essential contribution to producing such meaning. Indeed, beyond presenting his plastic number as a design tool, Van der Laan aimed to elevate it to the status of a philosophical principle, and this aim dominated almost all of his writings. The Belgian philosopher André Van de Putte described

In this paper, I will demonstrate that the plastic number is a practical design tool, by expanding upon Van der Laan’s attempts to elevate it to a philosophical principle. Van der Laan’s design methodology and philosophy, though reinforcing and even legitimizing each other in

The main question dominating Van der Laan’s life from his early youth onwards was, ‘How can I know things as they are?’ Knowing, as a continuous process of cognition between ratio and perception, for Van der Laan meant ordering, and architecture had the fundamental task of facilitating this process of ordering. In a letter to his biographer Richard Padovan, he defined the difference between ‘the natural spatial phenomenon as it is and the space as we humans perceive it’ (

The key to achieving this bond, this transformation of phenomena from the sensorial to the abstracting ratio, was the creation of a perceivable order in space, a spatial system governed by proportional relations. According to Van der Laan, one intuitively placed oneself in relation with one’s surroundings, reading the surroundings by relating the measurements of objects to each other. Relations were made through comparison and differentiation. When understanding a space through perception, one measured it, not with measuring equipment, but with one’s eyes. Natural space, however, consisted of endless continuous quantities that could not be measured as such. In order to measure, one had to abstract these continuous quantities into discrete quantities expressed in whole numbers.

Van der Laan defined the fundamental function of architecture through its direct connection with the process of cognition: to make space readable. To inhabit a space (

These theories of space and perception were more intertwined with Van der Laan’s Benedictine background than with the architectural scene of his time. After three years of architectural study in Delft, in 1927 Van der Laan entered the Benedictine St. Paul Abbey in Oosterhout, the Netherlands. The monks organised their life according to the rules of St. Benedict, which centred all thoughts and actions around the concept of ‘ora et labora’, a clearly defined repetitive succession of prayer through contemplation, Gregorian chant and work. Every day was rhythmically ordered around seven periods of prayer, but the other moments and modes of communication and silence or contemplation were also clearly ordered. All of this alternated with manual labour and study. Recreation was also orchestrated: outside walks were done in

His ideal was to maintain the spirit of the liturgical prayer in his daily activities (

In the beginning of the twentieth century, designing through proportion was very much alive. Architects such as Hendrik Petrus Berlage and Mies van der Rohe were inspired by the geometrical studies of Jan Hessel de Groot (

We are overwhelmed by nature and we are looking for artificial principles to dissolve this contrast, to again get a grip on space, to again understand and control it. Architecture in this sense becomes a necessary instrument for our intellect as well as an expression of a regained authority, of an understood space. (

Van der Laan linked proportion to the perceiving human beings by abstracting it from the way they perceived space, or more precisely, from their intuitive ability to distinguish different sizes. To establish that link, he went back further than Plato, towards the Pythagoreans. In contrast to Plato, whose proportion was geometrically based, the Pythagoreans believed that number was at the source of everything. Van der Laan identified this abstract number as the discrete quantity.

When Van der Laan started teaching in 1939, he referred to Aristotle to establish a foundation for this reading of space through number. For the concept of the yardstick, he took his reference directly from Aquinas’ reading of Aristotle’s

For in the case of a furlong (measure of length) or a talent (measure of weight) and always in the case of something larger, any addition or subtraction might more easily escape our notice than in the case of something smaller, so that the first thing from which, as far as our perception goes, nothing can be added or subtracted, all men make the measure, and they think they know this quantity, when they know it by means of this measure. (VDL notes, 1940).

From the Aristotelian notion of the indivisible number ‘one’, Van der Laan continued to develop his spatial framework based on sensorial perception. He defined the ‘one’ as the smallest perceptible yardstick, or the margin of space. This margin was not a fixed number, but a ratio relative to the scale of a building. It was the smallest ‘one’ that could still be perceived in that building: measures beyond it were not recognisable and thus discarded in the process of counting a space. When space is paced out (

The genealogy of the plastic number is quite ambiguous. Van der Laan claimed that already in 1928 he had come across the 3:4 ratio (

He selected 36 pebbles that diverged in diameter by 1/25th from each other in a continuous series (Van der Laan, AS V.5) (Fig.

Dom Hans van der Laan’s sorting test of a quasi-continuous series of 36 pebbles. Drawings by Dom Hans van der Laan, 1987 (VDLA).

When I repeated this process with my own students, however, other groups of pebbles, both smaller and larger than seven pebbles, were defined.

When reviewing Van der Laan’s first lecture series from 1939 to 1941, we find a definition of the plastic number, which until 1955 he called the ‘ground ratio’, as a hierarchical series of abstract measures that aimed to approximate as closely as possible the infinite and continuous series of concrete, natural measures.^{3}, where x equals 1.3247….

Dom Hans van der Laan’s comparison between the plastic number and the golden ratio. Charts based on Padovan (

Nevertheless, it was of no interest to Van der Laan to express this series as a sequence of irrational ratios. For him, it provided the tool to interrelate abstract measures into a hierarchical series, with 4:3 and 1:7 as its keys. Just as he expressed the concrete measure 1.3247… through the abstract measure 4:3, he translated the series into their abstract approximate equivalents. He developed a series of eight measures, which he called the ‘order of size’ (Fig.

1 4:3 7:4 7:3 3 4 16:3 7.

(

The series of the plastic number did not consist of fixed measurements, but of ratios that can be used to proportionally relate all manner of building scales. Within the eight measures that are bound by the multiplication factor 4:3, the outer limits relate as 1:7. Several relations are possible, while remaining within the series. The sum of two sequential measures is not the subsequent measure (as in the Fibonacci series, which approximates the golden ratio) but the one after that; for example, 1 + 4:3 = 7:3 (because 1 + 4:3 = 3:3 + 4:3 = 7:3). Furthermore, any measure in the series is equal to the difference between the fourth and fifth measures after that.

To achieve a finer grain of measures, Van der Laan added a ‘derived’ series with the same sequence as the first order of size, ranging from 1 to 7, which he called the ‘authentic’ series. Like the authentic measures, the derived measures are also interrelated through 4:3. Following the logic of Pythagorean means, an authentic measure lies in the arithmetic mean of two derived measures. A derived measure lies in the harmonic mean of two authentic measures.

6:7 7:6 3:2 4:2 5:2 7:2 9:2 6.

These abstract measures expressed as numerical ratios are only roughly related through 4:3. They do not constitute a geometric series, but a sequence of simple whole numbers, individually and in ratios, that demonstrate arithmetic coherence. The series is not entirely regular, but the deviations are minor.^{3}, has a proportional number that is only slightly smaller than 4:3. For Van der Laan, this meant that the plastic number resulted in a system that delivered an infinitely continuous series ruled by a single irrational ratio, 1.3247…, while as an arithmetic system it remained close to the simple whole numbers 3, 4, 7 and the fractions they can form, so that it was capable of answering to our limited perception and judgement (

To adjust the deviations, Van der Laan determined numerical values for the series. He did this for three successive orders of size, which he called the three ‘measure-systems’ or Series I, II and III (

100 132.5 175.5 232.5 308 408 540.5 716.

And its derived measure-system is

86 114 151 200 265 351 465 616.

The difference between an authentic and its derived measure is the authentic value in the lower measure-system: 716 – 616 = 100. In the design of buildings, the three measure-systems form an interlocking sequence of consecutive building scales. In the example in Figures

According to Van der Laan, designing with the orders of size as frameworks of measures for all building parts enabled a clear reading of the proportions of a structure with whole numbers, since they could be perceived through counting. He aimed to introduce a type of architecture that had as its goal the combination of abstract thinking and sensorial perception, an architecture where ‘number and measure meet’ (

I compare it with time, where the ‘now’ without duration separates the duration of the past from that of the future. And also we as ‘reasonable beings’ can differentiate that notion from experienced time, to which we give a duration with a beginning and an end = hours, days and weeks. Those times unroll themselves against the background of our objective knowledge of time that we make graspable through our festivities as we make natural space habitable through our architecture. In the endlessness of time we make times with a beginning and an end as our seasons, and in the borderless space we make spaces between walls. (

With the series of the plastic number as an underlying framework, Van der Laan developed a design methodology based on numerical proportions. This manner of working was more in line with Wittkower’s view of Alberti’s generation of ratios as a theory of music and of Palladio’s fugal system of proportion, than with the geometrical constructions of Hendrik Petrus Berlage or the

Van der Laan studied several monuments, believing that they embodied the plastic number. The Parthenon and Hagia Sophia were his first objects of analysis. By 1936, Van der Laan had marked the Parthenon as essential study material for understanding the plastic number (

It is magnificent to see how in the Parthenon, in which the whole pyramid is followed, all values of form and measure softly and naturally flow into each other. It is as if in a mighty manner everything is taken into account whilst placed gently so as not to break the subtle and all comprising balance. (

Dom Hans van der Laan, studies of the plastic number proportion in the Parthenon, 1940 (VDLA).

To prove his theory of the plastic number through the proportions of the Parthenon, Van der Laan looked for a potential expression of the 4:3 ratio. He found it in a specific point within the capital, which divided the lower and upper parts into 1149 cm and 382 cm, respectively: ‘a symmetrical hinging point in the vertical sense, a point from carrying to being carried, where the capital becomes rounded’ (

In 1945, Van der Laan started teaching a class for practicing architects in Breda and ‘s-Hertogenbosch, the Cursus Kerkelijke Architectuur (Course on church architecture, CKA). Although the aim was to introduce the architects to church design and restoration, Van der Laan focused on his architectural theory as a means for creating liturgical space. For this purpose he developed study material to explain how the ratios 4:3 and 1:7 could be implemented. From this material, two essential elements can be deduced.

First, Van der Laan emphasised an intrinsic relationship between what he called ‘mass and space’, meaning that mass and space related as 1:7, the outer limits of one order of size. He believed that measures beyond this ratio, for example 1:8 or 1:9, resolved in walls that were too thin, so the mutual nearness was in danger of dissolving. Walls thicker than this appear heavy in his opinion, and are in danger of relating to the space as a form (Van der Laan, AS XI.4). To realise this perceptible and pleasing range of scales architecturally, Van der Laan defined the ‘spatial cell’ as an intimate space for one person between 3 and 5 metres wide. During the first CKA lecture series, which lasted from 1945 to 1955, the capital served as a module for the spatial cell. Van der Laan explained this in a sketch, where the relation between the capital and the spatial cell is at least 1:5 (Fig.

Dom Hans van der Laan, relation between the capital and the spatial cell, as part of the gallery or side aisle, through its three dimensions. In his architectural designs, this relation was mostly 1:7. VDL, CKA II, 19 Nov. 1955–14 Jan. 1956–1955, p. 7 (VDLA). Text and scales added by author.

(

Cursus Kerkelijke Architectuur (Course on church architecture), student work: a study of Butler’s Mshabbak, presumably 2nd year, student name unknown, 1953. Cees Pouderoyen Private Archives, Nijmegen. Original drawings from Butler (

The abacus. Photograph added to the

Examples of churches built by the students of the Cursus Kerkelijke Architectuur (Course on church architecture) (VDLA). (

Second, Van der Laan composed larger spaces from the spatial cell. He used the Hagia Sophia to illustrate his concept of spatial overlap, which he called superposition (Fig.

In the CKA Van der Laan also presented the basilica building type as the ideal model for new churches. For him, the narthex, apse and side aisles, which he called ‘marginal spaces’, were necessary to introduce the spatial cell and relate a large, monumental nave to a smaller, more intimate human. For example, the apse and the nave would relate as 1:7. Taking the example of a spatial cell or narthex of 3.5 m deep, the length of the nave would be 24.5 m.

Van der Laan highlighted the mass-to-space relationship through ornamentation such as mouldings or capitals, emphasising the tectonic expression of the structure. In the example of the spatial cell of 3.5 m, the capital that defined this space was set at 0.5 m. All columns, wall pieces and ornaments ranged between 0.5 m and 3.5 m, all set in proportion to each other through the plastic number series. As such, all measurements, from the overall space to the spatial cell, and from the spatial cell to the capital, could be hierarchically interrelated by three successive orders of size. This sequence of proportions can be seen in a 1953 study of the Byzantine Mshabbak Church in Syria, made by one of Van der Laan’s students (Fig.

The students analysed other Early Christian basilicas, redrawing them according to the series of the plastic number in order to study its application. As their sources they used drawings by Butler (

From 1945 onwards, a vast production by Van der Laan’s students came out of this systematised way of working (Fig.

This lack of acceptance can be partly ascribed to Van der Laan’s own attitude. He himself criticized other approaches to proportion that were ongoing in his day. In 1939, he wrote to his younger architect brother Nico: ‘I flipped through your book on Ghyka, […] I would not recommend you to read these types of books. You lose yourself in mathematical jokes that are quite pleasant, but with no value to reach a harmonious insight. […] I believe we have to do it ourselves’ (

At first glance, the system of the plastic number shows striking resemblances to the Modulor. It too is conceived as a mathematical series, and moreover has two interwoven series, the authentic and the derived order of size, like Le Corbusier’s red and blue series. If Van der Laan was influenced by Le Corbusier’s writings on how a series could be implemented in architecture, however, he never admitted to it. Nevertheless, it is clear that the plastic number was developed before the Modulor. Although Van der Laan claimed to have discovered the plastic number in 1928, there is no proof of this date (

Van der Laan never compared the plastic number with the Modulor in

Van der Laan felt misunderstood by his architectural colleagues as well as by his fellow Benedictine brothers. But for him this was part of the difficult road he had to travel, and he continued to develop his theories as a personal calling (

The crypt of St. Benedictusberg Abbey, Vaals, 1958–61. Photo by Coen van der Heiden (2008).

As in his earlier work, Van der Laan designed a composition of numerous in-between spaces such as galleries or porticos, each of them integrating the scale of the spatial cell. But in the crypt, the monumental and elementary architecture is further expressed through a rough formal language, dominated by heavy-looking walls and galleries of stone or concrete. Because of the lack of ornamentation — typical details such as plinths, frames or inclined windowsills — the building parts are defined by sharp lines separating mass and space. Windows are rhythmical openings with the same dimensions inside and outside. Lintels and thresholds are continuous concrete elements that provide horizontal articulation. The only material finishing that accompanies the bare concrete is roughcast with plaster, and wooden boards painted in complementary grey colours. Guided by the articulated series of openings, daylight illuminates the spaces with varying intensities, and creates patterns through a pronounced play of light and dark shadows. The light plays over the rough topography of the wall surfaces, bringing the architecture to life. Van der Laan likewise designed the furniture in complementary colours and all the liturgical objects as parts of a whole. This architecture does not rely on religious symbolism for the production of meaning. Instead it thrives on a spirituality implied through its affective qualities.

Van der Laan designed this new elementary architecture through a more dynamic implementation of his proportional number series. His drawings changed, becoming abstract patterns of lines denoting distinctions between mass and space (Fig.

Design sketches by Dom Hans van der Laan for church and atrium of St. Benedictusberg, 1956 (VDLA).

Roosenberg Abbey, Waasmunster, 1975. 3-D model by author.

The capitals, impost blocks and frames (as seen in Fig.

The approach of Van der Laan’s book

(

Van der Laan’s diagrams showed a field of experience around a person on three scales: work space — walking space — field of vision (Fig.

The three orders of size of Roosenberg Abbey in detailed centimetres. Authentic measures are in white, derived measures in grey. The measures incrementally grow with the 4:3 ratio. Every order of size is 7 times larger than the one below. The order from 49 cm (thickness of the wall) to 351 cm (cella size) is used for dimensioning wall segments, window openings and columns. The order from 351 cm to 2513 cm is used for spaces and the convent wings. The order from 2513 cm to 17,991 cm defines the position of the building in the terrain. Reconstruction by the author.

Van der Laan based the second series of diagrams on the scales of the larger spaces. Just as the wall thickness served as the module for the cella in a 1:7 proportion, the smallest space served as a module for the whole (Fig.

Generic models showing continuous superposition of space: cell — gallery — hall. VDL, CKA V, Vijfde les over de architectonische ruimte, 4 March 1967 (VDLA).

Through this superposition of galleries, he aimed to secure the concept of mutual nearness between the whole space and the wall: the wall related to the gallery, and the gallery to the whole space. Van der Laan applied this cell-to-gallery relation in the designs of his buildings. In the crypt of Vaals, for example, side aisles are formed by a series of cells (Fig.

Dynamic spatial superposition. Source: Design drawing of Dom Hans van der Laan, 1959 (VDLA); defined rhythms added by author.

In each cell the mass-to-space (wall thickness-to-cell width) ratio of 1:7 has the dimensions 0.6:4.2 m. This cell is not a closed entity. It seems to blend with the surrounding space. All the elements together evolve through the movements and changing perspectives of the visitor. The same dynamics appear in the typical halls in Roosenberg Abbey (Fig.

As a third series of diagrams, Van der Laan introduced Vitruvius’s five intercolumniations to define the density of openings in a column row (Fig.

(

Van der Laan regarded the wall as a composition of wall pieces with orthogonal open and closed parts. He included every wall piece, not only the columns but also the horizontal concrete bands of bordering elements, such as thresholds and lintels, into a composition of measures within the order defined by the wall thickness as the basic unit (Van der Laan, AS XI.7). Through the repetition of openings and columns, each wall defined a distinct column spacing. As Van der Laan explained, ‘This repetition in the wall of open and closed parts to the rhythm of the smallest spatial unit is a universal architectural phenomenon; at all times and in all places, walls display rows of doors or windows, of columns or piers’ (Van der Laan, AS XI.6). As with the superposition in plan, the pier spacing did not cohere with a logic of construction. Van der Laan composed the different bay dimensions with the intention of expressing certain dynamics of spatial atectonic rhythms.

The result of this articulation of rhythms is prominent in, for example, the library in St. Benedictusberg, Vaals (Fig.

Library at St. Benedictusberg Abbey, Vaals. Photo by the author (2008).

The realisation of the crypt in 1961 and the church in 1968 at Abbey St. Benedictusberg remained unnoticed in the architectural community for a decade. Only from 1971 onwards did Van der Laan’s elementary architecture begin to draw attention, thanks to two Belgian journalists, Anthony Mertens and Guido van Hoof. They wrote an article accompanied by an interview in the

Until the end of his days, Van der Laan remained true to his quest: for him the plastic number was not just a means for the creation of architecture (as a working method or a means of creating beauty), but the end goal of architecture itself. The plastic number for him was not a design tool, but a philosophical tool. The framework that Van der Laan formulated in

Nevertheless, what

The author declares that they have no competing interests.

AS

CKA

Cursus Kerkelijke Architectuur (Course on church architecture)

VDL

Dom Hans van der Laan

VDL A

Van der Laan Archief St. Benedictusberg, Vaals (Van der Laan Archives St. Benedictusberg, Vaals)

VDL C

correspondence of dom Hans van der Laan, Van der Laan Archives St. Benedictusberg, Vaals

VDL L

unpublished lectures by dom Hans van der Laan, Van der Laan Archives St. Benedictusberg, Vaals

Dom Hans Van der Laan, in a letter to his brother Nico, 21 April 1940 (

Van der Laan explicitly used the term number, not ratio, because he intended to refer to Mocquereau’s ‘nombre musicale’.

This term was used to describe Van der Laan’s architecture in the Italian journal

Many architects and members of particular institutions visit Van der Laan’s abbeys in Vaals and Waasmunster, as can be seen in the archives. For example, in the last three years, Roosenberg Abbey in Waasmunster (1974) has been visited by several national and international universities, including ETH Zurich; University of Venice, Dep. of Architecture; University of Valladolid, Dep. of Architecture; Università della Svizzera Italiana, Accademia di Architettura di Mendrisio; and Leibniz Universität Hannover, Fakultät für Architektur und Landschaft Institut für Entwerfen und Gebäudelehre.

The Belgian philosopher André Van de Putte described

Tectonics is the study of how building elements are assembled. Ornamentation is then introduced as an emphasis of the tectonic structure, for example the capital of a column. With the term ‘atectonic approaches’, I refer to the legacy of Gottfried Semper, whose influence occurs in the works of, for example, Hendrik Petrus Berlage. The emphasis is here on the enclosure of space and spatial effects. Ornamentation is then related to patterns (Semper’s ‘dressing’) and spatial figures. Le Corbusier and Mies van der Rohe used an atectonic, meaning spatial, approach in their architecture. See, for example, Max Risselada,

Van der Laan differentiates continuous quantities from discrete quantities. A continuous quantity is infinite and can take any value, such as 3,6834… or 3,576…. The golden ratio 1,618… is a continuous quantity. A discrete quantity must take certain values, such as 1, 2, 457, for example, for shoe sizes or a number of people. These are also called integers, whole numbers or commensurable numbers. They are applied and abstract. Van der Laan defined the continuous quantities as natural or concrete, and the discrete quantities as abstract.

St. Thomas Aquinas,

Van der Laan often used the term ‘abstract number’ to refer to ideal, discrete numbers, as opposed to concrete, natural or continuous numbers: ‘The abstract number we use to count discrete quantity is perfectly knowable, since the unit on which it is based is absolute: it is the individual oneness of the things that we count, the smallest indivisible whole’ (Van der Laan, AS V. 4).

St. Thomas Aquinas,

I have conducted this experiment annually since 2008 with my master’s students from Sint-Lucas School of Architecture, Brussels-Ghent.

Van der Laan introduced the term

Starting from 1 as the smallest measure, this series is: 1 1.3247… 1.7548… 2.3246… 3.079… 4.079… 5.4039… 7.158… … By comparison, the Fibonacci series consists of the numbers in the following integer sequence: 1 1 2 3 5 8 13 21 … The series is an approximation of the mathematical definition x + 1 = x^{2}. The sum of two adjacent terms in the series equals the next term in the series.

The Pythagorean means are as follows: 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. 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. The geometric mean is the middle term between two others that has the same ratio to the one term as the other has to itself (Van der Laan, AS VII.12). Arithmetic mean h of two lengths l and w: l – h = h – w; geometric mean: l/h = h/w, harmonic mean: (l–h)/(h–w) = l/w.

For example: the authentic measure 7:4 × 2 = 14:4 = the derived measure 7:2.

The abstract measures are:

1 4:3 7:4 7:3 3 4 16:3 7.

1 1.3333… 1.75 2.3333… 3 4 5.3333… 7.

When comparing each part of these series with the 4:3 multiplication factor, the following deviations occur: The third measure (7:4) already deviates slightly (4:3 × 4:3 = 16:9, which is 64:36 instead of 63:36 = 7:4) and would be 1/36 larger than 7:4. The fifth measure (3) deviates more (7:3 × 4:3 = 28:9 instead of 27:9 = 3) and would be 1/9 larger than 3. The eighth measure (7) has the same deviation (1/9) as the fifth and needs to be reduced by 1/9 to end at 7 (16:3 × 4:3 = 64:9 instead of 63:9 = 7). This means that the multiplication factor is not exactly 4:3, but it nevertheless shows limited variations of it.

Deviations when interrelated with the multiplication factor 4:3 are:

– Third measure: 4:3 × 4:3 = 1.7777…. This does not equal 1.75, but deviates 0.0277… from it.

– Fifth measure: 7:3 × 4:3 = 3.11111….. Deviation from 3: 0.1111….

– Eighth measure: 16:3 × 4:3 = 7.111111…. Deviation from 7: 0.1111….

Van der Laan considered these deviations to be minor, meaning that 3, 4, 7, etc., could be used to describe the measures of the series. In the translation into the numerical values of the three measure-systems, these become more exact, e.g. 7 then is 716 (700 + 14 + 2).

The architect Hans van der Laan (junior) is the son of Nico van der Laan, Dom van der Laan’s younger brother.

Original drawings and references can be found in the VDLA.

For example, Sint Antonius Church, Groesbeek (arch. J. G. Deur and C. Pouderoyen, 1948), Holy Eligius Church, Oostburg (arch. F. Mol and J. Brugman, 1949), and Catharina Church, Heusden (arch. Nico van der Laan, 1951).

This number is deduced from the research conducted at the Van der Laan Archives Sint Benedictusberg, by J. M. M. van der Vaart, 2005 and 2007. Within the list of 371 participants, one has to make a differentiation between students who followed the three-year post-academic course and the participants in the open lecture days, Saturdays, which were open to the public. The official document produced by the CKA treasurer J. J. Van Dillen (VDLA) shows that between 1950 and 1967, 88 new students were admitted, of which 27 left with a charter. There was an average of 20 students per year. In the open days between 1955 and 1967, there were 316 participants.

It is not clear to which of Ghyka’s publications Van der Laan was referring. Two possible examples are Esthétique des proportions dans la nature et dans les arts (

The 1957 international Syracuse competition, which selected Michel Andrault and Pierre Parat’s monumental cone-shaped church in glass and concrete and not the more classical horizontal basilica with forecourts by Van der Laan’s pupil Jan de Jong, was the final proof for Van der Laan that the general opinion did not favour his approach (Van der Laan 1957).

In 1939, Van der Laan started a lecture series in Leiden, the Netherlands, for which he took notes in small notebooks (VDLA). For the date of the patent, see Le Corbusier (

For example, Van der Laan claimed that the golden section was two-dimensional, whereas the plastic number was three-dimensional. He based this claim on mathematical definitions. The golden ratio is defined by x^{2} = x + 1, while the plastic number is defined by x^{3} = x + 1. This reasoning is first exposed in

Maritain was a close friend of Pieter van der Meer de Walcheren (1880–1970), and like Van der Laan, a Benedictine monk at Oosterhout. Van der Meer de Walcheren set out to help Van der Laan in the recognition of his work. Maritain wrote, ‘L’auteur s’appelle Van der Laan et le livre Le Nombre Plastique. Je suppose que le génie vaut mieux que tous les nombres d’or, mais au fond il n’y a pas d’opposition’ (J. Maritain, letter to Jean Labatut, 26 September 1961 (Labatut Papers, Firestone Library – Princeton, Box 7/Folder 4)). This document was pointed out to me by Rajesh Heynickx.

Van der Laan had read the book in 1966. He explains this in a letter to Richard Padovan on 26 October 1983 (VDLA). He wrote to Padovan that he had received the book in 1966 from Granpré Molière.

Richard Padovan translates the word ‘nabijheid’ as ‘neighbourhood’, with the approval of Dom van der Laan himself. Nevertheless, the author here prefers the term ‘nearness’.

Van der Laan’s definition of eurhythmy and symmetry is as follows: ‘When the measures of a system are realized in the squared forms of the mass they can be proportionally related to each other in two ways. The form of the mass is fixed by the ratios between its various dimensions, which were referred to in antiquity by the name eurhythmy. But the size of the forms is fixed by its relation to that of other forms, and ultimately to that of the unit. Two three-dimensional sizes stand in a threefold linear ratio to each other; that is to say the three dimensions of one concrete datum stand in a ratio to the corresponding dimensions of the other. Here, it is not the several dimensions of a single form that are compared with each other, as with eurhythmy, but the corresponding dimensions of two distinct forms. This the ancients called symmetry, not in the sense in which the word is used at present, to mean the identity of two opposite halves, but in the sense of the proportion between the sizes of the parts of a building, from the smallest up to the whole’ (Van der Laan, AS IX.6).

Vitruvius defined five column-intervals by relating the intercolumniation (the space between the columns) to the diameter of the columns. Van der Laan did not use this intercolumniation, but related the width of the column to the bay-rhythm (intercolumniation + column width).

For more in-depth research in Van der Laan’s application of the three measuring scales into his architecture, see Voet (

Most notably, Haan and Haagsma (