Benjamin W. Betts and his computable approach towards morphogenesis

Abstract

Morphological theories in architecture are more popular than ever partly due to the rapid developments in computational techniques. This rapid development, however, is more fruitful when it is informed of the philosophical and ethical bases of such techniques. The neglected historical case of Benjamin W. Betts, an architect of the late Victorian Britain, is an early example of the effects of philosophical presuppositions on computable theories of morphogenesis in arts and architecture. The aims of this study are to shed light on his motivations and sources of inspiration, and also to present an algorithm for his procedure. We used the only book available on the Betts' works in addition to archival materials to conduct the research. Betts was under heavy influence of Idealism and Oriental thought in his symbology and the morphogenetic procedure he designed. The Python/Grasshopper algorithm for Rhino presented in this paper produces two dimensional Betts' diagrams that can be used as an educational tool or an opportunity of enjoying the beauty of mathematical forms, and the immense variety they offer thanks to parameterization.

Keywords

fourth dimension architecture computational morphogenesis geometrical psychology

Morphogenesis is about how forms appear, evolve, and die. Architects have always been inspired by the processes of change in patterns and forms in organic nature. The first systematic investigation of biological morphogenesis according to mathematical principles was done by D’arcy Wentworth Thompson (1860–1948) in his grand opus “On Growth and Form” published in 1917.¹ In this book D’arcy Thompson used a mathematical approach to describe and explain the evolution of natural forms, and subjected this evolution to mathematical and physical laws. This geometrical vision of morphogenesis had a remarkable effect on early modern architects that understood its importance like Le Corbusier, László Moholy-Nagy, Mies van der Rohe, Richard Buckminster Fuller and others. It also inspired Rene Thom’s “Structural Stability” published in 1975² that had a great impact on morphological studies in architecture and urbanism based on Thom’s Catastrophe Theory.³

The concept of morphogenesis can also be used for a paradigm shift focused on high performance architecture,⁴ and it can also be the central concept for new theoretical frameworks in architecture.⁵ Morphogenesis is related to the concepts of evolution, organic architecture, biomimetics, evolutionary thinking in design and generative design algorithms.

In recent decades with the rapid growth of computational techniques, morphological concepts are more popular and more powerful than ever as a tool for creating variations and rapid prototyping.⁶ Computational morphogenesis, based on biological aspirations, has many applications in domains of architecture⁷ and urbanism.⁸ These applications are directly inspired from the evolutionary processes in nature, and therefore have paved the way for the emergence of “bio-digital aesthetics”⁹ in an interdisciplinary dialogue. This paper adds to this lively dialogue, the neglected historical case of Benjamin W. Betts, that offers valuable opportunities of study. It presents one of the earliest attempts at procedural step-wise models of morphogenesis; and the algorithm itself can be used to produce parametric objects coming from a Victorian era’s theory. “Geometrical psychology” of Betts, which was published 20 years before the publication of “On Growth and Form,” presents an interesting case study of the architects’ efforts, not just to use but to invent morphological theories of their own to approach morphogenesis from an algorithmic point of view.

Benjamin W. Betts developed a computable approach to morphogenesis, a characteristic that was lacking in non-computable theories of his contemporaries. For example, Rudolph Steiner (1861–1925) utilized the concepts of morphogenesis to establish isomorphism between the patterns in mind and the patterns in nature and ornaments.¹⁰ Claude Fayette Bragdon (1866–1946) also linked the morphogenesis of patterns of human mind to spatial patterns of architecture¹¹ and four-dimensional objects.¹² But Betts differed from Steiner and Bragdon in that his theory was readily computable, and thus can be translated into an algorithm.

How did Betts come up with his ideas of morphogenesis in architecture? How can we develop an algorithm to draw two dimensional Betts' diagrams? The first part of this paper consists of a biography and an analysis of the Betts' ideas. In the second part a theoretical description of Geometrical Psychology, and the Python/Grasshopper algorithm will be presented.

Biography of Benjamin W. Betts (1832–?)

The only book about the works of Betts is what Louisa S. Cook has assembled and published with commentary and interpretations; 'Geometrical Psychology or The Science of Representation, an abstract of the theories and diagrams of B.W. Betts' in 1887.¹³ Some fragmented thoughts of his are also included in the Mary Everest Boole’s book “Symbolic Methods of Study,” published in 1884.¹⁴ Archival material from National Archives of New Zealand has also been used in this research.

Searching for Mathematical Principles of Design

Benjamin W. Betts was a British architect who was unsatisfied with the architecture of Victorian era like many of his contemporary architects and critics.¹⁵ He believed that the arbitrariness of architectural style is a questionable assumption, and there must be a method of architectural form generation based on mathematics, but which does not blindly imitate natural forms.

Betts was born in 1832, around a decade before the period of (1845–1860) whichGideon calls the period of looking for principles.¹⁶ One can, however, trace back the roots of this obsession in 19th century at least to Schinkel, the German architect. In the period of (1820–1830) Schinkel was seeking an architecture without style. His investigation, influenced by German Idealism, was based on the idea that architectural forms as objects of our consciousness should reflect the absolute principles, but this Idealism was also the reason for his failure based on what he called the “error of pure radical abstraction” rooted in his rejection of history.¹⁷

In the mid-19th century Britain, when Betts was in his twenties studying Decorative Arts, the battle of styles was raging, and arguments for and against particular styles were often interwoven with religious and nationalistic sentiments. This was especially the case for Neo-Gothic and its passionate defenders A. W. N. Pugin (1812–1852) and John Ruskin (1819–1900). Christian architecture, they believed, had a correct style that was essentially different from pagan works of art.

Between 1835 and 1851, Pugin was on his road to find the “true principles,” examples of which can be found in his books like “The true principles of pointed or Christian architecture,”¹⁸ and his principles for architectural ornament.¹⁹ Ruskin, too, spent the better part of his life crusading against poor taste, machine aesthetics and flawed architecture compared with his own version of principles reflected in his works like “The Seven Lamps of Architecture”.²⁰ But even these greatest hands and brains of the Gothic revival were not satisfied with the ultimate results, and it was both logical and moral to focus more on the principles of design instead of a particular style.¹⁷

These architects and critics were also obsessed with the concepts of truth and beauty.²¹ Therefore, mathematics would seem a natural choice, for them, to unify abstract principles, truth and beauty in a coherent system and propose it as a logical canon. But this was not a popular idea at the time, partly because of the problematic relationship of architecture and mathematics in the Victorian era that had roots in the beginning of the 19th century. Betts took the idea of abstracting principles of design, and used mathematics as the foundation of his work.

The Birth of Geometrical Psychology

In order to find the proper principles of design Betts started a journey on the borderline of genius and madness. He quit everything. He quit his career in which he had a “considerable promise”; his homeland, family and friends, and sailed away. Betts spent many years in the East, especially in India, isolated from civilization. What he found in his years of solitude, he believed, was the true laws of form, a computable morphological theory that could be applied both to organic and psychic forms. Unable to find a proper way to share his vision, and not willing to return back home, Betts decided to migrate to New Zealand. Maybe he imagined that the new home could offer him a more peaceful place to work and study than the old one.

Sources of Inspiration

His symbology had philosophical and spiritual roots in Idealism, Christianity and Buddhism. This diversity of the domain of religious beliefs was a reflection of the diversity and complexity of religious movements in Victorian Britain. One could find alongside the increasing number of Positivists a number of artists who were still grounded in firm Christian values using religious scriptures as the central theme of their arts. The Oriental influence, however, was also strong as it manifested itself not only as a peripheral influence but as a motivation and a source of inspiration.

Betts was influenced by Esoteric Buddhism that seems natural due to his long stay in India. He also believed in psychic powers, and suggested that his Science of Representation was a male (alpha) science which focused on physical senses, and it needed a practical female (omega) counterpart called the Science of Determination, with a focus on the psychic senses, and he expected that “it will be chiefly the task of women to develop it.”

Apart from the oriental influence, Betts was inspired by Idealism and the idea of morphogenesis in the fourth dimension, both with deep influences on the formation and formulation of his theory.

The Effect of German Idealism

The Idealist influence in the works of Betts can be seen in his choice of mathematical notations and their symbolic meanings, in his use of the concept of isomorphism and the idea of stagewise evolution of the human psyche.

Betts’s reasons to choose a mathematical system were inspired by Johann Gottlieb Fichte (1762–1814). Although Fichte never compiled a coherent mathematical philosophy his core concepts can indeed form a coherent theory of mathematics and geometry.²² Betts was inspired by Fichte in his treatment of geometry in The Science of Knowledge (Wissenschaftslehre) published in 1794.²³ Betts found out that Fichte has suggested geometrical elements like lines and circles to represent different modes of consciousness. He happily took the idea and built upon it.

Idealism also gave Betts the concept of isomorphism, and compatible principles of modeling the process of human mental evolution. The Idealist psychologists of the time were working on developmental models of human mind. Their investigations were based on the following principles, all of which Betts included in his theory. First of all, there should be distinguishable stages of development. These stages should model the flow of the mental processes and developmental stages in such a way that addition of senses and faculties are successive parts of the whole evolutional model. In other words, the development of senses and faculties should be a bottom-up process. It should be noted that this idea was not limited to the Idealists of the time. For example, W. B. Carpenter (1813–1885), a famous rationalist physiologist, believed that in order to understand human psyche one has to follow the “successive complication” of senses and faculties beginning from the lowest ranks.²⁴ Idealism also asserted that there must be some kind of isomorphism between structures of models and the processes modeled by them.²⁵ Therefore, Betts thought he should present a geometrical model of human mental growth in progressive stages, one built upon the other, starting from bare consciousness and developing towards development of senses, with the condition that the resulting model should be isomorphic with the structure of human mind. Based on these conditions, it seemed that the growth of form of consciousness should be isomorphic to growth of natural forms. But for Betts, the investigation of the mathematical principles of mental morphogenesis extended his theory into to the fourth dimension.

Morphogenesis in fourth Dimension

Betts was a part of the group interested in the fourth dimension in the second half of the 19^th century. Although the notion of a fourth dimension was present in scientific texts even before the 19^th century, the concept became more popular after 1854 when G. B. F. Riemann (1826–1866) proposed his n-dimensional manifold.²⁶ This mathematical demystification of the fourth dimension influenced a wide range of disciplines, and shaped the “tradition of 4th dimension.”²⁷ Betts learned about the fourth dimension from James Hinton. James Howard Hinton (1822–1875) was a surgeon and writer who used the concept in his theories, for example, in “Life in Nature” published in 1862.²⁸ But the real influencer of the time was his son James Charles Howard Hinton (1853–1907) who had a significant effect on the “hyperspace philosophy” between 1870 and 1920. Charles Hinton applied the idea of a fourth dimension in multiple fields, and also suggested a method of visualizing the fourth dimension of the “tesseract” by projection, in a series of books and articles beginning with “What is the Fourth Dimension?” published in 1880.²⁹

Based on these teachings, Rudolph Steiner (1861–1925), Austrian architect, utilized the concepts of morphogenesis to establish isomorphism between the patterns in mind and the patterns in nature and ornaments. He based his art and architecture on esoteric interpretations of space and time including definitions for various types of spatial planes corresponding to different aspects of human existence. The proper architectural form, he believed, should originate from the soul, produced by a consciousness which has undergone a morphogenetic process to reach beyond the three-dimensional familiar world into the spiritual realms of higher dimensions. He wrote a number of books, and designed buildings like Second Goetheanum that reflected his idea of transcendental capacity of morphogenesis. Claude Fayette Bragdon (1866–1946), American architect, linked the morphogenesis of patterns of human mind to spatial patterns of architecture. Bragdon related the progress of human mind to its abilities in spatial perception, and argued that perception of higher dimensions by means of visualization frees human kind from the three-dimensional world that is full of egoistic elements.³⁰ The idea that human mental evolution has something to do with increasing dimensional complexity, led Bragdon to design two dimensional ornaments based on axonometric projections of four-dimensional objects that he called “hyper solids” in his “Projective Ornament” published in 1915.

Betts distinguished himself from Steiner and Bragdon by presenting a computable approach. He invented a mathematical model and symbology of his own; and he utilized the fourth dimension in his morphological theory before 1887, while Steiner and Bragdon used the concept after 1900. Betts' Geometrical psychology presents an interesting case study of the architects’ efforts to invent morphological theories of their own, and to design procedures that are among the first endeavors to approach morphogenesis from an algorithmic point of view.

Critical response

Although New Zealand was a much smaller and less complex society than Britain, the new home still had its own challenges. The architectural trends in New Zealand during the 19th century were more or less reflections of what was happening in Britain, but they were also influenced locally by the process of rapid urbanization and socio-cultural unrests and contradictions. Betts found a job as a Trigonometrical “computor” of the Survey Department in Auckland, and kept it till 1887. The job provided a stable stream of income for him to spend more time on his theory: a morphological theory of human psyche represented by geometrical diagrams called Geometrical Psychology or “Science of Representation.”

Betts wrote a letter including his theory and diagrams to John Ruskin. Ruskin, in addition to his high rank in architectural criticism, was also a member of SPR (Society for Psychical Research) in London.³¹ So, Betts thought, that his system will appeal to Ruskin. Because it was both a mathematical system to generate ornaments, and also a model that can explain the gradual development of psychic powers in human beings. Ruskin rejected the ideas of Betts, and wrote him in a letter: “creation of art is a spontaneous action, not limited by mechanical and strict rules of mathematics.” The theory of Betts, as the next section will explain, was not about “rules of creation of art”; it was about rules of morphogenesis of human mind which can be isomorphic to natural forms and architectural ornaments, and at the same time a representation of human spiritual journey through life. Moreover, the method developed by Betts was not deterministic as Betts himself tried to explain:

‘The laws of mathematics are absolute and final within themselves, they are certain so far as any knowledge can be certain, but the fundamental law of the Science of Representation is that of the undeterminateness of Form; hence absolutely uncertain.’³²

After that cold rejection, Betts had little hope that any famous critic would like to comment on his work. So, fueled by the belief that a true work is always a combination of male and female forms, just like in his theory, he sent a letter to his sister in England, along with a manuscript that was not a technical description of his system, but rather an emotional expression of what they meant for him. But the problem was his sister could not understand the theory. So, she decided to show the letter and diagrams to Mary Everest Boole (1832–1916) the wife of George Boole, the founder of Boolean algebra, and a secretary to James Hinton.

Mary Boole found that Betts was on the same path as George Boole whose endeavor to express logical thinking led him to use an algebraic language.³³ She also found the diagrams beautiful and interesting, and started a correspondence with Betts. Although Mary Boole criticized Betts’s unsophisticated mathematical techniques, like his naïve use of numerical series, she nevertheless tried her best to introduce Betts to the scientists she knew. She presented the ideas of Betts to James Hinton and William Spottiswoode, the President of the Royal Society (from 1878 to 1883). Hinton and Spottiswoode showed interest in Betts’s ideas, but apparently, they did not have the time to study them carefully or comment on them critically. If Betts could have presented his ideas in person to these prominent scientists, he could have communicated his own ideas better than anybody else, but that was the inevitable price to be paid for Betts’s voluntary isolation in New Zealand.

Nevertheless, Betts remained an active member of the local scientific community. He was known as a local scientist as it is evident from the Auckland Star’s tribute to his Chromographe machine dated February 9th 1887 (Figure 1).³⁴ There is also an announcement in the New Zealand Herald dated July 4th 1887 in which T. F. Cheeseman the secretary of the Auckland Institute, informed the members of their monthly meetings in which a paper by Betts called “New experiments on the nature of colour,” was due to be read (Figure 2).³⁵

Figure 1.

Betts was known as a local scientist as it is evident from this tribute of the Auckland Star, February 9th 1887.³⁴

Figure 2.

Auckland Institute’s announcement of a reading of a paper of Betts, July 4th 1887.³⁵

Betts did not use his ideas in design of buildings and ornaments, but he was not totally disconnected from architecture either. He patented a number of architectural inventions including door and window systems called “self –balanced doors and sashes.” He also invented an improved method of wiring or tying corks so as to prevent them being forced out of the bottle by the pressure from the inside, and a device for regulating the flow of liquids from a reservoir. These inventions tell us, as he himself notes, that practicality of architecture was important to him, but apart from his Chromographe that uses his theory of forms and colors, we could find no ornamental designs of him whatsoever. Betts himself did not publish anything. There are neither any pictures of him, nor any marriage or death records available in the national archives of New Zealand and the newspapers of the time. The works by Betts are rarely known in the world of arts. An exception is the works of Miss Shelley Simpson of New Zealand who has made artworks based on the “Chromogaphe” machine, and also built his patented door systems. The ideas of Betts are also referred to in design thinking methods and idea mapping by W. E. Newman.³⁶

Geometrical Psychology or Science of Representation

Geometrical Psychology, as the name suggests, is a geometrical representation of evolution of natural forms including the forms of human consciousness, designed to represent the human mental evolution from the lowest ranks to his communion with the divine will. It begins in two dimensions and extends onto four dimensions of space.

Geometrical Language

This theory is represented by geometrical elements that follow an algorithm of stagewise development starting from two dimensions and ending in four dimensions of space. Betts used a geometrical language to describe the evolution of human mind. He writes

“Human intelligence is demonstrable by Geometric forms as Symbols of Thought in a definite and, to a certain extent, an absolute manner, in a series of evolutions.”³⁷

The elements of the geometrical system devised by Betts are lines, circles and points which are shown in (Table 1).

Table 1.

The elements of geometrical psychology.

Element	Description
Point	Center of ego
Line	Alpha—Positing, soul
Circle	Omega-determining-matter
Angular measure	Comparison and differentiation
The outermost circle	a non-persistent limit of possible consciousness
Smaller circles	Different modes of sense consciousness
Lines/radii	Inward/outward sensation/Real activities
Dotted lines	Ideal activities
Contour	Experience/Memory/Unites the sensations
Male forms	Intellect/Quantification/Positing
Female forms	Emotion/Qualification/Determining

As shown in (Table 2), Betts assigns positive and negative signs to his stages ofdevelopment of consciousness alternatively. The positive stage is an objective and the negative stage is a subjective plane of existence and behavior. As the process of evolution of consciousness proceeds the representing forms increase in dimensional complexity.

Table 2.

The stages of morphogenesis.

Stage of evolution	Description	Dimension	Motive	Polarity
0	Animal sense-consciousness	2	Survival	-
1	Rational sens consciousness	2	Self gratification	+
2	Lower morality	2	Self-control	-
3	Psychic activity	3	Work	+
4	Sacrifice of personal will	3	Union with universal will	-
5	Spiritual activity	4	Union with the absolute	+

Stagewise Evolution of Forms

Betts began by animal sense-consciousness which has a representation in twodimensions. These leaf shapes represented a form of consciousness in which no persistent sense of ego is present (Figures 3 and 4).

Figure 3.

Alpha (male) forms.

Figure 4.

Omega (female) forms.

The stage of rational sense-consciousness comes next. It is the beginning of thehuman mental development. This stage, with a positive sign, is also represented by two dimensional diagrams in which leaf forms come in pairs of male and female forms. In this stage of “self-gratification” the male form has a positive sign with the angle of the apex smaller than the right angle, and the female form has a negative sign with an angle greater than the right angle (Figure 5). By help of a parameterization of the algorithm of Betts, one can compare the effects of manipulating the number of senses (number of circles) and initial values for starting the arithmetical series. It is evident that the shape of the diagram virtually remains the same for all these manipulations for this stage of development.

Figure 5.

Alpha and Omega forms begin to take the shape of leaves.

In the next stage with a negative sign called “lower morality” Betts introduces the class χ of his diagrams in which the progressions are not uniform because the element of will is added. Here humans begin to work according to their will, and self control begins to develop. The diagrams are three dimensional, and they resemble the corollas of flowers. The male ones are trumpet shaped flowers, and the female ones are bell shaped (Figure 6).

Figure 6.

The stage of lower morality for male and female forms.

The stage of psychic activity follows that of lower morality with a negative sign (Figure 7). Humans begin to “sacrifice” their personal will in order to unite it with “universal will.” Here, the compound scales appear again. The difference between these scales of progression can be either interpreted as pleasure or pain. The form of the corolla of these three-dimensional flowers follows no determinate rules. An irregular corolla shape, like that of a leaf, represents an uneven development of cognitive faculties because it is unbalanced.

Figure 7.

The stage of psychic powers.

Extension into the fourth dimension

The last stage is called “intuitive knowledge” with a positive polarity. The diagrams of this stage are four dimensional, and they belong to the “occult plane.” He writes

‘…we finally arrive at the fifth, or ground of science, or perfect knowledge, which I have stated to be accomplished in the reduction of all human form to a number and a harmony as the element of the sphere music of a Universe.’³⁸

At the time there were several methods of visualizing the fourth dimension, that is, themethods suggested by W. I. Stringham,³⁹ Charles Hinton, Edwin A. Abbott,⁴⁰ and others that were either based on three-dimensional cuts or projections. Betts used a method similar to projection to visualize these forms, and by the same method he arrived at the result that the form of matter in the fourth dimension should be crystalline:

‘He takes a pair of his antithetical forms of the third ground and draws them in opposite directions, so placed that their obverse forms overlap, and by combining these obverse forms by lines through their salient points he gets various shapes of crystals, differing according to the scales of the corollas he uses. Hence, he infers that matter on the higher plane will be crystalline; and when solidity is merged in a more transcendent objectivity probably matter will be no longer resistant.’⁴¹ (Figures 8 and 9

Figure 8.

The stage of intuitive knowledge, two dimensional projection of a four-dimensional form.

Figure 9.

The stage of intuitive knowledge, representation of a two dimensional projection of a four-dimensional form.

)

Philosophy of Colors

The colors he used for his diagrams were chosen according to his philosophy of colors. Betts believed that “a true science of Form or Life would proceed from an analysis of the laws of Light as revealed in color”,³⁸ and he patented a “Chromographe,” which produced different colors according to various forms of his diagrams, to study his theory on colors (Figures 10 and 11).

Figure 10.

Chromographe Patent Details (National Archives of New Zealand-photo by Shelley Simpson, used with permission).

Figure 11.

An artwork produced by Chromographe. Betts believed that Omega (female) forms manifest in a deep blue (artwork by Miss Shelley Simpson of New Zealand, used with permission).

“He cuts out the forms in cardboard or zinc. The most pronounced Alpha form possible produces waves of a beautiful crimson color. The corresponding Omega form a deep blue; slightly modified forms waves of orange and violet; while any form in which indeterminateness predominates produces chiefly waves of green, which he regards as the color of infancy and incompleteness.”³⁸

Betts tried to infer the laws of form and beauty by universal laws of mathematics, which turned out to be a computable theory. His “science of representation” of human thinking process expressed by the geometrical techniques and numerical series is one of the earliest examples of procedural and computable theories of morphogenesis.

The python/grasshopper script

Here we present a Python algorithm and a tool for Grasshopper (for Rhino 5) to generate some classes of the Betts' diagrams. The algorithm works for two dimensional diagrams with single progression series(i.e., A, H or G). First, we explain the overall Grasshopper scheme (Figure 12), andthen we proceed to the Python algorithm (Table 3). Python was chosen because it is a rapid prototyping scripting language, and Grasshopper can both integrate and visualize the Python algorithm in a 3D modeling environment which may be useful for future three-dimensional extensions of the algorithm (Figure 13).

Figure 12.

The Grasshopper scheme.

Table 3.

The Python algorithm.

1	#Parabetts
2	#An algorithm to draw the diagrams of B. W. Betts
3	#Works for single scale two dimensional representations (Figures 1–17 in Cook,1887)
4	#Ver 1.0 Sep 2020
5	—
6	# Arithmetical progression function
7	Def Nth_of_AP(a, d, N)
8	—
9	Return (a + (N - 1) * d)
10	—
11	# Geometrical progression function
12	—
13	Def GP(a, r, n)
14	geo=range(n)
15	For i in range(0, n)
16	curr_term = a * pow(r, i)
17	geo[i]= curr_term
18	Return geo
19	—
20	s = start +1 #set the initial value
21	d = cd
22	z = 0
23	—
24	f = range(int(scale)) #list for line length scales
25	g = range(int(scale))
26	Ang = range(int(scale)+1) #list for angular scales
27	Angr = range(int(scale)+1)
28	—
29	—
30	# Arithmetical progression
31	If prog==1
32	For i in range(int(scale))
33	Rcd = Nth_of_AP(s, d, i+1)
34	z = z + rcd
35	f[i] = z
36	—
37	# Harmonical progression
38	—
39	If prog==2
40	For i in range(int(scale))
41	Rcd = Nth_of_AP(s, d, i+1)
42	z = z + rcd
43	g[i] = 1/z
44	g = list(reversed(g))
45	z = 0
46	For i in range(int(scale))
47	z = z + g[i]
48	f[i] = z
49	—
50	# Geometrical progression
51	—
52	If prog==3
53	For i in range(int(scale))
54	Rcd = GP(s,factor,i+1)[i]
55	z = z + rcd
56	f[i] = z
57	—
58	Div = f[-1]
59	f.insert(0,0)
60	—
61	# Compute the angular scales
62	For i in range(int(scale)+1)
63	ang[i] = (f[i]/div * 180 * phi)
64	angr[i] = f[i]/div
65	—
66	# Reverse the line lengths to have the obverse form
67	if x
68	f = list(reversed(f))
69	Else
70	For i in range(int(scale)+1): #Angular calibration
71	ang[i] = ang[i]/phi
72	—
73	# Reverse the scale, and compute the Omega(female) forms
74	—
75	If omega and x and prog==1
76	g = f[0]
77	For i in range(int(scale)): #Compute the lengths
78	Rcd = Nth_of_AP(s, d, i+1)
79	f[i] = g
80	g = g – rcd
81	ang=list(reversed(ang)) #reverse the angles
82	For i in range(int(scale))
83	ang[i] = ang[i] + (180-phi*180)
84	—
85	If omega and x and prog==2
86	p = f[0]
87	For i in range(int(scale))
88	f[i] = p
89	p = p - g[i]
90	ang=list(reversed(ang))
91	—
92	If omega and x and prog==3
93	p = f[0]
94	For i in range(int(scale))
95	f[i] = p
96	p = p - GP(s,factor,i+1)[i]
97	ang=list(reversed(ang))
98	For i in range(int(scale))
99	ang[i] = ang[i] + (180-phi*180)

Figure 13.

Evolution and morphogenesis of a male form with arithmetical progression from 7 to 100 senses/mental faculties. The seemingly chaotic behavior in the middle and reorganization into the rebirth of the original form/Experience are interesting.

The grasshopper scheme

Input and output variables

At the left-hand side of the Python component are the control variables which are fed into the main Python algorithm, and the outputs control the evolution of forms. Input variables include: (prog) the progression type (i.e., arithmetical, harmonic or geometrical), the (start) value at the beginning of the series, the common difference (cd), the (factor) for geometrical progression, (phi) which determines the scale of angles, the number of terms (scale) which Betts denotes by alphabet letters (i.e., F is 6), real activity (x) which if turned off gives the obverse form, and (omega) which if turned on gives the omega (female) form (the default is alpha (male) forms). These controls after passing the main algorithm result in two lists of outputs. The first one (ang) determines the angular scales, and the second one (f) determines the lengths of lines and the radii of circles. These are the variables that will be processed by the rest of the scheme.

The components

First, the angles (ang) are converted into radians (by the Rad component) to feed the Vector Rotation component which needs both positive and negative values for both halves of the form. The output vectors then determine the line tangents (drawn by the Line component) or directions of the lines which start from the center point (Center component), and their lengths are governed by the (f) output. The Circle component draws the desired number of concentric circles with radii of (f) from a fixed center determined also by the Center component. The end points of these lines, determined by the End component, mark the intersection points with the circles, and are fed to the Interpolate Curve components to draw two halves of the final form resulting in a smooth contour.

The python algorithm

The algorithm inside the Python Script component, begins with definitions of two functions (AP() and GP()) which compute the arithmetical and geometrical series (lines 6–18). The algorithm then defines the initial set of lists, and computes the desired series determined by the (prog) variable (lines 30–56), and then according to the resulted progression computes the angular scales (lines 62–64). A conditional statement then checks the Boolean variable (x) and reverses the list of line lengths for the obverse form in case the Real Activity toggle is turned off (lines 66–71). To compute the omega forms the loops go through the series backwards: starting from the maximum value in the series (div) to find the radii of circles (lines 73–99). The last step is to reverse the list of angles for the omega forms. The (angr) variable (line 27) is the actual scale Betts uses in his table,⁴² and is useful to check if one has computed the scales correctly.To obtain the omega forms with (phi >1), that is, an angular domain greater than 180°, it is better to turn off one set of Lines and Curves components to have the desired diagram near what Betts has drawn, because as he mentions two curves overlap in this condition. Otherwise, it gives overlapping curves which may or may not suit the application.

For example, to obtain (Ω F A 0.5 φ 0.5) change the number of terms to 6 (F is the sixth letter), omega toggle to true, the arithmetical progression with common difference 0.5, and phi to 0.5. Normal Betts diagrams all begin with the value of 0 (zero) for start. For the obverse form set the Real Activity toggle to false. For the alpha form turn the omega toggle to false. An omega form only appears when the Real Activity toggle is true. This is because obverse forms cannot have omega (female) forms.

The code is not limited to alphabetical scales (A–Z) which cannot go further than 24, and works with start values that the user can set with any interval desired.

By manipulating the start values one can obtain a huge number of forms. These forms may not have a one-to-one correspondence with our mental activities, as Betts himself could not do that, but they give us an opportunity of enjoying the beauty of mathematical forms, and the immense variety they offer thanks to parameterization.

Suggested improvements and extensions

The Python algorithm can be improved, and made more flexible by having functions and two output sets of angular and length scales for compound series of progression. The algorithm can be extended to include also the philosophy of colors proposed by Betts. A more robust algorithm can be written in C for better performance and portability. Because the nature of the algorithm is modular, it is easy to edit or add other modules or other types of progressions to the algorithm.

The third dimension can be added by defining the normal vectors of circles tangents to the specified curve of growth. This would be an interesting extension because of the three-dimensional nature of morphogenetic processes in architecture and urbanism. One could then experience with different settings and resulted 3D objects that can be used in various ways. Three-dimensional objects can also be obtained by cutting or projecting four-dimensional diagrams if one desires to test the theory to its limits.

Applications

The script can be used as a design tool or as an educational tool. As a 2D design tool it is a rather limited but fun script to explore. Many families of patterns can be used as tiling or shell patterns. The flower shapes and trumpets depicted in Cook’s book are the simplest classes of all possible three-dimensional shapes. This is why the algorithm would be more useful if it can be extended into three or even four dimensions, because then it can be used to explore the morphogenesis of families of possible architectural or urban forms. In this regard, the main weakness of the theory is that it does not describe the rules governing the interaction of a network of its proposed entities. As such, concepts like “emergence” of form which are based on the interaction of parts within a complex system cannot be applied to it.

As an educational tool, this script can help students to have a better grasp of the dynamics of different geometrical settings and arithmetical progressions in courses related to mathematics or geometry, or it can be used as a part of any introductory historical course on evolution of parametric design. As the fourth-dimension gains popularity again in the pedagogy, an extended version of the algorithm can provide opportunities for learning some spatial qualities by providing three-dimensional cuts of the four-dimensional objects.

Conclusion

Morphogenesis has always been an interesting concept for architects. After thepublication of D’Arcy Thompson’s “On Growth and Form” in 1917, architects learned to understand morphogenesis from a mathematical point of view, and the book inspired many architects of the early modern period from Le Corbusier to Fuller. Benjamin W. Betts compiled a computable theory of morphogenesis that could be applied to natural forms and architectural ornaments alike. Betts, like Steiner and Bragdon, used the concept of morphogenesis in correspondence with evolution of the human mind, and extended its domain beyond the familiar three-dimensional world. But unlike his contemporaries he presented his ideas in a computable theory.

This article developed from a curiosity about a mesmerizing diagram on the Internet into full-scale research because in addition to the theory, the story and its protagonist were inspiring. The process challenged our historical understanding of development of computable morphogenetic theories. We also learned that the theory of Betts despite its charm was disconnected from the history of architecture in the sense of what Schinkel would call an “error of pure radical abstraction,” partly because it was never materialized in physical form. Therefore, development of new morphogenetic theories is more fruitful when the processes are informed of the actuality and materiality of architecture.

This research reinforced the ethical justification of the use of computational morphogenesis techniques and algorithms in service of sustainable urbanism and architecture by extending their historical horizon. This was done through analyzing a historical case of the influence of ethical and philosophical systems on the formation of new morphogenetic theories in architecture. In this case, Betts offers a vision of the transcendental capacity of morphogenesis in the constructive dialogue between architecture, mathematics and human love and imagination.

Footnotes

Acknowledgments

The writers show their gratitude to Miss Shelley Simpson of New Zealand for her generous and kind cooperation in sharing the documents and information regarding Betts, and also her permission to use her artwork in this paper.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is supported by Yazd University.

Author note

This paper was extracted from the first author’s PhD dissertation in Yazd University, Iran.

ORCID iDs

Hossein Soltan Rahmati

Mohammad Hossein Ayatollahi

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