According to Christopher Alexander, beauty arises out of the wholeness, which is defined mathematically as a recursive structure, and exists in space and matter physically, and reflects in minds and cognition psychologically.
Beauty can be found both on and under the surface, and in the structure, and in the realization of function. Consider furniture: some veneers and lacquers are aesthetically pleasing in their own right, but we can also admire the elegance and craftsmanship of the elements of construction, both individually and holistically. Then we can go further and admire how well the construction is suited to its intended function. Sure, beauty can "arise out of the wholeness", but not always and not only.
@ Karl Pfeifer Yes, I agree with you that there are two kinds of beauty: surface beauty and structural beauty. The former is said to be in the eyes of the beholder, but the latter is measurable and quantifiable; see the mathematical model of beauty:
Article Wholeness as a Hierarchical Graph to Capture the Nature of Space
Now question is, given the two kinds of beauty, which one is more important or dominated? My answer is the structural or objective beauty; see the mirror of the self test:
To supplement my previous answer, the mirror of the self test I referred to is available here: http://hig.diva-portal.org/smash/record.jsf?pid=diva2%3A805296&dswid=nmWATopWindow
The mirror of the self test is NOT to seek inter-subjective agreement psychologically, but to seek the existence of the wholeness physically or mathematically. The following paper explored the physical and mathematical aspects of the wholeness:
Article A Complex-Network Perspective on Alexander's Wholeness
Music aesthetics can be explored with music appreciation. It has multiple dimensions and may provide more insights or directions to the question.
@ Makarand Velankar Despite of multiple dimensions, structural aspect is the dominated effect. For musics, it is melodic variation that evokes a sense of beauty, just as different sizes of bends (recursively defined) that make a coastline beautiful. Article A Smooth Curve as a Fractal Under the Third Definition
I'm far from an expert in this field, but I'm here to learn. I'm fascinated by Alexander's hypothesis, and just discovered your valuable work. As an amateur I can say I learned a lot of new math vocabulary today and I'm very interested in topology and it's use in our perceptions of space and data. Do you think your head/tail breaks analysis is the sort of mathematical revolution Alexander was looking for to define wholeness? Or is it still an approximation similar to, but more accurate than Santiago's degree of life calculations? Either way, it seems like a pretty important discovery for statistical analysis of space and data which is characterized by fractal patterns.
I think I might agree with you and Alexander that beauty is structural, but after trying to explain it to my fellow landscape design students and teachers, I realize how difficult it is to convey this concept.
Here: http://www.rainmagazine.com/archive/2015/living-structure-and-cognition
Greg Bryant does a nice job explaining the problem with teaching the idea that "wholeness/beauty is an inherent structural property of reality", and that it might help instead to begin by teaching how to feel this faculty or perception in ourselves. I see your attempt to do this by linking the mirror of the self test, but today people are often so distracted from their own feelings, and so indoctrinated to see discrete objects as wholes, that it's very difficult to convey the feeling of using the body as an instrument to detect structural beauty and wholeness, despite the fact that everyone is using this faculty every single day. We see wholes intuitively, and recognize the fractal nature of reality even through the lens of objectification
i.e. Subatomic particles < Atoms < Molecules < Cells < Tissues < Organs < Organisms < Populations < Communities < Ecosystem < Biosphere... < Universe
@ Trevor Lohr Many thanks for your fantastic and insightful questions!
My work helps bridge fractal geometry and Alexander's living geometry through a series of concepts or tools I developed such as head/tail breaks (ht-index), natural streets, natural cities, scaling law, the third definition of fractal, and beautimeter. I truly believe that the living geometry is more profound than fractal geometry, since the former is towards creation or design, while the latter is limited to understanding or explanation. The math model of wholeness I developed is exactly what Alexander was looking for I believe, because it is recursive. Other life or beauty models if non-recursive may work to some extent, do not capture the true meaning of life or beauty or wholeness.
I agree with you on the difficulty of conveying the concept of wholeness, because the (mechanistic) way of looking at things is wrong. By "wrong", I meant it is wrong in perceiving beauty or structural beauty. It is right for much of what we human beings achieved over the past 3 or 5 hundred years. Our ways of looking the world is fragmented rather than holistic. In this regard, I agree with you again that meditators, yogis, monks and similar others have better sense of wholeness, since we ordinary people (85%) tend to see things analytically rather than holistically. This is why Alexander conceived an organic world picture in contrast to Descartes' mechanistic one. However, we ordinary people can be trained to see things holistically just like meditators, yogis, monks and similar others.
I believe the degree of beauty topologically calculated is in agreement with human's reflection; see the empirical study conducted by one of former student: http://hig.diva-portal.org/smash/record.jsf?pid=diva2%3A805296&dswid=6082
I tend to say that beauty is 90% (or by a majority) accounted by the living structure, 10% (by a minority) by something else. According to Alexander and my model beautimeter, all human beings, even animals, have the same degree of beauty. For example, a tiger's face is living structure, so is a human's face. My way of analysis is based on fractal or living geometry. I tend to say any Euclidean shape (even a rectangle shape with Golden ratio) is ugly in terms of the underlying structure. However, we should not completely reject Euclidean geometry, since without it we cannot see far more smalls than larges. In other words, we must use Euclidean geometry to measure things first, and thereafter we are able to see whether there are far more smalls than larges. For creating beauty, yes, we must rely on the objective ways of analysis and understanding. BUT, for things that have the same degree of structural beauty, we need to use something else (10%) to account for surface beauty. For example, as said above, all human beings have the same degree of structural beauty, so we need to examine surface beauty to differentiate one people from another. What do you think?
In my opinion, beauty is manifestation of symmetry, so beauty of whatever system is under analysis, it is traced down to the symmetries in the system, which again is manifestation of the symmetries of the components that make up the system, all the way down to symmetries of leptons and quarks, and possibly the hidden underlying sub-realm and sub-sub realms not fully understood or unknown to modern science till date.
@ Trevor Lohr
Herewith I add my responses - both questions and answers - to your insightful comments:
[Trevor] Santiago's degree of life calculations
[Bin] What do you mean by Santiago's degree of life calculations? Do you mean Salingaros' degree of life calculations?
[Trevor] i.e. Subatomic particles < Atoms < Molecules < Cells < Tissues < Organs < Organisms < Populations < Communities < Ecosystem < Biosphere... < Universe
Dear Bin Jiang ,
could you please be more precise what do you mean with structure and surface?
What is structure and what is surface in the example you elaborate in the paper Article A Complex-Network Perspective on Alexander's Wholeness when comparing the wholeness of the two carpets?
If the mathematical model - beautimeter, that you have developed succeeds to measure the wholeness, why you still need the mirror of the self test to prove which one has more degree of life?
@ Aurora Saidi Thanks for your question!
By structure, I meant wholeness in which individual sub-wholes form an interconnected whole, being a living structure. By surface, I meant those perceived as individually fragmented pieces on an analytical or mechanistic perspective. Taking the Sierpinski carpet for example, if one considers it is just some different sized squares nicely allocated in a square space, you see the carpet on the surface. If instead, one considers it is an interconnected whole of different sized squares at different levels of scale, with far more smalls than larges, forming a scaling hierarchy, you see the carpet in the deep structure, thus leading to a living structure.
Wholeness is a living structure that can be measured mathematically, and the living structure is able to reflect in our minds and cognition psychologically. These two aspects (mathematical and psychological) are consistent. However, before the math model was developed, we had to rely on the human beings to detect the degree of life. After the math model, we can use the human beings to verify if the math model is correct. Herewith a related study:
http://hig.diva-portal.org/smash/record.jsf?pid=diva2%3A805296&dswid=1058
Put it in general, a square is more beautiful than a circle structurally. This is because a square is with five centers - the four corners and the center of the square, while a circle is with two centers - the circle border and the center of the circle. More importantly, with the square, there are far more smalls than larges, yet with the circle, there is NO far more smalls than larges, given these two centers. I am working on a paper currently, and shall share it with you as soon as it is well shaped.
NOTE that I am talking on beauty in terms of structure, rather than in terms of something on the surface. On the surface, you may prefer circle to square, but structurally, a square is more beautiful than a circle. One structure more beautiful than another is a matter of fact - governed by two fundamental laws: scaling law and Tobler's law - rather than an opinion:
Presentation Scaling Law and Tobler's Law for Characterizing Asymmetry in Geography
Dear Bin Jiang
Thank you for your answers. I am very familiar with Christopher Alexander's theory of wholeness. More or less, the phenomenon of structure and surface is also explained in your papers, but what concerns me is - can we use the mathematical model that you developed to analyse the degree of wholeness of more complex examples (beside the circle, square or Sierpinski carpet)? Could it be proved mathematically that one of the two carpets has greater degree of wholeness and that 90% of their structure is similar, the difference is only on the surface? (picture attached)
Do we still need "the mirror of the self test" because till now there is not yet developed a mathematical model that can fully capture the complex nature of wholeness in things?
@ Pieter de Kock
I cannot agree with you any more!
Human feeling remains very sensitive, and it cannot be fully replaced by any objective measurement. I should have said "largely replace or replaced ...". Or to paraphrase Alexander, 90% of our feelings are shared, and idiosyncratic parts account for only 10%. This is the message I wanted to convey with this thread of discussions.
@ Aurora Saidi
To reflect on your questions, herewith my brief feedback:
[Aurora] ... can we use the mathematical model that you developed to analyse the degree of wholeness of more complex examples (beside the circle, square or Sierpinski carpet)?
[Bin] My short answer is year, but the model can be further improved; see one of our recent studies:
Article Geographic Space as a Living Structure for Predicting Human ...
[Aurora] Could it be proved mathematically that one of the two carpets has greater degree of wholeness and that 90% of their structure is similar, the difference is only on the surface?
[Bin] Largely yes, unless it is disapproved; see Alexander's book The Nature of Order, the first volume's Appendices.
[Aurora] Do we still need "the mirror of the self test" because till now there is not yet developed a mathematical model that can fully capture the complex nature of wholeness in things?
[Bin] I tend to say yes, unless it is approved to be no. Even now there is a math model of wholeness, the mirror of the self test remains of great use. I would not use "fully", but "largely", to take a lesson from another discussion in this thread.
Bin Jiang
The paper (Geographic Space as a Living Structure for Predicting Human Activities Using Big Data) you suggested, as far as I could see, is about the living structure of cities.
Wouldn't it be interesting to compare the mathematical models of the two carpets mentioned above in the discussion, if it hasn't been done already?
Or are there any other similar examples (as the carpets) already analysed (beside those on the appendixes of The Nature of Order) in order to have a clear idea of how your mathematical model works?
@ Armir Ferati
Very good questions! What you raised deserve further research.
Herewith I list my publications on wholeness or living structure:
Article A New Kind of Beauty Out of the Underlying Scaling of Geographic Space
Article Wholeness as a Hierarchical Graph to Capture the Nature of Space
Article A Complex-Network Perspective on Alexander's Wholeness
Article A Topological Representation for Taking Cities as a Coherent Whole
Article Geographic Space as a Living Structure for Predicting Human ...
I have a couple of more papers under writing or review, and I share with you as soon as they become a bit well-shaped.
Dear All,
Beauty cannot be defined as a particular characteristic of some structures. It is a feeling that a persohn can experience when particular structure in all kind of domains are satisfactory. Take the example of good food. Good food can be a lot of different things. One may exclude with objective critera many type of food but one can't find objective critera that exist in all good food and would actually defined what is good food. The beautifull is an aesthetic experience that has some objectivity that comes through the aesthetic response of large group of people over long period of time. It is the outer manifestation in consciousness that the experience is positively transforming the person experincing it.
I am beginning the readingof ''The Nature of Order'' and I can't at this point comment on the opinion of Christopher Alexander.
Regards,
- Louis
@ Louis Brassard
Goodness of food is to a large extent subjective, while goodness of space is largely a matter of fact rather than opinion. Here the goodness of space is the notion of beauty that Alexander defined and developed in The Nature of Order. There are fundamental laws - scaling law and Tobler's law that govern beauty or goodness of space; one can refer to https://twitter.com/binjiangxp/ where many tweets are for understanding why beauty is structural or objective. I will be delivering a one-day tutorial on goodness of space:
Poster Goodness of Space As a Matter of Fact – Rather than Opinion ...
Dear Bin,
I do not see why the domain of food would not be 90% the same for all humans. There is no contradiction between what I said in my previous post about the beautiful and what I understand so far from the perspective of Alexander. Thanks for the reference to your paper. I will have a look.
Regards,
- Louis
Dear Louis,
The domain of food is a subjective matter (varies from people to people), while the domain of space is largely an objective matter. Let us use an example that Alexander often used, to see if it makes sense to you. When a group of kids say "we all love MacDonald!", it is very much an inter-subjective agreement. However, when a group of doctors say one has got a cancer, it is very much an objective judgement based on a series of diagnosis. The former example is an opinion expression, while the latter as a matter of fact to a great extent.
Thanks and cheers.
Bin
Beauty is deep in structure rather than on the surface. Do you agree upon?
Yes, because what is visible is temporary (i.e. if will fade away when time passes by) & what can't be seen on the surface is more long lasting & valuable.
Dear Bin,
Doctors received more or less the same empirical based type of education. They agree on how to come to agreement and so it is an intersubjective practice. There is no strict opposition between subjective and objective. The later is the subset of subjectivity we can agree, that does not depend on personal peculiarity. Every medical doctor is a different person as far as medicine goes they are very similar to each other and this make their intersubjectivity, objective. The kids unfortunate enough to grow up into a society where MacDonald has preaded, usually get addicted to this type of food and here they come to developed a common taste and appreciation for this type of junk food. So most of them come to agree and this subjectivity become in that way objectified by the very process of cultural uniformisation. But adults having the chance to have developed into a traditional food culture will agree tha McDo is junk food and even people of other culture will very quickly agree that some traditional food is good to them. But those having strong traditional negative biais against porc, insects, dog, etc etc will not often break these early learning. But I would contend that in general traditional food will received a high rate of intercultural approval. This is just a hunch and so this would have to be validated by some empirical studies.
Regards,
- Louis
Dear Louis,
My key message is the difference between subjective opinion and objective fact. Beauty lies in deep structure, so it is an objective fact rather than an opinion. Now that it is a fact, so 90% of our feelings evoked by beauty are shared, regardless our faiths, ethics, and cultures.
Cheers.
Bin
The structure is recursive if the shape of the whole recurs in the shape of the parts. In other words, the structure is recursive if there is a relation of identity between the shape of the whole and the shape of the parts. So, If beauty arises out of the wholeness, and if wholeness is defined mathematically as a recursive structure, then beauty arises out of certain relation of identity.
*********
There are many, so to speak, kinds of beauty like rhyme, golden ratio, symmetry, rhythm, etc. Analysis of these specific forms of general beauty shows that all of them are kinds of relations of identity:
• Rhyme is a relation of identity between one of two or more words or phrases according to the final sounds;
• Rhythm is a relation of identity, according to the time interval between the beats;
• The golden ratio is a relation of identity between the ratio of the whole to the larger part and the ratio of the larger part to the smaller;
• Symmetry is an identity relation between two sides or halves;
• Anaphora is a relation of identity, according to the initial phrase or word in consecutive phrases, clauses, sentences, or verses. (e.g., “It rained on his lousy tombstone, and it rained on the grass on his stomach. It rained all over the place”);
• Parallelism is a relation of identity according to the syntactic forms of two or more clauses sentences, or verse lines. (e.g., “The bigger they are, the harder they fall”);
• Assonance is a relation of identity, according to the vowels between neighboring or words in close proximity to one another. (e.g., “sweet dreams”, “hit or miss”);
• Anadiplosis is a relation of identity between the last and the first word of two neighboring phrases or sentences. (e.g., “rely on his honor—honor such as his?”);
• Epanalepsis is a relation of identity between a phrase or a word used at the beginning and the end of a sentence. (e.g., “Only the poor really know what it is to suffer; only the poor” or “The king is dead, long live the king“);
• Meter (in literature) is (1) relation of identity between feet, according to their structure; and (2) relation of identity between verses according to the number of feet (and syllables at the same time) they have;
• Etc.
This led us to an assumption that beauty in its essence is a relation of identity and the beauty of an object is the totality of identity relations it contains. English poet Coleridge in his “On Poesy or Art” essay writes: “…pleasure consists in the identity of two opposite elements, that is to say sameness and variety. …This unity in multeity I have elsewhere stated as the principle of beauty” (Coleridge, 1836, Lecture XIII).
Thanks @ Nikola Stojkoski for the insightful comments!
Two reflections (1) parts are not arbitrarily identified, but organically or naturally detected, (2) parts or sub-wholes (better termed as centers by Alexander) are defined at different levels, thus what you listed above can recur at different levels. Eventually the recursive structure can be represented as a complex network:
Article A Complex-Network Perspective on Alexander's Wholeness
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