This is a supplement reading to my work 生成畫荷.

It’s not so much in the similarity between compositions, but the lack of certain compositions across all compositions that gives rise to an artist’s aesthetics in composition as a totality. Similarity between compositions is kind of like consistency in mathematical systems. For similarity or consistency with regards to aesthetic ideals, one can imitate a finite number of artworks. This is a matter of competency, and not autonomy. So I looked up lotus paintings by Daqian.

The lack of certain composition is related to both the completeness and incompleteness in mathematical systems. Completeness is the generative nature of composition that the composition statement or its negation is provable or determinable from the axioms. Axioms are like rules or basic building blocks that define any given mathematical system or aesthetic system. It is in my view, that it is the incompleteness of a nearly complete aesthetic system that defines an identity of an aesthetic system.

Kurt Godel and Alan Turing

Perhap it is a good time now to reference Godel’s incomplete theorem here to illustrate some of my thought process. It goes something like:

Any sufficient consistent system is necessarily incomplete. And complete systems cannot be consistent.

In Roger Penrose’s discussion with regards to computation (machines) and in the form of Godel-Turing argument, according to Penrose, Godel and Turing had came to essentially opposite conclusions with the same evidence.

Turing: … in other words, then, if a machine is expected to be infallible, it cannot also be intelligent. There are several theorems which say almost exactly that. But these theorems say nothing about how much intelligence may be displayed if a machine makes no pretence at infallibility.

Godel: … it remains possible that there may exist (and even be empirically discovered) a theorem-proving machine which in fact is equivalent to mathematical intuition, but cannot be proved to be so, nor can be proved to yield only correct theorems of finitary number theory.

Borrowing their arguments for intelligence, mathematical intuition and appropriating them in aesthetics, we get the following augmented arguments:

Fictional Artist Alan: if a machine is expected to be infallible (in composition), i.e. imitations of (existing) composition of mine, it cannot also possess their own (aesthetic) identity. But these theorems say nothing about how much aesthetic identity may be displayed if a machine makes no pretence at infallibility.

Fictional Artist Kurt: it remains possible that there may exist an art creating machine which is equivalent to aesthetic intuition, but cannot be proved to be so, nor can be provide to yield only correct compositions of my aesthetic theory.

Penrose concluded with the summary:

Godel’s argument was that there is a ‘loophole’ to the direct use of the Godel-Turing argument as a refutation of computationalism, namely that mathematicians might be using an algorithmic procedure which is sound but which we cannot know for sure is sound. So it was the knowable part which Godel thought was a loophole and the sound part which the sound part which Turing settled on.

In my view the above reasoning applies to computational aesthetics too. In that there are computations in aesthetics, and using Godel-Turing arguments to refute otherwise is not sensible. However, it is in the Godel-Turing arguments, that we are directed to find aesthetics in computation, of ourselves and in computations alike.

河圖 Hetu — River Map

In trying to read Daqian’s lotus compositions. There wasn’t any trivial model that came to mind. There was definitely calligraphy elements that shaped Daqian’s composition. In this work, I wanted to experiment with a smaller set of computation systems than calligraphy, which would basically be an exercise of a lifetime. But I also wanted the system to give as wide of variation in composition given the number of bits to be studied.

I was reminded of I-ching, a compact system, that is generative in its formulation, and is applied to many different applications. The basic unit of I-Ching is a yao, (爻), that is 1-bit binary. (— ) vs (- -), yin and yang. Then it is cascaded, 2 bit, it becomes si-xiang, four elements (四象). 3 bit, is ba-gua, eight gua (卦). For two sets ba-gua, we get the standard I-ching or 6-bits and 64 states.

The typical I-Ching representation is a grid of 8x8. To reduce the scope of the composition parameter space, I needed a different representation system. Since if I want to compose I would need large multiples of the 64 state space drawn in space.

There was two derivatives of ba-gua, hetu and luoshu, that places the 8 guas in a grid. My intuition was to choose hetu for this experiment. Maybe it was because of the pronunciation of he is the same in hetu 河圖 and in hehua 荷花. Or maybe it was because of the way hetu is arranged in a locally asymmetrical but globally symmetrical way that I was drawn to it, a curiosity in the exploration of compositions. While luoshu is in a perfect square of a sudoku variety.

The ten states in the hetu system is ordered into sub-states from 1–10 as follows on the sketch below. Globally the hetu system has a 4 edged star topology. With 2 to 3 layers of edges and nodes depending on how you view state 5 and state 10. Locally, the nodes of the global, is expanded to the number of the given ordered state. The symbolism between the states in relation to ba-gua and wu-xing is drawn in the table on the right.

I felt this was the right mix of manageable state space and output variation. At 64 states, it is already stretching our limits in recalling of semantically and visually meaningful differentiation of heterogeneous entities. Yet by manipulating with 1 digit shifts, the 2 sets of ba-gua, 6 sets of yao is transformed into 4 sets of ba-gua as follows:

[y0,y1,y2,y3,y4,y5]

=> [y0,y1,y2]+[y1,y1,y2]+[y2,y3,y4]+[y3,y,4,y5]

=> [bg0]+[bg1]+[bg2]+[bg3]

Though the 4 sets of ba-gua are not independent, i.e. this is still really a 6 bit systems, not a 12 bit one. But by way of the above manipulation, it does procedurally determine 4 states to transverse through the hetu coordinate system. This makes more expansive expressions.

畫荷 — Drawing Lotus

For every generative token, two sets of ba-gua is drawn. And the composition is drawn, with fixed points to the 4 interdependent sets of ba-gua drawn. The points within a state are randomly selected. A lotus may be drawn or it might not be drawn. But each composition can be used to infer its own meaning, categorically 64 of them corresponding to I-Ching. Or infinitely many compositions are drawn, lotus or not.

I could have chosen to have a lotus drawn, everytime. I did not. I did tilt the the odds slightly towards lotus and lotus-like outputs. This was my choice of composition. The lowered probability of the other 59 states to be drawn, was a choice. But what was drawn in the end, was chance. What gens and erates are drawn, lotus or not. — 生成畫荷

Application of I-Ching

In this composition the origining coordinate kun 坤 (earth ground), shines above the second organizing coordinate gen 艮 (mountain). Which is random draw by chance, by fate, by hash. It is interested our ancestors, found meaning in symbolism relationship in nature, applied to living in society. So this gua (arrangement) converged into warning or recommending the person who drawn this gua to be humble. Since ground should not be at higher altitude than mountains. This is just one of the 64 gua 卦, i.e. interpretation of I-Ching, completely built from 2^6 combination of yaos 爻. It's fascinating, we do not need more than 64 scenarios to tell what may we face in life. We end with all 65 minted 生成畫荷.