BRIDGES - 2010

This paper, in well-formed English with almost no mistakes or typos (other than the bibliography, which needs to be better formatted), gives an overview of Saxon's explorations of certain extremely simple, self-referential or hierarchical, 2-D patterns or geometries, typically in only one or two colors, that he has been making since he was a teenager in the late 1970s.  His drawings are clearly mathematical explorations, and he understands some of the mathematical implications (convergence of summed areas) of his hierarchical explorations of squares and triangles.  But I've got a lot of misgivings about the paper, and I'm sorry to say that I don't think it should be accepted.

The title and abstract immediately grate, where he refers to himself with the authority of the third person.  He then posits that he has created a "coherent theory" (presumably encompassing the "poly-universe" of the title), but there's no real theory at all in the paper.  Much of the rest of the abstract makes overblown statements that smell of an unwarranted theory of everything and sacred geometry.  But the drawings presented seem to me to be too rudimentary, austere, and obvious to support such a theory.  Or is the entire abstract a quotation from Beke?  Either way, it's really not appropriate, and raises questions.

Unfortunately, instead of educating himself on well-known terminology known the world over, Saxon has created his own language to describe his thoughts, but he has used mathematical words, like "poly-dimensional" that are mathematically inaccurate, and serve more to confuse than to illuminate.  For example, the sentence "By the deliberate fusion of the black tone of the different-scale squares our eyes are stimulated to see one poly-dimensional square, in contradiction to mathematical laws."
What mathematical laws are contradicted? I don't believe the author knows, because he's using his own non-mathematical terms made up of mathematical words.

In an attempt to distinguish his work from early rudimentary fractal illustrations of 30 years ago, he speaks of his "'auxiliary plane' technique".  What it is I can't figure out.

Given that he knows the word "fractal", and knows that some art historian "discovered" his work as fractal (which is not completely true, Figs. 1, and 8-11 are clearly not fractals), I find it strange that he doesn't also know about, e.g. a Cantor set in 2 dimensions.  This is particularly troubling because the art historian he refers to, who has written some kind of treatise in Hungarian on Saxon that Saxon has no doubt read, does talk about the Cantor Set (see Google translation URL below).

The title's "Footless Chair" image is nothing more than a stacked set of space-extruded approximations to a 2-D Cantor set.  And of course, it would be footless only in the limit, but the construction doesn't converge vertically; the tower would be infinitely tall because each iteration is a "floor" of the same height.  If it were truly footless, one would be unable to climb up and sit in it, so there's a mixed metaphor there.  There was a construction of this type that properly converged, first drawn in the 1980s as the cover of a SigGRAPH proceedings.  And it was far more clever and interesting (it was titled "HierArchy" or something similarly amusing).

Another thing that bothered me a lot was that all the figures are labeled with media (e.g. "Oil on Wood"), and yet they all appear to have been drawn very exactingly by computer (diagonal and circular lines look like perfectly anti-aliased pixels, etc.).  So the illustrations are not of his paintings, which I would have liked to see.  Perhaps this is because of knowledge that the proceedings are in Black & White.

A book/article by the art historian Perneczky that Saxon cites, in Hungarian, is on the web, and a Google translation was helpful:

But it appears that much of the content of this paper is rehashed from this earlier or other reports, which shows a little bit of some paintings in yellow.  Many of the same illustrations are used.

I'm also not feeling confident that the art historian Saxon relies upon is all that knowledgeable, or that this is a particularly scholarly endeavor.  If it were, there would be nothing new being reported.  For instance, when I first saw Fig. 1, I immediately thought of a print by Alan d'Arcangelo, a pop-artist in the 1960s known for painting highway signs, consisting of nested diagonalized squares, with little graphic abstract variations here and there.  Unfortunately I can't find on the web the particular print I'm thinking of (which I own) to demonstrate my recognition.

The last two figures of the paper have little aesthetic interest or mathematical merit that hasn't been explored numerous times by others for the last half-century (at least), usually by high-school students.

I am hampered by not having access to any of the exhibition essays that the author uses as his primary references.  But this is yet another indication that the paper seems to me more a gallery PR piece than a scholarly endeavor.

In short, too much self-aggrandisement and not enough content.  I have learned nothing new from this paper, and I'm pretty sure that this will be true for others in the field. PC only: My grain of salt: I am terribly biased against papers whose authors present themselves in the way this paper does, starting with the Title.  It is peculiar for an artist to use some contemporary art "historian" as his primary reference to the importance of his own work.  Party of two.  Yes, artists get to pontificate about their own brilliance sometimes, but it helps if their art has more aesthetic merit than I find Saxon's rudimentary drawings to have.  There's no mystery, mathematical or aesthetic, to them.

I would love to see how large or comprehensive the books cited really are, but it appears they are (invariably inflated) exhibition pieces, which often use "artspeak" to sound deeply important.

I feel bad, because Saxon is plainly interested in many things that Bridges folks are interested in, myself included.  So feel free to outvote me for humanitarian reasons.


Review: Content Review:

This paper is very appropriate for the Bridges Conference.  It combines mathematics with the art of story-telling.  Strengths of the paper include its general appeal and its simplicity.  It provides necessary definitions and notation that is easy to follow.  The examples given in the transition matrices and the impact matrices help to illuminate the mathematics behind the never-ending story possibilities.  However, I would have like to have seen a better clarifying example at the end of section 3.2.  In addition, the paper seems to “set things up” well for the reader to understand how the process works, but it then only briefly refers to its prototype and ends rather abruptly.  I suspect space limitations play a part in this abrupt ending and I also suspect that the presentation of this paper at the conference will include some visuals of the “prototype in action” which will be very interesting to the audience.  

Thus, because the paper is interesting and readable and makes a nice connection between mathematics and literature I recommend to “accept” (score of 2) this paper.  I did note a number of minor grammatical corrections that need to be made to make the reading of this paper smoother.  I’ve noted those grammatical suggestions in detail below.

The “Nitty-Gritty” Grammatical Editorial Suggestions:

In Section 2, paragraph 4, the line, “Based on this discussion, we provide the following definitions by extending Forster’s categorization,” needs the word definition to be plural.  Then heading for the succeeding definitions that following also needs the word Definitions to be plural.

In Section 2, paragraph 5, the line, “For instance, if there are two statements that follow each other as …” needs the word follows to be changed to follow.  Further in that paragraph, the second to last sentence should begin, “As a result, …”

In Section 3, paragraph 3, the second sentence should read, “Different rule sets for events can lead to different mathematical approaches.”  For the rules that follow, should “un-predictableness be unpredictability?”  Also, the first sentence describing “logical continuity” should read, “Many states of a character last for a while.”  The last sentence of that same description should read “For example, surprise should not last long.”  For the rule “personality consistence,” the first sentence of the description should read, “Each character has its own personality, which should have a significant impact …”  The last sentence in that description should read, “For example, a happy person is probably happy most of the time, and a sad person may rarely be happy.”

In Section 3, paragraph 4, the second sentence should read, “To illustrate how the Markov …”

In Section 3, paragraph 5, in sentence 2, the word labeled is spelled incorrectly.  The third sentence of this same paragraph should read, “Each vertex in   corresponds to a state … labeled by  , which gives the probability that the state …”

In Section 3, paragraph 5, the last sentence should read, “For example, by comparing the above two personality matrices, we can easily tell that the first character,  , seems happier than the second character,  .”  Note that I think the authors ought to explicitly how this is evidenced in the comparison of the matrices.

In Section 3, paragraph 6, the second sentence should read, “This problem can easily be solved by letting users enter any number by assuming that the higher the number, the higher the transition probability.”  The next sentence in this same paragraph should read, “Then, the matrix can be …”

In Section 3, paragraph 6, the fourth sentence should read, “Thus, we can …”

In Section 3, paragraph 7, The last sentence should read, “Also, since surprise never lasts for a long time, the transition probability from surprise to surprise should be small.

In Section 3.1 paragraph 1, the first sentence should read, “The personality matrices are useful for development of individual characters who are not affected by other characters.

In Section 3.1, paragraph 4, the second sentence should read, “Therefore, for example, in the first matrix, an angry character,  , has 0% probability of making the other character,  , happy, has a 10% probability of making   sad, a 40% probability of making   angry, and a 50% probability of making  scared.”  The fifth sentence in this same paragraph should read, “If   becomes happy, then   most likely becomes sad.”  The sixth sentence should read, “Of course, the impact in these examples is not commutative, i.e., character  ’s impact to character   may not be the same as the impact of   to  .”  Note that I question whether the last sentence in that paragraph is necessary.

In Section 3.1, paragraph 5, sentence 1 should read, “To implement … where   are positive real numbers that add up to 1 …”

In Section 3.1, paragraph 6, sentence 1 should read, “The sociability parameter and closeness vector of a character  are closely …”  The second sentence in this paragraph should read, “A very social persona has a relatively high value of s, so …”  The third sentence in this paragraph should read, “On the other hand, a less social person has a smaller value s ….”

In Section 3.1, paragraph 7, the fourth sentence should read, “More specifically, …”

In Section 3.1, paragraph 8, the last sentence should begin, “Nevertheless, all these…”

In Section 3.2, paragraph 1, the fifth sentence should read, “Another advantage of emotional states is that there is more than one expression for any given emotional state.  The sixth sentence of this same paragraph should read, “For instance, a sad person can cry or can have a sad face, a social smile or a neutral face.”

In Section 3.2, paragraph 2, sentence 6 should begin, “In this figure, …”  The next sentence should read, “Other sets of images are created using photographs of real people.”

In Section 3.2, paragraph 3, sentence 3 should read, “On the other hand, another person may show the anger.”  The next sentence should italicize the word “expressiveness.”   Note that this paragraph ends with an example, “So if e = 1, ….”   This is not a very clear example.  I recommend it be clarified. PC only: None Time: Mar 14, 22:33


In keeping with the spirit of being constructive, I have spent a great deal of time to salvage the paper so it is
publishable.  The author still has work to do on the figures.  If the substantial revisions I suggest are not carried out, I
strongly recommend that it be rejected.


I like the prose style in the paper, and I admire the mosaics. I'd like to see this paper in the proceedings. Now, on to the nitpicking!

I have a few nits to pick with the title, and with the central thesis, which hinges on the very subjective word "attractive" and uses the hedge word "often": I should start by saying these nits are quite minor and I'm not entirely convinced the paper needs editing. It's a bit more of a post-read kibitzing session. But here they are.

1) "one can *often* produce much more *attractive* mosaics with a smaller set of tessarae ... or a smaller number of shades of gray ...." (emphasis mine) The word "often" is a hedge word which can make a statement you're not fully comfortable with more comforting. The statement has the same level of truth whether the word is used or not (and my internal Strunk-meter says to omit it for this reason). I presume the author used it because he is aware that that there are certain constrained tile sets that are fully incapable of producing attractive mosaics (for example, a single tile). An attractive mosaic requires a well designed set of tiles, constrained or no.

2) The problem with the word "attractive" is that it is quite subjective. The author finds his constrained mosaics attractive. I find them attractive. But I don't think it's demonstrable that they are attractive, because the the word is so fuzzy. What does "attractive" mean, in this case?

a) For some mosaic viewers, part of attraction is the fidelity of the mosaic to the original. In this case, I don't think that's what the authoris getting at, as the black and white tile sets tend to produce more edge noise than set with a larger number of gradient colors.

b) The computatability of the mosaic effects the level of attraction, but more for the constructor than the viewer.

c) I assume the author is talking about the basic aesthetic response - does the viewer like the result? Are they aesthetically pleasing? However, it's possible the author is talking about any and all of the above, in which case he should make it clear. It's clear (and I've personally experienced that) constraints are a useful crucible for the production of art. They can have a great effect on the productivity level of the artist. To say that the results are more aesthetically pleasing implies that they can also have a great effect on the viewer of said art. Maybe so - certainly if they help the artist get stuff accomplished that would otherwise not be accomplished, they have an effect on the viewer. Maybe they make the art more aesthetically pleasing as well. I'm not convinced one way or the other. It depends on the constraints. Essentially, the paper is a survey of these two methods, which are interesting in and of themselves, preceded by an unprovable thesis statement, which is really more of an observation. The methods are worthy of survey, whether or not the central thesis statement is provable. Because the paper is a survey of the two methods, it doesn't really even need a thesis statement. These methods are an interesting outgrowth of other methods, used by the author, which are also quite constrained. Ultimately, it is up to the viewer to decide whether the mosaics are more attractive than equivalents produced with larger tile sets. Many of them will indeed decide so. I did. What makes these pieces interesting is the design of the constrained tile sets themselves. The design of the tiles is part of the attraction of the piece. I suspect, a randomly chosen, arbitrary set of constraints would not necessarily produce more attractive results. However, in these cases, what actually was the constraint? Was it the black and white tile set itself, or the desire to produce a well designed black and white tile set?

3) Similarly, the title (a condensed version of the thesis statement) also relies on a hedge word, "sometimes", which can make almost any statement true. Yes, sometimes less is more. Sometimes less is less. Sometimes more is less. Sometimes more is more. I think we've covered all bases! Here are some alternate suggestions: Constrained Mosaicking: When Less is More or the more prosaic: "Two constrained mosaicking methods" (the equivalent of "The princess and the seven little men").

4) My spell checker says mosaicking is not a word. Let's ignore it, it's cute, especially with the K in there.

5) I find it very interesting that both methods, particularly the second one, are significantly computationally less intensive than some less constrained methods the author has used. I wonder if it bothers the author when a method turns out to be computationally easy? Does it make the image less or more satisfying knowing how much computation underlies its generation? I recently produced a piece of music which required a great deal of underlying computation, and it made me inordinately proud of the result, but I felt compelled to explain it to everyone so they would suitably appreciate it. Ultimately, I prefer art which is self-explanatory, where its "attractive" qualities are right in front of you. These methods seem to have that effect. I think the fact that these methods produce *sufficiently* attractive results with significantly less computation is interesting, and might be a more provable observation to base the thesis statement on, should a thesis statement be necessary.



Review: I'm a little bit on the fence on this one.

The paper could use a thorough editorial pass for English correction. I'd be happy to provide this pass, but I'd prefer to do it on the original, rather than providing a separate list of edits - there's probably a few dozen needed, although the paper is otherwise quite readable.

The bulk of the paper is about the problems presented by Pythagorean tuning, and the problems caused by the various compromises (including equal temperament) that have been devised to deal with them. This is an interesting and apparently well-researched section, although the perceptual issues were not written about by Stevin, the subject of the article, himself - he was primarily concerned with the mathematical aspects of finding a simple common representation for the interval of a half step. In other words, he was more concerned with mathematics than perception. The author does indeed address this incongruence in his concluding section, "Courage or Ignorance," concluding that while Stevin didn't write about the perceptual problems caused by then in-vogue tuning systems and the need to "conquest" more and more keys, "he must have recognized the importance of the problem."

I'm not so sure, but in the same concluding section, the author makes a good point that the Pythagorean and the Stevin system are trading in two different kinds of simplicity that are mutually incompatible. This observation alone makes the paper interesting reading, I think.

I'm not entirely comfortable with the author's use of words like "Courage", "brave" and "shameless" to describe writings on music theory, but what the heck.

Finally, the references are a bit scant, over-relying on online sources.

Go to top