The Tessellations of Craig S. Kaplan
Craig Kaplan’s tessellations are of a degree of magnitude below the leading lights in the field, with pass marks in six of the ten desired attributes for quality tessellations, of which although a relatively high assessment is tempered by these not being essentially weighted towards the more important categories, 1 and 2 below. Indeed, on the more important categories his tessellations fail, with criteria as listed in the introduction:
(1) X The inherent quality of the motif (silhouette and articulation)
(2) X Showing the whole motif (excluding ‘heads’)
(3) ✓ Excludes ‘breathing room’ and overlaps tessellations
(4) X The number of tessellations in the body of his work
(5) ✓ Variety of motifs
(6) X A tendency to the more difficult to achieve motifs
(7) ✓ Excluding wireframes
(8) ✓ Contrasting colouring of tessellations
(9) ✓ Rendering
(10) ✓ Borderline
Kaplan produces his tessellations by what he terms the ‘Escherization’ process, by computer, with only a modicum of artistic assistance by the individual (in contrast to most computer programs, where the roles are reversed). He shows 21 tessellations, derived from his thesis, ‘Computer Graphics and Geometric Ornamental Design’, of which Escherization forms a relatively small part, as Chapter 4.5. As such, although he shows his tessellations in a clear manner, as straightforward plane tilings (in contrast to some other artists), the merits of the tessellations are not easily discussed as otherwise may be thought, in that Kaplan purposefully strives for confluences and connections (albeit at the expense of intrinsic quality), of which although laudable in aim, is not necessarily conducive to producing high quality tessellation art. (Likely, without this restriction he imposes upon himself, he would show better quality tessellations.) In this he is unique, in that none of his contemporaries pursues this matter, at least to the above extent. Therefore, it is an open question as to whether his tessellations should be put marked up or down for this; as one can argue both pro and against this. However, to me at least, the quality issue is by far the more important, and so his tessellations are, reluctantly, marked down.
For the sake of straightforward comparisons, reference below is made to the top tessellators in the field, with Bailey, Bilney, Crompton, Escher, Nakamura, Nicolas and Scalfittura, of which for the sake of for brevity this group of artists as a body is referred to below as Bailey et al. who show their tessellations in the most typical way.
In various places below I pose questions concerning Kaplan’s tessellations that are left unanswered. I emailed Kaplan about these points, but had no reply. Therefore, although obviously not ideal, of necessity speculation is sometimes offered instead, with the qualifier that this is open to change. Although this is not ideal, circumstances dictate, and so I qualify these comments with the proviso above in mind.
Each of the categories above is discussed in more detail:
(1) Recognisable in Silhouette (Failure)
A major shortcoming in Kaplan’s tessellations is that they are nearly all not immediately recognisable in silhouette, the most important aspect of tessellation. Indeed, only Tea-sselation and Strange ‘Tractors (with a bit of imagination) can be said to be recognisable in silhouette, with the rest broadly non-descript. However, on rare occasions, of specific circumstance, this can be overlooked, as occurs with Tux-ture Mapping, of which the silhouette is most impressive, despite being non suggestive.
Of note and concern is the general lack of typical silhouettes – why no quadrupeds in the most typical sideways position (with the possible exception of Pigs in 2-Space)? Why no insect types as seen from above? These omissions is all the more puzzling as Kaplan makes reference to this, page 180, but seemingly neglects this aspect, leaving it to ideas for subsequent work. Contrast this with Bailey et al who show their tessellations in the most typical way.
The articulation is particularly poor in many, with most possessing essentially lower quality surface embellishment. Notable poor examples include The Owl and the Pussy Cat; Rembrandt and Mrs Van Rijn, A Walk in the Park, albeit this is qualified by pleasing confluences.
The aspect of recognisably of the motif, as seen in silhouette, is fundamental to the premise of a quality tessellation, and cannot be overstated enough. Lesser artists struggle with this concept (inexplicably so, given its simple premise), and fail to recognise its importance and unfortunately delude themselves as to equating interior life–like interior detail surface detail with a exterior outline that is articulated.
(2) Shows the Complete Motif (Limited Success)
A major shortcoming in Kaplan’s tessellations is that although he does indeed include the ideal whole-bodied tessellations, in percentage terms this is far too low, as he includes a noticeably high proportion of lower intrinsic worth of ‘heads’ (this including a ‘head and shoulder’), no less than 7 out of 21, 33% of his oeuvre: Escher’s Escher Escherized, Dogs; Dogs Everywhere, Bubbles the Cat, Gödel, Bach (Braided:) An Eternal Escherization, Dogs and Cats Living Together, Pentalateral Commission and ‘head and shoulders’ Rembrandt and Mrs van Rijn (7). Contrast this with Bailey et al, with these artists essentially excluding such types as this category is lacking in any challenge of worth, being all too easy. Again though, this concern of mine is qualified to a limited extent by his favouring confluences, but this should not excuse such a high percentage. Such bias towards the lower worth here is to be frowned upon, at least to this extent. Ideally, one’s oeuvre should consist of at the very least say 90% of complete motifs, and ideally much higher percentages; indeed, if not too their essential exclusion, with exceptions being recognizable portraits. That said, if these are shown with a specific aim in mind, namely of a recognisable portrait here, of Escher, and so examples of this type are indeed acceptable (in contrast to a generic head). However, this is the only worthy example of the type.
Lesser artists frequently do not understand the difference of intrinsic worth between the two types, and undertake such ‘head’ only examples (on account of their less challenging aspect), and unfortunately delude themselves as to equating these with the more challenging whole body motif.
(3) True Tessellations - Excludes ‘Breathing Room’ and Overlaps Types (Success)
Kaplan’s tessellations and are in effect all instances of the ‘true’ type as defined by mathematicians, i.e. a tiling without spaces or overlaps, and so pleasingly excludes the invariably inferior breathing room/gap and overlaps types favoured, unfortunately, by many. That said, he does on just one occasion, with Tea-ssellation, employ what I term as ‘natural’ breathing room, as part of the object, to better portray the inner space between the handle and pot, which to all extents and purposes is of the ‘no breathing room’ type. As such, I have no real qualms about this whatsoever, and so these are thus regarded as a bona fide tessellation.
Lesser artists frequently include ‘breathing room’ types and overlaps types (on account of their less challenging aspect), and unfortunately delude themselves as to equating these with the more challenging ‘true’ type.
(4) The Number of Tessellations (Failure)
Kaplan shows 21 tessellations, which is comparatively few, and of note is that his output is considerably lower than with most other artists. For comparison, Nakamura has 268, and Escher 137, albeit both of these are padded somewhat, in contrast to Kaplan’s. Given such a relatively low number, one might expect that these would be of relative high quality, on the premise of few in number, but high in quality. However, this is not so, with the tessellations being poorer quality to Bailey et al.
Lesser artists will typically show a large number of inferior examples, and consider (incorrectly) that such large numbers outweighs quality. It does not. One tessellation of true worth is superior to hundred (or more) decidedly inferior examples.
(5) Variety of Motifs (Limited Success)
Kaplan shows just 14 different motifs, which again is decidedly lower than Bailey et al. For comparison, Bilney has 39, Escher 32, and Nicolas 32. That said, I consider 14 a fairly reasonable number, although somewhat on the low side. One redeeming feature here is that some of these are of motifs/creatures not usually shown, such as teapots, tractors, pen nib, roses, but this is militated by their general inherent lower quality as well as interest per se, beyond confluence. For example, is a pen nib and rose of any real interest per se? However here, he has purposefully been selective, in that he is looking for combinations of motifs of a punning nature, such as paying tribute to Roger Penrose’s tessellations (pen and rose), and so I am not unduly critical here. Lesser artists frequently shy away from undertaking variety, showing simpler to achieve birds and fish to the exclusion of variety, and unfortunately delude themselves as to equating these with the more praiseworthy variety of motifs.
(6) Challenging Motifs – Primarily Human Figures (Failure)
Kaplan’s oeuvre is noteworthy in many ways in that this is to the almost complete exclusion of the more challenging (and aesthetically desirable) human figures (I disregard the ‘amputation’ types here, of ‘heads’ and ‘head and shoulders’), with just one example of the genre, Twisted Sisters, and furthermore this is lacking in silhouette, with voluminous clothing. Quite whether this omission is purposeful is unclear. Did Kaplan try out such a motif? Perhaps Escherization struggled here. This is fairly telling, as it would appear that Escherization would have difficulty in achieving this particular motif. However, one could argue that this oeuvre does indeed contain challenging motifs, such as tractors and teapots, and so he should not necessarily be marked down for this. However, I still feel that ‘challenging’ should include human figures, and so I thus mark him down, albeit with qualification. Lesser artists frequently shy away from such examples (on account of their challenging aspect), or when shown are best described as ‘mutants’, with disjoint elements, such overly long arms or legs, or anatomical inconsistencies, preferring the simpler to achieve birds and fish.
(7) Excluding Wireframes (Success)
Pleasingly, Kaplan excludes the wireframe type, a favoured type of the inferior tessellator. Lesser artists frequently show wireframe examples for no good reason (from which one can only conclude is that they do not understand the issues).
(8) Contrasting Colouring of Tessellations (Success)
Kaplan’s tessellations as a premise are coloured in contrasting colours in a largely consistent way, given the demands of his style of presentation, i.e. of true-to-life motifs, of a photorealistic nature. However, on occasions this rule is not invariably used, but is broken; however, it can be seen that Kaplan knows what he is doing here, in contrast to other rule breakers, who show none of his subtlety. Although again, due to the format used, such as with photographic examples, this is occasionally ignored, instead differentiation is emphasised with decided outline. Occasionally this rule is overlooked for the sake of greater symmetry, such as Penrose Chickens, where essentially two colours, of a contiguous nature are used. An especially pleasing feature is that of the use of tints on appropriate occasions. Specifically, this refers to Escher’s Escher Escherized. Due to the nature of this tessellation, with a substantial part of the heads of the same beard/hair colour where these are contiguous, a ‘natural’ colouring this would result in the heads being not readily discernable. To overcome this, Kaplan has used different tints to better emphasise each tile. I consider this a very pleasing solution to this (often encountered) problem of a head type tessellation, and one that I have not seen elsewhere. Certainly, Escher did not employ this device, albeit admittedly he had no real to do so. Although only a relatively minor subtlety, this attention to detail is most gratifying. Although I would not object to a ‘one colour head’ throughout here, and with a white border, this small nuance enhances the tessellation.
(9) Finished Rendering (Limited Success)
Again, this is somewhat difficult to assess, due to similarities in style. However, three distinct types of rendering can be discerned:
A Plague of Frogs, A Walk in the Park, Dogs; Dogs Everywhere, Escher’s Escher Escherized, Gödel, Bach (Braided): An Eternal Escherization, Pen/Rose tiling, Busby Berkeley Chickens, Rembrandt and Mrs van Rijn, Strange ‘Tractors, Twisted Sisters, Dogs and Cats Living Together, Isohedral Chickens (13)
The Owl and Pussycat, Pentalateral Commission, Wiener Dog Art, Pigs in 2-Space (Hamm the pig), Sketchy Dogs, Bubbles the Cat (6)
Tea-sselation, The Complete History of Computer Graphics, Tux-ture Mapping (3)
Kaplan’s style, as regards finish, varies to a considerable degree, from photorealism to simple cartoon-like; although with a strong predisposition to photographic. This is not likely through choice, but rather due to the nature of the Escherization process, in which it would appear natural to import photorealistic images. Likely, his use of photographic rendering is purposeful, given his predisposition to manipulating existing pictures. It would be interesting to know of his views on this. However, for all the apparent desire for photorealist rendering as the pinnacle of achievement is not necessarily the best means of presentation, in terms of identifying the motifs immediately. Too much detail only serves to make discernment more difficult. The adage ‘less is more’ comes to mind here.
Lesser artists sometimes render the motifs in too much detail, believing this to be superior to a more simplistic rendering. As a rule, generally a simplistic finish is to be preferred, as too much detail hinders a clear interpretation of the motif.
(10) Borderline (Success, Emphatically)
Kaplan invariably employs a borderline, of either black or white lines, as detailed below:
Black lines: A Plague of Frogs, Gödel, Bach (Braided): An Eternal Escherization, Tea-sselation, The Owl and Pussycat, Penrose, Pentalateral Commission, Twisted Sisters, Wiener Dog Art, Pigs in 2-Space, Sketchy Dogs, Bubbles the Cat, The Complete History of Computer Graphics, Dogs and Cats Living Together (13)
White lines: A Walk in the Park, Dogs; Dogs Everywhere, Escher’s Escher Escherized, Isohedral Chickens, Busby Berkeley Chickens, Rembrandt and Mrs van Rijn, Strange ‘Tractors, Tux-ture Mapping (8)
Of note is the dichotomy, with a decided favouring of black lines (13) to white (8). However, such choices of black and white are not dictated to by apparent favouritism or whim, but by circumstance. Kaplan here has been most circumspect, in which he choose the most appropriate borderline. Observe that a black line is used where the colours are mostly light and a white line where the colours are mostly dark. The reason for this is in giving the best contrast between the tile and border. Note that this occurs in all 21 tessellations, and so thus cannot be accidental. On the aspect per se of borderline, occasionally, ‘strong’, single colours are used that don’t strictly require a borderline e.g. Pentalateral Commission, Wiener Dog Art. Even so, even in such instances, he nonetheless uses this feature, and so one can surmise that he considers a borderline highly desirable, if not essential. Indeed, on both choice of colour for the borderline, and borderline per se where it is not necessarily a requirement, he invariably uses one. Indeed this is one of the strengths here, in that on all occasions he uses this optimally, whereas most, but not all other artists, at best only pay lip service to this aspect. Without doubt, he understands the issue here, in that a borderline is a decided boon to recognition. Undoubtedly, this is secondary to the quality of the tessellation itself, and so of less importance to other, more fundamental issues, but as detailed above is a decided gain if correctly employed.
Lesser artists invariably omit this feature, not understanding the reasons for its general desirability, namely that of aid in discerning the motifs.
Discussion on Escherization
Of note is the ways and means Kaplan goes about designing his Escher-like tessellations, in that it is noticeably different from just about everyone else in the field. As such, he eschews the usual use of pen and paper, and instead uses a computer, and furthermore in a decidedly advanced way, with what he calls his ‘Escherization’ program. Note that this is not the electronic equivalent of pen and paper (as with other users and computer programs, who use such as Tess or TesselMania!), but rather to approach the problem with such a tool in a different way. Here, he begins with a predetermined motif in mind (rather than an arbitrary motif that in essence ‘evolves’ to a more specific one during the design process of most other people), and then ‘plugs’ this into Escherization, resulting in a optimised motif emerging over a range of different (isohedral) tiling systems. Most people use, in his terminology, a ‘forward tiling process’, of an arbitrary tiling tile, from which the artist, upon seeing a potential or rudimentary motif performs a series of refinements for the sake of a more realistic motif. However, Kaplan starts from the opposite end, and begins with a motif, from which the motif is then essentially adapted into a tiling tile (based upon any one of the 93 isohedral types, of which this is reduced to 45 different ‘types’). The method he uses is not a matter of what I would term as 'design', as he in effect takes an arbitrary shape and instructs the computer to adapt this into a tessellating shape. This is very interesting indeed, in that this would apparently take a whole lot of drudgery out of the design process in the ‘traditional’ way, in that one can see across a whole swathe of tiling systems as to whether of not the motif is acceptable as regards inherent quality, this being in contrast to the traditional way, of much trial and error. As such, this can be regarded as the Holy Grail for producing tessellation art; if a program can be written to accomplish this, any need for human effort would be made redundant. Indeed, progress is being made along these lines. However, the tessellations produced by Escherization are not yet of the highest ranking. Many are lacking in inherent quality. When Kaplan’s tessellations are examined, most have shortcomings, to greater or lesser extent. Of note is a preoccupation with a lower category motif of difficulty, of heads (although whether this is a purposeful choice is unclear, or just an overlooking of the worthy types is unclear), although this is militated to a degree by in one instance of a specific portrait. Such a category is barely of consequence. Bailey et al completely disregard these. Other, more worthy whole-bodied motifs are used, but these are in general also lacking. However, as Kaplan himself states, pages 166 and 181, Escherization should not be looked upon as an entity in its own right, but in that it can be used to see which motifs ‘show promise’, upon which the human can then take over and refine the initial output. However, to me, this procedure is somewhat of a compromise. If testing ability, man against the machine, it should be one or the other, rather than a hybrid. For example, in the chess world, when the computer program Deep Blue plays opponents in this ‘man against machine’ challenge, no collaboration is entertained.
Pleasingly, aside from the end product of Escherization, Kaplan shows the source motif, and from this one can then judge the efficacy of this procedure by direct comparison. As can be seen, the verisimilitude is very good indeed, albeit tempered by the choice of material, in which the silhouette is general lacking, being essentially formless. To me this is somewhat poor practice, as why not in effect challenge Escherization with more recognisable silhouettes? Surely it would have been a simple matter to import such material; it surely makes no sense to omit, as one picture is in this sense like any other picture.
Kaplan also points out further work to be done, such as weighing various aspects of the outline, in terms of criticality. As he (correctly) points out, page 180, some aspects are more critical than others. Apparently, Escherization lacks the ability to make such fine distinctions. Such matters should be addressed, this being a fundamental issue.
Of note is that all the above is based upon ‘Escherization 2000’. It would indeed be interesting to see how this has developed in the decade after this.
Undoubtedly, Escherization still has some way to go before it can compare with the top humans in the field. An open question is to whether it ever will. An analogy perhaps can be drawn with that of a different field, music. Although it is possible to reduce musical notation to a mathematical analysis, and thereby with the potential to examine ‘all possibilities', there appears to be no shortage of new music being composed by humans, of whatever genre. A possible similar situation may also arise with tessellation – the variety of motifs, of differing poses, of differing symmetry systems may prove impractical to unite to a computer program, at least for now, in the relative short term, however this is defined. Certainly, although I am critical of some of the tessellations produced so far, I really am most impressed with the potential of this. At the very least, it is an aid to the artist, in that ‘promising avenues’ can be explored, and so concentrate one’s efforts to one of more potential success, rather than wasting much time by going down blind alleys as is usual in the traditional way of designing.
Kaplan shows a reasonable number of examples (in total, 21 tessellations), of both one- and two-motifs:
One-motif, pages 143-147, with 11 tessellations
Two-motif tessellations, pages 154-156, with 10 tessellations
No one number is apparently preferred. No matter, the relevant question is the quality aspect, in that are the one- and two-motifs on an approximate par as to quality? As such, I slightly favour the one-motifs, but the difference is not great. Confluences and Combinations One especially pleasing and interesting aspect to his tessellations is one of striving for confluences and particular combinations, as well as various puns, an aspect most people ignore. This at least shows true understanding of the issues, in that the ideal is for the tessellation to be themed in some way, and so make sense. All too often, artists ignore this with unlikely combinations and/or with ridiculously different scales. Kaplan often alludes in his tessellations to his many interests in mathematics and computer graphics, as well as tributes to various pioneers, with many of the tessellations reflecting this background:
(i) Computer Graphics: with teapots, with Tea-sselation (in many incarnations, including hyperbolic geometry), referring to the ‘Utah teapot’. http://en.wikipedia.org/wiki/Utah_teapot Rabbits (Bunnies), referring to the ‘Stanford bunny’. http://en.wikipedia.org/wiki/Stanford_Bunny and in a different context http://www.cgl.uwaterloo.ca/~csk/projects/semiregular/Observe that both these motifs/aspects are also united in a single tessellation The Complete History of Computer Graphics
(ii) Computer Animation: with a pig motif, Pigs in 2-Space (Hamm the pig), refers to the pig in Toy Story
(iii) Computer Systems: with the Linux mascot, Tux-ture Mapping http://en.wikipedia.org/wiki/Tux Tux refers to the Linux mascot, of which the tessellation is quite superb; indeed, it could easily be mistaken for the actual Tux logo
(iv) Escher: with a tribute to the pioneer in tessellation, M. C. Escher, with Escher’s Escher Escherized. The merits of this are detailed above
(v) Mathematics: with reference to Penrose tiling, with Pen-Rose tiling (devised by Roger Penrose) and Strange ‘Tractors (strange attractors) (vii) Musicians and Logicians: with reference to Bach and Gödel, and also acknowledgement in the title of the work to Douglas Hofstadter’s book Gödel, Escher, Bach: An Eternal Golden Braid, with Gödel, Bach (Braided): an Eternal Escherization
Such confluences make for a pleasing change from the usual nonsensical combinations. However, it must be said that these are mostly militated as to their quality, as these are, for the greater part lacking, excluding Escher’s Escher Escherized and Tux-ture Mapping.
Although I consider many of Kaplan’s tessellations to be generally lacking, there are indeed some of true worth:
Unfortunately, shortcomings, if not outright failings permeate in his work:
The main failing is that they, for the most part, fail the all important silhouette test, and so are essentially tiles with surface embellishment.
Particularly poor as a category (with the one exception detailed above) are the ‘head’ and ‘head and shoulders’ type tessellations (the latter being even more arbitrary than heads), of which by their inclusion are a detriment to the oeuvre, especially so in these high percentages (q.v.). In normal circumstances, I would assert that the author does not understand the issues of producing quality Escher-like art, or at least recognises it, but this is Craig Kaplan, a mathematician of renown we are talking about, of which it must be taken as a given that he does. But then why include so many of these…? When these are tested, as a silhouette, the resulting shapes are unrecognisable as heads. Particularly poor is Rembrandt and Mrs van Rijn, both arbitrary amputations, and furthermore of different scales, which again jars. Better would be to have the motifs of the same scale.
Noticeably weak in content are his two-motif tessellations albeit with purposeful confluences, such as Dog and Cat Living Together, Rembrandt and Mrs. van Rijn, The Owl and the Pussycat. Although a decided effort has been made to compile sensible confluences (in contrast to most other artists, who disregard this aspect, showing disjoint motifs) these are, however, for the most part, let down by their quality. ‘Natural confluences’, as here, do not override the quality aspect. Although a pleasing concern, and indeed such examples should be encouraged, this should not be at the expense of quality, as occurs time and time again here.
Strange ‘Tractors inadvertently shows clearly the unsuitability of inanimate objects, in that the wheels, in this instance, are forced into gross distortions (albeit another inanimate one, Tea-ssellation, a teapot, is good, albeit this lacks interest on a human level). That said, as there are only inanimate examples, I am not unduly harsh on this here.
Noticeably weak are the Penrose tiles (albeit at least Kaplan undertakes these challenging tiles, something which most artists ignore), and are let down by their quality. None of these are in essence worthy, at least as judged by higher standards. Certainly, some are fun in a light-hearted way, with various puns and side references to Penrose, but this is at the expense of quality, something of which such novelty can’t be overridden. The Penrose tiles can be said to test one’s mettle, in that here one is more restricted in choice of tiles. In effect, the artist is presented with a set of tiles, and challenged to produce something of worth. However, as ever, the quality issue arises. Although a pleasing concern, and indeed such investigations should be encouraged, this should not be at the expense of quality.
Overly Detailed Rendering
Many of these I have concerns with as to their rendering, namely with the photorealist types. These are decidedly awkward to discern. These include Strange ‘Tractors, A Plague of Frogs, A Walk in the Park. Although undeniably ‘clever’, I consider such presentations as over engineered – a simpler approach is better.
To me, many of these are just novelties, undertaken for the sake of a punning name, for example, Pen/Rose Tiling being one of many. The rose and nib are unrecognisable in silhouette (indeed the nib more nearly resembles a bottle).
Exclusion of Human Figures
As detailed above, the general exclusion of human figures is somewhat to the detriment of the oeuvre here, as ideally he would show more than just the one, Rembrandt and Mrs van Rijn, and furthermore this is of relative poor quality, just surface embellishment. However, he is not alone in this, in that other respectable tessellators, such as Crompton and Bilney also show very few. However, there is a world of difference in their other respective efforts!
All this preoccupation with lower worth types is somewhat surprising, as it can be seen by his writings that he truly understand the subject, as on page 180 he recognises the different views of different creatures that should be presented, echoing Escher.
Is he better than Escher? Well, no. The examinations below should show this:
Inferior in the ‘silhouette test’ with Escher (no figures available)
Fewer numbers of tessellations, 21 than with Escher (albeit Escher’s 137 was padded to some extent, whilst Kaplan’s are not), of generally inferior quality
Less variety of motifs with Escher, 14 against 32
Less challenging motifs (i.e. human figures) than Escher, 1 against 4
Kaplan, or more exactly his Escherization program, is what I term as a middling tessellator, neither particularly good, nor bad. Unquestionable, he lacks the quality of Bailey et al. Aside from the superb Escherization program aspect, puns and confluences per se do not outweigh tessellations of inherent quality. However, this is not to say that his work is entirely worthless, as he is very far from the opposite side of the spectrum, as with the execrable efforts of Jinny Beyer. There is indeed the odd example of true worth here, but this is masked somewhat by the inferior types permeating throughout his work.
Created 14 October 2010. Revised and enlarged 19 August 2011