As such, given that the in situ tiling by its very nature must serve as the model, this is thus examined and described. The most obvious observation is that it consists of one symmetrical pentagon, with two opposite angles of 90°, with four sides the same length, the fifth side (base) being longer. Further to this, when viewed as a tiling, subsidiary convex hexagons intersect at right angles. A further feature of this is that of collinearity when the subsidiary hexagon is extended. However, the exact condition as regards to angles has still to be established; I have two possibilities, from Macmillan or myself, either of which would appear likely
As such, the studies are based on the premises above, albeit with relaxed conditions; the number of pentagons, symmetry, angles, the side lengths, subsidiary concave hexagons, all being unrestricted.

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1 Pentagon

Given the in situ serving model, as detailed, I now analyse how the features of this can be ‘relaxed’, beginning with a single pentagon (larger number of pentagons are considered in succeeding chapters). Just for general reference, the properties of the serving model are:

One pentagon

Symmetrical

4, 1 sides (four short, one long)

Two 90° opposite angles

As a tiling, subsidiary convex par hexagons at right angles

As a tiling, lines of ‘nearest neighbour collinearity’

Special Cases

1. Equilateral pentagon (114° 18’, 131° 24’)

This possesses most of the properties of the serving model namely, symmetrical, two 90° opposite angles, subsidiary convex par hexagons at right angles. Lacking are the 4: 1 ratios and a long base and collinearity. However, although in normal circumstances such ‘omissions’ could be considered a drawback, in this instance it is not. Arguably, with the greater symmetry, this ‘improves’ upon the serving model, at least in an abstract sense.

Aesthetically this is very pleasing indeed, and arguably this is the ‘best’ pentagon in the series, not only here of the 1 pentagon category but the series as a whole, in that it is the most ‘basic’ of all the pentagon types, in that it possesses the most symmetry, namely of all five sides the same length, of which I consider the overriding issue as regards aesthetics. (The other examples in the series here can be described, at least in comparison, as weaker, with different side lengths). Indeed, this is my own personal favourite. Such aesthetics probably account for its popularity as the frequently stated serving model, likely from Gardner’s account. However, it does posses one relative shortcoming, in that it does not possess collinearity.

As can be seen, it is not necessarily a requirement that the sides be of different proportions, as in the 4, 1 of the serving model. However, this is very much the exception to the rule; this is the only 5, 0 pentagon possible.

As can be seen, aside from this, the subsidiary features remains the same, with two opposite 90° angles and convex par hexagons intersecting at right angles.

Also, of very much of minor importance, is that upon a ‘broad, casual glance’, it has the pleasing appearance of a ‘faux’ regular pentagon tiling, despite this being known not to tile.

2. Dual of the 3². 4. 3. 4 (120°)

This possesses most of the properties of the serving model, namely symmetrical, two 90° opposite angles, 4, 1 ratios, subsidiary convex par hexagons at right angles. Lacking is a long base and collinearity. Again, as above with the equilateral, likely on account of its aesthetics, many authors cite this as the defining model.

Aesthetically this is very pleasing indeed, and arguably, the second ‘best’ pentagon in the series, due to its underlying background, namely that of the dual of the semi-regular 3². 4. 3. 4 tiling. However, I consider that it pales aesthetically, in relative terms, as when compared with the equilateral, in that the major shortcoming is the sides lack those all of the same length. But that said, due to its underlying source it still has much aesthetic value. Furthermore, it has ‘pleasing’, round figure, ‘aesthetic’ interior angles, two of 90°, and three of 120°, something which the other ‘core value’ examples lack. In contrast to the equilateral pentagon, the sides not of the same length, with four of these being the same, with one decidedly ‘short’ in relation to the other four, described as of the ‘4, 1 short’ type (in contrast to ‘4, 1 long’, to others which is also possible).

3. Cordovan, by Buitrago and Iglesias (112.5°, 135°)

This possesses most of the properties of the serving model, namely symmetrical, two 90° opposite angles, 4, 1 sides, subsidiary convex par hexagons at right angles and a long base (the latter in relative terms). Lacking is collinearity of sides. However, it does indeed possess collinearity aspects, but of intersection rather than sides. When the sides are so produced it results in the ‘nearest neighbour’ pentagon in a like orientation intersecting at a nearest vertex.

Aesthetically, this is pleasing in regard as to the general resemblance to the serving mode, and is one of the strongest ‘round figure’ pentagons, and indeed in particular on account of its collinearity intersection. However, in matters of collinearity I still consider the side collinearity as a superior aspect.

Surprisingly, this particular pentagon is very little known, with apparently only Buitrago and Iglesias having written about this, in relation to their studies of the Cordovan proportion, hence the title.

4. Bailey (105°, 15°)

This possesses most of the properties of the serving model, namely symmetrical, two 90° opposite angles, 4, 1 sides, subsidiary convex par hexagons at right angles and a long base (the latter in relative terms). Lacking is collinearity of sides. However, it does indeed possess collinearity aspects, but of intersection rather than sides.

Aesthetically, this is pleasing in regard as to the general resemblance to the serving model, and is one of the strongest ‘round figure’ pentagons, and indeed in particular on account of its collinearity intersection. However, in matters of collinearity I still consider the side collinearity as a superior aspect.

This is somewhat alike in premise to the Cordovan example above, and essentially echoes its premise, in that it possesses the property not of collinearity of sides but of intersection, as when the sides are produced it results in the ‘nearest neighbour’ pentagon in a like orientation intersecting at a far vertex.

Amazingly, I believe this pentagons intersection properties is completely unknown, it arising as a result of my (hence my attribution) systematic search of pentagons with 5° increments. Certainly, I have not seen this quoted elsewhere.

5. Squared Intersections (108° 43’, 143° 13’)

Aesthetically, this is pleasing, but not in regard as to the general resemblance to the serving model; indeed, the pentagon is fundamentally different. This example, based on the reflected stick figure premise, with convex pentagons can be described as a special case, in that it possesses not two 90° angles, but three, with side ratios of 3, 2, and with convex subsidiary hexagons intersecting at right angles. Furthermore, and most interestingly, it is yet another instance of a one pentagon type possessing ‘nearest neighbour’ collinearity (and, I might just add, newly discovered at the amazingly late date of 19 October 2011! Are there yet others?). Another, pleasing aesthetic feature of this is that a Greek cross tiling can be discerned upon judicious selection, this further adding to its aesthetics.

This possesses some of the properties of the serving model, namely symmetrical, subsidiary convex par hexagons at right angles, collinearity. Lacking are the 4, 1 ratio, in that here the sides are 3, 2, and the two 90° angles, where here there are three.

The examples above can all be described as of angle considerations per se. Aside from such examples, one can also have examples based on unit intersections, which can have interesting connections, of which although these are also described as by angle, are, in general, more of a generic type.

Firstly, I show some examples of what can be determined as ‘special cases’, followed by ‘general cases’.