Exploring non-regular polygons (posted 4/5/08)
The following studies seek to extend the methods developed in “Uniformity, Variety and the Beauty of Polygons” (see the listing on this website) to non-regular polygons. The aim is to discover a viable way to rank the systemic beauty of such figures on the basis of the multiplicity of uniformities they contain.
1. Non-regular pentagonals analyzed
The regular pentagon sports a perfectly concentric, endless series of exact look-alikes (self-replications) of diminishing size. The tilted (or inverted) pentagons and pentagrams that alternate in scale with the vertical pentagons and polygrams (call them “alternators”) are also entirely regular and their centers coincide exactly with the centers of the title forms. The diagonals of the first set coincide with diagonals of the alternators and mark the midpoints of the pentagon's sides. In all these ways the whole suite of forms makes maximally efficient use of the foundational points and intersections relative to any irregular version of itself. Every intersection serves multiple users. Figure 1.
Stretched, compressed or skewed pentagons, like all such polygons, invariably fail to some degree in these respects. But more than this, they also fall short of the regularity of the most regular of their imperfectly regular brethren. Even-sided regular polygons can be stretched without change of angles, producing a fairly robust regularity score. The highest achievers among polygons in this respect are the quadrilaterals, specifically the rhombus and the rectangle, which score 233 according to the system set forth in “Uniformity, variety and the beauty of polygons”. The suite of diminishing concentric forms in a rhombus, it will be recalled, divides into alternating rhombi and rectangles whose form remains constant as the size diminishes. The diagonals of all the members of the suite intersect at a common center at all size levels and in general observe the economy of intersections characteristic of perfectly regular polygons. Their suites are also endless.
Standing below the rhombus and rectangle among the quadrilaterals is the truncated isosceles triangle with top side equal to the sides. It falls just short of its betters in respect of sides (3 out of 4 are equal) and angles, but suffers a drastic fall in symmetry, retaining only the lowest grade of reflective symmetry. As a result its BU score is 150. One next step further down is the parallelogram, or stretched rhombus, with a score of 133. Below that come the set of truncated isosceles triangles with only two sides equal, which scores 100.
Stretched, compressed or skewed odd-sided polygons never equal the degree of uniformity of the rhombus and rectangle. The reason is that there is no way to stretch or compress them without changing their angles. Here are three examples of stretched or compressed pentagons. The inserts show the non-coincidence of the diagonals. The nested outline forms deploy the forms so to show how the angles shift in the sequence of title polygons and polygrams. Figure 1. Basic uniformity scores (BUS) are also noted. Figure 2.
In the raised pentagon A the first pentagram, being stretched vertically, produces intersections that skew the first tilted pentagram so that its points do not fall at the midpoints of the sides of A, as is evident from A1, B1 and C1. In consequence the diagonals of both parent and tilted pentagram intersect 3x3 (see inserts) instead of at a common center. Essentially the same divided alignment occurs when the first tilted pentagon is drawn so as to connect the midpoints, as in A3, and of course the diagonals of the one set fail to pass through the intersections of the sides of the points of the other. The members of the resulting suite of pentagons and pentagrams fail to replicate their predecessors, as is shown by the other diagrams, in which the forms are aligned so as to make clear the failures of congruence.
The other odd-sided polygons when stretched, compressed or skewed suffer the same loss of uniformity. We can take the pentagon as exemplary of the entire set in this fundamental respect. Such difference as there is between prime and non-prime odd-sided polygons need not detain us.
2. Non-regular hexagonals analyzed, as exemplary of even-sided polygons whose sides number more than four.
The great divide between the odd and even-sided polygons registers powerfully when we study non-regular sets. As with quadrilaterals, so with all even-sided polygons, exactly two preserve either the equality of all the angles or all the side-lengths of the regular polygon. As a result the network of connections, like that of the rhombus and rectangle, retains the regular alternation of two forms as the sequence diminishes in size, the diagonals of all members of the suite share a common center, and the network attains high economy. For hexagons the two are the rhombic and the stretched or broadened hexagons. Their symmetry is reduced, compared with that of the regular hexagon, since fewer of their diagonals are axes of reflective or angles of rotational symmetry. They BU score of 228 is comparable to that of the rectangle. Figure3.
Let us next look at the interior geometry of the stretched and rhombic hexagons. I start with the stretched hexagon because investigation shows it to have by far the most regularity. In fact it is the only hexagon that can compare with the best of the highest scoring quadrilaterals. Here is what we find. Figure 4.
B-D and C-E give the pairs of alternating hexagons and hexagrams, distinguished both by form and orientation, following the lead of the rhombus and rectangle. The fact that all are hexagramic does not really distinguish them from their quadrilateral counterparts: we have two common names, rhombus and rectangle, for the quadrilaterals, and only the single word for the hexagons. That is a merely semantic difference. The essential similarity is that the forms remain constant as their size diminishes. They also diminish by a fixed ratio, one-third or one-half, as is shown by B-F.
The interior geometry of the rhombic hexagon is radically different in itself and in relation to the rhombus. Its change of angles from 6/6 to (4/4; 2/2) has a marked effect on the sequence of interior forms, whose proportions undergo incessant alteration with change of size. No hexagon or hexagram within it (or beyond it) has all sides parallel to those of another. Efficiency of use of the intersections also sharply decreases. While all the forms share a center and are individually as symmetrical as the whole (as shown in 5C), their diagonals are not aligned with those of their fellows and no longer pick out the midpoints of the sides of their neighbors in the sequence. Yet the sequence here is endless and concentric. Figure 5.
In view of this failure of constancy of form we must give the rhombic hexagon a lower overall uniformity score than we assign the broadened one. (Note that for even-sided polygons the broadened and raised form is one and the same, differing only in orientation).
The next step downward in uniformity is taken when free rein is given to diversity of sides and angles. In particular when all symmetry is lost, so is concentricity of the central suites of forms, as is the case in B of Figure 6.
Nothing in B exactly matches up with anything. Geometrically such patterns are disasters.
3. Non-regular octagonals analyzed
Let us turn to octagonal examples. If our findings with regard to them conform to the findings in regard to hexagons, we may take their features to be common to all even-sided polygons with more numerous sides. We start with a reminder of the interior structure of the regular octagon, as our essential reference. Figure 7.
Then consider the three top scorers among octagons. Figure 8.
The BU scores of the rhombic and broadened octagons are slightly lower than those of their hexagonal counterparts because of the lower credit given for symmetry. Single or duple symmetry is lower rated because the optima are higher due to the number of sides. It is natural to wonder whether this relativizing of symmetry is valid. Further study is indicated.
But when we consider the interior structure of the broadened and rhombic octagon with the 8/3 stellations highlighted we seem to find confirmation of the lower overall uniformity of these octagonal forms in relation to their hexagonal counterparts. Figures 9.
The broadened octagon has nothing like the impeccable uniformity of contained forms of the broadened hexagon, as comparison of Figures 9 and 4 makes plain. The diagonals connecting points p - p +3 of the stellations do not lie parallel with the sides of the octagons larger or smaller than themselves. Their tendency is to narrow the rhomboids progressively with each diminishment of scale. Within three stages the rhomboid falls entirely within the rectangle, as shown in F; an alternative endgame is possible, shown in G, where the surviving space is an angled rhombus, its upright companion a fully contained square, as in the center of G. In both cases one of the two forms is too small to combine with the other so as to produce the octagon needed to carry the sequence further. Thus the suite comes to an end.
For its part the rhombic octagon in the following diagrams seems to hover at the edge of regularity. Figure 10.
But the matter requires closer inspection than these small figures allows. Thus we must move on to another, closer look. Figure 11.
B, D and F simply reorient A, C and E so as to make the figures' symmetry evident. From the reoriented figures one can see that the rhombic octagon is an evenly squashed octagon. Its preserving the 4/4, 2/2 arrangement of angles ensures its retention of 7/1, 8/2 symmetry and the partial parallelism that underlies this symmetry.
In these diagrams the true disorder of the interior structures becomes evident. In B, for instance, the smallest octagram becomes so irregular that it no longer encloses an octagon. The area inside of it becomes a distorted square circumscribing a diagonal rectangle. The latter is the vestige of the form that would provide the other four sides of the normal octagonal area. It has shrunk so much that it no longer reaches outside of its companion quadrilateral. As to the sequence of polygons, D shows how very distorted they become at the third stage. This corresponds, of course, with the degradation of the octagram just cited. Diagram C show that the degradation in the sequence of “alternator” polygons follows the same course as that of the primary ones. The interior of the third stage octagram is again reduced to a quadrilateral rather than the normal octagonal. The sequence of title forms therefore comes to an abrupt end. This is as radical an irregularity as can afflict a polygon.
Yet on the other hand the A/B diagrams show that at least one of the contained suites is endless, which is something of a puzzle.
Does every irregular polygon show itself subject to such a fatality if the sequence is long enough pursued? This needs to be investigated. Leaving that question hanging for the moment, let us next consider the more typical skewed octagon – typical in not having either all angles or all sides equal, therefore with zero symmetry. The 8/2 stellation is featured. Figure 12.
B gives the outlines of the interior octagons oriented for easy comparison of their shapes. It seems apparent that the sequence of octagons will not come to an end. One is encouraged to think this by the following, since the innermost octagram never seems to become less rotund. Figure 12.
4. What difference does multiplying sides make in a broadened even-sided polygon?
Let us glance at the broadened dodecagon to see what if any difference the number of sides makes in such even-sided polygons. Figure 13 .
The dodecagons and dodecagrams survive only through the third stage. Thereafter we have only a broadened hexagon. This is fully consistent with the findings regarding the octagon. I will assume that this means that even-sided polygons with a larger number of sides will produce the same pattern. Next we should test the rhombic dodecagon to see whether a similar conclusion can be drawn regarding its consistency with the rhombic octagon. Figure 14.
The lesson is that essentially the same pattern is continued by higher number rhombic, even-sided polygons. The polygons in the central sequence become skewed as to lose some of their sides. In the dodecagon the pattern determined by the 12/3 stellation terminates at the fifth stage (as in A-C) , and the one determined by the 12/4 stellation lasts only to the second stage (as in D). The dark figure in A-D shows the point at which the number of sides falls below twelve.
Interestingly, the 12/4 pattern is more regular than the 12/3 example in the center. Let us check out the 12/5 pattern, to see if there are any relevant changes. Figure 15.
The answer is no, the series quickly terminates, the dodecagon losing four of its sides by the third stage, as shown in A. Also as before there is the option of the consistently 12/5 stellations from the intersections of the points of the next larger set, as in B.