Appendix I, Detailed Treatment of Analytical Symmetry

The Rise to Dominance of Abstract Methods

Neglect is far from being a novel phenomenon in the field of geometry. Of all the areas of mathematics, this field has been most susceptible to changing tastes and the vagaries of fashion from age to age. Geometry reached its zenith in classical Greece as a synthetic physical discipline (pure geometry), but fell to its nadir at about the time of the fall of Rome. With the blossoming of the analytical approach, following the innovations of René Descartes, (1596-1650) and Pierre de Fermat (1601-1655), geometry in the 17th century stood on the threshold of a new era, only to be all but ignored by research mathematicians for almost 2 centuries more, as it fell into the shadows of the ever-proliferating branches of analysis.

Geometry's sudden rediscovery and revival, and the ushering of the "Heroic Age," came chiefly with the dawn of the 19th century, following the efforts of Charles Julien Brianchon (1785-1864) and, particularly, Jean-Victor Poncelet (1788-1867), the effective founder of projective geometry. But the early-to-mid 19th century proved to be a period of often bitterly contesting forces in physical geometry, with Jakob Steiner (1796-1863), generally regarded as the greatest geometer of 'modern times,' effectively championing synthetic methods, and Julius Plücker (1801-1860), undoubtedly the most prolific of all analytical geometers, struggling to advance analytical approaches. In the end, both camps-in fact the entire discipline of physical geometry-were eclipsed. Compelling new forces had come into play.

To trace the origin of the emerging and currentl"Abstract Age," we must turn back to the first third of the 19th century and some of the most brilliant achievements in the field of mathematics. The stage was set, on the one hand by Evariste Galois (1811-1832), with his theory of groups, and on the other by Nicolai Ivanovitch Lobachevsky (1793-1856) and János Bolyai (1802-1860), with their studies (about 1830) in non-Euclidean geometry.

Building within the discipline of Galois' group theory, Felix Klein (1849-1925) and Sophus Lie (1842-1899) achieved results leading to Klein's remarkable general definition of "a geometry" in 1872, together with his advocacy of the Erlanger Program, a new, and ultimately exceedingly fruitful, program of abstract geometrical study. [Recall in this connection, Ruark's assertion about my animal behavior studies, "It seems to me you stand in somewhat the same position as Klein when he wrote the Erlanger program. After 53 years the consequences of that single paper still serve as a current stimulus to research." I'll have more to say about this assertion later. JLK]

Because the studies of Lobachevsky and Bolyai compelled geometers to adopt radically different viewpoints, they led to a profound re-examination of the foundations of geometry. The repercussions were so great that the foundations of virtually all fields of mathematics came under critical scrutiny. The most influential figure in the resulting postulational treatment of Euclidean geometry was David Hilbert (1862-1943). His Grundlagen der Geometrie, in 1899, broke down the last barriers to wide acceptance of the view of a purely hypothetico-deductive nature of geometry and firmly rooted the same method in virtually every branch of mathematical endeavor.

As a consequence of these achievements, the research interest of academic mathematicians turned increasingly from the concrete to the abstract and from specifics to the greatest possible generalizations (one of Grothendieck's great achievements). The major efforts of Euclidean geometers now became channeled into foundation studies, with almost 1,400 articles devoted to this field alone from 1880 to 1910. This turn of events prompted Bertrand Russell's (1872-1970) facetious comment that "mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."

With such concerted efforts along abstract avenues over a span of 5 generations, analytical research in physical Euclidean geometry inevitably fell into a state of neglect. Most 20th-century mathematicians are exposed to this field only as students in beginning courses, in which analytical geometry often is covered only as needed to prepare them for the calculus. They usually gain the impression that analytical geometry, like arithmetic and trigonometry, is a 'closed book.' The few researchers whose efforts continue to touch this discipline employ highly abstract, generalized approaches. [In this regard, consider a comment to me by Dieudonné (the mentor of Grothendieck) in a letter dated March 27, 1982, "....mathematicians at higher levels have stopped paying any attention to plane geometry and special curves; what they are after is properties of general algebraic curves and their invariants, stemming from the work of Riemann."]

Useful perspectives are provided in the 1981 presidential address of Sir Michael Atiyah to the Mathematical Assoc. of America, Titled, "What is Geometry/"

Of all the changes that have taken place in the mathematical curriculum, both in schools and universities, nothing is more striking than the decline in the central role of geometry. Euclidean geometry has been dethroned and in some places almost banished from the scene. The battle between geometry and algebra is like the battle between the sexes. It's perpetual. It's an ongoing battle. And it really is a battle in the sense that these are two sides of the same story, and you've got to have both sides present. It's the kind of problem that never disappears. It will never be dead and it will never get solved. The dichotomy between algebra, the way you do things with formal manipulations, and geometry, the way you think conceptually, are two main strands in mathematics. The question is what is the right balance?

Revising and Rejuvenating Treatments of Analytical Geometry

This is the background against which the present treatment of analytical geometry must be viewed. Instead of being reinvigorating periodically through continued scrutiny, reassessment, and revision by academic researchers, this fountainhead was all but cut off generations ago, leaving the subject virtually in limbo (with consequences that cannot be rectified by more course requirements and homework or better-qualified instructors.

An aim of the present work is to supply materials and suggest new approaches for the revision and rejuvenation of analytical geometry at all levels of instruction. The contributions of Chapters I and V-VIII (of Structural Equation Geometry) are virtually entirely new. Coverage of the rectangular and polar coordinate systems (Chapter II and III) emphasizes interesting and conceptually important aspects and relations that often go unmentioned in the classroom or have escaped previous notice. Most of Chapter IV, The Bipolar and Hybrid Polar-Rectangular Coordinate Systems, also consists of new material. It includes the first in-depth analysis and elucidation of the bipolar system and its equations and curves, and the only explicit recognition and treatment of the hybrid polar-rectangular system, which plays a key role for derivations in non-Cartesian coordinate systems.

Topics are presented in a form suitable for an advanced undergraduate text, including 155 Problems. Most of these also cover new ground. Specific major recommendations for course revisions, emphasizing applications to beginning courses are given below and at the end of Chapter IV. Instructors of courses at all levels will find abundant additional pertinent subject matter in the main text.

A strength of the proposed materials and methods is that the same approaches that introduce greater rigor, systematic organization, and more penetrating insights, also provide much needed substance to which students can better relate. Not surprisingly, these approaches run counter to the mainstream of the last century of mathematical endeavor, namely, in the direction of providing more concrete and specific material, rather than abstractions and generalizations.

The New Subject Matter

Details of how I came to re-examine the subject matter of Euclidean geometry were given earlier. I found the field to be in so great a state of neglect that examining it from new viewpoints took on the aura of excitement and anticipation of unearthing and inspecting an ancient treasure, piece-by-piece. By default, physical Euclidean geometry had become the frontier to a vast unknown; almost any step from the beaten track trod on virgin ground. Even for the line and the circle-curves that presumably could hold no secret-new properties were discovered (previously unknown equations for their circumpolar symmetry). The following are some of the highlights of these new findings.

The equilateral hyperbola as a curve of demarcation of Cassinian ovals and self-inversion at 90°

Nineteenth century mathematicians were amazed that over 100 years elapsed after Giovanni Cassini (1625-1712) studied his famous ovals in 1680 before it was realized that the equilateral lemniscate belonged to the same group (Fig. A-1b). In fact, it is a curve of demarcation. Much more surprising is the fact that an additional 180 years passed before I discovered that, incredibly, the common equilateral hyperbola also is a curve of demarcation of Cassinian ovals (Fig. A-1a). Yet this remarkable relation could have been guessed 100 years ago. Furthermore, confirming such a guess also would have led to the discovery of a new type of self-inversion, namely, self-inversion at 90°, in which Cassinian monovals inverted about their centers yield curves identical with, but rotated 90° from, the initial curve (Fig. A-1a).

This remarkable circumstance comes about as follows:

When the ensemble of Cassinian biovals of Fig. A-1b enclosed within the equilateral lemniscate (markedly darker curve that demarcates them from the outer monovals) are inverted conventionally (at 0° or 180°) about their common centers (crossover point in lemniscate), they self-invert (maintain their positions), as in Fig. A-1a  (not to same scale). Inversion of the lemniscate in the same way produces an equilateral hyperbola which, instead of enclosing the monovals, then bounds them, one group of monovals within each arm, as in Fig. A-1a. A remarkable circumstance is that the external monovals of Fig. A-1b also self-invert, but at 90°, producing the interior monovals of Fig. A-1a, becoming bounded between the arms of the equilateral hyperbola, as their curve of demarcation. Thus, only by self-inversion at more than one angle does the ensemble of Cassinian ovals of Fig. A-1b maintain its integrity in Fig. A-1a.

Eccentricity as a visible property

Though the above topics are not commonplace, the next example concerns a subject discussed weekly in thousands of classrooms-the eccentricities of conic sections. The eccentricity of a conic section has been found to be a 'visible' property, in much the same sense that a focus is visible. If the position of a focus is known, the eccentricity also can be illustrated. One simply draws or indicates a tangent-line to the curve at any of its points of intersection with a latus rectum (a chord through a focus at 90°). The magnitude of the slope of this tangent-line is the conic's eccentricity (see Fig. A-2).

Structure rules-new analytical tools

The above examples concern highly specific matters. The next finding has very much broader implications: a major new avenue for the analysis of curves and relations between curves has been discovered. Until a few years ago, virtually all the common equations of analytical geometry were coordinate equations for construction rules of curves. These are explicit rules for plotting curves in specified coordinate systems. However, they do not come to grips directly with structural properties of curves.

The recent work has disclosed powerful new tools for studying, characterizing, organizing, and classifying curves; their structure rules (the topics of Chapters VII and VIII). Structure rules are derived by employing only single reference elements, rather than the two or more reference elements of a coordinate system. Since structure rules are independent of coordinate systems, they do not give specific information about how to construct curves, or of the curves' appearances, but they do give specific information about other aspects of structure.

The only familiar examples of these largely hidden aspects of structure are the classical symmetries about points and lines, which also are the simplest cases. Because structure rules take their simplest form for the classical symmetries, namely, expressing the equality of two distances, the corresponding underlying structures of the curve are readily perceptible and were the first to be discovered. However, even the most complex of the new structural features can be rendered visible by plotting their structure rules in the rectangular system. Most of the resulting structural curves are of unfamiliar appearance and previously were unknown.

As a highly symmetrical example, leading to a simple structure rule, I use an ellipse and a structure rule about its traditional focus. Thus, the 180° structure rule of the centered rectangular ellipse (construction rule, b²X² + a²Y² = a²b²) about its traditional focus is 2aXY = b²(X + Y), which is, itself, the construction rule of an equilateral hyperbola. This is one of the rare instances in which the structure rule of a curve about a point is not of greater degree than its rectangular construction rule. Thus, for the  90° transform about a focus, the degree of the structure rule is four, while for a general  point on the X or Y axis (for the 180° transform), it also is four, as opposed to two at the focus. But the 90° transform (structure rule) about a general point on the X or Y axis is of degree sixteen (see Table below), as opposed to four at the focus.

Structure rules, inherent structure, and the classification of curves.

Structure rules cast about single points in the plane of curves are preeminent for analysis. Their great diagnostic power derives from the fact that they describe and define the long sought-after inherent structural properties of curves. Classical attempts to probe these properties analytically led up a more or less blind alley to the "intrinsic equations." Like the latter equations, structure rules about points provide unique characterizations. But unlike the relatively sterile intrinsic equations, each general structure rule of a curve can be resolved or decomposed into the components or specific structure rules that contribute to a unique characterization. None of these general structure rules previously was associated with its curve, not even the 4th degree structure rule of the line or the 6th degree structure rule of the circle about a point in the plane (see Table below).

Structure rules about points make it possible to classify curves on the basis of the structural properties that they possess in common, and to construct hierarchical trees of relations, which are essentially stepping stones to a deeper understanding of structure. The following is an example of the procedure for constructing a hierarchical tree:

Many known curves are characterized by a given specific structure rule, A, about a certain point; of these, a number also possess the same structure rule, B, about the same point, and some of the latter group also possess structure rule C. Of the curves that possess structure rules A, B, and C about the given point, some also have points in their planes for which structure rule D applies, and a fraction of these, in turn, are characterized by structure rule E about the same 2nd point, etc. Thus, large groups of curves whose members share single properties can be subdivided sequentially into smaller groups whose members share greater numbers of properties, leading, ultimately, to unique categorizations in which the identities of the single curves are implicit.

Structure rule analysis and a coherent theory of exceptional points

The points in the plane of curves of greatest interest to pure and applied mathematicians, as well as to physicists, astronomers, and engineers, are the "exceptional" points, such as vertices, foci, double points, points of infinite curvature, and the poles about which curves self-invert. In the past, points in these different categories were defined in diverse and essentially mutually exclusive ways.

The incisiveness of structure-rule analysis about points is illustrated by the fact that it provides a common analytical foundation for detecting, defining, and hierarchically ordering points in all the above-mentioned categories (as well as exceptional point-continua, including lines of symmetry and certain asymptotes. Briefly stated:

The exceptional points in the planes of curves are the locations for which the curves' structure rules have the lowest exponential degrees.

Examples are given below for the parabola, ellipses, and hyperbolas in rectangular coordinates.

Curve	Parabola				Ellipses				Hyperbolas
Transformtion°		90°	180°	0°		90°	180°	0°		90°	180°	0°
locus point in plane	32*	20	8	8	32*	20	8	8	32*	20	8	8
x-axis	16	16	4	4	16	16	4	4	16	16	4	4
y-axis	16	16	4	4	16	16	4	4	16	16	4	4
asymptotes	-	-	-	-	-	-	-	-	32*	-	6	6
incident point	32*	12	-	-	32*	12	-	-	32*	12	-	-
a-vertex	16	8	-	-	16	8	-	-	16	8	-	-
b-vertex	-	-	-	-	16	8	-	-	-	-	-	-
LR-vertex	32*	8	-	..	-	32*	8	-	-	32*	8	-
Focus	4	4	2	2	4	4	2	2	4	4	2	2
Center	-	-	-	-	8	4	1	1	8	4	1	1

A dash (-) indicates that the degree is unknown or that the structure rule is trivial or non-existent

* indicates maximum value

New applications and knowledge of polar equations

Some of the most productive and elegant analytical procedures of Structural Equation Geometry emerged as by-products of methods for deriving the structure rules referred to above. Among these are intriguing new uses for polar equations (see Chapters III, VII, and VIII in Structural Equation Geometry). An example is the procedure for defining and deriving the locations of the foci of conic sections and Cartesian ovals. Until now this was done primarily by constructions about two points (see Fig. A-8) or a point and a line. A lack of knowledge of these procedures frustrated classicists for over 200 years in their attempts to place the study of Cartesian ovals on a secure analytical footing.

Unfortunately, the polar coordinate system and the properties of its curves and equations have been no less neglected than other planar Euclidean systems. The paucity of attention is evident from the absence of a prior explicit treatment and definition of the exponential degrees of polar equations; nor had any prior attention been given to the standardization of equations in the polar system. Instead, polar equations merely were used in their simplest possible forms for the convenience these forms conferred. In consequence, there has been only a vague appreciation of the significance of the great differences that exist between the polar equations of differently-positioned congruent curves-- matters that goes to the heart of structural equation geometry.

The Devil's curve as a construction from conics

The Devil's curve (see Fig. 5-7), first studied in the mid-18th century by G. Cramer, is much employed as an example in analytical geometry, for example, in the study of asymptotes. It also finds uses in presenting the theory of Riemann surfaces and Abelian integrals. However, as used in the past the curve was known only through its equation; close affinities with conic sections were neither known nor suspected. Structural equation geometry revealed the Devil's curve to be the product of a simple construction from two central conics, the equilateral hyperbola and its elliptical equivalent, the circle. Thus, a useful curve of previously unknown origin now can be derived from common classroom curves and discussed in terms that beginning students can understand (see Fig. 5-7).

(X⁴ +Y⁴ ) - 2(R²+ a²)X² + 2(R ²-a ² )Y² = 0     Devil's curve

The most intriguing aspect of the construction of the Devil's curve is that it is the unique case in which reciprocal derivations yield identical structure rules. The curve is generated either as the 90° structure rule (transform) of an initial centered circle, overlapping a centered equilateral hyperbola as reference element, or as the 90° structure rule of an initial centered equilateral hyperbola, overlapping a centered circle as reference element. Obviously this construction defines a previously unknown family of Devil's curves (for other angles of transformation). Calling this the 90° Devil's curve, one may ask, what would a 30° Devil's curve look like, a 180° Devil's curve, etc. Would any of these curves also result from reciprocal transforms? The answer to any of these questions is unknown, but accessible to any inquisitive geometer.

Quartic equivalents of central conics

The next example concerns a hitherto unexplored topic of compelling interest: discovering the 4th-degree equivalents of central conics. Based on very close analytical homologies, these have been identified as the central Cartesians, whose only previously known representatives are Cassinian ovals (Fig. A-1b).

(X² + Y²)² + B²X² +C²Y² + D⁴ = 0    central Cartesians

The host of engaging known properties of central conics provide but a pale hint of the multi-faceted properties of their 4th-degree relatives. Some of the most compelling of these-all previously unknown-conclude Chapter VIII. In the domain of these properties alone, there probably remains to be discovered more than the totality of the known properties of conic sections.

A striking example concerns the inversions of central Cartesians to curves having true centers, a true center being the point of intersection of two orthogonal (at 90° to one another) lines of symmetry. In all previously known instances of this phenomenon, the poles for such inversions were the true centers themselves, a feature that was assumed to be essential. For example, the equilateral hyperbola and the equilateral lemniscate invert to one another about their true centers. Central Cartesians have proved to be far more versatile, with properties that greatly broaden our perspectives. The biovular members not only invert to 3 different curves that possess true centers-rather than only one-they do so about 5 different poles-rather than only about the true center (see Structural Equation Geometry and Curves and Symmetry, vol. 1).

Cartesian and Cassinian ovals related by inversion

Despite the fact that Cartesian and Cassinian ovals are two of the most famous 4th-degree curves, it was not known before my studies that Cassinian ovals are inversions of a subgroup of Cartesian ovals, the parabolic Cartesians (so-called, because their bipolar eqs. take a rectangular parabolic form), and vice versa. Accordingly, the immensely interesting central Cartesians, mentioned and formulated above -- the 4th-degree equivalents of central conics -- also are the inverses of Cartesian ovals. As a concomitant of this finding, limaçons (see Fig. A-8d1,d2) are found to stand in precisely the same relation to central conics (inversion about their foci) as Cartesian ovals stand to Cassinians (and other central Cartesians). It seems evident from these findings that the names "Cartesian" and "Cassinian" are destined to play a much larger role in the future of physical geometry than they did in its past.

(X² + Y²)2 - d²(X² - Y²)/² + d4(1/16 - C²) = 0            Cassinians (Fig. A-1b)

     d and C have the values that occur in the bipolar eq. of Cassinians, uv = Cd2

Unprecedented new types of curves

The new findings of greatest general interest emerged from a systematic study of curves tin simple but previously unstudied coordinate systems consisting of two of the three elements-a point, a line, and a circle. Even when we consider only unconventional manifestations of conic sections, a wealth of unprecedented new curves are commonplace in these other systems. These include: (a) 2-arm hyperbolas and parabolas with arms of different sizes; (b) closed curves consisting of segments of dissimilar, as well as similar but different-sized, ellipses (including circles) and hyperbolas; (c) open and closed curves formed from segments of two or more parabolas; and even (d) open and closed curves that include segments of both ellipses and hyperbolas Figs. A-3-6).

Directrices defined

The same studies revealed that the general analytical significance of directrices extends far beyond their known applications in the construction of conic sections. Many other curves have elegantly simple construction rules involving directrices, for example, limaçons and Cartesian ovals. A directrix-line of a curve is defined in Chapter V as a line-pole for which there exists a point-and-line (or circle-and-line) construction rule that lacks a constant term. In this connection, many other interesting yet unexploited constructions of conics involve line-poles that are not directrices. Circles also can be directrices. A circle pole is a directrix if there exists a point and circle (or line and circle) construction rule of a curve that lacks a constant term. By these criteria, directrix lines occur in Figs. 5-2a,b1,b2, 5-3a,b, and directrix circles in Figs. 5-3a,b and A-3a1,a2. Both lines and circles occur as directrices in Fig. A-6a-d.

The new coordinate system for deriving construction rules

Inasmuch as the polar and rectangular coordinate systems are preeminent for almost all analytical studies, it did not come entirely as a surprise to find that a hybrid system that combines some of the advantages of each achieves preeminence over the parent systems for certain types of analysis. Thus, by using the polar r and the rectangular x, the hybrid polar-rectangular coordinate system results. This system provides the key equations for deriving construction rules of curves in coordinate systems that include a point or circle among their reference elements, and for identifying the "point-foci" of curves in these coordinate systems. Point foci are the particular points about which the degrees of the construction rules of curves reduce from their values when cast about arbitrarily-located reference elements.

Although, in essence, the hybrid polar-rectangular coordinate system has been in use for hundreds of years, it was unrecognized. It is the implicit coordinate system of classical equations that relate distances from points of curves to arbitrary points in the plane-equations of great practical importance. This system has the potential to become the third-most important coordinate system for studying the properties of curves.

Inversion, immediate closure, and a "natural" scheme of classification

A startling finding within an old field concerns a basic property of the inversion transformation (see below). It is no coincidence that inversion has been mentioned several times. In my estimation it stands alone among geometrical transformations-in beauty, in fascination, and in its many unique properties.

Inversion long ago proved to be of great use to the mathematician-in the theory of differential equations, in proving difficult geometrical theorems, in deriving "inverse theorems," and in classical geometric constructions. Carathéodory (1873-191115) regarded inversive geometry as the best avenue of approach to the theory of functions, and H. A. Schwartz's (1843-1921) knowledge of this field provided the foundation for many of his celebrated successes. Inversive geometry also can serve in place of projective geometry, as a common foundation for Euclidean, spherical, and hyperbolic geometries.

Inversion is no less important to the physicist and engineer. Some of its most notable applications occur in potential theory, the theory of elasticity, and the theory of special relativity. These depend upon such important features as the invariance of the n-dimensional Laplacian under inversion, the fact that inversion makes it possible to set up Green's function in closed form for an n-dimensional sphere, and most intriguingly, the fact that inversion brings singularities of functions at infinity into the finite plane, where they can be examined and dealt with.

In view of these many important and long-standing applications, one might have expected that the intensive studies of the inversion transformation by classicists had thoroughly plumbed its depths, and that it could hold no secret in a fresh examination. Quite the opposite proved to be the case; many of inversion's basic properties, including those that are most beautiful and fascinating, proved to have been overlooked and unsuspected.

Attention here is confined to an astonishing new property of inversion that provides the foundation for the first 'natural' paradigm for classifying curves. Consider any initial curve inverted about all points in the extended Euclidean plane (the Euclidean plane plus the ideal line at infinity), to yield a first-generation ensemble of inverse curves. The new property-termed "immediate closure"-is that if any one of these first-generation curves also is inverted about all points in the plane, no curve not already present in the first-generation is obtained. In fact, the ensemble of curves obtained in this way possesses the same members as the first-generation ensemble itself.

The broad implications of immediate closure for classifying curves are evident. By virtue of this property of the inversion transformation, any given initial curve defines a unique closed group of related curves. No other non-trivial type of transformation provides this facility.

Asymptotes of inverse curves

Inverting a curve about a point incident upon it gives rise to an inverse curve that possesses at least one asymptote. Classicists overlooked the elegantly simple and beautiful relationships that determine the number and locations of these asymptotes in the plane of the inverse curve, which provides the last example:

The asymptotes of inverse curves are the inverses of the circles of curvature of the initial curves at their points of inversion.

Ultimately, these many new findings, new techniques of analysis, and broad new perspectives on the properties of curves and the ways in which curves are related to one another, can be expected to have far-reaching practical applications. At the moment, however, they are largely unknown to the mathematics community. This book presents the new results in a readily assimilated form and directs attention to their most obvious, immediately-accessible, and urgent applications, namely those for the classroom. [I believed the last sentence at the time the words were first written, but my son, Christopher, who is exceptionally well accomplished mathematically, didn't find the results to be in such "a readily assimilated form." As I go over my geometry books now, I have to agree. JLK]

New Approaches, Viewpoints, and Paradigms

Structural simplicity-the thread of continuity

More attractive ways to present standard topics in elementary analytical geometry, and more incisive viewpoints and paradigms, were suggested in the course of studies in structural equation geometry. The guiding theme at this early stage is to try to stimulate and develop the interests of all students in geometry. Only the most gifted and imaginative ones usually are reached with traditional approaches.

One lays a foundation for achieving this goal by providing a thread of continuity, interest, and anticipation from topic-to-topic-giving students a view of the "forest," also, rather than only of the "trees." To achieve this, the concept of the structural simplicity of a curve is introduced. Structural simplicity is merely a logical extension of classical ideas of symmetry to embrace other aspects of structure, with emphasis on the fact that symmetry of position and symmetry of form are equivalent concepts. This equivalence has been placed on a readily-grasped analytical footing by assessing the simplicities of position and form of curves in terms of the simplicities of the corresponding construction rules.

Two examples drawn from construction rules help to clarify this approach. The first is a special case that will be somewhat familiar, since it involves differences between the construction rules of circles plotted at different locations in the rectangular coordinate system (Fig. A-7). Circles with the least structural simplicity (symmetry of position and form) and the most complicated construction  rules (most numbers of terms and parameters) are centered at an arbitrary point in the plane (5 terms, 4 parameters). Circles with somewhat greater structural simplicity and slightly simpler construction rules are centered on a coordinate bisector (5 terms, 3 parameters). With each increase in the locational symmetry (and structural simplicity) of a circle, its construction rule simplifies. The simplest and most symmetrically positioned circles, with the simplest construction rules, are centered at the origin (2 terms, one parameter). Of course, the point circle (a circle of null radius) has the greatest structural simplicity and the simplest construction rule (X2 + Y2  = 0).

The 2nd example represents the general situation. It concerns lines in the bipolar coordinate system, that is, lines in a system consisting of two point poles (pu and pv) a distance "d" apart Fig. A-8). The simplest and most symmetrically-positioned bipolar line is the midline, which has the simplest first-degree construction rule, u = v, and for which all points are equidistant from the two poles. The next-simplest and most-symmetrically-positioned line is the mid segment that connects the two poles, along which only one point is equidistant from the two poles (as is true of any line that intersects the midline). It has the more complicated but still very simple first-degree construction rule, u + v = d .

Less-simple and less-symmetrically-positioned lines, with simple first-degree construction rules are lateral rays that extend in opposite directions from the poles, u - v = d, for which no point is equidistant from the two poles. These are followed by lines normal (perpendicular) to the bipolar axis, with more complicated construction rules of second degree, also for which no point is equidistant from the poles, and by the entire bipolar axis, itself, with a still more complicated construction rule of third degree. Lines with the least structural simplicity and symmetry of position are randomly positioned and have the most complicated construction rules, which are of fourth degree.

The concept of structural simplicity is emphasized by highlighting, at every step in the presentation of elementary analytical geometry, the correlations between the simplicities (or symmetries) of the positions and forms of curves and the simplicities of their corresponding construction rules (with respect to given reference elements).

Dimensional uniformity of equations

Breaching the barrier between the average elementary student and the use of algebraic equations in the primary objective. One approach to achieving this is to write equations in rigorous fashion. The familiar quadratic equation in the distance variable x, Ax2 + Bx + C = 0, provides a convenient though very simple example.  In this conventional form, even though each of the three coefficients, A, B, and C has a different dimension (1, 2, and 3, respectively), this multidimensionality is masked. Rewritten in dimensional balance, the equation becomes, Ax2 + B2x + C3 = 0 (or A0x2 + Bx + C2 = 0), where the exponents of the capital letters, like those of the variable, x, indicate their distance dimensions; the latter must sum to 3 (or 2) for each term.

Maintaining dimensional uniformity of equations involves only a very modest change in the way they are written. Yet it provides a valuable new pathway for students to gain familiarity with equations, even without understanding fully what the equations mean. Students can immediately begin exercises and problems in balancing equations dimensionally and detecting dimensional imbalance. These are precisely the kinds of down-to-earth contacts needed in early stages to increase the appeal of working with equations. And the students who are likely to find these exercises most rewarding are the ones most in need of stimulation and encouragement.

Maintenance of dimensional uniformity of equations provides a valuable aid for detecting and tracking down errors in algebraic derivations, particularly those of structural equation geometry. Its importance was recognized by the eminent mathematical physicist, Arnold Summerfeld, who once took his colleagues to task for the "bad habit"--as he put it-of being careless with dimensional balance. The power of "dimensional analysis," an analytical approach that employs the principle of dimensional uniformity for the derivation of specific physical laws from general initial assumptions is, or used to be, well known to engineers and physicists.

Using more meaningful terms than "equation"

Another effective aid for the average student is to employ more specific and meaningful terms than "equation" wherever possible. When the word "equation" is used in classical analytical geometry, for example, it usually refers to a construction rule. Accordingly, the term "construction rule" should be employed in these cases. It is counterproductive to use a vague general term to which the average student relates poorly, when a specific term that is meaningful to everyone is available.

Introducing and comparing coordinate systems

Tradition also leads us to begin the study of coordinate systems with the rectangular system. Following this route, most students are introduced to three new concepts simultaneously, at a juncture where one would suffice. Thus, in addition to the new concept of a coordinate system, the average student also meets the unfamiliar ideas of measuring distances in a negative sense, and in measuring distances perpendicularly to lines. Neither of the latter concepts is needed to introduce a coordinate system. All that is needed for this purpose is the idea of measuring distances from a point, with which even the least-informed students already are familiar.

Nor does any benefit result from introducing coordinate systems with the rectangular system. Not only is this system far from being the simplest conceptually, its properties are highly exceptional. And, though its most interesting and practical features are precisely those in which it is atypical-degree-restriction, singularity of loci, and uniqueness of identity-traditional presentations have not even recognized these features as being exceptional; in fact, the level of awareness of them is so low that I was the first to name them. Since the rectangular system is introduced and employed in an entirely 'matter-of-fact' manner, it fails to stimulate the interest and imagination of many students.

The bipolar coordinate system-the introductory system of choice

Because it combines the merits of being intriguing and being the simplest coordinate system, the bipolar system is the best introductory system. Geometry instructors are familiar with its rudiments from the constructions of ellipses and hyperbolas about two foci Fig. A-8c1, c2. These constructions point the way to other productive uses.

Though the bipolar system is not a powerful analytical tool, it provides an ideal framework for introducing a coordinate system through the route of constructional puzzles. These are presented in common language for students to solve independently, with chalk and string, pencil, compass, and ruler. One begins with the simplest constructions. For example, find the shapes and positions of the paths covered by a browsing rabbit when it moves in such a way as to:

(a) always be at equal distances from its two burrows, which are separated by a distance "d" (the construction rule of the midline, u = v, where v and u are the distances from the two burrows; Fig. A-8a);

(b) keep the sum of its distances from the two burrows equal to the distance between them (u + v = d); the axial line connecting the two poles (Fig. A-8b2, left);

(c) keep the difference between its distances from the two burrows equal to the distance between the poles, u - v = d); the axial rays extending outward from the poles (Fig. A-8b2, right); and

(d) always keep its distance from one burrow equal to twice its distance from the other one, u = 2v, the circle of Apollonius (Fig. A-8b1).

Such simple puzzles can be relied on to stimulate interest while preparing the way for more intriguing constructions. These first exercises also introduce the idea of simplest constructions and present a line segment and a ray as the outcome of very simple construction rules, which is not possible in the rectangular system.

Familiarity with these simple constructions provides the foundation for a student to proceed to the next level of complexity-the conventional constructions for central conics. These are presented as logical extensions of the simplest constructions. Eccentricity is introduced in these extensions by modifying the construction rules given above to u + v = d/e and u - v = d/e. For the former, letting e = 1 yields the connecting line-segment, letting e be greater than 1 yields no locus (introducing the concept of a construction rule that cannot be fulfilled), but letting e be less than 1 generates ellipses whose eccentricities are defined by the value of e employed.

Similar considerations apply to the construction rule u - v = d/e, but now the axial "rays" open to become the arms of hyperbolas (Fig. A-8c2) of eccentricity > 1. The larger the value of e, the closer the locus approaches the midline; the smaller the value of e-but not less than 1-the closer each arm approximates to a ray; while for values of e less than 1, no construction exists. [The common definition of eccentricity, in terms of focus-directrix construction rules, also is conveniently introduced here, with the intriguing revelation that two entirely different types of constructions take simple forms when expressed in terms of e.]

Some may wish to limit their treatment to the first-degree construction rules of conic sections, others will want to show how simple modifications of linear construction rules lead to limaçons (Fig. A-8d1d2), and that the most complex first-degree construction rules produce Cartesian ovals. In any event, the simple second-degree construction rule,  u2 + v2 = d2, should not be overlooked. Since this construction produces a circle, and is equivalent to the Pythagorean theorem, the resulting circle is most appropriately referred to as the circle of Pythagoras.

In concluding the treatment of the bipolar system, one draws attention to some key properties of general applicability: the line, ellipses, hyperbolas, etc., have first-degree construction rules only when located in the particular highly-symmetrical positions of the above constructions. All departures from these positions lead to more complicated construction rules that are of higher degree. Departures also produce curves with multiple segments, as a result of reflections in the coordinate line(s) of symmetry. These very unfamiliar circumstances are important because they are typical of the vast majority of coordinate systems:

In general, the more symmetrical the position of a curve with respect to coordinate elements, the simpler and/or the lower the degree of its construction rule and the fewer the number of segments it possesses.

The rectangular and polar coordinate systems

It is against this background that the rectangular system is best introduced-not in the present-day manner, as a matter-of-fact practical system, but as a most extraordinary one with properties that differ remarkably from those of almost all other systems. Practically speaking, it is the only system in which the degrees of construction rules remain unaltered when curves are moved from symmetrical to asymmetrical positions-less-symmetrical positions merely require more complicated constructions rules (Fig. A-7). Moreover, the rectangular system makes possible simple expeditious analytical solutions to the previously studied bipolar "puzzles."

Unlike circumstances in the bipolar and most other coordinate systems, linear construction rules in the rectangular system cannot code for a multiplicity of curves; they generate only lines (an aspect of degree restriction). Furthermore, only a single representation of a curve is produced, no matter where its position (singularity of loci), and all such representations are similar to each other (uniqueness of identity). Following the usual introductory exposition of the rectangular system, it is most instructive to derive the rectangular construction rules that correspond to the "puzzles" of the linear bipolar equations. This will illustrate the simplifications and reductions in degree that occur when the coefficients of the coordinate distances, u and v, are of equal magnitude,

The same comparative approach is most effective for introducing the polar coordinate system-presenting it against the background of the properties of the bipolar and rectangular coordinate systems and highlighting its differences and similarities. Treatments of transformations between rectangular and polar construction rules should emphasize the basis for the frequently differing exponential degrees of equations in the two systems, and for the existence of equations of different degrees for differently-positioned curves, correlating these degrees with the curves' positional symmetries (structural simplicities).

Other coordinate systems

The curves of the rectangular and polar coordinate systems are the conventional curves of classicists. The bipolar system adds types that are unconventional only in consisting either of multiple segments, by virtue of reflections, or only of single arms or ovals. When one combines in one coordinate system a line-pole of the rectangular system and a point-pole of the polar system, or either of them together with a circle pole, new factors come into play; extraordinary curves that are unprecedented in classical experience now become commonplace (Figs. 5-3, A-3-6). Selected examples from these systems are useful at the high-school level to broaden the students' perspectives and provide additional constructions as puzzles.

Characterizing curves by eccentricity

Eccentricity also provides an ideal, heretofore overlooked basis for identifying curves inverse to conic sections. The inversion transformation has the key property of preserving the magnitudes of angles, such as angles of intersection and tangent angles. Since each conic section can be characterized by an angle related to its eccentricity, for example, the angle between asymptotes, so also can any curve obtained by inverting a conic section. Accordingly, any curve inverse to a conic can be said to have the same eccentricity as the initial conic.

Since a number of well-known curves are the inverses of conic sections, they can be characterized by eccentricities. Some of these curves and their eccentricities are the cardiod (e = 1), the cissoid of Diocles (e = 1), the equilateral strophoid (e = √2), the equilateral lima�on (e = √2), the equilateral lemniscate (e = √2), and the trisectrix of Maclaurin (e = 2). limaçons parallel conic sections in that their eccentricities also are the magnitudes of the slopes of tangents to the curves at the level of their double point. In fact, it also was recognized that the eccentricity of the initial conic enters directly into the conventional polar equation of the inverse limaçon in the form r = a - bcosθ = a(1 - ecosθ).

An example of the pertinence of identifying such curves in terms of eccentricity comes from comparisons of the structural simplicities of conic sections and limaçons. Employing the various criteria of structural equation geometry, the conic sections and limaçons with the greatest structural simplicity fall into pairs with reciprocal eccentricities, notable, 0 and ∞, √2/2 and √2, 1 and 1, and 2 and ½.

Eccentricity and the exceptional lines of conics

Once the topic of eccentricity has been introduced, students can be given very simple relations involving eccentricity that are not being used to full advantage. One, in particular, makes it easy to remember the positions of all the exceptional normal lines in the planes of central conics, namely, the minor and conjugate axes, the directrices, the vertex tangents, and the latera recta. In terms of e and a, these lie at the easily remembered distances from center of 0, ±a/e, ±a, and ±ae, respectively (or, in units of a, 0 ±1/e, ±1, and ±e).

Systematic nomenclature for position

Another significant avenue for engaging the interest of students lies in employing systematic nomenclature for the locations and orientations of curves. The need for this was not felt in the past because there never was more than a vague appreciation of relations between a curve's position with respect to reference elements and the degrees and complexities of corresponding construction rules (the structural simplicities of the first example). Since these relations lie at the heart of structural equation geometry, a systematic nomenclature for position becomes desirable. But, again, advantages in the classroom far outweigh practical needs.

Just as adherence to dimensional uniformity provides another avenue for the student to relate to equations, so also does the employment of systematic nomenclature for position provide a rich new avenue for relating to curves. And, just as balancing equations dimensionally provides new opportunities for exercises, independently of the meanings of the equations, so also does describing and deducing the locations and orientations of curves, independently of the significance of the curves.

A few examples of positional nomenclature for the rectangular coordinate system suffice to make its use clear: a general ellipse is positioned arbitrarily in the plane; a general bisector ellipse has its center upon a coordinate bisector; a general 0° -axial ellipse has its center incident upon a coordinate axis,  with its major axis parallel to the x-axis; a 45° -bisector ellipse including the origin is a bisector ellipse that passes through the origin and has its major axis at 45°  to the x-axis; a 45°-bitangent ellipse has its major axis oriented at 45° to the x-axis and is tangent to both axes; etc. Students can be expected to find exercises based on providing descriptive names of this nature for curves in given positions, and positioning curves in accordance with given descriptive names, to be appealing.

Systematic nomenclature for curves

The systematic naming of curves is another topic that received very little attention from classicists, principally because of a paucity of comparative studies. Thus, no adequate scheme of classification existed to render obvious the utility of systematic nomenclature. Yet simple, systematically applied names provide still another important means for the student o relate to curves. This topic receives detailed treatment in Symmetry, An Analytical Treatment, but some very simple examples drawn from the present work suffice to illustrate one of the approaches.

We saw above how an individual limaçon, the inverse of a conic section about a focus, can be identified in terms of the eccentricity of the initial conic. But what of the names of the 3 groups of limaçons so obtained? Some possess 2 loops, some consist of a single closed segment or oval, and one, the cardiod has a cusp. No classical designation exists for these groups. They are most appropriately named for the parent conic group from which they derive., as hyperbolic limaçons, elliptical limaçons, and the parabolic limaçon, and similarly for all such groups of curves obtained in other non-incident inversions of conic sections. For the inverse curves about the vertices, the corresponding abbreviated names are hyperbola vertex cubics, ellipse vertex cubics, and the parabola vertex cubic (the cissoid of Diocles).

Algebraic calculations

A pertinent point should be made for those who employ Structural Equation Geometry as a text. Some of the calculations are formidable to the uninitiated. The situation is reminiscent of that during the revival period of analytical geometry. Thus, in 1813, Gergonne (1779-1851) observed that "the prolixity often is repulsive" for the calculations involved in some theorems and constructions. Plücker's many improved procedures went a long way toward eliminating such difficulties, at the same time highlighting the elegance of the analytical method, properly employed.

Similar complaints concerning algebraic calculations are voiced today, even by experts, with the consequence that much effort is expended to develop new computer programs and very little effort is expended to find more productive avenues of algebraic analysis. One of the great fascinations of structural equation geometry lies in achieving and understanding the genesis of the ultimate simplifications and degree reductions that frequently materialize in the course of its algebraic procedures. These can be expected whenever relations involving a high degree of structural simplicity are being studied. Some type of calculations that appear to be most formidable on first encounter, yield readily to appropriate techniques. In fact, the analyses mentioned above that frustrated classicists for over two centuries-the analytic geometry of Cartesian ovals-became elegantly resolved.

Improving the style of mathematics texts

Lastly, the traditional style of elementary mathematics texts leaves something to be desired. Many such books contain page after page of bare equations and plain letter-labeled constructions and line-drawings, more after the style of a research paper than an elementary text. Name labels for curves and figures are used very sparingly, and significant features, such as asymptotes, vertices, foci, directrices, and double-points are not identified with word-labels. Today's student cannot be expected to give high priority to material presented in such austere fashion (in competition with the many more overtly appealing subjects at hand).

Elementary textbooks would much better serve the student and the subject if curves, figures, constructions, and significant features were accompanied by a generous use of name-labels and descriptive word-labels, clearly "tagged" to the identified material by light lines. Every significant equation should be identified by immediately adjacent text, and adjacent step-by-step explanations should be given for all algebraic derivations.

In addition, there should be abundant examples of figures and curves of each type studied. Where appropriate, these should be arranged in sequence, emphasizing changes in form with progressively varying modes of origin or construction. Although the present work is not intended as an elementary text, its almost entirely unfamiliar contents have made similar approaches desirable, leading to many examples of the recommended innovations.

I give a lecture on structural equation geometry

I believe the innovations discussed in the foregoing will greatly improve the appeal of algebra and geometry to beginning students. In Nov., 1984 an opportunity presented itself to explore this matter in an indirect fashion. I received a letter from Deans Clarence Hall and John D. O'Connor of UCLA inviting me to participate in an NSF-funded Project for Renewed Incentives for Science Teaching in Los Angeles (Project RISE-LA) in the following summer. This seemed a good chance to try out some of my suggestions for improving the teaching of algebra and geometry in high schools, so I gladly accepted. In the following June I received a letter from Susie Hakansson, Director of the UCLA High School Mathematics Project looking forward to my presentation of "New Approaches in the Teaching of Analytic Geometry" to about 26 ninth-to-twelfth grade teachers one day in July from 10:40 to 12:00 a.m..

At the appointed time and place I gave a talk to the group mentioned. I don't remember exactly the material I covered but it surely included many of the points I made above. I do remember emphasizing the topics of employing dimensional uniformity of equations and using more meaningful terms than "equation" (e.g., "construction rule") as aids in presenting the material. The group seemed to be listening attentively for some 70 minutes when, at about 11:50, one of the group put up her hand and asked if I could start over, since she had no idea what I had been talking about. I did what I could in the next 10 minutes, but the basis for my apparent failure remains as much a mystery to me, today, as the talk apparently was to the group, then. It seems it is the instructors, steeped in conventional and traditional approaches, not the students, that will be the most difficult to reorient.