Euclid to Abstract Algebra.html

[For a geometric version of many points made in this article go here]

We live in the third great age of development of triangle geometry. Technology allows us to try out ideas and to communicate them quickly. Algebra, considered in a very wide sense, allows us to think in ways our forebears could not. This paper tells how Euclid both contains and leads to the abstract algebra of the modern era.

Synthetic geometry contains the inevitability of coordinatization. Indeed much of Euclid is really an algebra book, teaching us how to do algebra in a vaguely geometric context.

Our new algebra will assign an algebraic nature to a geometric object based on the manner of its construction. This algebraic nature will be invariant over all possible constructions of the object.

Coxeter's book on the projective plane and the Bourbaki history notes about geometry are good sources for the move of geometry to coordinates. Coordinates form an algebra and lead to higher algebra. Both Klein and Lemoine talked about "the abstract theory of the triangle," which is coming to fruition in our age. This page, done in Mathematica, is about algebraic techniques that are currently being developed in triangle geometry.

We will examine some of Euclid's postulates and propositions and discuss their algebraic ramifications. The links take you to Joyce's translation of Euclid, which for a geometer is solace for the soul, even as the loose terminology irritates. The important idea is that of incommensurablility and we begin with that.

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Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.

We can set a unit by opening our compass to a certain width and marking a distance on a line. There are then constructions to multiply or divide this distance by given amounts, none of which require the compass to be set to a different setting.

There are three circumstances where we can call a pair of numbers incommensurable. First is the one we expect–where a distance is irrational, requiring a new compass setting to that distance.

The second meaning is that some distances cannot in principle be measured in terms of a common unit because the distances are mutually independent. As an example, create a point on each side of an angle by drawing a circle centered at the vertex. This creates a unit on the angle sides (blue in the figure below). Now connect the two new points (red in the figure). This distance cannot in general be expressed in terms of the unit and is incommensurable with it.

The figure shows the incommensuable distance as a different color. We will say that the red distance has a different algebraic nature from the blue one. We will continue the convention of using color to indicate different algebraic natures. Lengths have a different algebraic nature if they are incommensurable.

Algebraically the red distance would be computed using the law of cosines. Cosine is a transcendental function, which means the computed number is almost always incommensurable with the ones it are computed from.

A third meaning of incommensurable uses randomness. If we set a compass randomly, constructed distances will be incommensurable with distances constructed with another setting.

Definition 2
Straight lines are commensurable in square when the squares erected on them are measured by the same area, and incommensurable in square when the squares on them cannot have any area as a common measure.

Definition 3
With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called rational, and those straight lines which are commensurable with it, whether in length and in square, or in square only, rational, but those that are incommensurable with it irrational.

It is time to explain some of the algebra of square roots. A square root and a rational number have different algebraic natures. For example 2 a + √2 b = 0, with a and b rational, can only imply that a = b = 0. We say that 2 and √2 are independent, their algebraic natures are incompatible. This would not happen if both numbers are rational as in 2 a + 3 b = 0, for which rational a and b do not have to both be zero. Their like algebraic natures lets them combine in a way that unlike algebraic natures cannot.

Fundamental Principle: If an irrational point is on a rational line, so is its conjugate. Example: (1 + √3, 2 + 2√3) is on the line y = 2x. So is (1 – √3, 2 – 2√3).

If a construction does not require the use of a compass or randomness, the algebraic nature of the constructed objects is the same.

Two points construct a line and two lines construct a point. The line and point so constructed have the same algebraic natures and exist with no extension of the computational field.

We note that the equation of a line is computed by adding, subtracting, multiplying, and dividing the coordinates. For example slope is computed using subtraction and division. These operations do not change the algebraic nature of the numbers. We show this by giving these objects the same color.

Using circles to construct geometric objects; field extensions and extra versions

A construction defines the distances with which we build our figures, thus creating the algebraic field in which we do our computations. Many constructions create points of the same algebraic nature as those from which the constructions began. The next paragraph gives an example where some points have a different algebraic nature.

If a line intersects a circle, the intersection will in general involve a square root not in the computational field of the line. Hence the computational field will be extended and the intersection points will both have a different algebraic nature from the original line.

A circle is constructed from two points. If the first two have the same algebraic nature so does the line between them, i.e., Its coefficients exist in the same field. The second intersection of the circle with the line has the same algebraic nature.

This has the result that many constructions do not require any field extensions and construct points of the same algebraic nature.

However if a line and a circle center exist in one field, the field will in general have to be extended for the intersections of the line with the circle.

Midpoints, perpendicular bisectors, and isosceles triangles; the intersection of two circles usually requires a field extension.

Book 1, Proposition 1.
To construct an equilateral triangle on a given finite straight line.

Proposition 11.
To draw a straight line at right angles to a given straight line from a given point on it.

In general the double intersection of two circles engenders a field extension in the form of a square root for the points of intersection. A very important corollary to this is that the line joining these two points, the radical axis, does not require this field extension. Hence to construct a midpoint of two given point no field extension is ever required. However, if a construction uses one of the two intersectionsnegative of a pair of circles (e.g., the 3rd vertex of an equilateral triangle), a field extension is required and the point so constructed has a different algebraic nature to those that created it. See the figure below.

The coordinate construction (algebraic) of a midpoint from two points requires addition and division, hence no field extension is required. The midpoint has the same algebraic nature as the points that determine it.

Perpendicularity requires that slope be changed to the negative reciprocal, a rational operation. The perpendicular bisector between two points has the same nature as the original points.

The vertex of the equilateral triangle is the intersection of two circles, in this case creating a number having the square root of 3. This root requires a field extension, so the vertex of the equilateral triangle has a different algebraic nature. A square root can be plus or minus, so the construction constructs two equilateral triangles. In general field extensions give multiple versions of an object.

The following picture shows the construction of the three propositions above. This construction begins with two points and the line segment between them, which have a given algabraic nature (blue). The vertex of the equilateral triangle, being the intersection of two circles, will be of a different algebraic nature to the endpoints of the initial segment, while the midpoint and the perpendicular bisector will have the same nature.

Consider the system of the two red points and the line that joins them. If we negate the square root that defines them, each point changes to a different point, the other one. The object defined by the two of the however, is invariant, as is the line that joins them.

To bisect an angle the red lines are constructed to be equal, and hence commensurable; but the blue line connecting their endpoints is incommensurable with the red lines. The bisector of the angle is constructed as the perpendicular of the blue line, and is thus of the same nature as the new line but a different algebraic nature than the initial points and lines of the angle.

A line intersects a circle twice, not once as shown in the above picture. The lines in the above figure should really be extended. On the two lines the circles create four intersections which leads to an extra bisector. Algebraic fullness often leads to extra versions of objects.

There are three algebraic natures in this picture: the original lines (black), the segments created by a circle drawn with a random compass setting (red), and the rectangle and bisectors (blue). Each side of the shown rectangle is constructed by the law of cosines from the red lines. The cosine is tanscendental, requiring a field extension. The cosines of the four angles determined by the red lines are rationally related, so all four sides of the rectangle have the same algebraic nature as the angle bisectors.

The inscribed circled is constructed from three angle bisectors, each of which is not constructible in terms of the others. This means that the incenter and incircle require field extensions and are incommensurable with the lengths and coordinates of the triangle. In the illustration, each bisector has a different color, illustrating that, in general, it is incommensurable with any given unit and with the other bisectors. The incenter is multicolored because it inherits the algebraic natures of all the angle bisectors.

A more modern way to say this is that the bisectors are determined from the edgelengths, each of which is computed as a square root so that the incenter and the incircle depend on three square roots, which gives the incenter a high level of algebraic complexity..

If a line connects two points, it inherits the algebraic complexity of those two points. If the coordinates of the two points are rational, the coefficients of the line will be also. Similarly a point determined by two lines will be rational if the lines are.

There is a rational structure of points and lines derived from any point or line in the triangle plane. The resulting structure can be seen here. This set of relationships among lines and points is as fundamental a structure as exists in triangle geometry. This is the projective structure of a point or a line in relation to a triangle.

Points have an algebraic nature determined by the coordinates used to describe them. First are the points with constant rational barycentric coordinates. This includes points such as the centroid of the triangle and the midpoints of the sides. These

and are the affine invariant background in which geometry occurs. These points are so fundamental that they are usually ignored.

Triangle geometry is usually done using either trilinear or barycentric coordinates. If you do not know these then skip this section.

Trilinear point coordinates are the distances to the sides of the triangle. Each coordinate thus requires a field extension independent of the geometrical situation. The algebraic nature of everything is all messed up. This makes fundamental considerations more difficult and limited.

Barycentric coordinate are affinely invariant, and those very coordinates express the underlying nature of the affine background of points. The algebraic nature of everything is clarified. Fundamental considerations are often easy.

Interestingly the objection to trilinear point coordinates does not extend to trilinear lines, where the sign ambiguity is desirable, particularly if line parameters are expressed in terms of triangle angles.

I tend to think that trilinear lines (expressed using angles) or barycentrics (expressed using distances) are fundamentally ok. Barycentrics would still be preferred as the coordinate machinery is well developed. No one has used trilinear lines since the late 1800's and this is not so well developed.

The vertices of the triangle and the edgelengths of the triangle have different algebraic natures, a fact that has profound and far reaching effects. The edgelengths are distances computed from point coordinates using square roots. Each edgelength requires a field extension and is algebraically different from the points that are the vertices, the lines that are the sides, and from the other edgelengths.

Geometrically incenter is constructed from angle bisectors, each of which is incommensurable with the sides. As shown above it has a complex algebraic nature that it inherits from the bisectors.

Algebraically the incenter has coordinates (a:b:c) where a,b,c are the edgelengths, which are square roots and thus irrational.

The algebraic and geometric worlds are true mirrors of each other, as they should be.

Geometrically: The incenter is defined as the point of concurrence of the angle bisectors. But we have seen that each angle bisector has a second version. Each vertex contains two bisectors. These 6 lines meet three at a time a 4 points, one of which is the incenter. The other three are outside the triangle and are the excenters and are equivalent to the original one.

Algebraically: Each coordinate of the incenter is a square root. The negated root will be a solution of the same equations and will have an analogous existence. There are 8 ways to negate the three sides, which reduce to 4, as pairs like +++ and --- are considered equivalent.

Geometrically: The circumcenter of a triangle is constructed from perpendicular bisectors of the sides and has the same algebraic nature as the points used to create the triangle.

Algebraically: The circumcenter has coordinates that depend on the squares of the edgelengths, thus rational in terms of the vertex coordinates, and requires no field extension.

The coordinates of the circumcenter and the incenter have different algebraic natures, corresponding to a fundamental difference in their geometrical natures. John Conway names the points with rational coordinates "strong" and those with roots "weak." The fundamental difference is two fold. Weak points have multiple versions while strong points are alone. Geometrically weak points behave differently from strong ones. Here is a web page (with movies) illustrating the fundamental geometric difference between the circumcircle (strong) and the incircle (weak).

Every introduction of algebraic complexity in the form of field extensions adds a new layer of geometric complexity.

Many points derived from the incenter or the incircles share their four-fold or quartile nature. Here is a picture of the classical weak points, all of which can be derived from the incenter or incircle. These are the Gergonne, Nagel, Mittenpunkt, Spieker, Bevan, Feuerbach, Clawson, and all their medials and conjugates. Each has coordinates that depend on a, b, and c, the edgelengths of the triangle. John Conway deems them "quartile." This picture shows the 4 Gergonne and 4 Nagel points as they create an interlinked structure named "desmic" by John Conway. See also this slideshow.

The Fermat point is created by erecting equilateral triangles on the edges of the reference triangle. The equilateral triangle construction uses the intersection of two circles, requiring a field extension as shown above. But these circles intersect twice, so there is a second equilateral triangle constructed at the same time. Two versions of everything. Two versions of Fermat points.

The irrational number introduced by this construction is √3, which of course can be plus and minus, algebraically giving the two versions.

John Conway calls this type of point fissile. The line connecting the pair of fissile points is almost always strong.

An lines, conics, cubics and other curves are described by equations. As such they exist in a certain computational field. If any point on it requires a field extension, then all versions of that point have coordinate will satisfy the equation.

A less general way of saying this is that if a weak point is on a strong curve, then all versions of the point are on that curve.

Examples: The isotomic Spieker point is on the GK line, so all its versions are as well.
The Spieker point is on the Kiepert hyperbola, so all four of them are.

Similarly if a weak curve goes through a strong point, then all versions of the line go through the same point.

Examples: The Nagel line IoNo goes through G, hence all 4 Nagel lines do.
The Feuerbach hyperbola goes through H, hence all four do.