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SIMPLICITY: CAS Geometry Distances Equidistances Constant Distances Algebraic Eqs. and Inequations Angles Taxicab Distance Fields   VERSATILITY List of Polylines   FLEXIBILITY: Elastic geometry Elastic Geometry Newton's Principia Tensegrities

Abstract   Since its inception, GeoGebra has been specifically designed to display the dual representation, both graphical and algebraic, of mathematical objects. In this presentation, as the central focus, I will show some procedures that exploit the didactic possibilities of this duality. These procedures, presented to students aged 15 or 16, are so simple, engaging and quick to create that they allow the students themselves to generate and use them from scratch... with great success! Despite their simplicity, we will see that they are so powerful that they enable us to delve into mathematical depths that are practically unapproachable in the high school classroom without the assistance of GeoGebra, ranging from algebraic structures (such as fields) to non-Euclidean metrics (like the taxicab metric).

Note: All the GeoGebra constructions linked on this webpage have been created by the presenter of this content. None of them, except for the Bubbles construction, make use of JavaScript programming.

The Author

In my 40 years of teaching as a High School Teacher, in the pursuit of fostering the interest of students, I have researched the relationship between Mathematics and other areas as diverse as Games , Perception , and Music . The advent of Dynamic Geometry brought new and significant opportunities to engage students and promote the creation of their own constructions.

 

My relationship with GeoGebra dates back to 2005 when I first encountered this program created by Markus Hohenwarter [7], although I had already worked with other dynamic geometry software. Two years later, in 2007, Professor Tomás Recio invited me to the International Center for Mathematical Meetings (CIEM , Cantabria) which brought together several Spanish high school teachers who were pioneers in the educational use of dynamic geometry. During that meeting, I defended the efficiency of GeoGebra [10] compared to other software like Cabri. One outcome of that gathering was the formation of the G⁴D  group, consisting of J.M. Arranz , J.A. Mora , M. Sada  and the author of this text .

 

Two years later, from the Ministry of Education of Spain, Antonio Pérez , who was then the director of the Institute of Educational Technologies (ITE, now INTEF ), entrusted me with the task of conducting courses for training Primary and Secondary Education teachers in GeoGebra [11, 13]. Additionally, I was tasked with creating a set of complete activities (including topic introductions, constructions to explore, and questionnaires) for students, categorized by subjects and levels, which we named the Gauss Project [14, 1]. Simultaneously, Tomás launched the first Spanish-language GeoGebra Institute, the GeoGebra Institute of Cantabria , of which I have been a Trainer since its inception.

 

Introduction

The main objective of this conference is to demonstrate the close relationship between algebraic and geometric procedures using GeoGebra. A significant portion of the time will be dedicated to presenting activities that can be approached by secondary education students through constructions carried out by themselves. Beyond sporadic use for exploring specific content, this type of construction gains its full didactic power in mathematics education based on competence acquisition.

In the first part of this presentation I will detail very simple procedures that harness the strong interconnection between geometry and algebra that gives its name to GeoGebra (hence the title of Principia) and enable secondary school students (around 15 or 16 years old) engage in mathematical explorations that are "in principle" beyond their reach, leaving the heavy algebraic and geometric calculations to the powerful tools of GeoGebra, much like we currently rely on calculators and spreadsheets for tedious arithmetic calculations.

 

In particular, the ease with which we can create parallel lines and circles will aid us in constructing a dynamic offset whose colorful trace allows us to visualize a wide variety of geometric loci. Simultaneously, the Computer Algebra System (CAS), applied to Euclidean distances, will facilitate the creation of implicit curves that conform to these loci. We will also see how to occasionally convert these implicit curves into algebraic equations and inequations.

 

We will expand the use of CAS to angles and also address other non-Euclidean distances, such as the taxicab distance.

 

We will conclude this first part with a reciprocal example, wherein resorting to geometry will aid us in visualizing and manipulating the concepts and properties inherent to the algebraic structure of a field.

 

In the second part, I will showcase some ideas for creating slightly more sophisticated yet equally captivating constructions, which can serve as models to be analyzed or modified by students.

 

Firstly, we will explore how GeoGebra's lists facilitate the incorporation of a substantial amount of information. As an example, we will represent the coastlines of continents in a single list, creating a template of the Earth.

 

Next, we will employ vectors to instantaneously modify (thanks to the GeoGebra scripts) the position of points according to our interests. These vectors can be used to combine repulsive forces (like particles of the same charge), attractive forces (similar to those used by Newton to formulate his universal law of gravitation), or simply reactive forces (as in elastic collisions).

 

Furthermore, we can use vectors to create an "elastic geometry". In this context, points do not possess a fixed location, but rather their position at each moment is the result of the application of the aforementioned forces. For instance, in elastic geometry, a point is not "on" a circle, but is inevitably drawn towards it, it has "its limit" in it.

 

Lastly, as an application of this type of elastic geometry, we will examine examples of tensegrities constructions.

 

SIMPLICITY: CAS Geometry

GeoGebra: Geometry and Algebra

Since its inception, GeoGebra has enabled the rapid and straightforward construction of geometric models, enhancing learning, teaching, and research. This ease of use contributes to the IKEA effect (we value more what we are capable of building ourselves, hence the success of classic construction games, such as Meccano or Lego). Here, I present one of the constructions that I showcased at the 2007 CIEM meeting [9], where I advocated for the use of this program in education. I have chosen this one because Markus popularized it by featuring it for years at the header of the official GeoGebra website .

 

Since then, GeoGebra has seen significant development. We can create countless models and applications related to areas such as Arithmetic, Equations, Functions, Isometries, Complex Variables, Statistics, Probability...

 

The variety of procedures is also immense: from slicing a hypercube into sections [15] to automatically proving a proposition [12, 17, 18, 28].

 

It is evident that what characterizes dynamic geometry is precisely its dynamism. Just like in most animals with a visual system, evolution has triggered a mental alert when any object or entity in our surroundings starts moving. Therefore, motion is a natural and excellent means to focus attention.

 

However, how can we easily visualize geometric loci without resorting to complex constructions or using cumbersome algebraic equations? We will see that one possible answer is to use the CAS.

Distances

The mathematical concept I will revolve around is a fundamental one: distance.

 

When placing a point in a space, the concept of distance to it behaves like what physicists call a "field": it doesn't manifest until we introduce another object into it.   We will employ two simple procedures to visualize geometric places related to distance: the creation of implicit curves and the use of dynamic offset with activated trace.

 

Classic Method: Sequences of Parallel Curves (Static Offset)

Using the UnitPerpendicularVector command (and its opposite vector), it's simple to create sequences of parallels to a line at progressive distances. For each line r, we find a pair of sequences:

 

Sequence(Translate(r,  k UnitPerpendicularVector(r)), k, 0, 20, 0.2)

Sequence(Translate(r, -k UnitPerpendicularVector(r)), k, 0, 20, 0.2)

 

 

Thanks to the CurvatureVector command and the Locus tool, we can generalize parallelism to many curves (offset ). If P is a point on curve c, the two parallel curves at distance k will be given by the locus of the points:

 

P ± k UnitVector(CurvatureVector(P, c))

 

Note that, in general, offset curves are not congruent with the original curve. In other words, parallel curves are not simple translations, except in the case of lines.

 

However, in the case of the circle (let's assume with center O and radius 4), whose offset is also a circle, we don't need the CurvatureVector command or the Locus tool, as it's sufficient to vary the radius of the original circle appropriately:

 

Sequence(Circle(O, 4 + k), k, 0, 20, 0.2)

Sequence(Circle(O, 4 – k), k, 0, 20, 0.2)

 

Furthermore, if we consider a point O as a circle with radius 0, we obtain a unique sequence of offsets centered on it:

 

Sequence(Circle(O, k), k, 0, 20, 0.2)

 

In summary, we can easily create sequences of parallels to lines, circles and points.

Dynamic Offset with Activated Trace

Now we will replace each sequence of parallels with a single dynamic parallel. As before, using the UnitPerpendicularVector command (and its opposite vector), it's straightforward to create parallels to a line at a given distance d.

 

For each line r, we find a pair of parallels:

 

Translate(r,  d UnitPerpendicularVector(r))

Translate(r, –d UnitPerpendicularVector(r))

 

Thanks to the CurvatureVector command and the Locus tool, we can generalize parallelism to many curves (offset). If P is a point on curve c, the two parallel curves at distance d will be given by the locus of the points:

 

P ± d UnitVector(CurvatureVector(P, c))

 

Note that, in general, offset curves are not congruent with the original curve. In other words, parallel curves are not simple translations, except in the case of lines.

 

However, in the case of the circle (let's assume with center O and radius 4), whose offset is also a circle, we don't need the CurvatureVector command or the Locus tool, as it's sufficient to vary the radius of the original circle appropriately:

 

Circle(O, 4 + d)

Circle(O, 4 – d)

 

Furthermore, if we consider a point O as a circle with radius 0, we obtain a unique sequence of offset centered on it:

 

Circle(O, d)

 

In summary, we can easily create sequences of offsets of lines, circles and points.

 

Also, we can create the intersection points of two objects and the corresponding locus. The problem with using the Locus command or tool is that in many situations (more complex than the one shown here) it's not possible to use it properly.

 

Since GeoGebra is a Dynamic Geometry program, we can not only move geometric objects at our will but also establish automatic animations [22].

 

To achieve this, we add a trace to the offset and choose a decreasing value of d (opposite to a slider "increasing once"). Note: alternatively, we can choose an increasing value for d (increasing once) and assign it a speed of -1 instead of 1.

 

By doing so, simultaneously offsetting a point and a line, for instance, we can visualize the parabola through color contrast.

 

The advantage of using offset over implicit curves, which we will see next, is that it allows us to pause the procedure's playback at any time and observe how the traces of the lines intersect. This helps us understand why these intersection points are part of the sought-after locus.

 

Implicit Curves from Definitions in CAS

A parabola can be defined as the locus of points in the plane equidistant from a line (directrix) and an external point (focus). Locating one point (the vertex) is easy, but how do we locate the others?

 

 

With GeoGebra, we can create a free point to explore the situation and mark those positions where both distances are equal. It's quite instructive but, after several exercises, it becomes tedious.

 

Alternatively, we can construct a generic point that defines the locus, but this construction will only work for this case or similar cases.

 

We can also create the implicit curve by defining an arbitrary point X(x,y) in the CAS View:

 

X:= (x, y)

 

the distance from X to the focus F:

 

XF(x,y):= Distance(X, F)

 

 

the distance from X to the directrix r:

 

Xr(x,y):= Distance(X, r)

 

 

and by equating both distances:

 

XF – Xr = 0

 

 

GeoGebra uses numerical algorithms to create this implicit curve, so small errors or omissions may appear in some cases.

Note: At least for now, GeoGebra does not represent equations of this type in three variables. That is, it recognizes x² + y² + z² = 16 as a sphere, but it does not recognize the equivalent equation (sqrt(x² + y² + z²))² = 16 as such.

 

Equidistances

Now we just have to use those simple tools to investigate a wide variety of situations with their assistance.

 

From now on, we consider the distances from an arbitrary point X(x,y) to A and B defined as:

 

XA(x,y):= Distance(X, A)

XB(x,y):= Distance(X, B)

 

to a line r as:

Xr(x,y):= Distance(X, r)

 

and to a circle c as:

 

Xc(x,y):= Distance(X, c)

Equidistance to Two or Three Points

By contracting the circles with the activated trace, at each point in the plane, the color of the nearest center remains, resulting in the perpendicular bisector.

 

The implicit curve of the perpendicular bisector of AB is given by the equation:

 

XA – XB = 0

 

In the case of three points, we can visualize the circumcenter [2] of the triangle they form.

 

Equidistance to a Point and a Circle

If the point is inside the circle, we obtain an ellipse, and if it's outside, a branch of a hyperbola.

 

The implicit curve of the ellipse or branch of the hyperbola is given by the equation:

 

XA – Xc = 0

 

 

Note that a construction similar to the one for the parabola also allows us to generate this locus.

 

Equidistance to Two Lines

By simultaneously moving parallels to two lines, with the trace activated, the color of the nearest line remains at each point, yielding the angle bisectors. In the case of three lines, we can visualize the incenter and the excenters of the triangle they determine.

 

As a specific case, we can visualize the medial axis of a polygon as the boundary (composed of segments and arcs of parabolas) between the surviving traces.

 

Similar to before, we can also easily visualize and construct the corresponding geometric locus of the angle bisector.

Equidistance Line-Circle and Two Circles

The locus of points equidistant from a line and a circle is generally composed of one parabola or, if they intersect, two parabolas.

 

The locus of points equidistant from two circles is generally composed of a branch of a hyperbola or, if they intersect, an ellipse.

 

Voronoi Diagram and Similar Maps

Although the implicit curve is faster to create and use, the offset method allows us to tackle problems that the implicit curve cannot address. For instance, if instead of applying the offset method to the equidistance between two points (the bisector), we apply it to several points, we obtain the Voronoi diagram (or Thiessen polygons).

 

Similarly, we can also create the map of regions closest to a collection of lines or circles.

Note: GeoGebra's Voronoi command does not color each region. A dynamic way of achieving this can be seen here .

Equidistance to Two Curves

We can apply the offset method to two curves as long as we can calculate the normal vector at each point of both.

 

In situations where calculating the normal vectors isn't feasible, we can always generate a heat map using the dynamic color scanner technique [3, 19, 26, 27, 31].

 

Constant Distances

Point-Point

When the sum of the distances from the points of the sought-after locus to points A and B is constant, we obtain an ellipse.

 

When the difference of the distances from the points of the sought-after locus to points A and B is constant, we obtain a branch of a hyperbola. (In the case where the constant is 0, we obtain the perpendicular bisector.)

 

When the product of the distances from the points of the sought-after locus to points A and B is constant, we obtain a Cassini oval . If the constant coincides with the square of half the distance AB, we obtain a Bernoulli lemniscate .

 

Surprisingly , when the quotient of the distances from the points of the sought-after locus to points A and B is constant, we obtain a circle. (In the case where the constant is 1, we obtain the perpendicular bisector.)

 

Constant Ratio Point-Line

When the quotient of the distances of the points of the sought-after locus to point A and line r is a constant k, we obtain a conic section, which will be an ellipse, parabola or hyperbola depending on whether the value of k (eccentricity) is less than 1, equal to 1, or greater than 1, respectively .

 

Constant Linear Combination

We can generalize the sums and differences to any linear combination of distances between points, point and line, point and circle, etc.

 

This leads to conic sections and quartic curves (Cartesian ovals, products of lines...), Pascal's limaçon , as well as higher-degree curves formed as products of the aforementioned curves.

 

Constant Sum to Three or Four Points

In the case of a constant sum k of distances to three points A, B and C, you simply need to input:

 

XA + XB + XC = k

 

To perform the offset, what we do is overlay the trace of ellipses Ellipse(A, B, (k–h)/2) with circles Circle(C, h), where h is a positive real parameter that decreases from the value of k to zero. The boundary points of color will then be precisely the points that satisfy:

Ellipse(A, B, (k–h)/2) = Circle(C, h)

which is equivalent to the sum of the distances from those points to A, B and C being exactly the predetermined quantity k (since XA + XB = k – h, XC = h). This way, we can display a 3-ellipse . In the case of four points the traces of two ellipses overlap, determining a 4-ellipse.

Note: An algebraic approach to this situation, also using GeoGebra, can be seen in this article [8] by Zoltán Kovács.

Algebraic Eqs. and Inequations

Equidistance to a Point and a Circle: Algebraic Equation

Previously, we directly defined the distance from a point X(x,y) to the circle c as:

 

Xc(x,y):= Distance(X, c)

 

 

With this, the equation for the points equidistant from a point A and a circle reduces to:

 

XA – Xc = 0

 

If the circle has a center O and radius s, we can redefine the equation as:

 

|XO – s| = XA

 

This redefinition allows us to visualize the two branches of the hyperbola. To achieve this, we transform the previous irrational equation into an algebraic equation by squaring it to eliminate the roots (reaching the following expression is straightforward, as it doesn't require grouping, simplification or cancellation steps, but it's also okay to assist students with limited algebraic resources in solving this small exercise—the result is worthwhile):

 

(XA² – XO² – s²)² = 4s² XO²

 

Moreover, algebraic equations have the advantage of enabling representation of the corresponding inequations without resorting to the offset method. For this, we simply define:

 

XA2(x,y) := Simplify(XA^2)

 

XO2(x,y) := Simplify(XO^2)

 

This way, we can introduce the inequations:

 

(XA2 – XO2 – s²)² < 4s² XO2

(XA2– XO2 – s²)² > 4s² XO2

Constraints on Inequation Representation

However, not always does the algebraic equation allow GeoGebra to represent the corresponding inequations. As shown in the official manual , this representation is limited to the following cases:

 

 

• Polynomial inequalities in one variable, like x³ > x + 1

• Quadratic inequalities in two variables, like x² + y² + x y < 4

• Linear inequalities in one of the variables, like 2x > sin(y) or y < sqrt(x).

 

When we find the algebraic equation corresponding to XA – XB = k, we obtain the same as the one corresponding to XA + XB = k:

 

4 XB2 XA2 = (k² – XA2 – XB2)²

 

This equation reduces to a quadratic in two variables, allowing GeoGebra to represent its corresponding inequations.

Note: The common quadratic equation of the ellipse and hyperbola is nothing but the general equation of a conic:

a x² + b x y + c y² + d x + e y + f = 0

where the ellipse and the hyperbola differ only by the sign of the discriminant b² – 4 a c.

However, the algebraic equation corresponding to XA XB = k doesn't represent a conic, so GeoGebra can't represent the corresponding inequations. On the other hand, the algebraic equation corresponding to XA = k XB becomes a conic once again, allowing GeoGebra to represent the corresponding inequations.

 

Equal Sums of Distances to Two Pairs of Points

If we input XA + XB = XC, the resulting locus corresponds to the intersection of a family of ellipses XA + XB = k, and a family of circles XC = k, as the parameter k varies.

 

In the case of four points, XA + XB = XC + XD corresponds to the intersection of two families of ellipses.

Note: Calculating the minimum and maximum values of k that ensure intersection is not straightforward. An approach can be seen in [6].

The case XA + XB = Xr is also presented, along with the representation of the corresponding algebraic equations.

 

Families of Curves

In conclusion, the field to explore can expand indefinitely. As final examples with distances, let's observe some results involving powers.

 

It is easy to demonstrate that the representation of XA2 + XB2 = k, with k constant, is a circle centered at the midpoint of A and B.

 

Note: The radius of that circle is sqrt(k/2 − (x(A-B)/2)² − (y(A-B)/2)²).

From this, we deduce that the locus where the sum of the squares of distances to several points is constant is a circle centered at the midpoint of those points.

 

Furthermore, taking D = XA2, we can observe that the real-plane representation of any polynomial p(D) is exclusively composed of one or more circles.

Note: This follows from the Fundamental Theorem of Algebra, since p(D) can be decomposed by factors (D − c), where c is a complex number. If c is non-negative real, then D − c = 0 corresponds to a circle with radius the square root of c. Otherwise, nothing is displayed.

Here, we also see that we can represent multiple curves of the same family, such as XAⁿ = XB and observe their behavior simultaneously.

 

 

 

Angles

The following is one of the images generated using the dynamic color scanner that I particularly like. The scanner has tremendous versatility, it can create a heat map for virtually any situation [3, 19, 26, 27, 31].

 

 

In this case, the first isogonic point I₁ is visualized by intersecting the loci that see each pair of sides of the triangle under the same angle.

Note: I₁ coincides with the Fermat point when the triangle's largest angle is not greater than 120 degrees; otherwise, the Fermat point coincides with the vertex corresponding to that angle. It can be calculated directly as the triangle's center X(13) :

 

I₁ = TriangleCenter(O, A, B, 13).

 

Taxicab Geometry

Indeed, as Magritte would say, Ceci n'est pas un disque (This is not a disk) , but we will see that it can indeed be the representation of a circle if we consider the Taxicab metric.

 

I will use the prefixes T and E to distinguish between the Taxicab metric and the Euclidean metric.

Note: Despite the fact that a T-circle has a square shape, Magritte would probably still assert -rightly- that the T-circle we see is only a representation, an image of the disk. However, furthermore, here, unlike what happens with a pipe, the represented disk is a mental abstraction (an ideal mathematical form) instead of something material, which makes the potential confusion even greater.

In Taxicab metric (or Manhattan metric), distances are measured horizontally and vertically, never diagonally. Thus, the T-distance from an arbitrary point (x, y) to point O is the sum of the horizontal and vertical differences, in absolute value, of its coordinates:

 

XO(x,y) := |x – x(O)| + |y – y(O)|

 

Unlike in Euclidean metric, GeoGebra does not have the T-distance command implemented, so we will have to formulate both the distance between two points and the distance between a point and a line "manually", providing both formulas to the students.

 

Just as GeoGebra renders a segment by fitting it into the pixel grid of the screen, we can imagine a diagonal segment composed of horizontal or vertical segments as small as we want: the T-distance between two points B and C will not change.

 

The T-distance between B and C will also be the same for any increasing or decreasing arc of a function whose graph goes from B to C.

Basic T-Constructions

If we fix a point O in the plane, we can consider the taxicab distance from the rest of the points to O.

 

As we have seen, the points that are T-equidistant from O form a square (with diagonals parallel to the axes). If the radius is r, the perimeter is 8r, so the ratio between the T-circumference and its T-diameter is 4 (instead of 𝜋).

 

By fixing another point I different from O, we establish an orientation O→I and a line. We will take the T-distance from O to I as the unit. We can continue to think of the T-lines as if they were E-lines, as only the way of measuring each segment changes. Remember that pixels force GeoGebra to draw lines composed of horizontal and vertical segments!

 

Given a point A on the line r, there exists only another point A' on this line at the same distance from O as A. This T-symmetric coincides with the E-symmetric point.

 

For two distinct points A and B, we can find all the points equidistant from them.

 

This T-perpendicular bisector does not coincide with the Euclidean perpendicular bisector.

 

By intersecting the T-perpendicular bisector with the line, we obtain the midpoint, which coincides with the Euclidean midpoint.

 

Perpendicular and parallel lines are the same as in Euclidean geometry, but the orthogonal projection of a point onto a line does not generally provide the nearest point on the line. (Moreover, "nearest point" is not uniquely determined when the line has a slope of 1 or –1.)

 

To perform a T-inversion , we move points A and I to the horizontal line passing through O, invert (x(A), y(O)) on the dashed E-circle with center O passing through (x(I), y(O)), and create similar triangles that guarantee the new inversion.

 

The T-inverse of A does not coincide with the E-inverse of A.

T-Angles

In the unit T-circle, we can define the T-radian exactly the same way we define an E-radian in the unit E-circle. To T-measure an angle, it's sufficient to measure the T-length of the corresponding (straight) arc on the unit T-circle. A T-circle has 8 T-radians.
 

Perpendicularity and parallelism are preserved under rotations, but, in general, T-distances are not invariant with respect to E-rotations... nor with respect to T-rotations! In fact, one of the peculiarities of T-distance is that it is sensitive to the orientation of lines: a segment, when T-rotated, no longer measures the same.

 

Cartoon of Mafalda, by Quino
"It's quite a puzzle, isn't it? How on earth
does time manage to round the corners on square clocks?"
 

The sum of the angles in any T-triangle is 4 T-radians. A T-triangle can be equilateral or equiangular, but it can never be regular.

Any E-square is also a T-square. But because the taxicab distance is not uniform in every direction, these two T-squares have the same perimeter (though not the same area):

 

The square on the left is also a T-circle.
The one on the right is not.

 

Trigonometric T-functions are much simpler than their Euclidean counterparts. For example, the T-sine function is not only non-transcendent but also piecewise linear. The T-tangent function is composed by piecewise E-hyperbolas.

Note: One possible expression for the T-sine function is: tsin(x) = 1 – 2 |1/2 - x/4 + 2 floor(1/4 + x/8)|. Thus, the T-cosine function can be defined as tcos(x) = tsin(x+2). The T-tangent function is a piecewise homographic function .

T-Equidistances

When contracting T-circles with the trace activated, at each point in the plane the color corresponding to the nearest center survives.

 

With multiple points, we can visualize the Voronoi diagram and compare it with the one corresponding to the Euclidean distance.

 

To analyze the equidistance point-line, we need to determine the distance from a point (x, y) to a line r: a x + b y + c = 0. This distance is (this formula is provided to students and can be directly introduced in the algebraic view):

 

Xr(x,y) = |a x + b y + c| / Max(|a|, |b|)

 

From the point-line equidistance the T-parabola arises, while from the point-circle equidistance the T-ellipse and the T-hyperbola emerge.

 

If we consider equidistance to the sides of a polygon, its skeleton and median axis arise. We can traverse it with a bitangent disk to verify this.

 

Finally, we can also find the T-equidistant path between two curves, either through offset (as shown here) or by generating a heat map.

Preliminary Constructions

The Field of the Points on a Line

Let's continue with our construction process of the structure. Now we will define the four elementary operations.

• Addition: To obtain A + B, we reflect O in M_AB obtaining a new point on r.
• Subtraction: To obtain A − B, we add A + B'.
• Multiplication: We create the product by constructing similar triangles, obtaining a new point on r.
• Division: To obtain A/B, we multiply A x B^–1. Division is not commutative.
• Order. The symmetry I' O I allows us to define a ORDER RELATION:

A ≤ O :⇔AI' ≤ AI                    A ≤ B :⇔A − B ≤ O

Structure. Based on everything aforementioned, the set of points on the line r, endowed with the operations of addition and multiplication as defined, constitutes a similar structure ("ordered field") to that of ℝ (real numbers). In fact, we can establish a bijection (isomorphism) between both structures:

(r, O, +, ×) → (ℝ, +, ×)

correspond each point P on r to the real number –OP if P<O and the real number OP if P≥O.

Note: Let us mark that we do not delve into the more intricate question of how to geometrically construct all the points on the line (completeness of the real line). We assume that every point corresponds to a number and vice versa. However, if we wish to restrict ourselves to points constructible with the mentioned operations, we can establish an isomorphism of those points (no longer spanning the entire line) with the field of the constructible numbers.

The points on a line are not the only geometric objects we can endow with the structure of a field. We can apply the same concept to any other set of objects that share the same definition, in which there is only one free point residing on a line. Here are a few examples:

VERSATILITY

List of Polylines

Lists allow a large amount of information to be grouped into a single object. In this example, humble polygonal lines, lacking the elegance found in many curves, make up for it with the versatility that their grouping in a single list provides.

The Static Closed Polyline

We can import large amounts of data into a spreadsheet. In this case, 6122 geographical coordinates from publicly available North American Cartographic Information Society data , once cleaned up in a word processor and properly grouped, become lists.

 

 

Each point list is called by the Polyline command, so we finally obtain a list of polygonal chains whose arguments are point lists. Unlike polygons, let's remember that the vertices of the polyline don't necessarily have to lie in the same plane. Through this process, we achieve a singular design: that of the Earth's coastlines. Thanks to this design, we transform an ordinary sphere into a model of the Earth's surface.

Coast = {Polyline((5; 3.142; 1.204), (5; 3.117; 1.211), (5; 3.067; 1.22), (5; 3.031; 1.219), (5; 2.975; 1.223), (5; 2.967; 1.216), (5; 2.981; 1.204), (5; 2.96; 1.199), (5; 2.93; 1.214), (5; 2.896; 1.212), (5; 2.863; 1.216), (5; 2.832; 1.215), (5; 2.809; 1.212), (5; 2.787; 1.217), (5; 2.79; 1.23), (5; 2.775; 1.237), (5; 2.74; 1.24), (5; 2.67; 1.236), (5; 2.624; 1.25), (5; 2.609; 1.26), (5; 2.452; 1.271), (5; 2.429; 1.264), (5; 2.441; 1.248), (5; 2.413; 1.25), (5; 2.4; 1.245), (5; 2.366; 1.251), (5; 2.336; 1.246), (5; 2.308; 1.254), (5; 2.291; 1.236), (5; 2.264; 1.242), (5; 2.242; 1.256), (5; 2.252; 1.264), (5; 2.244; 1.275), ...

With this model [16] we can construct a scenario in which we can engage in a multitude of activities. These activities range from observing geographical elements (pole, meridian, parallel, cardinal point, equator, tropic, polar circle, time zone...), to conducting analysis and measurements (octant, latitude, longitude, course, rhumb line, orthodromic distances, spherical triangle...).

Note: This model can serve as an excellent example of the collaborative spirit that characterizes the resources created by the GeoGebra community since its inception. Using it as a template, Chris Cambré published books on GeoGebra about map projections in Dutch and English, as well as another one about Mercator . Afterwards, Carmen Mathias translated both into Portuguese (, ). Closing the circle, I translated them into Spanishl (, ). This collaborative behavior reaches its peak in the GeoGebra forum .

 

If we now add the apparent solar orbit, we'll have an armillary sphere (or spherical astrolabe) that enables us to analyze the passage of time (years, days, hours, GMT, UTC, seasons, zenith, obliquity of the ecliptic, vernal point, equinox, solstice, celestial coordinates, analemma, celestial horizon, sunrise, sunset, the 88 constellations, the zodiac, the brightest stars...) [20].

Note: The Right Ascension and Declination data for each star have been collected from here.

 

 

The Self-Generated Static Polyline

At times, we can spare ourselves the task of importing vertices. The iteration of a script associated with an animated slider (we will delve into this procedure later) makes it easy to construct situations in which the list of vertices for a polyline gradually expands on its own. A typical example of applying this method is the generation of fractals, like the one illustrating the iterative process for creating the well-known Koch snowflake and its corresponding anti-snowflake [29].

FLEXIBILITY: Elastic Geometry

Elastic Geometry

Normally, a point either is or isn't in a specific position. However, through scripts and vectors, we can introduce flexibility, granting points the ability to move freely while attempting to maintain a certain relationship with other points.

Instead of fixing a specific position for each point, we will establish a relationship with the rest of the points.

Scripts and Vectors

Our goal is to achieve an equilateral polygon with all its vertices free. How can we close the polyline while keeping its vertices free (like a carpenter's ruler)?

Or, starting from a polygon: How can we construct a rhombus while keeping the fourth vertex free?

 

The solution lies in using scripts. For instance, a free point Q will always remain 5 units away from the free point P if this script is executed upon updating the position of P:

 

SetValue(Q, P + 5 UnitVector(Q−P))

 

and updating the position of Q executes the script:

 

SetValue(P, Q + 5 UnitVector(P−Q))

 

This way, in a rhombus, we can maintain the distance between vertices A and B while both points remain free.

 

We can also represent types of triangles (right-angled, isosceles, equilateral...) that retain their characteristics while any of their three vertices can be moved.

 

This method also applies to preserving angles rather than distances. Simply modify the vector applied to the point, using the appropriate rotation to readjust the angle. As an example, we can observe an equiangular pentagon with all its vertices free.

Here, a result is included (published in 2015) that serves as a beautiful example of the close relationship between geometry and algebra: "a polygon with n sides is equiangular if and only if e^2𝜋i/n is a complex root of the polynomial of degree n–1 whose coefficients are the lengths of the consecutive sides of the polygon" .

Linkages

An example in space. This construction is based on the aforementioned principle to maintain the distance (not the angles) between the vertices of a cube.

 

 

 

This way, an articulated cube is obtained, where its faces are not necessarily flat, but consist of hinges formed by two isosceles triangles. The possible positions of its vertices become difficult to analyze without resorting to this method [4, 5, 32].

 

Animated Slider Scripts

However, we cannot generalize this method to polygons with more than 4 sides, as adjusting the length of the fifth side can disrupt the already adjusted ones. We would need iterative, continuous readjustment, until achieving the desired result. Well, we can achieve this continuous readjustment by utilizing a script that is executed continuously, associated with the update of the value of an animated slider.

 

For instance, in the first scene, it might seem that P is a point on the circle (with a radius of 4 centered at A), but in reality, it is a free point. By dragging it, it will return to the circle because the script of the animated slider is executed continuously. This script is:

 

SetValue(P, P + u)
 

where:

 

u = A - P + 4 UnitVector(P − A)

 

We can add a coefficient (k) to control the speed: SetValue(P, P + k u)

If we add another vector, P will move towards the nearest intersection, even if there is no intersection!

By doing the same with 5 points and 5 vectors, voilà! We can now maintain equidistance among more than four points.

 

 

Fluid

By masking the polygon or the polyline with curves, for example using splines , we smooth the perception of the effect of the sum of internal forces (which preserve the distance of the points) with external forces (which apply a movement to the ensemble).

 

Amorphous

The achieved versatility is significant. For instance, we can literally make figures "jump through the hoop". Let's remember that at no point is the shape of the whole set determined; rather, it's a result of the vectorial addition corresponding to the position reached at each moment.

Funicular Polygon

In this example, we start with a thread without appreciable mass from which weights hang (the vertices of the polyline). The script of slider-associated animation causes these vertices to vertically move downwards, simulating gravity (external force), while the cohesive forces between neighboring vertices (internal forces of the thread) limit this movement.

 

As we can observe, after a few seconds, the polyline takes on the shape of the resulting funicular polygon . The vertices align themselves quite well with an ellipse. If we add more vertices, the alignment will come closer and closer to the theoretical parabola (infinite point loads uniformly distributed horizontally). The catenary is not far off either (it would be the result of removing the weight from the vertices and adding uniform weight to the polyline thread: the loads are uniformly distributed but not horizontally, rather along the curve, that is, separated by the same arc length instead of the same horizontal length).

Newton's Principia

If there's a mathematical branch where traditional algebra and geometry naturally blend, it's in analytic geometry, the core of dynamic geometry programs like GeoGebra.

 

One of the key concepts in analytic geometry is that of a vector. Its graphical representation as an arrow prompts us to think about movement, about dynamism. We will use vectors to create dynamic procedures, which are very simple yet incredibly powerful, allowing us to address various situations.

 

Vector Sum

The hands of an analog clock can serve as an illustration of vectors. The tip of the hour hand traces a circle, much like the tip of the minute hand. When we combine both vectors, we obtain a more intriguing path (this path intersects itself at 11 points, assuming the minute hand is longer than the hour hand, in 11 different directions, which means that within 12 hours, the vector sum of both hands coincides 121 times). If we also include the second hand, vector addition leads to an even more complex trajectory.

Vector sum open the door to the simulation of balances of forces. Thanks to vectors, we can simulate forces, whether attractive or repulsive, that compel a point to seek a relatively stable position (or path), one that is in equilibrium with respect to other forces or constraints.

Circular Confinement Between Repelling Points

We can imagine the points as charged particles that repel each other and are confined to the same circle. Each point is associated with the sum of the repulsion vectors from the other points. In this way, the points automatically readjust themselves, seeking the equilibrium position. As expected, this position always corresponds to the vertices of a regular polygon inscribed in the circle.

Square Confinement Between Repelling Points

If we replace the circle with a less symmetrical shape, such as a square, regular distributions are no longer possible in all cases. Nevertheless, the equilibrium position is still always reached at the perimeter boundary.

 

Spherical Confinement Between Repelling Points

If we transition from the circle to the sphere (the Thomson problem , a particular case of one of the eighteen unsolved mathematical problems proposed by mathematician Steve Smale in the year 2000), perfect regularity is no longer achievable since there are no Platonic solids with 5 or 7 vertices, for example.

 

Moreover, it is not even true that equilibrium is always reached in perfect regularity! In fact, with 8 vertices, the cube is not the configuration that achieves equilibrium. Note also that in most cases, polyhedra with triangular faces appear (but generally not equilateral, hence they are not deltahedra ).

Minimal Path

The vectors governing the movements of particles can be undefined. At each step, we can try various vectors and choose the one that best suits our objective.

 

 

 

In this example, a simulation of the well-known soap film experiment, the particles try various movements before settling on those that minimize the total length of the profile and, therefore, the total area of the surface (in this case, they head towards the Steiner points [23] of the four vertices). The angles of the sheets are 120º due to the uniformity in the direction of the Euclidean distance.

Orbit

Let's take a closer look at how simple it is, thanks to the continuously animated slider, to observe the elliptical motion of the Earth around the Sun without the need for infinitesimal analysis [20].

 

 

Note: This construction was made based on the suggestion of my colleague Julio Valbuena, who adapted the idea presented by Richard Feynman in his famous book The Feynman Lectures on Physics (volume I, 9-7, Planetary motions), see Bibliography.

We place the point S (Sun) at the coordinate center and a point T (Earth) with initial velocity vector v. If d is the distance TS and k is a positive constant, we have the gravitational force vector:

 

g = k/d² UnitVector(S–E)

 

Now we just need to introduce an auxiliary slider so that, whenever it is updated, it executes the very simple script:

 

SetValue(v, v + 0.03 g)

SetValue(E, E + 0.03 v)

 

And we already have elliptical motion! (Note that we haven't used any equations or geometric loci.)

Chaos

The previous construction explains the nearly perfect elliptical orbits of planets around the Sun. The "almost" is determined by the presence of other bodies. Fortunately for Earthlings, the orbits of the other planets are sufficiently distant not to disrupt our solar year too much. Here, we see what would happen if that weren't the case. This is a simplification (two bodies orbiting another in the same plane) of the famous "three body problem" . It is impressive to observe how such a simple construction can transform order into chaos [24].

Bubbles (using JavaScript)

We can use vectors to modify the movement of an object based on its distance from another object, allowing, for example, collision detection. The problem is that if there are many objects (n), the number of events will be high, as it grows with the square of the number of objects: n(n–1)/2. In these cases, it's best to replace the GeoGebra scripts with faster JavaScript code, as demonstrated in this construction. It shows elastic collisions, and you can verify thee conservation of total kinetic energy at all times.

 

Tensegrities

Elastic geometry allows us to find equilibrium situations between different forces. An interesting application is tensegrity structures, composed of bars and tensioned cables that hold them together.

Tensegrity

When we connect different vertices, we obtain the graph of a network [25]. But if the connections are made up of bars and springs, we can achieve that in certain positions, the tension of the springs balances out in a stable structure, called a tensegrity .

 

Here is an example in the plane. Because the rhombus is a parallelogram, the forces at each vertex cancel out, so the structure remains in stable equilibrium in any position, as long as the horizontal and vertical tensions are equal.

Tensegrity of a Triangular Prism (simplex)

Let's move into three-dimensional space. One of the simplest tensegrities starts with a right prism whose bases are equilateral triangles. We place bars along its lateral edges and cables everywhere else. If we tension the cables, tensegrity is achieved when one of the bases rotates exactly 150° relative to the other.

Icosahedral Tensegrity by Tensioning Cables

One of the pioneers of tensegrities, Buckminster Fuller, showed particular interest in this type of tensegrity. It consists of a structure formed by three pairs of parallel bars, perpendicular to each other, tensioned by cables. The whole structure constitutes a non-convex icosahedron, known as the Jessen's icosahedron , whose vertices do not occupy the same positions as in a regular icosahedron.

 

We start with bars attached in pairs. When tensioned, the bars separate until the direction of the resulting force aligns with that of the bar. The ratio between the length of each bar and each cable will then be exactly (≈1.63). Note that in a regular icosahedron, this ratio is the golden ratio (≈1.62). We can observe that the angle of the faces of Jessen Icosahedron is 90º.

Icosahedral Tensegrity by Extending Bars

We can achieve the same result by operating in reverse. Instead of tensioning the cables, we try to extend the six bars as much as possible. In this case, we start with a cuboctahedron. We place the bars and stretch them to their maximum length while maintaining the original length of the edges (the cables) of the cuboctahedron.

 

 

 

Conclusions

It is true that the current GeoGebra is much more complex than its first version from a couple of decades ago. We have so many procedures and commands that advance planning is necessary to decide, depending on the context, which of them we really need to achieve our educational objectives.

In this presentation, we have seen with numerous examples that GeoGebra allows us to tackle a wide variety of problems with minimal means. While CAS commands make it possible to transcend, in some cases, the difficulty of introducing algebra in secondary education, lists, vectors, and scripts add ease in representing flexible, dynamic, and interactive situations. And often, these situations are tremendously attractive, both in terms of aesthetics and in the sense that they invite us to explore our own constructions (let's remember the Ikea effect), which promotes the acquisition of mathematical competence.

 

 

 

Acknowledgments

While in this presentation I did not intend to follow the expositional lines set out by Tomás Recio in his 1998 book Cálculo Simbólico y Geométrico [30], I must acknowledge the enormous influence that this book, which I highly recommend, has had on my understanding of both Mathematics and its teaching. Additionally, I want to thank Tomás for his suggestions and clarifications on some of the points discussed here.

References

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[2] Arranz, J.M., Losada, R., Mora, J.A. and Sada, M. (2008). El cristo de la farola. Divulgamat, Royal Spanish Mathematical Society. GeoGebra Book: G⁴D en Divulgamat

[3] Arranz, J.M., Losada, R., Mora, J.A. and Sada, M. (2010). De luz y de color. Divulgamat, Royal Spanish Mathematical Society. GeoGebra Book: G⁴D en Divulgamat

[4] Arranz, J.M., Losada, R., Mora, J.A., Recio, T. and Sada, M. (2009). GeoGebra on the rocks. Dynamic Geometry Geometry Learn.

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[7] Hohenwarter, M. (2006). GeoGebra – didaktische Materialien und Anwendungen für den Mathematikunterricht.

[8] Kovács, Z. (2021). Two almost-circles, and two real ones. Mathematics in Computer Science 15, pp. 789–801.

[9] Losada, R. (2007). La chica en el espejo. GeoGebra Book.

[10] Losada, R. (2007). GeoGebra: la eficiencia de la intución. The Gazette of the Royal Spanish Mathematical Society, Vol. 10.1, pp. 223-239.

[11] Losada, R. (2009). GeoGebra en la Enseñanza de las Matemáticas. Ministry of Education. CD-ROM. ISBN: 978-84-369-4794-6.

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[13] Losada, R. and Álvarez, J.L. (2010). GeoGebra en la Educación Primaria. Ministry of Education. CD-ROM. ISBN: 978-84-369-4909-4.

[14] Losada, R. and Álvarez, J.L. (2011). Proyecto Gauss. Institute of Educational Technologies, Ministry of Education.

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[16] Losada, R. (2011). Modelos. GeoGebra Book.

[17] Losada, R., Recio, T. and Valcarce, J.L. (2011). Equal Bisectors at a Vertex of a Triangle. Computational Science and Its Applications - ICCSA.

[18] Losada, R. and Recio, T. (2011). Descubrimiento automático en un problema centenario. The Gazette of the Royal Spanish Mathematical Society, Vol. 14.4, pp. 693-702.

[19] Losada, R. (2014). El color dinámico de GeoGebra. The Gazette of the Royal Spanish Mathematical Society. Vol. 17 (nº 3), 525–547. GeoGebra Book: Color dinámico.

[20] Losada, R. (2016). La Tierra y el Sol. GeoGebra Book. English version: Earth and Sun.

[21] Losada, R. (2017). La percepción del tamaño. GeoGebra Book.

[22] Losada, R. (2018). Animaciones automáticas. GeoGebra Book.

[23] Losada, R. (2018). Autómatas. GeoGebra Book.

[24] Losada, R. (2018). Billares: orden y caos. GeoGebra Book

[25] Losada, R. and Mora, J.A. (2021). Redes y grafos. Las comunicaciones y la logística. Mathematics for a better world exhibition. Red DiMa, International Mathematics Day. GeoGebra Book: Redes y Grafos.

[26] Losada, R. and Recio, T. (2021). Mirando a los cuadros a través de los ojos de Voronoi. Puig Adam Society Newsletter, Vol. 112, pp. 32–53. Complutense University of Madrid. GeoGebra Book: Voronoi paintings.

[27] Losada, R. (2022). Mapas de c@lor con GeoGebra. SUMA Magazine, No. 102, pp. 43-57. GeoGebra Book: Mapas de c@lor con GeoGebra.

[28] Losada, R. and Recio, T. (2023). Inclinando la botella de Piaget con GeoGebra Discovery. Puig Adam Society Newsletter, Vol. 115, pp. 43-86. Complutense University of Madrid. GeoGebra Book: Inclinando la botella de Piaget con GeoGebra Discovery.

[29] Pérez, A., Sada, M. and Losada, R. (2021). Fractales, la Geometría del Caos. Mathematics for a better world exhibition. Red DiMa, International Mathematics Day.

[30] Recio, T. (1998). Cálculo Simbólico y Geométrico, Secondary Mathematics Education collection, Editorial Síntesis, Madrid.

[31] Recio, T., Losada, R., Kovács, Z. and Ueno, C. (2021). Discovering Geometric Inequalities: The Concourse of GeoGebra Discovery, Dynamic Coloring and Maple Tools. Mathematics 9 (20), 2548.

[32] Recio, T., Losada, R., Tabera, L.F. and Ueno, C. (2022). Visualizing a Cubic Linkage through the Use of CAS and DGS. Mathematics 2022, 10(15), 2550; https://doi.org/10.3390/math10152550. GeoGebra Book: Mecanismos. English version: Linkages.

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