Mohi-Al-Deen Technologies

Geogebra as research aid

Edward Chesky¹, Ivan Alexander², Mohideen Ibramsha³

1.        Edward A. Chesky Jr., Site Supervisor, Wackenhut Corporation, Hartford, CT – 06183, USA

2.        Ivan Alexander, Independent Researcher, USA

3.        Mohideen Ibramsha, Manager, Mohi-Al-Deen technologies, LLC,

Introduction: During the late 20th century and the early 21st century India became a power house in software development. We believe this happened because India embarked on an educational program producing a large number of B.E.s and M.C.A.s. Both these programs require mathematical skill. Both had emphasis on programming, developing a machine-executable version of a specification. Program development requires deductive reasoning. Now the field of software development is moving away from programming to developing specifications. Specifications involve generalization from examples. In other words the future software development requires skill in inductive reasoning.

One educational tool recently introduced is GeoGebra. This allows interaction using pictorial inputs and algebraic expressions. We give references to sources of GeoGebra system and also to articles explaining its use. Since the need of future is inductive reasoning, we give an instant of using GeoGebra in research. A challenging problem in geometry is to develop a procedure to convert a given circle to a square of equal area. The mathematical prodigy Ramanujam solved this. We have solved the inverse problem of finding a circle for a given square. It is further demonstrated that for functions with inverse, it is possible to use the value of the function and its inverse to get greater accuracy.

We hope that GeoGebra would find extensive use among our school and college students. We expect that our computer experts would develop the GeoGebra system and its extensions in multiple languages in conformance with the educational system in India.

Literature: Aykan Kursat Erbas [1] in a dissertation towards Ph.D. states:

The value of algebra goes beyond academic study. Algebraic literacy is also a medium for social justice and job and later opportunities.

Like Aykan, John Cyril Ridd [2] in his Ph.D. dissertation considers the contribution of algebra for ‘problem solving.’

Here the consequence of algebraic literacy is that not only are all questions involving cost price, selling price and percentage profit or loss all possible, but they can all be solved by the same approach: form an equation, substitute the information and solve. Because that general approach is applicable to an enormous variety of situations, the implications of algebraic literacy are so large as to become a fundamental change in a student's approach to mathematics, in particular to 'problem solving.'

Notwithstanding the importance of algebra, mass literacy in algebra is lacking. Marya D [3] gives the following explanation.

When industrialization was taking place, universal literacy became important for progress. What “literacy” meant for math, back then, was basic arithmetic. Now it looks like the civilization needs, among other things, mass algebraic literacy to progress and to solve its problems. Governments have been working toward that goal, by their usual means (influencing school curricula). Despite this mass algebra instruction, we don’t see mass algebraic literacy.

I think one of the reasons why we don’t see mass algebraic literacy is that algebra isn’t viewed as something you MESS with. Kids doodle or compose texts for their myspace, but they don’t MAKE (construct, create, build) any algebra entities, ever.

A system to ‘mess with algebra’ is GeoGebra. A brief description from the ‘about’ page [4] of the GeoGebra web is given below.

GeoGebra is a dynamic mathematics software for schools that joins geometry, algebra and calculus.

On the one hand, GeoGebra is an interactive geometry system. You can do constructions with points, vectors, segments, lines, conic sections as well as functions and change them dynamically afterwards.

On the other hand, equations and coordinates can be entered directly. Thus, GeoGebra has the ability to deal with variables for numbers, vectors and points, finds derivatives and integrals of functions and offers commands like Root or Extremum.

These two views are characteristic of GeoGebra: an expression in the algebra window corresponds to an object in the geometry window and vice versa.

Research is facilitated by a system that helps answer ‘what if?’ questions. GeoGebra is designed to mess with algebra. Thus using GeoGebra helps the students learn and practice inductive reasoning.

Research using GeoGebra: Algebra has infinite precision. However any computer system by necessity has limited precision. GeoGebra offers two views of the same object: a geometric view and an algebraic view. Because of limitations of the display device the geometric view is limited in accuracy while the algebraic equations suffer limited precision due to the limited precision of the numbers that are given as constants in the algebraic expressions. We urge the reader to experiment with the GeoGebra system to feel the limitations. In this paper we give the steps to create the objects using GeoGebra.

Circle à Square: Ramanujam [5] gave an intuitive procedure to draw a square of equal area to a given circle. In view of the difficulty in getting the above paper, we quote the procedure below (from page 367 of the article):

Let AB … be a diameter of a circle whose center is O.

Bisect AO at M and trisect OB at T.

Draw TP perpendicular to AB and meeting the circumference at P.

Draw a chord BQ equal to PT and join AQ.

Draw OS and TR parallel to BQ and meeting AQ at S and R respectively.

Draw a chord AD equal to AS and a tangent AC = RS.

Join BC, BD, and CD and draw EX, parallel to CD, meeting BC at X.

Then the square on BX is very nearly equal to the area of the circle, the error being less than a tenth of an inch when the diameter is 40 miles long.

We draw equivalent squares for circles with areas of 1, 1000000, and 1000000000000 respectively. A circle of radius  0.56418958354775628694807945156077 has an area of 1. The ‘Construction Protocol’ for this example follows.

We are unable to proceed now. Ramanujam has not given any instruction for fixing the point E on the chord BD. A review of the instructions reveals that there is no action taken on point M which bisects the chord AO. We presume that performing some action on point M gets point E.  We first try to see whether E is on the line perpendicular to the diagonal AB on the chord BD.

Since the area of the resultant square is 1.27, our assumption that point E lies on the perpendicular to the diagonal through M is wrong. We undo all the steps from 44 till 49 and try the case when E lies on a line perpendicular to the chord BD through point M.

We have not succeeded as the area is just 0.78. It is possible that BE equals BM. We try this option now after undoing steps from 44 on. This yields a square with unit area. Thus we complete the procedure given by Ramanujam as below.

Let AB … be a diameter of a circle whose center is O.

Bisect AO at M and trisect OB at T.

Draw TP perpendicular to AB and meeting the circumference at P.

Draw a chord BQ equal to PT and join AQ.

Draw OS and TR parallel to BQ and meeting AQ at S and R respectively.

Draw a chord AD equal to AS and a tangent AC = RS.

Join BC, BD, and CD.

Mark E on BD such that BE equals BM and draw EX, parallel to CD, meeting BC at X.

Then the square on BX is very nearly equal to the area of the circle, the error being less than a tenth of an inch when the diameter is 40 miles long.

The remaining steps of the ‘Construction Protocol’ follow.

Starting with a circle of unit area, we get a square of unit area. Still Ramanujam indicates that his method is not perfect. The precision used by GeoGebra for this example is not high enough to indicate the mismatch. We try with a circle of area 1000000. We need not rework the problem from scratch. In the GeoGebra file for the unit area circle, we redefine step 2. Instead of giving the radius as 0.56418958354775628694807945156077, we give the radius as 564.18958354775628694807945156077. We need to ‘zoom out’ to get the display right. The resultant square has an area of 1000000.08 giving 8-digit accuracy for Ramanujam’s method. Similarly for a circle of area 1000000000000 Ramanujam’s method gives a square of area 1000000084913.68 with 8-digit accuracy. We conclude that Ramanujam’s method gives an accuracy of 8-digits and this accuracy is not limited by the implementation of GeoGebra.

Square à Circle: We have developed the inverse method of getting a circle of equal area for a given square. Initially we describe the intuitive method. We work out examples for squares of areas 1, 1000000, and 1000000000000 respectively.

The steps to get circle from a given square are:

ABCD is the given square.

E, F, G, and H are midpoints of AB, AD, BC, and AC respectively.

Draw quarter circle e with center A passing through E, and F.

Draw quarter circle f with center B passing through G, and E.

Draw semi circle g with AB as diameter.

Point I is intersection of quarter circle e and semi circle g.

Point J is intersection of quarter circle f and semi circle g.

Draw circle h with center I and passing through J.

Draw circle k with center J and passing through I.

Point K is intersection of circles h, and k.

Draw segment i joining points D, and H.

Draw segment j joining points C, and H.

Draw circle p with center K and passing through H.

Point M is intersection of circle h with segment i.

Draw ray l from point M through point K.

Mark the intersections of circle p with ray l as point N (near point M) and point O (away from point M).

Draw ray m from point J through point O.

Mark the intersections of circle p with side CD of square as point Q (near M) and point P (away from M).

Point R is the intersection of circle k and segment j.

Draw segment n between points Q and R.

Point S is the intersection of segment n, and ray m.

Draw circle q with center H passing through S.

Circle q is the result.

We give the ‘Construction Protocol’ for drawing a circle of area 1 starting with a square of area 1 below.

Redefining the points A and B as (-500, -500) and (500, -500) gives the area of the square as 1000000, while that of the resultant circle is just 999924.59 giving at best 5-digit accuracy. Changing the points A and B to (-500000, -500000) and (500000, -500000) gives a square of area 1000000000000 and the equivalent circle of area 999924591399.52 again with 5-digit accuracy only.

Increasing the accuracy – concept: We propose the following hypothesis. Let functions f and g be inverse of each other. That is, y = f(x); and x = g(y). Given the finite precision of computers, we would rarely find x = g( f(x) ). Let y1 = f (x1). Calculate x2 = g (y1). Now define a new function g( f(x) )  = ( x + g( f(x) ) )/2. Mathematically this is always true. This is because simplifying the definition, we get g( f(x) ) / 2 = x / 2. However due to numerical errors, the value g ( f(x1) ) would be different from the value ( x1 + g( f(x1) ) )/2.

For example consider the drawing of the square given a circle (Ramanujam’s method) as function f, and the drawing of a circle given a square (our method) as g, then we define a new method of finding a circle for a square as follows.

1.        Find a square given a circle using Ramanujam’s method.

2.        For the resultant square, find a circle using our method.

3.        The new function defines a circle with average of the radii of the initial circle and the final circle as the more accurate result for the square.

4.        Use similar triangles and find the radius of the resultant circle for the given square.

Ramanujam’s method gives a square of area 1000000084913.68 for a circle of area 1000000000000. Our method gives a circle of area 999924591399.52 for a square of area 1000000000000. The circles of both methods are less in area than the corresponding squares. In this situation, finding the average improves the accuracy of our method while the result is worse than the accuracy of Ramanujam’s method. This is because the errors are non-compensating. To get compensating errors, one method should give a circle of area larger than the square. Since our method has less accuracy, we fine tune the intuitive method algorithmically to get a circle that is larger in area than the given square.

Modified square à circle: The resultant circle in our method has point S, (point of intersection of segment QR with ray JO) on the circle with the center at point H. If we rotate the segment QR with center at R in a clockwise direction, and consider the new point of intersection with ray JO, the area of the resultant circle would be larger. We continue with our method below.

We redefine the points A and B to be (-500000, -500000), and (500000, -500000) and find that the area of the circle s is 1000849918452.57. The area of circle s is greater than the area of the square. Thus, we need to identify a point between the points Q, and U such that the average of the circles has an area as close to that of the square as possible.

Increasing accuracy – demonstration:  We start with a circle; find the square using Ramanujam’s method; find a circle for this square using our method; select a point between the two points corresponding to points Q, and U and get a circle with area larger than the square; we find the average of the radii of the circles and draw the result circle. We compare the accuracy of the transformations. The ‘Construction Protocol’ is given below.

The radius of circle p₂ used in the above Construction Protocol was found after 24 iterations. The first value was 5500000 giving the area of the resultant circle as 5026550333806841, which was greater than that of the square. We tried 5550000 and found that the resulting circle had an area of 5026548593427671 less than that of the square. Since GeoGebra allows us to redefine the inputs given by us, we need to just make one change using the Construction Protocol and GeoGebra recalculates all the dependent values and we get the final answer. We have manually followed an approximate binary search. The search process is documented below. We see that the last row gives the area of the circle to be 5026548672566372 giving a 16-digit accuracy.

Area of square is 5026548672566372

Before demonstrating the derivation of a circle of area equal to a given square, we calculate the errors due to Ramanujam’s method and our method. The starting circle with a radius of 40000000 had an area of 5026548245743669. Ramanujam’s method gave a square of area 5026548672566372 which is larger by 426822703. Starting with a square of area 5026548672566372 our modified method gave a circle of area 5026549099389091. The square is smaller than this circle by 426822719. The resultant circle was drawn using the average of the radii of the two circles giving the area equal to that of the square. In terms of accuracy, Ramanujam’s method differs on the eighth digit, as also our method. However the average of the radii of the two circles gives 16 digit accuracy. We have demonstrated our hypothesis. It is hoped that other researchers and programmers would exploit the hypothesis in future.

Accurate square à circle: Let us find a circle with area equal to a square of side 60000000. The area of the square is 3600000000000000. Draw a right angled triangle AI₁I₂ where I₂ is the point of intersection of the result circle and a line perpendicular to the x-Axis through the point I₁. In the above triangle the side AI₁ is equal to half the side of the square and the side AI₂ is equal to the radius of a circle of equal area. We define point J₂ at (-30000000, 0) making the side AJ₂ equal to half the side of the given square. A perpendicular to x-Axis through J₂ intersects the side AI₂ at K₂. The triangle AJ₂K₂ is similar to triangle AI₁I₂. Thus the side AK₂ is equal to the radius of a circle with area of 3600000000000000. A circle with center A passing through point K₂ is circle t₂. The area of this circle is 3599999999999998.5 giving 16-digit accuracy the error being just 1,5 out of 3600000000000000. The steps from Construction Protocol are given below.

We have the option to set the number of decimal places to be displayed in Construction Protocol. We worked with 2 decimal places. The current version allows up to 5 decimal places to be displayed. Thus selecting 5 decimals to be displayed would invalidate suggestion 4 that follows. We demonstrate that there are problems requiring more decimal places than any fixed limit given in GeoGebra. Instead of starting with the initial circle with radius 40000000 let us start with a circle with radius 4000000. The independent variables and their new values are given below.

If we set the initial radius to 400000, we get the following values.

GeoGebra reports the radius of circle p2 as 55477.06488 because we have set the number of decimals to 5. In this problem itself, decreasing the radius of circle c to smaller and smaller values shall require more decimals to be reported. Thus it is required that GeoGebra reports the user input without any modification in the Construction Protocol.

Suggested improvements to GeoGebra: We have demonstrated that GeoGebra permits messing with algebra. The following suggestions are made based on our experience in writing this paper.

1.        We spent lot more time in copying the ‘Construction Protocol’ than in using GeoGebra to solve the problems. This is because the ‘Construction Protocol’ could not be copied; it could be printed alone. The ‘Construction Protocol’ should be made exportable to a document as a table.

2.        To avoid keying in the ‘Construction Protocol’ given as a table in a document, it should be possible to import a ‘Construction Protocol.’

3.        As a picture is worth a thousand words, to follow the steps involved in developing a solution, the ‘Construction Protocol’ of which was imported, we need to have the facility to execute a ‘Construction Protocol’ one step at a time.

4.        We performed an approximate binary search on the parameter ‘radius of circle p₂.’ It should be possible to specify a range for a number of desired input parameters and request the GeoGebra system to optimize some objective function.

In the next section we consider some techniques to modify the GeoGebra system.

Modifying GeoGebra:

GeoGebra could be downloaded from [6] freely. If there is to freedom change the GeoGebra system by any expert, there would be chaos. There is a forum to discuss possible changes to GeoGebra. The user forum is accessed through [7]. The forum has two sections, one dedicated to use issues, and the other dedicated to technical issues. Konrad Remus [8] discusses the possibility of using recursive functions in GeoGebra. From the discussion it is clear that the changes to the GeoGebra system are under the control of the system administrator.

We have suggested the possibility of exporting and importing the Construction Protocol to and from documents. GeoGebra community provides an online facility to upload GeoGebra programs, web pages and the like. The steps to be followed in uploading are described in [9]. Some of the uploaded materials are found in [10]. To facilitate the training in inductive reasoning, Markus Hohenwarter and Judith Preiner [11] have suggested methods to create dynamic web pages. Notwithstanding these online resources, we believe the modification of GeoGebra system to export and import the Construction Protocol is justified as the permanency of a journal article is orders of magnitude more than an online resource. An online resource becomes obsolete when the supporting technology becomes obsolete.

Conclusion: Computer code development is more or less automated. Thus future computer applications would be oriented towards developing the specifications of applications. Specifications development needs generalization requiring inductive reasoning. GeoGebra permits messing with algebra and geometry and helps the teachers in developing lesson plans to train the students in inductive reasoning.  We have demonstrated that GeoGebra is a research aid too.

Dedication:

Edward Chesky:

If you would be a real seeker after truth, it is necessary that at least once in your life you doubt, as far as possible, all things - Descartes
 
The journey that led me to pick up compass, ruler and computer, in an attempt to solve this problem of antiquity was long, hard and filled with doubt.  Doubt that what I had done was flawed, that I had wasted much time, that life had passed me by and that even my God had abandoned me.  Working in concert with other men of faith, yet from faiths different to mine, much of my doubt faded and what was left is what is included in this work.
 
I dedicate this work to the men and women of the United States armed forces that paid the ultimate price in order that we might live free and prosper. 

Ivan Alexander:

"It is so important for today's youth to understand that their vision of the future is what the future will be.  We are seeing the present through the visions of giants who preceded us, and they will see the future through their eyes.  May they do so in the spirit of greatness as of those upon whose shoulders we all stand."

Mohideen Ibramsha:

I dedicate this work - giving me immense pleasure in improving upon the work of the mathematical wizard Ramanujam - to my intellectual parents: Professor and Dharma Rajaraman.

References:

1.        http://jwilson.coe.uga.edu/Pers/erbas_ayhan_k_200408_phd.pdf p19 of 366.

2.        http://eprints.jcu.edu.au/1176/02/02whole.pdf p 81 of 222.

3.        http://www.squarecirclez.com/blog/21st-century-computer-algebra-literacies/937

4.        http://www.geogebra.org/cms/index.php?option=com_content&task=blogcategory&id=67&Itemid=63

5.        Ramanujam, Approximate geometrical constructions for p, Quarterly Journal of Mathematics, XLV (1914) 350 – 374.

6.        http://www.geogebra.org/cms/index.php?option=com_content&task=blogcategory&id=71&Itemid=55

7.        http://www.geogebra.org/forum/

8.        http://www.geogebra.org/forum/viewtopic.php?f=13&t=4913

9.        http://www.geogebra.org/en/wiki/index.php/Upload_Materials

10.     http://www.geogebra.org/en/upload/

11.     Marcus Hohenwarter and Judith Preiner, Creating Mathlets with Open Source Tools, The Journal of Online Mathematics and Its Applications, 7 (July 2007) Article ID 1574. http://www.maa.org/joma/Volume7/Hohenwarter2/index.html