Mathematical Curves with Turtle Geometry

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MATHEMATICAL CURVES WITH TURTLE GEOMETRY
FARKAS Károly, HU
Abstract: Mathematical curves can be generated by several programs. Why do we think that the best
way to use in the education the Logo? The key feature of Logo is turtle geometry. The turtle is an
object to which attributes and rules can be connected. The turtle movement is described in a relative
coordinate system. Logo usually uses the polar coordinate system instead of Descartes coordinates.
This way the drawing will be more syntonic, we can more easily identify ourselves with the motion.
This way, it is easier, more playful and more expressive to understand mathematical curves. Thus
Logo may contribute to the study of mathematics. The application of these algorithms, natural in the
sense of Piaget, that can be discovered by learning based on experience, are called real turtle
geometry. We shall present on the conference the generation of some curves with this new way.
Keywords: mathematical curves, turtle geometry, syntonic
MATEMATIKAI GÖRBÉK TEKNŐC-GEOMETRIÁVAL
Összefoglalás: Matematikai görbéket sokféle számítógépes programmal ábrázolhatunk. Miért
gondoljuk, hogy az oktatásban erre a legjobb a Logo? A Logo lényegi eleme a teknőc-geometria. A
teknőc egy objektum, amelyet tulajdonságokkal, viselkedési szabályokkal láthatunk el. A teknőcök
mozgását relatív koordinátarendszerben írjuk le, a Descartes koordinátarendszer helyett rendszerint
poláris koordinátarendszert használunk. Az ábrázolás ezen módja szintonikus, azaz könnyen
azonosulhatunk a teknőccel, könnyebben beleélhetjük magunkat a mozgásba. Ez a matematikai
görbék megértésének könnyebb, játékosabb, expresszív módja. A Logo ezzel segítheti a matematikai
tanulmányokat. Ezeknek a Piaget szerinti természetes, tapasztalatok alapján történő tanulással
felfedezhető algoritmusoknak az alkalmazását nevezzük igazi teknőc-geometriának. Előadásunkban
néhány görbe ilyen újszerű generálását mutatjuk be.
Kulcsszavak: matematikai görbék, teknőc-geometria, szintonikus
1 Introduction
Informatics classes in Hungary are typically involved with training for the ECDL exams.
This is not enough. We developed a methodical system, the Playful Informatics. Playful
Informatics, presents how IT tools can be used for the development of thinking. Our motto:
Instead of longlife learning, longlife playing. We (MicroWorlds Logo Group of the John
von Neumann Computer Society) think and claim that one of the most effective, therefore
most important field of computer applications is thinking development. At this conference
we demonstrate the use of Logo for mathematics by drawing circle and ellipses using
several methods. We use the Microworlds Ex Logo version.
2 Circles
2.1 To Draw a Circle with Logo First of All there is the Classical Method by Papert
Repeat 360 [forward 1 right 1]
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“When you walk in a circle you take a little step forward and you turn a little. And you
keep doing it.” From this description it is only a small step to a formal Turtle program.
TO CIRCLE REPEAT [FORWARD 1 RIGHT 1]
2.2 Circle with more Syntonic
I have learned the next algorithm from my pupils in the third (in primary school!) class:
Repeat 360 [pu fd 50 pd fd 1 pu bk 51 rt 1]
I think it is more syntonical.
2.3 Circle with the Cartesian Coordinates
to kör :r
make "x -1 * :r
körf
end
to körf
if :x > :r [körl]
setpos list :x sqrt ((:r * :r) - (:x * :x))
make "x :x + 1
körf
end
to körl
make "x :x - 1
if :x < -1 * :r [setpos list :x 0 stopall]
setpos list :x -1 * sqrt ((:r * :r) - (:x * :x))
körl
end
This algorithm is not Logo-like.
2.4 Intrinsic Circle
We have looked up an original algorithm. It was published first in Porto [1]
We use two actors: Adam and Eva. Adam is the masterpiece of creation, stands at the
centre, Eva is apart from him on the radius. We stick Eva to Adam, and Adam begins to
rotate.
to rotate
Eva stick "Adam
Adam, rt 1
rotate
end
to stick :a
; I stick the current turtle to :a turtle
towards :a
make "d distance :a
make "alpha ask :a [heading]
make "x ask :a [xcor]
make "y ask :a [ycor]
setpos list :x + :d * sin :alpha :y + :d * cos :alpha
end
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As far as we are concerned this model is the most syntonic, since for the small child the
rotation, the experience of circling, starts the ownership of that circle, which can be
touched.
3 Ellipses
After the circles, let us see the ellipses.
3.1. Abelson and diSessa
Our first algorithms again a classical, by Abelson and diSessa [2]
to ellipse :a :e
make "n 0
repeat 360[rt :n fd :a lt :n lt :n fd :a * :e rt :n make "n :n + 1]
end
It is syntonic and intristic as well. We can shorten it a little bit.
to ellipszis :a :e
make "n 0
repeat 360[rt :n fd :a lt 2 * :n fd :a * :e rt :n make "n :n + 1]
end
3.2 Ellipse move with Polar Coordinates
to pell :a :e
;e less than 1
make "r :a * (1 - :e * :e) / (1 + :e * cos heading)
pu fd :r pd fd 1 pu bk 1 + :r rt .5
pell :a :e
end
This method similar onto the second manner of the drawing of the
circle.
3.3 Ellipse in the Paradise
How can we draw ellipse in our microworlds “Paradise”?
There are the two actors: Adam and Eva. We call Cain,
Lucifer, and Lili as well. Lucifer and Lili rotate. Eva and
Adam ask them the y and x coordinates. Cain imitates motion
Adam and Eve, he walks through an ellipse.
to Lissajous
Lucifer, fd 1 rt 1
Lili, fd 1.5 rt 1
Adam, setx ask "Lili [xcor]
Eva, sety ask "Lucifer [ycor]
Cain, setpos list ask "Lili [xcor] ask "Lucifer
[ycor]
Lissajous
end
We need Adam and Eva only in didactic aspect. The model works without them too.
Picture 1: pell 100 0,75
Picture 2: Ellipse like Lissajous
curve
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3.4. Ellipse on Simple Way
It is possible to manufacture an ellipse in such a way that everything is a circle for his dot x
we multiply his coordinate by a constant, it y it had them beside leaving him invariably.
Let Adam be his motion continuously
fd 1 rt 1
Eva makes always
setpos list ask "Adam [xcor] (ask "Adam [ycor]) * 2
The result Adam circles, Eva moves on an elliptical orbit.
4 Conclusion
One major difference between turtle geometry and coordinate geometry rests on the notion
of the intrinsic properties of geometric figures. Scientific experiments have proved that
Logo pedagogy considerably enhances the development of creativity, empathy and fantasy,
makes pupils self-assured and helps goal driven thinking. In many schools, Logo is used
for drawing and desktop publishing. We can go further and use Logo in mathematics and
physics studies too.
5 References
[1] FARKAS, K.: Logo and native language. Intrinsic procedures of some curves. Porto,
2003. In: Proceedings of the 9th European Logo Conference, ed.: Correia, S.
[2] ABELSON, H. – DISESSA, A.: Turtle Geometry. The Computer as a Medium for
Exploring Mathematics. The MIT Press. 1986.
Reviewed by: prof. Ing. Veronika Stoffová, CSc.
Contact address:
Dr. Károly Farkas CSc.
Universitas Budensis
Bécsi u. 96/b Budapest, HU
tel: 00630 9403794 farkas.karoly@nik.uni-obuda.hu