I want to share with you a vision of learning and teaching Mathematics.   In this vision Mathematics is seen as an medium to share and express beautiful ideas.  In this vision mathematics is seen as an art form where the students and teachers become explorers of knowledge and where there is no losing, no bad grades but only encouragement and learning.  In my classes at Ithaca College I am gradually manifesting this vision and am continuously learning, changing and growing.  One of the courses that I teach is called Mathematical Art with Computers. The main too that I use in thsi course is a computer program called SeeLogo that can be freely downloaded from the web through www.ithaca.edu/seelogo

 My goal is to have this into an interactive fun lecture where we will be creating together art.

The first design that we will make is  the "Spiral of Life" Design that consists of a few lines rotated and repeatedly scaled.

How to make the spiral of life

1.  Use the line tool to draw lines that end up near the center of the screen (you can use other tools)
2. Type SPIN 60 IT  (or another number)
3.  Type RT 90 (or other numbers, they determine the symmetry)
4.  Type GROWBY 3% 1000 IT  (or other numbers)
To make other spirals type NEW choosename and do similar experiments

The Spiral of Life Design became very popular at Ithaca College and is being used both by college students and also by groups of younger kids who come to the college and are being taught by our students.  Once the steps are mastered one starts exploring the connection between the pictures and the numbers and be able to answer questions like:  How to created a Spiral that has 3 (or n) subsiprals n(called "babies")  How to rotate the "babies"  How to transform the "spiral" into a "crystal".    By being anle to save the spiral of life creating on a common server and demonstrate the creations with the entire class brings a "performance component" to teaching Mathematics.  It is interesting to observe how the choice of colors mirror students attitiudes and inner feeling towards Math.

The following is a quite untypical example which demonstrates a beautiful mathematical idea and also the syntax of SeeLogo:

LOCAL t
t=3
SIZE 3  [
 LOOP 20  [NGON t WAIT 1 COLOR White  [NGON t]  t=t+1]
]

Now I want to focus on a particular aspect of this program that can be used to explore mathematical functions:

Typically one chooses one or more (constant or variable) shapes that can be transformed in various ways using functions  The Transformation we typically use are:
Color, Location (Translation), Orierntation (Rotation) and Size (Scale)

All the transformation use a single global variable we call time and denote by the letter t.

Example of shapes are:  Line, Rectangle, Polygon, Circle, Star.
Syntax examples:  FD 50, MT 100 0, BOX 80 50, NGON 7, CIRCLE 30, STARSK 3 7

The color transfomration uses the typical R,G,B (Red, Blue Green) standards currently used where the values of the three variables range from 0 to 255.

The Syntax for:   Translation ==  TRANS x(t) y(t) [somthing]
Rotation == Rot f(t) [Something]
Size == Size f(t) [Something]

For the rest of the talk we will experiment together with examples.  If you are starting from scratch type:
DECLARE t
NEW show

Local r g b
t=0
LOOP 10000  [

r=127*(1+SIN (4*360*t))
g=0*(1+SIN (360*t))
b=0*(1+SIN (360*t))
 TRANS (t*100 - 100)  0  [COLOR rgb r g b  [SIZE (1)  [ROT 0  [circle 120]]]]
 t=t+0.0001]

We define the frequency of an event as the number of of times that the event occures during a show.

We will explore the answers to these questions together:

Questions:  1. How would you make the circles green instead of red?
2.  How to make the circles multi colored so that the red has a frequency of 2, the green a frequency of 3 and the blue a frequency of 5?
3.  What is the beginning and ending coordinate of the center of the circle?
4.  How to speed up the picture drawing by a factor of 2?
5.  How to add another circle that moves in the opposite direction?
6.  How to change the circles to boxes (or lines)
7.  How to rotate the boxes in a frequency of  5?
8.  How to change the size of the boxes linearly so they will double at the end of the show?
9.  How to change the size of the boxes so that they will range from 1 to 4  in a freqwuncy of 3 (starting and ending at 1)
10.  How to have the shapes move from the 4 corners of the screen and meet at the center?
11. How would you make the shapes go around a circular path or a spiral path?

Answer for # 11
 

LOCAL r g b

t=0
LOOP 10000  [

 r=127*(1+SIN (2*360*t))
 g=127*(1+SIN (3*360*t))
 b=127*(1+SIN (4*360*t))

 TRANS 200*SIN (360*2*t) 200*COS (360*2*t)  [COLOR RGB r g b  [SIZE (2+SIN (360*t))  [ROT 0  [CIRCLE 120]]]]
 TRANS 200*SIN (360*2*t+180) 200*COS (360*2*t+180)  [COLOR RGB r g b  [SIZE (2+SIN (360*t))  [ROT 0  [CIRCLE 120]]]]

 t=t+0.0001]

Further examples:

Two ships (equation of lines)
Touring ship (using matrices and tangent lines)