## Tubes ver. 070526

This Java applet creates pseudo-3D pictures based on the dynamics of quadratic polynomials on the complex plane. See "Tiles 1" and "Tubes" in Gallery. Here I don't claim any kind of mathematical meanings of Tubes. But it's just beautiful.

### +++ Buttons +++

Draw Julia-----draws tubes. Basically, the shape of tubes depends on the function f(z)=zz+C, where c is the complex number shown below. the parameter c is given in this way: there are three text boxes for the parameters r, p, and q. set l=R*exp(2*pi*i*p/q). Then L is the multiplier of a fixed point, so-called the alpha fixed point. Now we define C by L/2-L*L/4.

Try the check boxes "Period 2", "Checks Inside" and "*** Outside". The picture will change. "Thickness=1" gives the thickest tubes. try another thickness.

There are four kinds of colors to draw tubes: Inside 1 is used for the tubes inside. Inside 2 is also used for the tubes inside, but only when "Checks Inside" is checked. When "Tubes Outside" or "Binary Outside" is checked, Outside 1 is used for the tubes outside. Outside 2 is used also for the tubes outside, but only when "Binary Outside" is checked.

Magnification Square----First press this button, then you can draw a square on the window by dragging mouse. Next, press "Draw Julia" button to magnify the region in the square.

### +++ Petal and K +++

As you can see in the pictures in the Gallery, tubes has white and gray parts. Roughly speaking, one tube changes its color frequently if parameter Petal (must be an integer) is large. If the check box "Petal=q" is checked, petal is automatically set to be the same value as q. To draw tubes like tiles in the Gallery, it is recommended to check this.

I call K the Magic Number. In this applet K has two meanings corresponding to the cases of positive value and negative value.

In the case of K positive, K must be an integer. Then the applet calculates the K-th forward orbit of 0, and measure the distance between this point and the alpha fixed point. Tubes uses this distance to draw tubes. If R is smaller than 1, the distance tends to 0 as K increase. Try and see how tubes change.

In the case of K negative, K can be non-integer. Tubes uses -K instead of the distance described above. Try small negative values with "Black Outside". In general, it is effective to take K between -2 and 0.

### +++ Hints & Remarks +++

- R accepts negatives. p accepts non integers. But q must be non-zero integers.
- Try R= 0.8, p= 1, q= 3, k= 10. you will have a tiling-like picture. then change these parameters a little.
- Try R= 0.95, p= 1.05, q= 1, k= -0.4. you will have a coral-like picture. then change these parameters a little.
- For more general remarks, see Java Applet Instructions.

### +++ History +++

The original version of this program, "OTIS-L", was built to draw the figures of tessellation (tiling) of the interior of the filled Julia sets. By using the algorithm of this applet, Tubes generates tube-like objects.

----20070526 Added some check-boxes to choose colors.
Specialized for drawing pictures. (So some buttons related to dynamics like "Forward Orbits", "Fixed Points" are erased.)
Now one can change the size of pictures by changing the parameter juliaWindowSize in tubes.html. (Note that to change the size one must execute the applet off-line.)

(Last update: 2007/05/26)

Tomoki KAWAHIRA

Email: kawahiraAmath.titech.ac.jp (A = @)