CONTENTS
 
6.  Composite offset curves

On a previous page we learned that the quadratic formula for composite offset curves is:

y = ax2 + bx

Such numbers can be factored like this:

y = x(ax+b)

This factorization explains why all the integers on these curves are composite except for the first (where x is zero) and possibly the second (where x is one).

As an example, let's look at the first composite offset curve of angle 3/4. From the information on this page, we know that the coefficients of its corresponding quadratic function are

a = 4
b = 3

Therefore the function is

y = 4x2 + 3x

which tells us that the curve's integers can be factored like so:

x(4x+3)

This allows us to make the following table of the integers on the curve:


Index Factorization Integer  
0 0 x 3 0  
1 1 x 7 7  
2 2 x 11 22  
3 3 x 15 45  
4 4 x 19 76  
5 5 x 23 115  
6 6 x 27 162  
7 7 x 31 217  


For comparison, here's a screenshot of the same numbers on the spiral:    

Number wheel, figure 4  
  Vortex
To draw this curve with Vortex, enter "coe 4 3 0".
 
Figure 1      

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Copyright © 2003, 2007 Robert Sacks