However, noone should be satisfied by that. In fact, this number is perhaps the most fantastic aspect of this fractal. There are many many formulas that produce the same tree, but the number is always the same. It is said that mr. Mitchell Feigenbaum called home to his mother when he discovered this universality and said this was going to make him famous.

The famous value, comes when you compare the length of one part of the tree, that is a parts between the line divisions/bifurcations. See illustration at right. The first part is from -0.25 to 0.75, and has a length of 1.00. The next part is from 0.75 to 1.25, and has a length of 0.50. The relationship between the two lengths is 1.00/0.50=2.00. Now that is far from the Feigenbaumvalue, but the exact value springs up when you compare two parts as far right as possible, as long as x follows a periodic orbit. I have graphically found the values for the first 6 bifurcations:

Bifurc no.   Divides at    Length    This length/next length
------------------------------------------------------------
0             -0.25          -                -
1              0.75        L1=1.0         L1/L2=2.0
2              1.25        L2=0.5         L2/L3=4.25
3              1.3677      L3=0.1147      L3/L4=4.492
4              1.3939      L4=0.0262      L4/L5=4.6208
5              1.39957     L5=0.00567     L5/L6=4.536
6              1.40082     L6=0.00125     L6/L7=?
------------------------------------------------------------

The relationship at right in the table should converge to the feigenbaumvalue when you look at late bifurcations. I will not guarantee the validality of these figures in this table, as they are found in an awkward way by a graphical test&fail method. It is important to remember that by every bifurcation, one has to compute more and more iterations before an accurate value can be obtained. X needs more iterations to stabilize. If one chooses to few iterations before any points is drawn (see the programming page if you are not following), the bifurcations will appear earlier than they should. This is more important as you show a smaller and smaller portion of the full tree. (I have used up to 7000 iterations for the last bifurcations in the table).

It should be noted that the last value, 4.536 probably is erranous, because it should be bigger than the previous value. I will check these values more accurately as I get to it.