The Bifurcating Series **

John Gribbin wrote a fine book, "Deep Simplicity" and he discussed the Bifurcating Series.


Two dimensional graphs show chaos after 3 generations. I won't go on about it just now because I am far from home, writing this up at a Youth Hostel in Sydney, and I don't have my notes.


Dr Taimina is magically able to create shapes that the usual Euclidean Geometry cannot describe. It is to crochet that we turn.







Here is what 1, 2, 4, ........512, looks like.


It shows a butterfly shape when configured.



But.... there are at least 12 other possible configurations, and I need to draw them for a later post.











Now we have a model up to 1024 stitches on the outer row:-
4 spirals; very elegant.














Not only that, folks, it goes on to 16000 and 320000 or so:-















When crocheting I need to use the longer stitch, a treble or double treble crochet as this is easier to configure into complex spirals. The single crochet or half treble make for much more compact models, nice when making seashells.


OK, This shows that one can go on forever, the only constraining factors are the size of the room or the Universe, and the quantity of raw material available.


Where is the conundrum?
It'll be obvious when I post the graph of this series against hyperbolic axes graph.
What has happened regarding our ancestors back in time?

After 1000 years there have been 50 generations
and 10 to the power 15 ancestors, a quadrillion, clearly an impossibility.

1000 years ago there were only half a billion humans on Earth, or so.

In about 1400, 600 years ago, the two lines intersect,
graph in next posts,
so there can never be more than a billion ancestors.

I checked it out on the Internet, on Yahoo! Answers,
One person observed that the family lineages merge,
like branches and limbs and trunk of a tree.
This question has not been solved, not yet.


I would love to ask Anastasia or Volodhya.
They are mathematically absolutely fast in their speed of thinking.
See Book 6 of The Ringing Cedar Series, "The Book of Kin".
On pages 58, 59 Volodhiya observed that in a living system, 1+1=3.
mother + father make child .
There is clearly a new way of looking at maths.

Maybe we should not be lazy, and try to figure it out ourselves.