Fibonacci Equations

It has been a long time since I posted any new items. In a caravan with no electricity and no mobile phone reception there is no way of having my own computer, not even with wifi.
In the meantime the Australian Museum in Sydney was kind enough to show a display of mine for 2 weeks as part of the Alive! Exhibition on biodiversity, in September.

Summer is coming and the wild oats are up to 8 ft tall. I am having trouble looking after 4 acres of land by myself working only with hand tools. Pink and red bush roses are abundantly in flower after the unusual rains of winter and spring. Chocolate lilies ( yes they do have this perfume) are flowering in grassy woodland remnants.

Mindful of last year's devastating bushfires I am publishing the equations I have found in connection with the Fibonacci Series. If my caravan were to burn down then all my papers and models would be lost. I am only briefly noting them. There is a lot to document and it might be a slow process. So here goes:-

1. F(n) = F(n-1) + F(n-2)....this is well know; it is definition of the series
F30 = F29 + F28=317811 + 196418 = 514229

2. F(n) = 2F(n-2) + F(n-3).......this is used for seashell spirals
F17 = 2F15 + F14 = 2x377 + 233 = 754 + 233 = 987
F20 = 2F18 + F17 = 2x1597 + 987 = 3194 + 987 = 4181
F30 = 2F28 + F27 = 2x196418 + 121393 = 514229

3. F(1) + F(2) + F(3) +.....+F(n) = F(n + 2) - 1....this is sum total
F1 +....F10 = 0+1+1+2+3+5+8+13+21+34 = 88 = 89-1
F1 + F2 + ....F16 = F18 - 1 = 0+1+1+2+3+....+610 = 1596 = 1597-1

4. F(n) = 7F(n-4) - F(n-8)....this came up while working out how to crochet a whirlpool
F14 = 7x F10 - F6 = (7 x 34) - 5 = 238 - 5 = 233 = F14

5. F(n) = 2F(n-4) + 3F(n-3) .... further "unpacking" of the series.

6. F(n) = 5F(n-4) + 3F(n-5).... ditto of above

7. F(n) = 5F(n-6) + 8F(n-5).... ditto
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8. The following are spookily exact especially at the higher numbers, not always so at lowest iterations:-

F(n) / F(n-1)
F8/F7 = 13/8 = 1.625
F10/F9 = 34/21 = 1.6190476
F12/F11 = 89/55 = 1.6181818
F16/F15 = 610/377 = 1.6180371
F18/F17 = 1597/987 = 1.6180344
F20/F19 = 4181/2584 = 1.618034
F30/F29 = 514229/317811 = 1.6180339
F40/F39 = 63245986/39088169 = 1.6180339

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****This is Phi, a "transcedental number", 1.6180339887498
The golden mean is 1:618
or 2:3:5

****There is a most extraordinary website http://www.humanresonance.org/
 The author Alexander Putney has written a book on Phi.  One can read it online for free.

NB.  5/2013  Incredibly, this website has been stolen!  The author had to buy a new web domain.
 It is http://www.human-resonance.org
...................................................................................

F(n)/F(n-2)
F12/F10 = 89/34 = 2.617647
F20/F18 = 4181/1597 = 2.6180338
F40/F38 = 63245986/24157817 = 2.6180339

F(n)/F(n-3)
F12/F9 = 89/21 = 4.2380952
F20/F17 = 4181/ 987 = 4.2360688
F40/F37 = 63245986/14930352 = 4.2360679

F(n)/F(n-4)
F12/F8 = 89/13 = 6.8461538
F20/F16 = 4181/610 = 6.8540983
F40/F36 = 63245986/9227465 = 6.8541019

F(n)/F(n-5)
F20/F15 = 4181/377 = 11.090185
F36/F31 = 9227465/832040 = 11.090169
F40/F35 = 63245986/5702887 = 11.090169

F(n)/F(n-6)
F16/F10 = 610/34 = 17.941176
F26/F20 = 75025/4181 = 17.944271
F40/F34 = 63245986/3524578 = 17.944271

F(n)/F(n-7)
F10/F3 = 34/1 = 34
F11/F4 = 55/2 = 27.5
F17/F10 = 987/34 = 29.029411
F33/F26 = 2178309/75025 = 29.034441
F40/F33 = 63245986/2178309 = 29.034441

F(n)/F(n-10)
F26/F16 = 75025/610 = 122.9918
F36/F26 = 9227465/75025 = 122.99186
F40/F30 = 63245986/514229 = 122.99186

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F(n)/F(n-20)

F27/F7 = 121393/8 = 15174.125

F30/F10 = 514229/34 = 15124.382

F55/F35 = 86267571272/5702887

divide top and bottom each by 1000, ie 1.

=86267571/5702.887 = 15126.999

F60/F40 =
956722026041/63245986 = 95672203/6324.5986

= 15126.999

F90/F70 = 1779978866004714/117669030460994

= 17799789/1176.69 = 15126.9999

Amazing, isn't it, and elegant?

It is fascinating playing around with these numbers. There is intricate correlation indeed!

Cheerio from Tiiu Vanamois till later.



9. Half F(n) = 9[ 0.5 x 2F(n-8) + F(n-9)] - half F(n-12)
........ for knitting a half clam shell with wavy lips!

10. Half F(n) = 9[ 0.5x2F(n-7) + F(n-8)] - half F(n-13)
for half clam

11. There is also Half F(n) = 9[F(n-6)] - half F(n-12)

0.5 x 2Fn denotes knit 2 together to make 1.
These equations are not quite right yet but they are on the way.

These lines got placed oddly.
One day I might be able to elaborate more fully
but I think it is beyond the scope of a little blog.

One Big Family

This crochet model is the shape formed for 1, 2, 4, 8, 16, 32, 64, 128, 256, 512. That is the shape of the web of ancestors for one single person, for 200 years, or 10 generations.
Extend that out further into 20,000 years and more.....
Multiply by the number of people on earth now.
There will be merging of ancestors for siblings.
Every time two people are married, there is merging of ancestral lineages.For that matter, there would be merging of ancestors each time two people have sex, especially if there is a child conceived.

From the previous population graph, there will be a limit to the number of ancestors possible. Somewhere further along is the Common ancestor, the UR mother and the UR father. My guess is that there may be more than one seed person ancestor, one for each human type group, eg asian, african, aboriginal, european, maybe siberian or from Altai Mountains area. DNA and deep ancestry investigations are interesting.

Are we One big family or a number of different families?

More 12/6/2014.    It has taken me so long to publish this!

on its way....

Please click on image to enlarge it.

It seems as if the planet was seeded by a number of different races-----so I don't think we are related to everyone on the planet back in 1400s.........


The Tree of Life has many dead and broken branches.
Said by Charles Darwin.




So also has the Tree of Man.  Plague, wars, famine and climate change has trimmed the tree over aeons.  It is a miracle that we are still here.    It is possible that we might discover a way to consciously make this a happy world that can live forever.

An Important Message

The important message is - it looks like world population is doing a flip!

Click photo to enlarge to full page size, and click back to return to text.

.........

Some say that the megacities will save us and that the population of the world will stabilise at 9 billion in 2050

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PS. 27/8/2010 My friend tells me that the population graph is not clear.

The straight line to northeast is the 2, 4, 8, 16, 32 etc; this is ancestor line

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It intersects the population line and that is the point of maximum number of ancestors possible for one person. The common ancestors must be part of merging sooner in time, then merging to form limbs and trunk of human family tree(s).

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Another important message is in http://www.earthmothercrying.org/

Please especially read Chief Oren Lyon's lecture which he gave

at the 24th Schumacher Lectures in 2004.

Click on the ice is melting photo:

We need to discuss matters that are important.

Remember that the ice is melting.

There may be a reason why there are so many of us on earth at this time.

Click on Medicine wheel photo:

Message that the earth could heal instantly if every one on earth would put away their toys of war and join together in peace.

..........................conundrum.............

Please check out on Yahoo! Answers "How many ancestors did you have 1000 years ago?"

Joseph T. Chang, Professor of Statistics at Yale University has given thought in Science News Sept 30, 2004, regarding our most recent common ancestors.

When I change the parameters of the x axis, then the increase does not look so dramatic.

Sorry this is sideways. How to fix this I don't know. Internet has a mind of its own.

Here is data for the graph of Ancestors and World Population.

I would be glad of any comments , please send email to tmvanam@gmail.com

PS 27/8/2010 There was a quote from an article on collapsing societies, eg the Anasazi, ie "The rich reserve for themselves the right to die last". This is a cue to live symbiotically and sustainably on planet earth.

Heteronympha fluttered by **

This photo is such a beauty and it happened at a numinous moment in my garden.

Arid Australia and last February 2010 we had such a lot of rain!

There were butterflies floating through the garden and I wished I could have a photo of one.

As if it knew my thoughts, this Common Brown settled in a patch of sunlight near my right foot and I was able to put my new camera on macro and take this lovely shot. It felt like a blessing.

You can click on the butterfly photo to enlarge to page size, then print it. If you try to print using file, print preview, landscape, the right wing tip is cut off.

The knitted flowers can't be enlarged (photo is fixed).

Why do I put this in a blog on hyperbolae?

Answer is I can't resist the interplay of Heterodontus and Heteronympha.

(I know scientific names should be underlined, but I don't know how to do it.)

Plus there is a spin off from Heterodontus and it was quite unexpected:-

I used the long stitch, treble or double treble when crocheting the 5, 25, 125, 625 series.

What if I made short stitches, more rows, but tighter?

ie 1, then 2, then 3, then 5.

Here it is, knitted. It's a bit tricky, because it does matter if you knit into the front or back of the stitch. You need to play with it to get it right.

It looks like a tubular flower. Just sew it together, and make yellow stamens etc for inside, add green stem and leaves.

The plain side can be inside or outside the flower.

So a whole corsage was made, to give as a gift to a dear friend on her 60th birthday, Hello Jan!

Oh dear- the photo won't allow itself to come down to this level on the page and I don't know how to fix it, so let's leave it where it is.

Here are the instructions to knit these flowers:-

Cast on 25 stitches in a pretty colour.  Actually   correction 2/1/2015.  it is 23 to begin.

knit 2 together twice, knit one; continue to end of row in this pattern, makes 15 stitches; turn.

purl 2 together, purl 1; repeat till end of row, makes 10 stitches; turn.

knit 2 together, 5 times, makes 5 stitches,

break thread and tie on green thread, ot wool.

purl 2 tog, twice, purl 1, makes 3 stitches

knit 2 tog, knit 1, makes 2 stitches

purl 2 tog, makes 1.

Continue on making some chains to make the stem.

make a leaf by knitting 5, then 3, then2, then 1, then add it to stem, knitting both together, and doing some more chains to lengthen stem,

Finish off. Sew flower part together in colour. Sew green sepal part together.

Make more of these and have a bunch of pretty flowers.

I used number 12 knitting needles and 1 or 2 ply fine tapestry wool (English, Applemore) but you can use any yarn at all. You could crochet them also, I guess, working from 1 stitch and reading the pattern backwards, increasing, knit 2 together to get 5, 10, 15, 25. (5x 1, 2, 3, 5.)

I do hope this pattern is OK to understand. It is Fibonacci again, ie, 1, 2, 3, 5.

***2/1/2015 I have re posted this Pattern recently in "some people and the models they make"
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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.

Heterodontus Does Crochet **

In a future post I will publish the graphs I have made of the Fibonacci Series, with the y axis being an imaginary hypothetical hyperbola of units 10 to the powers 1, 2, 3, 4, 5, .....15.


The x axis is F1, F2, F3, ..... F(n).



What does the y axis look like?






It looks like a hyperbola when you begin to crochet it.





It configures into 5 spirals; this is one shape;

the blue and purple model.



there may be others.





This one is 10 stitches,

Next row has 10 stitches into each of the 10 stitches.

This makes 100 in row 2.

10 stitches into each of the 100 makes 1000 in row 3.





(I am working on 10 to the power 4 but it is in sewing machine cotton and very fine.

It would be too expensive to crochet the higher numbers with thicker yarn.)

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This is what it looks like.






I was reminded of spirals again -





when you fold the work in half it makes 8 spirals








This reminds me of the eggcase of Heterodontus portus jacksoni, but the real thing has only 4 spirals. I guessed that 5 was the base number, so I crocheted 5 chains.

In each chain I crocheted 5 stitches, making 25.

In each of thes 25 I crocheted 5, making 125.

Then the last row, I crocheted 5 in each of the125, making 625.




Fold it in half and sew it up and pad it

and there are 4 spirals!



I'll have some web refs next time for folk to look at of pictures of the real eggs of the Port Jackson Shark.

Plus I guess I might have to take a trip to Jervis Bay to try to find some eggcases on the beach.

I need to check out how the baby shark gets out- via the opening, one similar to the snailshapes?

Ideas, anyone? Let me know at tmvanam@gmail.com Thanks muchly.



I remember when I was a kid at Burnside Homes, we went to the Murdoch Holiday Home at Huskisson. After some big storms there were huge dead sharks washed up on the rocks!



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Not only does this remind me of shark eggs, but what about DNA, the double helix!

The bit of string hanging off is the first stitch. um...centromere...? or is that too far fetched?



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OK next post is 2, 4, 8, 16 etc. It describes populations and particularly

THE ANCESTORS. Here is a great conundrum and a big message!



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Here are the Bivalves **

Here is the New Pattern for Making a Bivalve.

Click photo below to enlarge to page size.

Choose file and print preview, print if you need it.

The white crocheted model is this exactly. 610.

The configured white bivalves are at the end of this section.

16/8/2010 correction:- I found it is not possible to knit the pattern by following it backwards. There are 4 new ways to make this; please see a future blog in August.

1......The Original Pattern is used to create the hyperbolic fan shape:
Instructions were given April 29, 2010 and

May 25, 2010

and May 26, Variations....

Original Pattern was posted May 19, 2010.

The green one is knitted.
It begins with 610 stitches and decreases are made until 1 stitch is left..




The red and blue one is crocheted.
It begins with 1 stitch; outer edge has 610 stitches.
They will configure into spiral seashell shapes just like all the previous models.


However in Feb 2010 I found you can configure them into bivalves:-

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2...... The New Pattern at beginning of section was used to crochet the next shapes:-

There are 2 ways of configuring it:-

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An Idea: One day I (or anyone) might make a giant clam with deeply wavy lips, made in hemp string, knitted with 1 metre dowelling "knitting needles" and covered with Roman hydraulic cement (fine powdered clay and cement with minimal water added until it forms a gel= great material for sculpture.) A pinky grey satin pillow could be included for clam's inside body, and a big "pearl "could be embedded in the shell!

***** NB. As far as I am concerned, none of this work of mine is copyright.

The Fibonacci Series belongs to Great Nature and thus belongs to anyone who is interested to use it creatively and happily. Enjoy!

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We have almost completed the work on Molluscs.

I still need to do a cross section.

I am awaiting permission from the publishers to use 3 photos of seashells with identifications, from "Australian Seashores in Colour" by Keith Gillettt and John Yaldwyn, a 1969 publication, dedicated to their mentors and colleagues Elizabeth Carrinton Pope and the late Frank Alexander McNeil, from the Australian Museum.

It seems the publisher AH&W Reed may no longer exist, so it is a problem to know what to do.

There is more work to do with vortexes and ripples, using the maths of Fibonacci series to make the models. The shapes have been made, they just need to be documented as they soon will be.

I would like to digress for a while, away from the Fibonacci series, to the Bifurcating series,

1, 2, 4, 8, 16, 32, 64, 128 etc. and to the Series which are Powers of 5 and of 10 which make very interesting models.

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