Pattern 3 for Imaginary Vortex to 196418. **

Click to enlarge. Arrow back to return.

Pattern 2 for Vortex to 4181 **

Click on image to enlarge. Arrow back to return.

Pattern 1 for " Fibonacci" Vortex to 233 **

Click on image to enlarge. Back arrow to return.

More Crocheting the Vortex **

The large brown crochet model might be 6765-

It shows various configurations, like a whirlpool, or frilly edges .


The diagram shows eddies and folding. One can play with models to discover connections with Nature and Processes. A vortex is a cone upside down. One can crochet or knit plain rows between the increase or decrease rows. The Patterns could be read backwards if knitting in which stitches are "knit 2 together". The number of plain rows determines if it will be a cone or ripples; see previous posts for the maths. When knitting, we can use circular needles, any number to accommodate the larger number of stitches, or a number of double pointed knitting needles can be used.

There may be umpteen websites on Vortexes; many are coming online recently, it seems. A year ago I didn't find much.


In Resurgence magazine May/June 2008 issue, page 71, there was a review by Chris Clarke on a book by Fritjof Capra on Leonardo's Science. Mr Clarke said "I have spent many hours on bridges looking at weirs and sluices, and I find that Leonardo's notebook drawings based on transformations of mathematical spirals, just do not look like turbulent water........ Concepts of energy and vorticity developed by Helmholtz do enable me to see..." ....this amazing world around us.

Crafting the Vortex; the beginning. **

click on image to enlarge once, click again to enlarge more; then "back" arrow to return.

Here the stitches are increased evenly along the row.

These little models are to 377, showing open hyperbolic shape and the vortex configuration.

The Pattern is - work twice in each of 3 stitches or loops,

then work once in the 4th .

Thus 4 stitches become 7, all along, to be iaw the Fibonacci Series.

It can be more complicated, as Tables 1, 2, 3, and the three diagrams in the next posts show.

Our Solar System

The orbits of the planets are like ripples. The purpose of this exercise is to see if there is any correlation with the Fibonacci series.

***Actually there is a Fibonacci Correlation that leaps out at us!
144 x 100000000 km = 89.4816 x 100000000 miles
233 x 100000000 km = 144.7862 x 100000000 miles
Moreover, 0.7862 / 0.4816 = 1.632475 = close to phi 1.618

1 km = 0.6214 miles.
1 mile = 1 / 0.6214 km = 1.609269

What if I try larger Fibonacci numbers?
4181 x 100000000 km = 2598.0734 miles.... = 2584 + 14.0734
2584 x 100000000 km = 1605.6976 miles.... = 1597 + 8.6976
14.0734 / 8.6976 = 1.6180785  and this is closer to phi!

The author of www.humanresonance.org has noted that our Sanskrit language Vedic forebears created the mile as a distance of measurement.

NB.  5/5/2013  This website has gone!  the author had to buy a new domain.
It is http://www.human-resonance.org

Please visit www.world-mysteries.com and read Mr Yurchey's research on the Solar System and Bode's Law which is in American miles, not kilometres.

Imaginary Giza-like Fibonacci Pyramids

Giza Pyramid

Make a "Fibonacci Pyramid"

Beyond Zero

"Emptiness is form; form is emptiness."

This is the supreme Mantra, or the Heart Sutra, the theme of which is emptiness.
Emptiness must not be confused with nothingness, so I read on a zen site.
How extraordinary for there to be such a distinction!
This sutra is said to be from the Prajna Paramita Hridaya Sutra,
600 scrolls of the Indian Mahayana text of 2000 years ago.


The Fibonacci series begins at zero. Something comes from nothing.
0 + 1 = 1. It is the beginning of existence. Binary mathematics.
Does it all come from the world of spirit?
The spirit pattern is "hingata" as they say in Japan.

Here is table of simplified fibonacci series, so

What if there were to be negative values of the Fibonacci Series?






Here is a graph and I am guessing there might be four states of existence.






What if dark matter be the unhappy spirits of the dead who died tragically, eg in war?






What about anti-matter? Could it be on the other side of the black hole?

****26/10/2012.    -Correction!  There is a big mistake in graph-  I should have seen it!
10 to the power minus n is actually 1 / 10 to the power n.   There will be discussion in later posts.


The Ancients must have known Something when they created the Fibonacci Series. I wrote previously that I guessed the origins to be Vedic.

Recently I searched online for "fibonacci, vedic" and there were more than 35 pages! I didn't explore any; there were too many. Very interesting.

The Golden Mean, 1.618, is called a transcendental number, from the Latin for "going beyond".

When I was a university student I encountered "The Rubiyat of Omar Khayyam" and I memorised many verses. Here are verses 28 and 29:-

"I came like Water

and like Wind I go.

Into this Universe

and Why not knowing.

Nor Whence, like Water

willy-nilly flowing

And out of It

as Wind along the Waste

I know not Whither,

willy-nilly blowing.

http://www.istanbulstreets.wordpress.com/












The equation 0=1+(-1) is said to do with anti-gravity. Dematerialisation to another realm.






The late Sir Lawrence Gardner gave a series of 3 lectures and he mentioned this at the Nexus Conference in the 1980s, and published in Nexus magazine.






I have also read it in Traci Harding's "The Dragon Queens".

***************************************************************************

*****NB   27/05/2013
Please see Update, "Mending Mistake #1",  12/12/2012, and thereabouts, for corrections and new discussion.  Thank you.

The Tip of a Drop

There is something strange going on at the tip of the triangle or cone when I try to sketch the shape IAW the Fibonacci series, at 0, 1, 1, 2, 3, 5, etc. Where is the true "0"?

It makes me think that there might be a singularity at this point.

Here is a movie of drops falling into bathwater, making music. I had hoped we could see the ripples and the drops splashing up. I will ask a friend what the notes are of the music.

There are some relevant websites I might note later.

"Fibonacci ripples" type 2.

1. Where the diameter is a fibonacci number. diagram 3





a) F11=89, radius=44.5, circumference=279.7

F12=144, r=72, C=452.6; F13=233, r=116.5, C=732.3; F14=377, r=188.5, C=1184.9

10 x radius F11=445 ="close to" C of F12, =452.6

10 x radius F12=720 ="close to" C of F13, =732.3

10 x 10 x radius F13=1165="close to" C of F14, =1184.9

So 10 x radius = "close to" Circumference of Fn+1 when the diameter is a Fibonacci number.

b) Intervals between ripples;- 188.5 - 116.5=72

c) 27.5+44.5=72 etc

d)Radius Fn+1/C, Fn. 144/279.7143=0.5148, 233/452.47=0.5148

Circumference Fn/radius Fn+1. 452.57/233=1.94236, 732.2857/377=1.9424

e) diam 1/diam 2. 233/144=1.618,

C1/C2=452.5714/279.7143=1.61797 ~1.618, r1/r2=116.5/72=1.618

*****************************************************

2. Where radius is a Fibonacci number. diagram 4

a) F14----radius 377----circumference 2369.7
---F15----610-----------3834.3
---F16----987-----------6204
---F17---1597-----------10038.3
---F18---2584-----------16242.3
---F19---4181-----------26280.6
---F20---6765-----------26280.6
---F21--10946-----------68803.4
---F22--17711------------111326.3

Circumference Fn+1------radius Fn x 10---------difference----------ratio
3834.3--------------------3770------------------64.3
6204----------------------6100-----------------104----------------104/64.3=1.6174183
10038.3-------------------9870-----------------168.3-------------168.3/104=1.618269216235.7------------------15970-----------------272.3------------272.3/168.3=1.617944126280.6------------------25840----------------440.6------------440.6/272.3=1.6180683
42522.9------------------41810-----------------712.9------------712.9/440.6=1.6180208
68803.4------------------67650----------------1153.4-----------1153.4/712.9=1.6178987111326.3----------------109460----------------1868.3=--------1868.3/1153.4=1.6198196

This is an extraordinary discovery!
It means that
an increment of "close to" is actually governed by The Golden Mean, 1.618 !

more:- Fn x 44/7 = circumference; higher numbers:-

F22--17711-----x44/7 =11326.3
F23--28657-----"-----=180129.7, -177110 = difference =3019.3,

F24--46368--------------291456, -286570 =4886------ratio 4886/3019.3=1.6182558
F25--75025------------471585.7, -463680 =7905.7-----------7905.7/4886=1.6180311
F26--21393------------763041.7, -750250 =12791.7-------12791.7/7905.7=1.6180335
F27--196418-----------123462.4, -1213930 =20697.4---20697.4/12791.7=1.6180335

Here is The Golden Mean , 1.618, playing a part.

***large fibonacci numbers are too much for calculator to handle.

b) intervals between ripples
1597-987=610, etc

c) 233+377=610, etc

d) r1/r2=1597/987=1.6180344
C1/C2=10038.25/6204=1.6180286


There are quite a few websites concerned with ripples and numbers.
Some are technical; some are religious, to do with Fibonacci numbers. I shall list some asap.

"Fibonacci Ripples" type 1.

1. Where the circumference of each ripple is a Fibonacci number.

diagram 1: Gazing at the diagram and calculating from tables in previous posting, one sees more relationships.

a) radius of C,F(n) = 10% of C,F(n+1),almost.

F8=21, radius =3.34 ~10%of F9. F9=34. radius=5.4~10% of F10, F10=55.

or 10 x radius F(n) nearly = Circumference F(n+1)

Diagram 2. For higher numbers.

F22=17711, radius=2.8 x 1000~=10% of F23 which is 28657 whose radius is 4.6 1000, ~10% of F24 which is 46368 whose radius is ~7.3 x 1000 which is 10% of F25 which is 75025.

Calculating even higher:- F30=832040, radius = 132 x 1000~=10% of F31 which is 1346269.

b) distance between ripples is close to a Fibonacci number

C,F10=55, r is 8.8. C,F9=34, r is 5.4; difference is 8.8-5.4=3.4=10% of F9.

C,F24=46368, r is 7.3 x 1000. C,F23 is 28657, r is 4.6 x 1000; difference is 7.3-4.6=2.7 x 1000 which is "close" to 10% of F23.
C, F30=832040, r is 132.04 x 1000. F31=1346269, r is 214.77 x 1000; difference is 214.77-132.04=82.73 which is ~=10% of F30.

c) distance between ripples. AB + BC = CD.

d) Circumference Fn/ radius Fn+1 = 21/5.4=3.8888.

( F8=21, r is 3.3, F9=34, r is 5.4.)

F21=10946,F22=17711, r is 2.9 x 1000, C,F21/rF22=10946/2.9 x 1000=3.77448 ~= 3.8

Radius Fn+1/C,Fn = 5.4/31=0.257148 ~=2.6.

e) C1/C2 = 17711/10946=1.618 = the Golden Mean.

Fibonacci Cones, Type 2.

Here we make cones each of which has a circumference which is a Fibonacci number. Invert the cone and it is a funnel shape. Example 3 was the first diagram made, and there was a mistake but I include it to show working out.

Notice the obvious relationships:-

Radius F(n) - Radius F(n-1) = Radius F(n-2)

or Radius F(n) + Radius F(n=1) = Radius F(n-2)

Collapse the cones to make a series of concentric circles

---- Theoretical Fibonacci Ripples.

******************************************

Example 2 shows shapes with higher Series numbers. Out of the corner of my eye I notice more relationships:-

fib no 1597 is approx 1.6 x 1000; half = 0.8 and so on, along rhs of page.

........0.8, 1.3, 2.1, 3.4, 5.5, 9, 14.5, 23.......

This looks like 1/10 of series ......8, 13, 21, 34, 55, 144, 233.

Another Equation presents itself! The mind maps the design and sees a pattern.

Our Equation is F(n) / 200 almost = F(n-11)

eg F24 = 46368, half = 23184; this is almost 233 x 100, ie F13.

Also F32 = 2178309, half = 1089309.5; this is almost 10946 x 100, ie F21

We can make all kinds of cones or ripples with radius, diameter or circumference being Fibonacci numbers.

Fibonacci Cones, Type 1.

These diagrams are all self evident and self explanatory. They show the harmonious relationships of numbers in the world around us.

Click on picture to enlarge; click again to enlarge further. Go to arrow "back" to return.

The Fibonacci christmas tree idea, posted December 2010, came from a woodworking project at the local Craft Workshop and Gallery where I have often volunteered. Young Alex and I worked together to make a wooden tree. We made a design, measuring by eye, lengths that were pleasing, and we used what we had available in timber and dowelling. It looked like a series of triangles. I wondered later if the Fibonacci series of numbers would fit in. It turned out the numbers fit exactly as the above diagram shows. Take a triangle and make a cone; it becomes 3 dimensional. The cone can go on forever with the relative dimensions as above. One can even double or triple etc the width or height etc so long as ratios remain same.

We can calculate the circumference using the equation C = 2 Pi times radius

or = Pi times diameter, where r or d can be a fibonacci number.

Being old-fashioned I like to use Pi = 22/7; the modern value is 3.14159

This is Archimedes' Constant, and one can search the Internet for history. I remember seeing small grove of Hoop Pines growing in the Cumberland State Forest Nursery at Pennant Hills, back in the 1980s. It was shady inside, with soft pine needles underfoot, and not a single weed. Outside the little forest it was brilliantly sunny and weeds were like a jungle.

Forests mean a lot to me. My earliest memory is of getting lost in a pine forest in Germany. My parents eventually found me. I was 3.

I am concerned at the wholesale destruction of forests worldwide, and even in Australia. We lose the intricate ecology; the understorey is valuable too. Some small trees in Tasmania have the best wood for making musical instruments. Pulping trees is a terrible waste, especially so because the Indian Hemp was maligned mostly by big business in the 1930s demonising the plant because it threatened their easy profits. I am grateful for Doug Yurchey who has written a most lucid article in http://www.world-mysteries.com/

We need to grow mixed forests again. Mother Nature needs us to do this.

I greatly admired Damien Lynch who began August Ethical Investments in the 1980s. He began EcoForests 2001. 93 hectares of mixed species of valuable timber native Australian trees were planted at Mill Creek near Stroud in the Hunter Valley, NSW. I looked it up on the Internet as I was writing this section. "Green is good, but hardly invincible". This is what I was saddened to find. Ecoforests went into liquidation in 2005, squeezed out by MIS. 120 investors lost 2/3 of a million dollars. NSW Forestry was subsidised by Govt to the tune of $14 million in 2007-2008. The wood is sold at $10 a ton. Private growers get $35 - $60 a ton. Plus we know that monoculture forests requires much pesticide; as any unhappy neighbouring farmer in Tasmania and elsewhere knows. Oh dear. We should know by now that the right thing should be done. It is a pity that heroes like Damien are failed by our society.

There is more.

30,000 hectares of native forest in the Tiwi Islands have been planted with an Acacia which has the potential to become a weed. Even these huge projects are bankrupt. Plantations of African Mahogany and Asian Teak being grown in the Kimberleys will be a terrible pest. I as a longtime bush regenerator see this with dismay.

For me, the only true Apology is to return the Land to its rightful pristine condition, with its own rich ecology. Maybe this is an impossible dream, but dream we must, if we want to have any kind of meaningful future.

I did find a great website, in Queensland in Ravensthorp...I think it is, Toowoomba way, they grow mature Hoop Pines in fields. Also Tuckeroo and Blueberry Ash and many more. A gigantic machine takes out a big root ball for transplanting the tree. I need to write more on this asap.

Fibonacci Series, Tables, of First 100 Numbers. ** **

These numbers are now correct and in accordance with general acceptance. I had been rather pedantic in making F(n)1 = 0. Usually they say F(0) = 0. etc.

All further equations will be described via these tables, and all previous equations remain as they were published at the time, unless one day I might edit all my work.