Monday, December 17, 2012

Real Sunflowers.

We all know that the sunflower spirals are skewed one way eg 13 and the other way eg 21,  or whatever Fibonacci number is displayed.  Next I will  study the Fibonacci angles. eg 137.5 degrees. 


This is a real Artwork.  It shows real Joy. 
 It shows how accurate is the eye of the artist, Abby Heathcote.


Wednesday, December 12, 2012

Geometry of a Theoretical Sunflower

Everyone knows that there are spirals with Fibonacci series relationships in the Sunflower.

I wish to explore the basic geometry and try to work out how it really might be so.

Here is a circle divided into 6, simply, with a compass.  Then each section is divided into 3 then 3 again.  This makes 54  which is close to 55, a Fibonacci number.  I was delighted to find that there are 54 in each circular row from inside to outside.  The base of the flower is like a huge placenta holding all those ovaries which will be fertilised to become mature seeds.

I've drawn simply diagrams for a range of numbers with bias to those close to Fibonacci numbers:-


It was interesting to find online as I searched New Scientist a video
"3D Vortex Illusion in a 100 Year Old American Quilt"
Instead of compass curves dividing concentric circles as in my diagram 1, there were straight lines radiating out from centre, and alternate "squares" were red and white.
See Friday Illusion, in  of 7 Dec 2012.

Powers of Further Phi.

I have calculated quite a few numbers.  One can see Correlations.  If one were to calculate the whole bang lot then we could create a grid pattern and see multiple correlations;  almost exact.

I tried to add these tables to the previous post, but errors occurred, so this is posted separately.

I also made an error in posting Powers of Pi too early;  it would have been more orderly to connect the Ouroboros post with the Zero post.  Too hurried.  Better late than never,  they say.

Powers of Phi

Curiosity impels me to study the Powers of Phi.

I am trying to discover for myself any more Equations and Correlations.
And here is one.  It is already known,
for I  saw it on Wikipedia after I had figured it out for myself.
Dr Ron Knott's Fibonacci website is famous.
You can find there "what's new" to do with powers of Phi.
He also distinguishes between Phi and phi ( 1.61803 and 0.61803, respectively.)

Here might be something new.
Imagine there are more Golden Means, not only Fn / Fn-1
There is Fn / Fn-2,  Fn /Fn-3, ........Fn / Fn-20,    etc.
I like to call them Phi One,  Phi two,  Phi three,  .... Phi Twenty.......etc.

Compare both Tables.  There are Correlations.
The numbers might be different if taken to 4 decimal places only.
.  I have published back in  "Fibonacci Equations", post Nov 21, 2010  part 8.
The ratio results use many Fibonacci numbers, 
showing how amazingly exact and consistent the values are, especially for larger numbers!

.eg  For Fn / Fn-10         F26 / F16 = 75025 / 610 = 122.9918
F36 / F26 = 9227465 / 75022 = 122.99186

I have to check my  numbers because there were errors in my early calculations for The Fibonacci Series.  We can check with Dr Knott's website-  he has published the first 500 numbers in the series.

Mending Mistake #1.

I mentioned a huge mistake in a graph I drew in  2010:-

10 to the power of  minus any number is not equal to minus 10 to a power,
 as I plotted the negative Y axis wrongly!

All Powers of 10 with a minus should be located between zero and one.

The negative Y axis should be -10,  -10 squared, etc

The second graph here is an amended one;  just imaginary
and most maybe somewhere wrong too!
Zero is the singularity,  the black hole....

****Dr Ron Knott has a famous Fibonacci site and he does use minus fibonacci numbers.

This table of Fibonacci numbers is simplified for easy plotting on a graph.
Observe the elegant regularity of the iterations  with regard to Powers of 10.

What is going on between 0 and 1? 
I tried to figure it out,
imagining mini-fibonacci series ie 0, 0.1, 0.2, 0.3, 0.5 etc.
Then mini mini f. series ie 0, 0.001, 0.002, 0.003, 0.005 etc.
Then continuing up to millionth.

I did draw graphs but it would take many sheets of paper up.
This is just simple playing.

For between 0.000001 and 1.0 the  " ultra fibonacci series"   might be
.000001,  .000002, .000003, .000013 near .00001 on vertical axis.
then .000144 near .0001 on axis, then .001597 near .001 on axis,
then .010946 near .01 on axis, then .121393 near .1 on axis
and .83204030 near 1.0 on axis1   If it were true it would be a miracle!
Just guessing.  The last mini fib no  relates to F30 which is 83204030.
Or F31 / 1000000  is next -  ie  1.3462931.
All squashed into the space near zero, the singularity.
And it goes on forever!

 Ten to the power  minus 33 cm.   
The Planck length;  you can see it on the Cosmic Ouroboros Diagram.
Said to have been chosen as our bearable limit to space in our known universe.
(Named after Max Planck who also has an Institute outside Munich named after him.)

It could go to 10 to the power minus infinity.  I am guessing forever.
Enormous Power between us and Vacuum.
Nassim Haramein has given much thought to all of this.  Worth a hearing on YouTube.

I once found the Vedic OM on a site
This has great creative drawing etc, sanskrit,  but no longer vedic om.
So, looking further,  one finds much on Internet on this subject. has t shirts etc for sale with the symbol on it.
Especially interesting is one with layers of AUM/OM,  as if floating in space.

The symbol reminds me of 0, 1, 1, 2, 3-  did the Ancients Know something?
Something going on around zero.....
I search google for "Vedic";  it is just fascinating;  an enormous realm to study..

What does Aum sound like?
See Jill Purce's YouTube video, "Just One More Breath".
Overtone chanting in a tower in Middleburg, Holland, in 1984.
Find more on
It is" based on harmonic series-  makes our own biology/geometry audible,
allowing us to hear in a sonorous way our own structure.
We are hearing what we are."
Jill is married to biologist Dr Rupert Sheldrake and there is more on YouTube
of great interest-  Look for Bruce Lipton also.
eg try "A Morphic Resonance of Love." Oct 25, 2009,  London.
A fine rendition of Hava Nagila is a real treat, with young
Merlin Sheldrake on accordion, with musicians in a group of four.
There are a great many people alive today and they are already helping save the world.

Powers of Pi

There is an unusual book, "Co-Evolution" by Alec Newald;  one can buy it from Nexus Magazine.  Alec's story is of "aliens" or "Other Visitors" taking him to the planet Haven where he met many friendly Beings and he learnt their ways.  I was interested that they used 12, and Pi squared.  They built houses that looked like snails!  There were buildings like pyramids.  Even the space craft were shaped like pyramids, spinning as they travel time and dimensions.

To me the lesson could be that these folk might be our future ancestors.  Their sun is dying and they are looking for another planet to call home.  In fact our Moon was their Home until a terrible dimensional mistake by their too clever scientists made it  a dead place.  Old beyond time!  A few of them managed to escape and go to Planet Earth in early Ages not quite habitable then, and not many survived.  Accomplished survival somehow.  To time travel backwards to try  to mend mistakes and change history is deemed to be useless because it might happen that they would have no familiar future to return to.  They are unable to reproduce like humans do,  their mouths are tiny holes to suck up nutrient (no smiles),  their heads are huge and eyes are huge,  etc.  You know the image.

The question to me is-- do we want to look like them in our future incarnations?

My feeling is That is what might happen if we continue to love technology without caring how it affects our health and humanity.  Of course I could be greatly mistaken-  or it could be true.....
The German techno wizards are making robots so real almost human even capable of deceit (Ref is a ABC Radio National "Future Tense" program.  What if these humanoid robots were to be the only ones to survive in future?  It makes their creation now to be a problem morally and ethically.  Humans and robots both deserve love.  It is a question of values beyond money and convenience.

The time is NOW and it is a singularity between past and future.
What we do now or what we decide not to do now will affect the possible future and it can mend the past.   What we do now could fix the sun on Haven, with Love, and bring back Life to our future Selves!
An Enlightened Person can achieve much through Love,  the strongest Power in the Universe.  If one person can do it, then potentially every one of us could be able to do so too. 
One might stress out thinking it impossible.
However there are a myriad possibilities and realities.  We choose what we are able to do,  and there is said to be a lot of help out there if we need it.  A Quantum Leap in Consciousness on the part of humanity and miracles might happen!   There might be a happy aspect to the End of the Mayan Calendar!  December 21,  only  2 weeks away,  and counting......

What kind of world could you imagine in which you could be happy and creative-  forever?

Anyway,  here are the Powers of Pi:-

Our Place in the Universe

In a recent post I mentioned The Cosmic Ouroboros.  
The Maya knew it as a symbol for the imminent closing of an age.
We are said to be facing the end of a 26,000 year cycle!
I feel it is an important image for us to see our place = in
the very centre of this Universe:-

I was worried about copyright laws, so I searched Google.
An Australian artist has also painted the Ouroboros.
and post of Thursday November 1, 2012
She writes that a similar diagram(as used by Nancy Abrams and
and printed in a recent Resurgence Magazine)
is in "Just Six Numbers" by Astronomer Royal Lord Martin Rees.
Thus I feel it is ok for me to draw my own version.

Ant and Human live in the yellow mid section of 1 cm.
I forgot to draw in the mountains and the Earth in relative size position
to the righ of humanst.  So this is how we fit in!

Most amazing is the microcosm,  with smaller and smaller particles.
I believe that the world of spirit lives in the minutest size realm.
The Ancients knew it!  In Shinto the prayer says
"Gokubi jissoo gengen shikai...."  which means
 "in the realm of infinitesimal particles"...
which are gen gen.....
I learned the Amatsu Norigoto Prayer while I was 20 years a member
of Sukyo Mahikari.  The Holy Founder received revelations  and I treasure
these Teachings which even touch on quantum science, eg neutrons , and high energy particles.
Anyway the revelations showed that many ideas had become wrong, back to front, over the ages.
eg it should be Taka Ama Hara ,  ie ama not maga.  The latter denotes evil gods...

Most fortuitously, last night I found some YouTube videos which are entirely apt!
One is from Nexus Conference, 2010.
Presented by Nassim Haramein, a self taught man,
with "The Schwarzschild Proton"  - all 1 hour and 30 minutes was fascinating.
Nassim even explained the Planck length!  Most unexpected were many of his insights.
Every proton is a mini black hole.  Two protons spin around each other at
the speed of light, according to his maths.  Much more, as you can imagine.
Space.  The Vacuum.  The Universe.  The Multiverse.  And the MicroCosm.
We are MADE of Light.  We can find out much more on
Nassim has a Youtube video,  Nibiru has come and gone.  We are floating in Grace.  Good.

I found this and related videos via   This is a most unusual and confronting website.
 It pays to persevere with it and look up Fibonacci and spirals and vortexes, Phi,  Fractals,  Mandelbrot;
  Jain or Vedic sacred geometry and mathematics.  Yes there are videos on Vedic Maths.

Responding Literally **

Mindful of the end of the Mayan Calendar December 21,  I decided to take literally Mother Nature's challenge quoted in the last post-  "to learn to follow the designs of her nature."

So I planted brown rice seeds in a pattern according to Fibonacci, as follows:-

**Please click on images, to enlarge them


Insert 3/4/2015   Nothing grew from the rice seeds!  Only couch grass and oats.  At least I found out some design in the numbers.

We are awaiting rain,  and there is hardly any, for the last two months only 8mm has fallen!
I only have rainwater in a tank from my shed 4 m x 4 m roof.  The tank is full but might be needed.
Here is photo of area planted, with containers weighted with dirt to mark the spot.

To the left is a clump of rye 5 ft tall from last  year's seeds.

This Pattern has shown me a larger bilateral symmetrical Pattern with Fibonacci numbers.
Every third number is even and the two numbers in between are both odd.
This means there is a central rib,  all the way up,  like so:-

fairy grass  22/10/2012  6 AM

a delight to see in my garden

Radiance indicates left edge of a fogbow
or white rainbow.  22/9/2012  6:27 AM

One can see such a sight once or twice a year;
sometimes the bow is more evident than this.

Pink Peach tree in blossom
to cheer us up.
16/9/2012  9 AM

Thursday, October 25, 2012

Errors Galore! And a Beautiful Solution

There was some problem with my gwt.xml module file on blogger and an error message kept coming up.  Fortunately, the specialist computer repair men fixed my laptop.
There was some IT incompatibility with new Internet Explorer.
They loaded Google Chrome correctly, with icon.
I had tried to do it myself and it crashed by itself, it is ok,  thanks, guys!

I found a big mistake in an early post, "Beyond Zero", to do with negative powers of 10.
Between 0 and 1 is half the entire Universe, including the realm of spirit.
 I plan to explore this theme when I can.
 I've just come back from Queensland and am now busy on my land.
With the warm weather and a bit  of rain there is huge growth of grass,  and snake(s)....
Look up the Cosmic Ouroboros picture, by Nancy Ellen Abrams and Joel R. Primack of Yale University.
It all connects!   Snakes and the Cosmos.
Actually the book is "The New Universe and the Human Future: How a Shared Cosmology Could Transform the World".
 Resurgence magazine has an article on pages 36 to 41 of the July/August 2012 issue.
You might find on YouTube some videos of Nancy speaking.


To celebrate the end of the Mayan Calendar on 21 Dec 2012 (and to defuse those
holocaust images)  I have planted 30 native trees, mainly Wattles-  silver, varnish, hickory wattles, etc.

Here is the beautiful solution:-

To further imagine our planet as surviving, may I share a valuable website
This is an initiative of a Maori lady, Dr Erena, married to a Swedish man.
Her full name means "hopping, flying, peace angel".
She is the Author of "Mother Earth's Declaration" to humans;
you can print the 2 pages off easily;  just press control and P together
Dr Erena asks us to read and read it and to find deeper and deeper layers of meaning.
 It took a day and a night of deep listening for her to hear it all and to write it down.

One sentence is apropos to the themes of this blog:-

"6.   Mother Earth declares that , you humanity, use square, linear, circular shapes and designs in your way of thinking.  Such thought factors are a stress factor, to the natural ecological patterns of Mother Earth, because these patterns separate and eventually stop the flow of life.  Mother Earth challenges you, humanity to learn to follow the designs of her nature.  Human pollution is not the result of what we use or do, it is mostly, caused by how we think."

(...what if the design is...hyperbolic!?)

 Dr Erena says that We can help Mother Nature, by saying out loud every day,
"I love you, Love.  I love you, Life.  I love you, Mother Earth."
That is all she is asking us to do.


 There are many errors in the care we give to our planet.
 One Australian writer, Eleanor Dark, said of the Land---
".....he saw them saying to the land, "Give! Give!" ....
and in her inviolable way   She will not Give until they cease to Rob. "

I have written some pieces on my other two blogs;  we  need to find new stories to save our world.
Maybe the children will show us, though many are stressed by the difficulties of life.  Siberian Wisewoman Anastasia of the Ringing Cedars Series  tells stories of a little girl making friends with the bacteria and fungi and asking them to clean up the ugly radioactive pollution etc.
And it can happen!  Imagination is good Power.

I just finished reading Sue Woolfe's "The Oldest Song in the World"  in which a young linguist lady goes to an Alice Springs Aboriginal Community to record the Oldest Song only known to one surviving old lady.  In the process she finds out about her own self.  Now I have on library loan her "Leaning Towards Infinity", about her mother who was an unknown mathematical genius.
There is another book I'd love to read, "Perfect Rigour" by Masha Gesson in 2009, about a modern Russian Mathematical genius, Grigory Perelman, who solved The Poincare Conjecture and refused the  $1 million Millennium Prize because  'mathematics is reward enough in itself".

 I am here in a cafe before taking taxi home to my caravan without electricity......  I do hope that when urgent work on my land is done then I can write up a lot of this biomathcraft with more models and pictures, and to do it with care, not hurry.

 I think I might call my land "Alloha",  because then I can go out in the mornings and say "Hallo, Alloha!"

Wednesday, August 1, 2012

Visuals for Fibonacci Series.**

Thanks to a multifaceted jewel of a maths blog,
we can see a lovely video by Cristobal Vila in the section on Fibonacci.

If you check it out via YouTube
there are adjacent videos by Vi Hart,  most engaging,
"Doodling in Maths, and Being a Plant".
She explains the Fibonacci angle which is 360 degrees divided by phi.
See also "Wind and Mr Ug", to do with Mobius strip.  Nice.


More Mermaids

Longtime no post.
Soon there will be some pics of more mermaids.

Meanwhile see,  and Body Sphere, dated 15 July 2012.  You need to click on 2012 under 2013 for transcripts and downloads of the program.
Mention is made of Lynne Sherr's book, "Swim",  Million dollar mermaid Annette Kellerman.
In Byron Bay, Aust, there is Kazzie Mahina of
There are young women in Victoria Aust. who dress up in specially made tails,
 Nerissa, Aradnia and Volitania,
as mermaids,  to teach water safety and marine ecology.
For example, there is video on with many more videos embedded.  eg Mermaid Melissa of Florida, and YouTube fame.    Her life is sheer hard work to save the oceans with love.
Fascinating.  There has been a story of a young man swimming with merfin (a single fin).

1/12/2013  as promised, more mermaids,  these were drawn by Paris whose mermaid was published at the end of the previous post.

And a merman!  Picture was found in a book about the Russian Artists of Palekh.

Newspaper clipping from "The Australian"Tom Bass sees the bronze mermaid he sculpted 60 years before;  it is in a garden alongside Sydney Harbour.


Wednesday, June 13, 2012

A Small Model, and a cute mermaid. **

This is a little "fibonacci cone"  with 21 stitches base circumference, then 4 rows half treble in 4 ply cotton, decrease to make 13 stitches, do 3 rows, decrease to make 8 stitches, do 2 rows, decrease to make 5 stitches, make 1 row in double crochet, decrease to make 3 stitches, make 1 row single crochet, decrease to make 2 stitches, finish off.  The work is done simply spiralling up keeping in line with  the beginning thread.
The concentric circles might be the shape as if seen from the base up the cone.  I had been mistakenly thinking of the "fibonacci ripples" from my blog posts of April 2011, but the numbers are different there.  I had then discovered for myself a number of equations,  eg the radius of a circle with circumference F(n) eg 21  is almost 1/10 of F(n+1)  ie  3.4 units.  True all along;
 Moreover the "almost", the incremental difference, when dividing the larger by the smaller, is phi, 1.618;  the Golden Rule,  etc.

A bit of a mess!  Experiment to see how it takes a layer of roman hydraulic cement.  Fine clay dust plus fine grey cement, equal quantities, with minimum water to create a kind of gel.  Apply with knife to dry crochet cone,  or it might work better with wet surface.
Make body by squishing dacron cushion filling material in cement mix, or you can use wool or whatever to give body.  Sticks for eyes.  This is a very rough model!

Finished model.  The next efforts might be a little more careful.

One notes that grey cement has a high proportion of fly ash from blast furnaces burning coal.  They will do anything to flog waste materials especially this which is in supermassive abundance in our civilised world.  Care about the fact that there is a high level of toxic heavy metals in fly ash which might harm plants, soil micro-organisms, insects, birds et al.  Next time I will use white cement which I think is only burnt lime.  The Romans used Vesuvius ash and fine clay, and their aqueducts still stand to this day.

My secret wish is to make a DIORMA of crocheted seashells with and without moulded bodies, with hyperbolic green seaweeds,  some Nautilus, maybe some Octopi (8 and 144, divisible by 8),  plus mobiles of mermaids, an example has been already drawn by a young lady with a confident hand:- 


On the subject of sculpture, please have a look at 
I heard a wonderful interview on LNL, Radio National, some time ago.  He was fascinated by mathematics when he was young.  He remembered watching a caterpillar walking in its own peculiar way.  Years later he MADE one, sculpted in wood and with a MOTOR.   You would be amazed to see the giant sculptures he's made,  one like a huge amoeba, with motors to move the body.

Wednesday, June 6, 2012

A Better Photo

Here is a better photo to show details of hands and feet and of my mother's lovely face.
That's me age 7 in lower left hand corner.  I must have had to stay still for awhile.
1954 and we had a family tragedy.  We children were put into the care of Burnside Homes for Children, Parramatta, Australia, for 8 years.  No more sculpture.

Two books have been written about Burnside by Kate Shayler, of Hazelbrook, NSW.  - "The Long Way Home"  and most recent, "Burnished" which describes life for a whole bunch of us, polishing floors, shoes etc.  My name in the books begins with A.

 Kate has her own website.

****"Burnished" has recently won
 the 2012 Global Ebook Award for Biography.

24/12/2015 More:

20 years ago my aunt inAaustria found this card in a shop in Linz.
She recognised it as a sculpture made by her sister, my mother.
"Trudi's Engel 1942".

Thursday, April 19, 2012

Tribute to Trudy.

My mother, GertrudEntrich,  was a gifted sculptress (bildhauerin).
Here are the few photos we have of her work from the late 40s and early 1950s.

Clay, plaster, wood, bronze, cement.
Mutti,  We honour and thank you and we wish to share your artistry with the world.
Australian artists TomBass. MirkaMora, MargaretOlley and JudyCassab were your contemporaries and might have been your colleagues if times had been kinder.
You departed this world aged 58 in April 33 years ago.
We remember you with love.


TomBass tribute ctd.

I think it was 1961.  Made of plaster with copper electroplated over all.
There is more story to Ethos and I need to ask for permission to write more.

I thank Dr Bass for this lovely statue which I love to visit.

Tribute to TomBass

One observes Seashells with wonder for they present as amazing sculptures of exquisite precision.  Therefore it feels appropriate to create a space for sculptors.  This is for the late TomBass who was still teaching into his 90s.  Here is "Ethos",  an angel in front of the Canberra Legislature.

to be continued...

Monday, March 19, 2012

On the Trail of the Fibonacci Snail **

This time I thought I had it right!  Then realised I had made it back to front.  So I stitched it inside out to make a correct right hand spiral.  It looks quite nice and I gave it away to a new friend as a gift.  Hello Kay!  The work took all of one afternoon and the next morning.  Meanwhile the garden outside is crying out to be weeded and the termites are attacking,  even eating some Resurgence magazines.  Yikes!  And so it will be another time when I can publish a proper knitting and crochet pattern.  Photos show the story so far:-

For the large snail,  pink is 89,  green is 144,  brown is 233,  purple is 55.
For the small snail it is white is 34,  purple is 55,  pink is 89.

Another one.  I tried to make a pointy shell where the central colmn is 9 times taller than the mouth radius.  It sewed  nicely up to the middle and then was impossible:  too bulky.  So next time there will be a fine thread section.  Yellow is 233,  grey is 144:-

Update,   Insert      22/8/2014
The previous model was a disaster.  
There is a new way of sewing this up!  And it looks heaps better.     .......
Plus,   the final row was added to,   using 0.75 mm crochet hook -
 a tighter row has the effect of gathering the loose shape into a neater edge.

Here we are!  
1.    loosely wrapping the shape to see how it might fit.  Begin from the very tip down,  back stitching so as to make a half turn of the piece much like a kind of Moebius strip,  gradually stitching easing it down to make a wider top side edge,  if you can understand what that means.....not easy to explain,     sorry.

2.  The inner side was originallymade thinner than the outer edge,  but it was still too bulky.
This time the thin area at left hand side top is tightly stitched to the very tip in a spiral,  then the space is increased between the top and bottom of the model's upper area..

3.  Padding was inserted as the model was stitched up,  not after.

4.  This is mentioned in a later post-   "the yellow and white model"  of 22/8/2014.

Fixed at last,     and looking more like a Real shell

Back to 19/3/2012:-
And then.  I made another one with the finely knitted section ok, as shown.
But the collar was too wide!  Next try will be more fine tuned.  The lip edge of collar will be wide and the inner apex will be extremely somehow the long tab at the end wasn't needed.....

Addition 5/5/2013    The following shows there to be mistake in the Pattern.
The top outside of the shell is hard, while the internal spiral column might be paper thin.
Further work is shown in post of 5/5/2010    

****22/8/2014------------  It could be restitched too!