Friday, September 26, 2014

A Hyperbolic Rainbow Torus, Algorithm 9. **

This Diagram shows a Torus of circumference 72,  and uses Algorithm 9
ie  3   6   9   18   (27)   36   (45)   54   (63)   72.

**Please click on image to see it better.

 My plan is to try to use 6 stranded embroidery cotton,  $1 each,  of rainbow colours,  and using a smaller crochet hook eg 1.  .....sometime...... the skeins would have to be rolled into balls, without tangling them!


There is another Pattern featuring the number 9 and half fibonacci numbers,  published 26/5/2011 when I was studying how to make Bivalve Shells.  It came to me that by using Algorithm 18 we can make a full fibonacci Pattern.  It tends to make "corners".  so this might become an "18 sided polygonic torus",  if such a thing might exist.  I can't wait to try it.  Never a dull moment.


A Hyperbolic Crochet Apple **

We can use any old numbers-  it doesn't have to be just Fibonacci numbers.

What if I choose 72?    It is half of 144, a fibonacci number
---but 72 can be divided into 12s and 9s,....

This is what the Apple looks like.  It is 9.5 cm wide and 8 cm high.

This is what the geometry looks like.  It is in bands of 3,  6,  9,  12,  24,  36,  48,  60, 66,  72.  then reversing 72 to 66 to 60 to 48 to 36 to 24 to 12 to 9 to to 6 to 3... and join them in the middle, inside.
-----5 ply wool and 8 ply wool was used,  with a 2.5 crochet hook.

**please click on image to see it better.

You might figure it all out from this Diagram of how to make this,  if you have been able to crochet the "55 torus "of the previous post.

It is interesting to see how the yellow line marking the end of each row  --- forms a spiral.

This one has been worked so that the back of the crochet work  makes the outside surface of the apple.  Keep the point of origin facing you as you work the crochet pattern around and around. 
Later turn the whole thing out the correct way before putting in the padding:-

The padding is 2 plastic bread bags.....but I had to undo the end to 21 stitches because more padding was needed-  28 cotton balls more-  to plump out the shape.   

You could glue in a wood stick and make a paper leaf...  
make a pear or orange or a banana by skewing the Design to make a different shape...


A Hyperbolic Fibonacci Crochet Torus **

Here follows mathematical modelling and geometry of the torus.
See Wikipedia for definition of "torus".

 **Also see previous posts here on this blog-   2/5/2014, 4/16/2014.

Here is a presentation of how to crochet a Torus.
It is an opportunity for me to use the Fibonacci + half Fibonacci Series of numbers as the basis, a handle, for elucidation and practical soft sculpture.  Includes the Algorithm -  make 4 to 5.
This model is 6 cm wide and 5 cm high.  The circumference is 55 units, a Fibonacci number:-

***********Please click on images to see them more clearly.

Simple Written Instructions on How to Crochet a Torus

1.  Begin with 1 chain;  1.5 size crochet hook, in UK Europe Aust (use appropriate size in USA)
Here Sullivan’s 4 ply equivalent crochet cotton in white was used;  blue for midline.
Tension:  10 stitches = 3.5 cm;  10 rows = 3.25 cm .  You don’t have to stick to this one….any will do.
Single crochet here is  “ hook into loop,  yarn around hook,  pull yarn thro,  yarn around hook,  pull thro 3 loops;  in US the name of stitch is different.  Or you can use any kind of stitch you like. 
You can even transcribe Pattern to enable knitting in exactly the same way.  With hyperbolic shapes of large numbers eg 377 or 610 one needs to work the knitting backwards from wide frilly edge to narrower areas.
***Any kind of yarn or appropriate hook is ok to use.
2.  increase 1 to make 2 loops
3.  make 2 to 4
4.  make 4 to 8;  as we begin to crochet in a continuous circle please mark the end of each row with a different coloured length of yarn eg pale blue as here, weaving it from front to back and vice versa after each row.  If you keep the beginning point with some good cm of  yarn behind your work as you go,  then the right side (front) of work will become the outside surface of the torus. 
5.  make 8 to 13,  ie  crochet twice in next 3 loops, do 1, crochet twice in next 2 loops, do 2.
From here on, we are doing increases in some rows in accordance with the Fibonacci and Half Fibonacci numbers…
6.  make 13 to 21,  ie  (crochet 2x  in next 3 loops, do 1)  2 times.  crochet 2x in next 2 loops, do 3.
7.  crochet 1 plain row of 21 loops (or stitches)  (no increases). 
8.  make 21 to 28,  ie  crochet 1x into next 2 loops, crochet 2x in next loop, (3 makes 4,) 7 times.
Begin next quarter of Torus:-
1.  do 1 plain row of 28
2.  make 28 to 34,  ie  crochet 1x into next 3 loops, crochet 2x into next loop, (4 makes 5), 7 times.
Ie  it would be 28 to 35,  so omit last increase, to make 28 to 34.
3.  do 1 plain row of 34
4.  make 34 to 44,  ie  make 4 to 5,  all 5 times.  Ie you have made 25 out of 20,      with 14 left to make 19.  so make 3 to 4, all 4x,  which is now 44 in the row .
5.  make 1 plain row of 44
6.  make 44 to 52,  ie   crochet 1x into next 7 loops, crochet 2x into next loop (8 makes 9), all 4x. ie you have made 36 out of 32,  with 12 left to make 16.  So make 3 to 4, all 4x, which is now 52 in row.
7.  make 1 plain row of 52.
8.  make 52 to 55,  ie crochet once into 16 loops, crochet 2x into 1 loop (17 makes 18), all 3x,  so 51 becomes 54,  leaving do 1 = end.   This is the last row of second quarter of torus.  The work should be curving nicely.
MIDLINE = using blue, do 1 row of 55 stitches.
Begin second half of Torus.  Here the shape is made by decreasing in some rows
1.  make 1 plain row of 55
2.  make 55 to 52,  ie crochet 16, skip 1, crochet 1, (18 makes 17),  all 3x,  so 54 becomes 51;  do 1.
3.  make 1 plain row of 52.
4. make 52 to 44,  ie do 6, skip 1, do 1. (8 to 7), all 6x,  ie 48 makes 42, leaving 4 to be decreased to 2.
5.  make 1 plain row of 44.
6.  make 44 to 34,  ie do 3, skip 1, do 1, (5 to 4), all 5x,  ie 25 makes 20, leaving 19 to make 14.  So do 2,  skip 1, do 1 (4 to 3),  all 4x,  so that 16 made 12,  leaving 3 to be decreased to 2.
7.  make 1 plain row of 34.
8.  make 34 to 28,  ie do 3, skip 1, do 1, (5 to 4), all 6x,  ie 30 makes 24, leaving do 4.
This is the end of the 3rd quarter of the Torus.
Begin last quarter of Torus.
1.  make 1 plain row of 28
2.  make 28 to 21,  ie do 3, skip 1, do 1, (5 to 4),  all 4x,  ie 20 makes 16, leaving 8 to make 5 by – (do 1 skip 1, do 1 and do 1, skip 1, do 1) then skip 1, do 1.
3.  make 1 plain row of 21 *********
At this stage we need to put padding inside the model!  One can use cotton wool or wool fleece…..
Make sure that the beginning length of yarn is coming up through, inside the middle.
4.  make 21 to 13,  ie  do 1, skip 1, do 1, skip 1, do 1, skip 1, do 1,  (7 to 4), all x 3. You have made 21 to 12,  so   Adjust one at the end, ie don’t skip one at end.

5.  make 13 to 8,  ie do 1, skip 1, do 1, skip 1, do 1, skip 1, do 1,  (7 to 4), all 2x.  Adjust one at the end ie 6 to 4.
6.  make 8 to 4
7.  make 4 to 2.
8.    Make sure that the work is curved so that the length of yarn at the beginning of the work is coming up out of the tiny hole made by the circle of 2 loops!  Tie, and tuck in the ends.
You have completed a model of the Torus with circumference 55.


You can copy this Pattern, 
using -------highlight text etc-------control and "c" to copy--------control and "v" to paste  onto  a new word page-------edit and improve font----save on your computer---and print.
Good luck with your crocheting---or knitting!

Supporting Schemata:-

Pi is   3.14159 26535 89793.....
a transcendental has  been calculated by computer to 12 trillion decimal places!

2 Pi is  6.28318........

Whereas Phi is 1.6180339 8874 98948....

See how the original thread comes up through the tiny 2 chain hole-  join them at central zero point;  tuck in  to form the dimple.   You can see where each row ends by the light blue line weaving upwards. 


This work has not been without its failures and first I tried to calculate the arcs for each number band-  2, 4, 8, 13, 21, 28, 34, 44, 55-     arc =  x degrees / 360 x Circumference....
 result was sometimes 4 rows between increase or decrease show may indicate how to crochet a cow horn,  or a unicorn horn,  or a banana with skew....or a pear...or a Sufi whirling dervish hat,  or a wizard or witch's hat ( try 377)!

Ridges show on working crochet from front and back,....  
versus working all the time from one side only,  in a continuous spiral, shows an even surface and it is the one used in this model.

Tension samples
How wide is 10 stiches
How high is 10 rows?


next is a Hyperbolic Apple = algorithm 12-------- not Fibonacci:.........


A Discussion of the Torus

I hope this might be an ongoing discussion regarding the torus.
Two previous posts in this blog  began recently on this topic-  on 4/16/2014.

The last 3 posts are uploads from YouTube and are most in tune with this blog on Hyperbolae.
Thrive" says" the Universe is a Torus Factory."  Torii are everywhere in Nature-   and in Consciousness.......

Indeed the big news of recent times is-  the BRICS Nations-  Brazil, Russia, India, China and South Africa-  are using torus plasma free energy -for antigravity cars and space travel.  India has announced a successful Mars project,  China has a hyperbomb which is 100 times more powerful than the atom bombs of WW@.  In addition, you can find story on YouTube about the BRICS Nations forming a new multibillion $ Bank independent of the IMF.  Free energy could bring World Peace..  Nicola Tesla's inventions have led to  Iran's Dr Keshe's energy generator based on torus,  based is Belgium............

One wonders what the future may hold.......we shall see!