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 number......it 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 challenges........at 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 rows.......photos show results....it 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:.........
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