biomathcraft **

Hyperbolic Crochet Snail or Seashell

The model is crocheted in half treble, 1 ply wool, starting with one stitch and increasing according to the series F(n) = 2F(n-2) + F(n-3), as given here.
Rows alternate, so that the increasing stitches are at the beginning of each row.

This creates a classic hyperbolic shape. It is interesting to see that the beginning and the end are in close to each other

To make up the model, fold in half. It configures neatly into a left hand or right hand spiral. Most shells in nature are right hand spirals.


Continue sewing the two sides together of the last row, while softly padding the spiral with wool fleece or cotton wool as you go.


Fibonacci Series

F1 .............0
F2 .............1
F3 .............1
F4 .............2 =(2x1) +0
F5 .............3 =(2x1) +1
F6 .............5 =(2x2) +1
F7 .............8 =(2x3) +2
F8 ...........13 =(2x5) +3
F9 ...........21 =(2x8) +5
F10 ........34 =(2x13) +8
F12 ........89 =(2x34) +21
F13 .......144 =(2x55) +34
F14 .......233 =(2x89) +55
F15 .......377 =(2x144) +89
F16 .......610 =(2x233) +144
F17 .......987 =(2x377)+233
F18 .....1597 =(2x610) +377


This series holds true forever along the Fibonacci series. As you can see, there are series within series.
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The sum total
F1 + F2+…….Fn = F(n+2) minus 1
This is useful to figure how many stitches are made from one skein of cotton or wool, and to plan accordingly how much material is need to complete a shape, various colours.
eg. F1 + F2+… F13 =376 =F15 minus 1.
One skein made 13 rows, that is 376 stitches.
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I am guessing that a hyperbolic shape is one which begins with one point and continues on in an orderly fashion until it becomes a fan shape with an exceedingly frilled edge. The definition of hyperbola in Wikipedia is hard for me to get my brain around.

One can also knit this shape but it needs to be started at the outer edge, eg 233 or 377 or even 610 stitches or more , so long as it fits on the knitting needles and follows the Pattern as in photo .