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f56r - an inverse hyperbolic spiral?
Following on from my recent letter to New Scientist, I've been emailed by a
number of different people asking for more information, with some offering
their own observations etc.
The most interesting observation came from Stan Tenen, who some of you may
know from the Meru Foundation (http://www.meru.org).
If I understand it correctly, Meru is essentially based around two
(1) that the Hebrew alphabet was preceded by a universal gesture language
(which some people equate with the language of Babel), and that the forms
of Hebrew letters are in some way a codfication of that gesture language; and
(2) that the letters of the alphabet can also be thought of as being
multiple views of a single non-linear curve - the "light in the meeting
Over the last 15 years, he has been trying to systematically make these two
hypotheses concrete: and believes that the non-linear curve involved is the
inverse hyperbolic spiral... a particularly clear example of which is on
Voynich f56r (which Dana provisionally identified as "Palm, Chamaedorea
elegans"?). Stan's comments on this illustration:-
The shape of the stalk is that of the inverse or hyperbolic spiral.
This particular spiral, which is not a log or a golden type spiral,
appears throughout Leonardo's notebooks, throughout the
illustrations in the Book of Kells, and most prominently, under
the "Eye of Horus".
The two florets at the bottom have a total of 33 spikes. There are
33 tetrahedra in a unit-turn tetrahelical column, and there are 33
vertebrae (including fused) in the human spine.
The total number of flame-like "leaves" is 18, but there are also 2
more complicated florets, plus the 2 at the bottom, for a total of
22, which is the number of Canaanite letters. There are also 22
faces exposed on each of the 3 ribbons of a 33-tetrahedron
The base has a pineapple-pine tree type structure, with 9 or 10
spikes. Either number could also allude to the
Hebrew-Greek-Arabic alphabet system.
The 18 flame-like leaves reverse-taper, with the larger leaves
near the center of the spiral, and the smaller leaves closer to the
vertical, linear part of the "stem". This is a characteristic of the
reciprocal spiral, where angles get larger, the closer to the center.
The equation for the reciprocal or hyperbolic spiral is r x theta = 1.
The reciprocal or hyperbolic spiral is almost completely
un-noticed by scholars, who consider all spirals to be the same.
But in fact, it's very particular, and totally different than any other
spiral, because it moves smoothly from circular motion,
strangling the origin, to become asymptotic to a straight line (in
this case, vertical).
It's the number of parts, the flame-shaped leaves, and the shape
of the bigger leaf-blossoms, that matches the features of the
model I've been researching. So, the exact geometry is not as
important, particularly since it's my guess that the geometry
was known in slightly different contexts at different times, and
in different cultures.
But there's no other structure besides the unit-turn tetrahelical
column that has 33 volumes and 3 ribbons of 22 faces. Also,
the 18 flame-shaped leaves could be a reference to Islamic
sources, where although the alphabet contains the full 27
letters, the basic shapes number 18 (if I remember correctly
-- someone should check this).
So, I'd assume that the drawing on Voynich page 56 was
passed down through a Moslem source, at least at some point.
He also strongly asserts that the construction of this spiral is based
around a mathematical approximation to squaring the circle (based on
inverse numbers), which may have been known to the Egyptians - there's a
proper discussion of this on the meru.org website.
The obvious question to ask here is: have any of you seen this same inverse
hyperbolic spiral in any other documents of the time (or earlier)? I'm
thinking specifically of the Arabic Dioscorides etc?
Cheers, .....Nick Pelling.....