Fibonacci Trees

Several of the Sareoso diagrams feature Fibonacci trees. One of these is the Zuhamua Aldi diagram shown below:

diag16sm

The diagram shows a tree or vine shape (the name of the diagram means “tree/vine of Aldi”, and the label ILAR means “pea”), so perhaps this is a pea plant.

450px-snow_pea_flowers

Snow Pea Flowers (by Bmdavll via Wikipedia)

The shape shows the possibilities of branching by dividing into two at each stage, so that the shape could grow by doubling: 1, 2, 4, 8, 16, 32, 64. But these possibilities are not all realised – in this diagram, the unrealised possibilities are represented by little spiral tendrils (by which a vine takes hold of its supports).

This results in slower growth, according to the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21. Growth according to the Fibonacci sequence is very common in nature, for example in spirals, branching and flower patterns. There is a good overview of the Fibonacci sequence and the way it appears in nature at Ron Knott’s Fibonacci Numbers and Nature page.

The way the structure is represented in the diagram isn’t random: it follows a simple rule. Each branching-point is represented by one of six two-letter codes or “doubles”. For example, at the base of the tree is found the double AT.

A double has the possibility of branching into two others, which have to start with the same letter as the first one finished with. For example, AT can be followed by TA or TE. Branchings which revert to the previous double are not allowed though. For example, there is no problem with AT-TA-TE, but AT-TA-AT is not allowed because it goes back to AT.

Out of all the branches, there are only three that ‘bear fruit’ in the diagram. These are branches which go through all the six doubles exactly once, without repetition.

The Sareoso diagram Erregaluak shows all three of the ‘vines’ from the Zuhamua diagrams together, radiating out from the centre. Instead of the double letters, shapes are used, with the outer shape equivalent to the first letter, and the inner shape equivalent to the second one. The diagram Zuhaitza Bizi gives a similar tree, but using three shapes instead of two (adding a middle shape). This shows similarity to the tree in Chapter 6 of the Book of Jubilee.

 

Feast of Santa Agueda

Today and tomorrow is the feast of Santa Agueda in the Basque country. This is my arrangement of the song for classical guitar.

Mandelbrot Set

The Sareoso diagram Ogi Almendra Orotasun looks like the Mandelbrot set, and in fact the name of the diagram Ogi Almendra is a bit of a word play, since it means Almond Bread, as does Mandel Brot.

The Mandelbrot Set is defined by very simple maths, but is endlessly complex. If you look at one part of it in detail, you see different shapes, similar but different, and the detail keeps on going forever. Who knows what we might find as we zoom in!

Here’s an example of some detail you can see (there are many other videos which look at different parts of the set, all looking different). It reminds me of a voyage through space!

 

What is the Mandelbrot Set?

Mathematics is used to determine if a point is in the Mandelbrot set or not. There is a particular simple calculation that is applied over and over again to the point. Points that are in the Mandlebrot set don’t move very far from the centre as this process goes on. Points that are not in the Mandlebrot set keep on moving further away as the calculation is repeated. (See the notes at the end for some more detail on the calculation).

In the Ogi Almendra Orotasun diagram, the dark blue area is the main part of the Mandlebrot set. The light blue surround and the other colours are points which are outside the Mandlebrot set – the colours show how quickly the points move away from the centre.

diag02sm

Ogi Almendra Orotasun

All the detail and complexity of the set is in the border.


Some Geometry

In the diagram, the two largest parts of the set are the heart-shaped cardioid area at the top, and below it the smaller circular area. It turns out that these have some interesting relationships. A cardioid can be drawn as the locus of a point on a circle as it rolls around another circle.

It turns out that the two circles that make the Mandelbrot cardioid are the same size as the circle below it, as illustrated below on a modified version of the diagram:

ogi-almendra

 


Notes: The Mandelbrot Set Calculation 

The calculation used to figure out if a point is in the Mandelbrot Set is simple:

  1. Start with the point you want to test as the value.
  2. Multiply the value by itself, and then add in the point again.
  3. Repeat step 2 with the new value.

In a one-dimensional example, I would test the point at 1.5 by working out:

Calculation 1: 1.5 x 1.5 + 1.5 = 3.75
Calculation 2: 3.75 x 3.75 + 1.5 = 15.5625
Calculation 3: 15.5625 x 15.5625 + 1.5 = 243.6914…

You can see that the value is rapidly getting bigger and bigger, so this point is not in the Mandelbrot set.

Testing the point at 0.1 gives:

Calculation 1: 0.1 x 0.1 + 0.1 = 0.11
Calculation 2: 0.11 x 0.11 + 0.1 = 0.1121
Calculation 3: 0.1121 x 0.1121 + 0.1 = 0.11256641

If I keep going for 40 more calculations I get 0.11270…, so you can see that the value is staying small, meaning it is in the Mandelbrot set.

In fact the borderline is at one quarter (0.25), which the dip of the cardioid in the picture!

The actual calculation uses two dimensions of course, so it is a little bit more complicated than the one-dimensional example, but only a bit – mathematicians use a scheme called complex numbers to work out the calculation.

 

 

An Extract from Plato’s Parmenides

Here is an extract from Plato’s dialogue “Parmenides”.
The topic is the nature of ‘the one’.

Parmenides proceeded: If one is, he said, the one cannot be many?

Impossible.

Then the one cannot have parts, and cannot be a whole?

Why not?

Because every part is part of a whole; is it not?

Yes.

And what is a whole? would not that of which no part is wanting be a whole?

Certainly.

Then, in either case, the one would be made up of parts; both as
being a whole, and also as having parts?

To be sure.

And in either case, the one would be many, and not one?

True.

But, surely, it ought to be one and not many?

It ought.

Then, if the one is to remain one, it will not be a whole, and
will not have parts?

No.

But if it has no parts, it will have neither beginning, middle,
nor end; for these would of course be parts of it.

Right.

But then, again, a beginning and an end are the limits of
everything?

Certainly.

Then the one, having neither beginning nor end, is unlimited?

Yes, unlimited.

And therefore formless; for it cannot partake either of round or
straight.

But why?

Why, because the round is that of which all the extreme points are
equidistant from the centre?

Yes.

And the straight is that of which the centre intercepts the view
of the extremes?

True.

Then the one would have parts and would be many, if it partook
either of a straight or of a circular form?

Assuredly.

But having no parts, it will be neither straight nor round?

Right.

And, being of such a nature, it cannot be in any place, for it
cannot be either in another or in itself.

How so?

Because if it were in another, it would be encircled by that in
which it was, and would touch it at many places and with many parts;
but that which is one and indivisible, and does not partake of a
circular nature, cannot be touched all round in many places.

Certainly not.

But if, on the other hand, one were in itself, it would also be
contained by nothing else but itself; that is to say, if it were
really in itself; for nothing can be in anything which does not
contain it.

Impossible.

But then, that which contains must be other than that which is
contained? for the same whole cannot do and suffer both at once; and
if so, one will be no longer one, but two?

True.

Then one cannot be anywhere, either in itself or in another?

No.

Further consider, whether that which is of such a nature can have
either rest or motion.

Why not?

Why, because the one, if it were moved, would be either moved in
place or changed in nature; for these are the only kinds of motion.

Yes.

And the one, when it changes and ceases to be itself, cannot be
any longer one.

It cannot.

It cannot therefore experience the sort of motion which is change of
nature?

Clearly not.

Then can the motion of the one be in place?

Perhaps.

But if the one moved in place, must it not either move round and
round in the same place, or from one place to another?

It must.

And that which moves in a circle must rest upon a centre; and that
which goes round upon a centre must have parts which are different
from the centre; but that which has no centre and no parts cannot
possibly be carried round upon a centre?

Impossible.

But perhaps the motion of the one consists in change of place?

Perhaps so, if it moves at all.

And have we not already shown that it cannot be in anything?

Yes.

Then its coming into being in anything is still more impossible;
is it not?

I do not see why.

Why, because anything which comes into being in anything, can
neither as yet be in that other thing while still coming into being,
nor be altogether out of it, if already coming into being in it.

Certainly not.

And therefore whatever comes into being in another must have
parts, and then one part may be in, and another part out of that
other; but that which has no parts can never be at one and the same
time neither wholly within nor wholly without anything.

True.

And is there not a still greater impossibility in that which has
no parts, and is not a whole, coming into being anywhere, since it
cannot come into being either as a part or as a whole?

Clearly.

Then it does not change place by revolving in the same spot, not
by going somewhere and coming into being in something; nor again, by
change in itself?

Very true.

Then in respect of any kind of motion the one is immoveable?

Immoveable.

But neither can the one be in anything, as we affirm.

Yes, we said so.

Then it is never in the same?

Why not?

Because if it were in the same it would be in something.

Certainly.

And we said that it could not be in itself, and could not be in
other?

True.

Then one is never in the same place?

It would seem not.

But that which is never in the same place is never quiet or at rest?

Never.

One then, as would seem, is neither rest nor in motion?

It certainly appears so.

Neither will it be the same with itself or other; nor again, other
than itself or other.

How is that?

If other than itself it would be other than one, and would not be
one.

True.

And if the same with other, it would be that other, and not
itself; so that upon this supposition too, it would not have the
nature of one, but would be other than one?

It would.

Then it will not be the same with other, or other than itself?

It will not.

Neither will it be other than other, while it remains one; for not
one, but only other, can be other than other, and nothing else.

True.

Then not by virtue of being one will it be other?

Certainly not.

But if not by virtue of being one, not by virtue of itself; and if
not by virtue of itself, not itself, and itself not being other at
all, will not be other than anything?

Right.

Neither will one be the same with itself.

How not?

Surely the nature of the one is not the nature of the same.

Why not?

It is not when anything becomes the same with anything that it
becomes one.

What of that?

Anything which becomes the same with the many, necessarily becomes
many and not one.

True.

But, if there were no difference between the one and the same,
when a thing became the same, it would always become one; and when
it became one, the same?

Certainly.

And, therefore, if one be the same with itself, it is not one with
itself, and will therefore be one and also not one.

Surely that is impossible.

And therefore the one can neither be other than other, nor the
same with itself.

Impossible.

And thus the one can neither be the same, nor other, either in
relation to itself or other?

No.

Neither will the one be like anything or unlike itself or other.

Why not?

Because likeness is sameness of affections.

Yes.

And sameness has been shown to be of a nature distinct from oneness?

That has been shown.

But if the one had any other affection than that of being one, it
would be affected in such a way as to be more than one; which is
impossible.

True.

Then the one can never be so affected as to be the same either
with another or with itself?

Clearly not.

Then it cannot be like another, or like itself?

No.

Nor can it be affected so as to be other, for then it would be
affected in such a way as to be more than one.

It would.

That which is affected otherwise than itself or another, will be
unlike itself or another, for sameness of affections is likeness.

True.

But the one, as appears, never being affected otherwise, is never
unlike itself or other?

Never.

Then the one will never be either like or unlike itself or other?

Plainly not.

Again, being of this nature, it can neither be equal nor unequal
either to itself or to other.

How is that?

Why, because the one if equal must be of the same measures as that
to which it is equal.

True.

And if greater or less than things which are commensurable with
it, the one will have more measures than that which is less, and fewer
than that which is greater?

Yes.

And so of things which are not commensurate with it, the one will
have greater measures than that which is less and smaller than that
which is greater.

Certainly.

But how can that which does not partake of sameness, have either the
same measures or have anything else the same?

Impossible.

And not having the same measures, the one cannot be equal either
with itself or with another?

It appears so.

But again, whether it have fewer or more measures, it will have as
many parts as it has measures; and thus again the one will be no
longer one but will have as many parts as measures.

Right.

And if it were of one measure, it would be equal to that measure;
yet it has been shown to be incapable of equality.

It has.

Then it will neither partake of one measure, nor of many, nor of
few, nor of the same at all, nor be equal to itself or another; nor be
greater or less than itself, or other?

Certainly.

Well, and do we suppose that one can be older, or younger than
anything, or of the same age with it?

Why not?

Why, because that which is of the same age with itself or other,
must partake of equality or likeness of time; and we said that the one
did not partake either of equality or of likeness?

We did say so.

And we also said, that it did not partake of inequality or
unlikeness.

Very true.

How then can one, being of this nature, be either older or younger
than anything, or have the same age with it?

In no way.

Then one cannot be older or younger, or of the same age, either with
itself or with another?

Clearly not.

Then the one, being of this nature, cannot be in time at all; for
must not that which is in time, be always growing older than itself?

Certainly.

And that which is older, must always be older than something which
is younger?

True.

Then, that which becomes older than itself, also becomes at the same
time younger than itself, if it is to have something to become older
than.

What do you mean?

I mean this:-A thing does not need to become different from
another thing which is already different; it is different, and if
its different has become, it has become different; if its different
will be, it will be different; but of that which is becoming
different, there cannot have been, or be about to be, or yet be, a
different-the only different possible is one which is becoming.

That is inevitable.

But, surely, the elder is a difference relative to the younger,
and to nothing else.

True.

Then that which becomes older than itself must also, at the same
time, become younger than itself?

Yes.

But again, it is true that it cannot become for a longer or for a
shorter time than itself, but it must become, and be, and have become,
and be about to be, for the same time with itself?

That again is inevitable.

Then things which are in time, and partake of time, must in every
case, I suppose, be of the same age with themselves; and must also
become at once older and younger than themselves?

Yes.

But the one did not partake of those affections?

Not at all.

Then it does not partake of time, and is not in any time?

So the argument shows.

Well, but do not the expressions “was,” and “has become,” and “was
becoming,” signify a participation of past time?

Certainly.

And do not “will be,” “will become,” “will have become,” signify a
participation of future time?

Yes.

And “is,” or “becomes,” signifies a participation of present time?

Certainly.

And if the one is absolutely without participation in time, it never
had become, or was becoming, or was at any time, or is now become or
is becoming, or is, or will become, or will have become, or will be,
hereafter.

Most true.

But are there any modes of partaking of being other than these?

There are none.

Then the one cannot possibly partake of being?

That is the inference.

Then the one is not at all?

Clearly not.

Then the one does not exist in such way as to be one; for if it were
and partook of being, it would already be; but if the argument is to
be trusted, the one neither is nor is one?

True.

But that which is not admits of no attribute or relation?

Of course not.

Then there is no name, nor expression, nor perception, nor
opinion, nor knowledge of it?

Clearly not.

Then it is neither named, nor expressed, nor opined, nor known,
nor does anything that is perceive it.

 


From the translation by Benjamin Jowett.
http://www.sacred-texts.com/cla/plato/parmeni.htm

Baga Biga Higa

brujas_arena

 

This is the song that the witches sing when they meet together:

 Baga, biga, higa,
laga, boga, sega,
Zai, zoi, bele,
harma, tiro, pun!
Xirristi-mirristi
gerrena plat,
Olio zopa
Kikili salda,
Urrup edan edo klik …
ikimilikiliklik …

Here is a famous Basque musician singing it:

Translation:
1 Baga (bat) 2 biga (bi) 3 higa (hiru) 
4 laga (lau) 5 boga (bost) 6 sega (sei)
7 zai (zazpi) 8 zoi (zortzi) 9 bele (bederatzi)
10 harma (hamar) tiro pun! – (bang!)
Xirristi-mirristi (onomatopoeia for slipping down the throat, or the sound of a crackling fire)
gerrena (BBQ/spitting) plat (dish), 
Olio Zopa (oil soup) Kikili salda (chicken broth)
Urrup edan (drinking at once) edo (or) klik (swallowing) 
ikimilikiliklik (glug glug glug) – a bit like ‘abracadabra!’

 

and here is a video of the 3 forces dancing to this song!