I will comment on only three earlier points made by Namadev:
- Namadev writes: 'I still fail to understand why a triangle such as the one I've drawn with 12 dots on each of its 3 sides isn't an equilateral triangle.
Simply, that the depiction expands the dots on one of its sides and renders it a right-angled triangle.
And equilateral triangle is also an equi-angular one: its three internal angles each have 60°.
The one depicted by Namadev has an angle of 90°, and two of 45.
It is this specific depiction which allows for both the visual impact of the 'incomplete' pyramid form, as well as the possibility of having the
square of 16, and
four depictions of the triangular base-four tetracti.
If the 78 points are arranged in equilateral-equiangular shape (ie, if they become a true equilateral triangle), the visual imagery presented vanishes.
Nonetheless, and as I have mentioned before, representations of triangular numbers as diagonally truncated squares (as right-angled triangles, to make what I mean more readily obvious) is quite acceptable.
- Namadev writes: 'You suggested the tetrahedron when I had a preference for the Octahedron'
It
seems to me that stated preference for the octahedron is based on its visual similarity to the desired shape - and that only of its 'top' four-sided pyramid shape, rather than the platonic solid as a whole.
The reason I mentioned the tetrahedron is because the 'open-shaped' 22 dots do, in fact, make a tetrahedron if the open space is joined.
To see what I mean, I suggest printing the depiction, cutting-out the 'open-pyramid', folding each diagonal to ease movement, and actually joining the edges of the opening. A tetrahedron will have been formed, and the 2-dimensional figure transformed to 3-D.
- square root of Phi and relations to various numbers in Tarot
Namadev mentions the relation of the square root of Phi ('racine of Phi') and various important numbers found in Tarot.
Firstly, it can be pointed out the Root Phi - specifically:
- [(1+[5^(1/2)])/2]^(1/2) = (approx.) 1.272
For the sake of interest, and though I realise this is more generally known, Phi itself is:
- (1+[5^(1/2)])/2 = (approx.) 1.618
The numbers given are important in various considerations, but especially given that there are a total of 220 implements represented in the pips (as mentioned above and in other threads):
- each suit has 1+2+3+4+5+6+7+8+9+10 = 55, making a total of (given there are four suits) 220
This is of course quite marvelous in terms of its mathematical connection to the 22 Atouts.
Namadev also points out that the 16 courts, if added together in such addition, gives us the triangular number (base-16) of 136. What I personally fail to see here is the grounds on which such addition, in the case of the courts, is warranted. It
suggests that one of the courts has a value of 1, and another of 16 - but that is another matter, for of course one may play with these numbers and see what emerges.
Adding the 136 to the 220 does indeed give us 356, which, if divided by the 22 Atouts, makes (approx.) 16.18, and if divided by a tenfold aspect, and by a number also used in the generation of the 356 (ie, 220), then an approximation to Phi is given: 1.618.
It should be pointed out that any sequence which uses an additive method of generation (such as the famous Lucas or Fibonacci series) will result in adjacent numbers having close approximations to Phi when one is divided by the other. Given that 356 divided by 220 is an approximation of Phi, it will also be the case that 220 divided by (356-220) will also approximate it - though not as closely.
Of greater interest, for me at any rate, is that the 220 is a very close approximation of the Golden Angle (360 / Phi = approx. 220 degrees).
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All this, though highly interesting, shows and demonstrates that numerous mathematical and geometrical considerations are inherent within the Tarot and there to be discovered, without, in my view, implying that they were there as guiding factors in its creation or development.