2. Mathematical Constants
• Numbers that arise in different areas of reality
for no apparent reason
• Mathematical constants are used in a variety
of calculations
• It is not clear how they arise or where they
come from
• Three of the most relevant mathematical
constants are Phi, Pi and Euler’s number
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3. Phi = 1.618033981
• Known as the “Golden Ratio”, this number appears
many times in nature, architecture, geometry and
even in the human body
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4. Pi = 3.14159…
• The most famous number in mathematics, Pi originates
in the ratio of the circumference to the diameter of a
circle, but it keeps showing up in statistics, cosmology,
electro-magnetism, etc.
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5. Euler’s number = 2.7182818284…
• This constant is used to calculate and simulate any
phenomenon that involves growth over time
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6. Golden Spiral
• Phi, Pi and Euler’s number come together to
form one of the basic forms used by nature
to develop many configurations; from,
microscopic chemical structures to
individual organisms, biological systems and
even galaxies:
the Golden Spiral
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7. Golden Spiral in Nature
• This geometrical figure is found in the sprouts
of ferns, sun flowers, marine life forms, ram
horns, the human body, hurricanes and even
galaxies.
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8. The Spiral in History
• The spiral was used to by ancient
civilizations. For instance, at the megalithic
temple of Ggantija in Malta, the Great Stupa
of Sanchi, and the Tibetan Dharma Wheel
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9. The Golden Spiral and the Fibonacci
Numbers
• The Fibonacci numbers are a series that forms
by adding each consecutive natural number to
the one before. That is:
– Natural Numbers:
• 0 1 2 3 4 5 6 7 8 9...
– Fibonacci numbers:
0+1=1 1+1=2 2+1=3 3+2=5
5+3=8 8+5=13 13+8=21 21+13=34
And so on...
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10. The Golden Spiral and the Fibonacci
Numbers
• The Fibonacci numbers are also a fundamental
part of the non-material elements by which
nature organizes matter into systems of
growing complexity. They also appear in
biological settings, such as branching in trees,
phyllotaxis (the arrangement of leaves on a
stem), the fruit sprouts of a pineapple, the
flowering of an artichoke, an uncurling fern
and the arrangement of a pine cone's bracts.
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11. The Golden Spiral and the Fibonacci
Numbers
• It is possible to make a logarithmic spiral that
converges towards a golden spiral based on a
set of rectangles whose sides equal the length
of consecutive Fibonacci numbers:
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12. The Golden Spiral, the Fibonacci
Numbers and Pascal`s Triangle
• The Fibonacci numbers connect the Golden Mean and the
Golden Spiral to another fundamental fractal through a
mathematical concept known as Pascal’s Triangle:
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13. The Golden Spiral, the Fibonacci
Numbers and Pascal`s Triangle
• One of the many properties of Pascal’s
Triangle is that the Fibonacci numbers can be
obtained from it by adding the numbers in
diagonals:
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14. The Golden Spiral, the Fibonacci
Numbers and Pascal`s Triangle
• It is the Fibonacci numbers that connect the
three fundamental mathematical constants
contained in the Golden Spiral (Phi, Pi and
the Euler Number) to the basic
tridimensional structure known as the
tetrahedron.
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15. Pascal’s Triangle and the Tetrahedron
• If we substitute the even numbers with zeros
and the odd numbers with ones in the Pascal
Triangle we get the following configuration:
1
1 1
1 0 1
1 1 1 1
1 0 0 0 1
1 1 0 0 1 1
1 0 1 0 1 0 1
1 1 1 1 1 1 1 1
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16. Pascal’s Triangle and the Tetrahedron
• This triangle made out of smaller triangles, is a
fractal known as the Sierpinski Gasket and is
our next non-material component of the
objects we find in the physical world:
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17. • If we fold the blue triangles on the white triangle
we get a triangular pyramid, the basic three-
dimensional structure known as tetrahedron:
Pascal’s Triangle and the Tetrahedron
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18. The Tetrahedron: elemental tri-
dimensional shape
• Just as the triangle is the elemental two-
dimensional figure as it can be
constructed by linking three dots on a
piece of paper; the tetrahedron is the
elemental three-dimensional figure that
arises from connecting four dots in a
three dimensional space
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19. Tetrahedral Molecules
• The relevance of this shape in the self-
organizing process of nature becomes evident
at the atomic level, where the molecules of
several solids are shaped as tetrahedrons
Diamond molecule
Silica molecule
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20. Quartz Crystals
• the actual shape of the silica crystal (also
known as quartz) resembles the shape of a
tetrahedron:
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21. Building Blocks of the Planet
• Silicate minerals constitute 90% of the Earth’s
crust and are also the largest and most
important class of rock-forming minerals;
which means that silicate minerals are the
basic building blocks of this planet. When six
silicate mineral molecules are tied together to
form beryl their structure exhibit a
configuration like:
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22. Building Blocks of the Planet
• The six pointed star, represents the Anahata or
heart chakra, which is related to love, empathy,
selflessness and devotion.
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23. Sri Yantra: The Universal Mind
• All the previous geometrical figures and
mathematical constants come together in the
Sri Yantra:
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24. Universal Spirit
• The Universal Spirit “...manifests itself as a
craftsman, producing itself as an object
through its work, as bees build their cells”
(Hegel, 1985, p. 405).
• “The Spirit, which is under the will of God and
serves as his instrument, is that by which all
beings in the Universe are moved and
directed” (Trismegistus, 1998, p. 147).
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25. Universal Spirit
• our Universe has a Spirit, this Spirit has a Mind
or self-consciousness, and this Mind has a will.
We live in a Universe that speaks to us
through symbols, mathematical magnitudes
and geometrical shapes. The signs of the
language of the Universe can be read through
philosophy, religion or science but all of them
lead to the same end: God.
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