Grimoire for the Apprentice Wizard

Course Six: Spectrum, Part 2 273

Carbon

Graphite

Diamond

such as methane, which is tetrahedral in

shape. The electron bonds make an angle

of approximately 109°, which is the most

fundamental angle in chemistry.

Solids can be thought of as networks

of atoms built using these bonds as a glue to hold the

atoms together. Bcause we ourselves are carbon-based

life-forms, consider, for example, solids built entirely

of carbon atoms. We know of two variations: diamond

and graphite. The natural arrangement that obeys the

symmetry of the local bonds, 109°, is a diamond, a

three-dimensional tightly-bound network of carbons.

The angle related to the bond in graphite is also close

to this natural angle. In graphite the carbon atoms

form sheets of strongly-glued atoms that do not inter-

act much with those in the next layer. This small differ-

ence in the chemical bonding results in vastly different

properties; compare the prices of the two versions!

This we knew for centuries. It was therefore a

great astonishment when a third form of carbon—C 60 —

was discovered in 1996. Called the Buckyball in honor

of Buckminster Fuller (1895–1983), its geometry is ex-

actly as drawn by Leonardo da Vinci (1452–1519), with

12 pentagons and 20 hexagons. The actual arrange-

ment is the same as the sections of a soccer ball! Two

other modifications can also be prepared by increas-

ing or decreasing the number of hexagons. Surpris-

ingly the most perfect of the three is the most likely to

be formed.

With these new carbon molecules, nanotubes can

be constructed, providing fibers hunderds of times

stronger than steel. Applications will transform our

world! We could even build an orbital elevator....

Lesson 5: Sacred Geometry

“Let proportion be found not only in numbers

and measures but also in sounds, weights,

times, positions, and whatever force there is.”

—Leonardo da Vinci

In Nature, we find geometrical patterns, designs, and

structures from the tiniest particles to the greater Cos-

mos. These are also symbolic of the underlying meta-

physical relationship of the part to the whole—“As

below, so above; as within, so without.” It is this prin-

ciple of underlying oneness that permeates the geo-

metrical architecture of all form in its myriad diversity.

These principles of interconnectedness, inseparabil-

ity, and union provide us with a blueprint for the sa-

cred foundation of all things and a continuous re-

minder of our own relationship to the whole Universe.

Life itself is inextricably interwoven with geo-

metric forms, from the angles of atomic bonds in

the molecules of the amino acids, to the helical

spirals of DNA, to the spherical prototype of the

cell, to the first few cells of an organism which

assume vesical, tetrahedral, and star (double)

tetrahedral forms prior to the diversification of

tissues for different physiological functions. Our

human bodies on this planet all developed with a

common geometric progression from one to two

to four to eight primal cells and beyond.

Almost everywhere we look, the mineral in-

telligence embodied within crystalline structures

follows a geometry unfaltering in its exactitude.

The lattice patterns of crystals all express the

principles of mathematical perfection and rep-

etition of a fundamental essence, each with a

characteristic spectrum of resonances defined by

the angles, lengths and relational orientations of

its atomic components.

—Bruce Rawles, Sacred Geometry

The Fibonacci Sequence

Discovered in the year 1202 by Italian Leonardo Pisano

Fibonacci (1170–1250), this is a very important series

of numbers in which each one is the sum of the two

previous ones: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.... The

Fibonacci numbers go on like this infinitely. Any two

consecutive numbers in this series, expressed as a ra-

tio or fraction, define practically all ratios and rela-

tionships found in Nature—that is, 1:1, 1:2, 2:3, 3:5,

5:8... or 1, 1/2, 2/3, 3/5, 5/8....

If you know how to look, you can find the Fi-

bonacci sequence in pinecones and poems, sunflow-

ers and symphonies, ancient art and modern comput-

ers, family trees, and the stock market. This sequence

is the mathematical Key of the Universe!

Fibonacci ratios appear in the ratio of the number

of spiral arms in daisies, in the chronology of rabbit

populations, in the sequence of leaf patterns as they

twist around a branch, and a myriad of places in na-

ture where self-generating patterns are in effect. The

sequence is the rational progression towards the irra-

tional number embodied in the quintessential Golden

Ratio, or Golden Mean. This most aesthetically pleas-

ing proportion, called phi, has been utilized by numer-

ous artists since (and probably before!) the construc-

tion of the Great Pyramid.

H

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H—C—H

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H

Carbon 70 Buckyball Nanotube

6. Spectrum 2.p65 273 1/15/2004, 9:31 AM