Geometry of Genesis I

The story of Genesis chapter I can be read as a geometric meditation in the following way.  First get out your compass and prepare to draw some circles.

We follow Aryeh Kaplan's lead on reading sacred texts in the imperative so as to treat them as meditative verses, which he used to give an interesting translation of Sepher Yetzirah.  All verbs are given in present tense, since we want to understand the creation story as if it were taking place in front of us with our geometric tools.


The word simply translated "God" is actually "Elohim" in the original Hebrew text of Genesis.  But the word "Elohim" has the "-im" suffix, making it plural.  Esoteric traditions understand the "Elohim" to be a rank of spiritual beings who inhabit the Sun and hold the knowledge of the geometric laws of creation underlying the forms of the physical world.  When reading the following rendering of Genesis I, please remember that this is to be understood.


To make this into a meditation, take your compass and a blank sheet of paper, and draw a circle to represent the Divine Monad as the center of the process.  Then for each of the 6 Days of active creation  you will draw another circle as shown below.  Recite the text as you work through the construction process.

Begin Meditation:

Before the Beginning (Resheit), the ineffable One existed without a second.

Berasheit

Day One

In the beginning the Elohim created the heaven and the earth.

And the earth was without form, and void; and darkness was upon the face of the abyss. And the Spirits of the Elohim are vibrating upon the face of the waters.

And the Elohim are saying, Let there be light: and now there is light.

And the Elohim are seeing the light, that it is good: and the Elohim are dividing the light from the darkness.

And the Elohim are calling the light Day, and the darkness they are calling Night. And the evening and the morning are Day One.

Day Two

And the Elohim are saying, Let there be a firmament in the midst of the waters, and let it divide the waters from the waters.

And the Elohim are making the firmament, and dividing the waters which are under the firmament from the waters which are above the firmament: and it is so.

And the Elohim are calling the firmament Heaven. And the evening and the morning are Day Two.

Day Three

And the Elohim are saying, Let the waters under the heaven be gathered together unto one place, and let the dry land appear: and it is so.

And the Elohim are calling the dry land Earth; and the gathering together of the waters they are calling Seas: and the Elohim are seeing that is good.

And the Elohim are saying, Let the earth bring forth grass, the herb yielding seed, and the fruit tree yielding fruit after his kind, whose seed is in itself, upon the earth: and it is so.

And the earth is bringing forth grass, and herb yielding seed after its kind, and the tree yielding fruit, whose seed is in itself, after its kind: and the Elohim are seeing that it is good.

And the evening and the morning are Day Three.

Day Four

And the Elohim are saying, Let there be lights in the firmament of the heaven to divide the day from the night; and let them be for signs, and for seasons, and for days, and years:

And let them be for lights in the firmament of the heaven to give light upon the earth: and it is so.

And the Elohim are making two great lights; the greater light to rule the day, and the lesser light to rule the night: they are making the stars also.

And the Elohim are setting them in the firmament of the heaven to give light upon the earth,

And to rule over the day and over the night, and to divide the light from the darkness: and the Elohim are seeing that it is good.

And the evening and the morning are Day Four.

Day Five

And the Elohim are saying, Let the waters bring forth abundantly the moving creature that hath life, and fowl may fly above the earth in the open firmament of heaven.

And the Elohim are creating great whales, and every living creature that moves, which the waters brought forth abundantly, after their kind, and every winged fowl after his kind: and the Elohim are seeing that it is good.

And the Elohim are blessing them, saying, Be fruitful, and multiply, and fill the waters in the seas, and let fowl multiply in the earth.

And the evening and the morning are Day Five.

Day Six

And the Elohim are saying, Let the earth bring forth the living creature after their kind, cattle, and creeping thing, and beast of the earth after their kind: and it is so.

And the Elohim are making the beast of the earth after their kind, and cattle after their kind, and everything that creeps upon the earth after their kind: and the Elohim are seeing that it is good.

And the Elohim are saying, We are making humanity in our imagination, as likeness of us: and letting them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creeps upon the earth.

So the Elohim are creating humanity in their imagination, in the imagination of the Elohim they are creating humanity; male and female they are creating them.

And the Elohim are blessing them, and the Elohim are saying unto them, Be fruitful, and multiply, and replenish the earth, and subdue it: and have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moves upon the earth.

And the Elohim are saying, Behold, I am giving you every herb bearing seed, which is upon the face of all the earth, and every tree, in which is the fruit of a tree yielding seed; to you it shall be for food.

And to every beast of the earth, and to every fowl of the air, and to everything that creeps upon the earth, wherein there is life, we have given every green herb for food: and it is so.

And the Elohim are seeing everything that they are making, and, behold, it is very good. And the evening and the morning are Day Six.

Day Seven

Thus the heavens and the earth are becoming finished, and all the host of them.

And on Day Seven, the Elohim are finishing their work which they are making; and they are resting on Day Seven from all their work which they are making.

End Meditation

There is no circle drawn on the Seventh Day, as this is the Shabbat, the Day of Rest.  

Notice how the emanation of the six circles around the central circle proceeds in diametric pairings.  This is similar to the description of the creation of the Cube of Space as described in the Sepher Yetzirah.  It tells of an extension of a central point into the six directions of space, and their order is Above, Below, East, West, North, South.  These are the first six Doubles in the Hebrew alphabet.  Finally there is Tav, described as the "Palace of Holiness in the middle".  Tav is attribued to the planet Saturn, which is a fitting place for the Seventh Day.

More on this in future posts.

Introduction to the Pythagorean Mathematical Philosophy & the Quadrivium

The philosophical tradition surrounding the name of Pythagoras derives its mistique from how little we know of its origins, while at the same time can claim the mathematical development and scientific advancements as verification of its basis in wisdom and truth.

The Pythagoreans divided their teaching into a four-fold system called the Quadrivium, consisting of Arithmetic/Number, Harmonics/Music, Geometry, Cosmology/Astronomy. Nicomachus of Gerasa, well known as being the greatest Neo-Pythagorean of his time, can help us understand the meaning of this division of their teaching:

"Things, then, both those properly so called and those that simply have the name, are some of them unified and continuous, for example, an animal, the universe, a tree, and the like, which are properly and peculiarly called "magnitudes"; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called "multitudes", a flock, for instance, a people, a heap, a chorus, and the like.

"Wisdom, then, must be considered to be knowledge of these two forms.  Since, however, all multitude and magnitude are by their own nature of necessity infinite-- for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end, but proceeds therefore to infinity-- and since sciences are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing either with magnitude, per se, or with multitude, per se, could never be formulated, for each of them is limitless in itself, multitude in the direction of the more, and magnitude in the direction of the less.  A science, however, would arise to deal with something separated from each of them, with quantity, set of from multitude, and size, set off from magnitude.

Again, to start afresh, since of quantity one kind is viewed by itself, having no relation to anything else, as "even", "odd", "perfect", and the like, and the other is relative to something else and is conceived of together with its relationship to another thing, like "double", "greater", "smaller", [etc], it is clear that two scientific methods will lay hold of and deal with the whole investigation of quantity; arithmetic, absolute quantity, and music, relative quantity.

"And once more, inasmuch as part of "size" is in a state of rest and stability, and another part in motion and revolution, two other sciences in the same way will accurately treat of "size", geometry the part that abides and is at rest, astronomy that which moves and revolves."    

 -- Nicomachus of Gerasa: Introduction to Arithmetic I - ch. 2-3

Another clear description of the way the Pythagoreans divided learning into the Quadrivium is given by the late Neo-Platonist Proclus:

"The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold.  A quantity can be considered in regard to its character by itself or in its relation to another quantity; magnitudes as either stationary or in motion.  Arithmetic, then, studies quantity as such; music the relations between quantities; geometry [studies] magnitude at rest, spherics [studies] magnitude inherently moving.  The Pythagoreans consider quantity and magnitude not in their generality, however, but only as finite in each case.  For they say that the sciences study the finite in abstraction from infinite quantities and magnitudes, since it is impossible to comprehend infinity in either of them.  Since this assertion is made by men who have reached the summit of wisdom, it is not for us to demand that we be taught about quantity in sense objects or magnitude that appears in bodies.  To examine these matters is, I think, the province of the science of nature, not that of mathematics itself."   

 -- Proclus: A Commentary on the First Book of Euclid's Elements - Prologue I ch. 7

 The study of arithmetic was given before anything else, so fundamental was the doctrine of Number for their philosophical school.  Nicomachus can help us see why arithmetic must be studied first:

"Which then of these four methods must we first learn?  Evidently, the one which naturally exists before them all, is superior and takes the place of origin and root and, as it were, of mother to the others.  And this is arithmetic, not solely because we said that it existed before all the others in the mind of the creating God like some universal and exemplary plan, relying upon which as a design and archetypal example the creator of the universe sets in order to their proper ends; but also because it is naturally prior in birth, inasmuch as it abolishes other sciences with itself, but is not abolished together with them.

"So it is with the foregoing sciences; if geometry exists, arithmetic must also needs be implied, for it is with the help of this latter that we can speak of triangle, quadrilateral, octahedron, icosahedron, double, eightfold, or one and one-half times, or anything else of the sort which is used as a term by geometry, and such things cannot be conceived of without the numbers that are implied with each one.  For how can "triple" exist, or be spoken of, unless, the number 3 exists beforehand, or "eightfold", without 8?  But on the contrary 3, 4, and the rest might be without the figures existing to which they give names.

"Hence arithmetic abolishes geometry along with itself, but is not abolished by it, and while it is implied by geometry, it does not itself imply geometry.

"And once more is this true in the case of music; not only because the absolute is prior to the relative, as "great" to "greater" and "rich" to "richer" and "man" to "father", but also because the musical harmonies, diatessaron, diapente, and diapason, are named for numbers; similiarly all of their harmonic ratios are arithmetical ones, for the diatessaron is the ratio 4:3, the diapente that of 3:2, and the diapason the double ratio; and the most perfect, the didiapason, is the quadruple ratio.

"More evidently still astronomy attains through arithmetic the investigations that pertain to it, not alone because it is later than geometry in origin-- for motion naturally comes after rest-- nor because the motions of the stars have a perfectly melodious harmony, but also because risings, settings, progressions, retrogressions, increases, and all sorts of phases are governed by numerical cycles and quantites.

"So then we have rightly undertaken first the systematic treatment of this, as the science naturally prior, more honorable, and more venerable, and as it were, mother and nurse of the rest."    

 -- Nicomachaus of Gerasa: Introduction to Arithmetic I - ch. 4-5

It is recommended that a deeper understanding of the philosophy of the Quadrivium is gained by meditation on these quotations from the mathematician-philosophers.

We shall be exploring the Quadrivium and related areas in future posts.

Sacred Geometry

I have been learning some very interesting things about Sacred Geometry lately.  Sacred Geometry is a mystical mathematical science which studies Geometry through meditation and contemplation of mandalas.  Geometric shapes are considered to be archetypes in the mind of God. When we contemplate these shapes we bring our mind closer to God.

It seems that one of the most important introductory concepts in Sacred Geometry is the progression from the Vesica Piscis to Metatron's Cube.  I thought it would be a good idea to follow this idea through for my first post.

We begin with the Vesica Piscis.  The Vesica Piscis is obtained from the intersection of two circles of the same radius, such that the center of one circle is on the circumference of the other.

The Vesica Piscis is the basis for the construction of just about every well-known Sacred Geometry archetype. Try to find how many Vesica Piscis shapes appear in the following diagrams.

We begin with the Germ of Life:

This becomes the Seed of Life:

One of the most important archetypes is the Flower of Life:

This becomes the basis for the Qabalistic Tree of Life:

This all culminates in the Fruit of Life:

The Fruit of Life is the basis for Metatron's Cube:

Meditation on Metatron's Cube will help unlock your clairvoyant capacities.  It shows how 3-dimensional forms emerge out of 2-dimensional forms.  This prepares our consciousness for experiencing fourth-dimensional awareness.

Metatron's Cube is a container for the five Platonic solids.  Each of the Platonic solids has a correspondence with the Elements (Tattwas).

Tetrahedron (Fire):

Hexahedron/Cube (Earth):

Octahedron (Air):

Icosahedron (Water):

Dodecahedron (Spirit):

Here is a template for meditation that can be done with Metatron's Cube:

Invoke the archangel Metatron.  Stare at the first picture of Metatron's Cube for 5 minutes (more if you want to). Close your eyes and imagine yourself to be the flat image of one of the Platonic solids embedded in the image, as shown above.  Feel the shape as encompassing your entire being. Then imagine yourself expanding outward into the three dimensional solid, i.e. imagine that you are a Cube, etc.  Feel the elemental quality associated with the solid. Only do this for one of the solids at a time.  Close the meditation by imagining your Higher Self as a four-dimensional being emerging from your three-dimensional physical body.

These are just some of the most introductory things that can be done in Sacred Geometry.  I am currently investigating some other areas of Sacred Geometry, especially how it relates to meditation and magic.