Turn Inside Out
Ring Around the Holy
Now, like Alice before Wonderland, it's time to meet the White Rabbit—time to leave the ordinary world of 4-square geometry and discover the higher order, higher dimension mysteries of sacred geometry and mystical math. Hold tight to your cortex, it's time to take an inter-dimensional flip trip.
has 12 vertexes and 20 faces,
with five 3-sided faces
meeting at each point, or vertex.
It's geometric opposite—
has 12 faces and 20 vertexes,
with three 5-sided faces
meeting at each point, or vertex.
Now, imagine adding a 13th point at the center of the Icosahedron, and attaching it to all the other vertex points with flexible, stretchable rubber bands. With this, you can turn this rigid geometric form inside out. Then, the center becomes the periphery, and edge becomes core. Turning outside and inside out. 12 + 1 = 13 is the geometric key to unlock this topological transformation of spacetime through a higher dimension.
This "imaginary" operation is hard to imagine, since we're stuck making 2-D diagrams in 3-D, 4-square reality, but it creates a familiar and useful universal figure. This hollow ring is the donut of everyday American culture.
To the mathmetician and physicist, this is the "torus"—the geometry of magnetism. In poplular culture, this ring on the third finger is the symbol of commitment and connection in marriage and family.
Thus, 12 + 1 = 13 is a geometric key to unlock 3-D space and open into higher dimension geometries. Turning inside out—in-folding—is a convenient, universal way to store information ("in-form") in fractal packages. This turning in trick of time, space and geometry has many important applications in biology, physiology and psychology.
In water, this universal flowform is the shape of fluid in motion within a container. And since living water is water in motion, and since every living body is a water body, we are each encoded with and attuned to this rolling rhythmic geometry.
In math, this infolded space has negative curvature—space turning and folding and curling in on itself.
In physics, Albert Einstein's Theory of Relativity taught us that spacetime has a negative curvature—folded inward on itself.
In biology, this is the shape of a red blood cell, full of hemoglobin, rich in magnetic iron.
In metaphysics, it's a halo—sign of divine dimension and intention.
Now it's time for Alice to follow the White Rabbit into this extraordinary hole. First, the hole.
The Donut Hole
Vesica & Ovary
Divide a donut in half by slicing plane AB through opposite sides of the ring:
In side view, this cross section appears as two circles, whose radius r is the ring's inner dimension. These two circles are separated by diameter D of the donut's central hole.
Now, reduce the diameter of the hole in the donut ring unto d=0. The inner edge of the two circles just touch, and the hole vanishes. The result is more like a swollen bagel than a donut.
A cross section of this donut-without-a-hole looks like this:
However, the donut can continue to shrink until diameter goes past zero and becomes negative. Few of us can imagine a space with a negative diameter. Yet, this overlap of the inner edge of the donut ring forms a curious and bizarre space where the donut hole once was.
For example, in this diagram, d = —r:
A pointed, oval chamber—a football shape—appears in this negatively curved space as the donut folds in on itself.
In Sacred Geometry, this almond-shaped space is the Vesica a sacred enclosure found in stone circles, catherdrals and gourd rattles. It is also called "God's Eye." Within this place, time and space are altered.
In biology, this geometry is apple. Instead of an empty hole at the center, the apple has five ovaries in its core—an infolded space where DNA condenses and coils into seeds to instruct the next generation.
Plants use variations on the PHI geometry to create ovary, seed and fruit.
The Fourth Dimension
Physical World = PHI-Cycle Whirled
The fourth dimension is time, which arises because all 3-D bodies are in motion, and the most basic movement is spin. The physical world is bits of matter spinning on axes and in orbits. In the biological realm, living matter grows over time in orderly, progressive geometries. Earth's spin creates alternating day and night—a smooth curving circle, broken by an invisible International Date Line north-south through the Pacific Ocean—Earth's wrinkle in time.
Ancient China recognized life's perpetual growing motions manifest as PHI-cycle Nature. After the Unifying Principle of Yin & Yang, Oriental Medicine's second law is The Five Elements." The universal dance of opposing dualities becomes a 5-step cycle.
A Creative cycle (outer pentagon) defines a positive order, while a Control cycle (inner star) depicts oppositional relationships. All natural phenomena and their qualities are classified in this order.
In biology, plants harness PHI-cycle geometry in photosynthesis to transform sunshine into sugar. Chlorophyll invites light into a 5-step water splitting cycle. Four photons of light strip four protons and four electrons from two water molecules, releasing a molecule of oxygen.
Solar fire split's Earth's water to liberate electrons, the common currency of chemistry. A chloroplast is a PHI spiral vortex of enzymes, minerals and water, where beams of light become spinning charge (electrons) racing around 5- and 6-carbon rings (sugar), common currency of biology.
Summation & Growth
The Fibonacci Series is another way to derive PHI, named for the Italian mathmetician in the Renaissance Era who rediscovered and explored this mathematical principle. In this number series, each number is the sum of the preceding two numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
In algebra, this formula is written:
Xn = Xn-1 + Xn-2
Any number in this series diuided by its preceding number is a ratio that gets closer and coser to PHI as the numbers get larger and larger. In the symbolic shorthand of calculus, the mathematics of change:
The mathematics of this 4-D complexity is calculus, which analyzes the rate of change by taking snapshots in time—frozen moments in the flow of events.
The purpose of this Summation Series is made obvious when seen as a geometric diagram:
So, Fibonacci's Summation Series creates a PHI spiral—the universal form used in Nature—fromflowers to cyclones to galaxies—pattern and proportion of living, growing things. Whirling bodies emit particle trails and wave energy tails like the PHI.
In biology, a plant unfolds its leaves in Fibonacci intervals. The PHI spiral is the geometry of growth.
Medieval Italy's Renaissance was born and bloomed amid the rediscovery of PHI.
Indeed, to "repent" is to return to righteous, rhythmic harmony with the living order of the universe.