Charles Gilchrist Mandalal: Gallery #11
The Golden Mean Rectangle and The Phi Ratio

LDV Phi Ratio Geometry: by Charles Gilchrist
A beautiful print on fine watercolor paper is now available of this popular Gilchrist Mandala based on Leonardo Da Vinci's famous masterpiece. Charles personally creates these prints. The paper size is 13 inches by apox. 19 inches. The price is #39.95, plus $14.00 insurance, shipping, and handling anywhere in the continental USA.
The low wholesale price is:
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One print - only $39.95 - plus $14.00 I/S/H
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This print is shipped flat
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The Golden Mean Rectangle and The Phi Ratio
An Interview With The Artist - Number 11
by Leslie Page
Sacred Sequence: Number 11
L.P. "Tape running, here we are again, Charles, continuing our ongoing conversation on Sacred Geometry and all the unbelievable connections it has to our everyday lives. I'm amazed, Charles, I truly am. This is our eleventh tape and I promised you our subject would be your choice this time. What are we going to talk about today?"
C.G. "Thank you, Leslie. Generally, when I take on a really bright student, like yourself, I begin by letting them ask the questions that first come to their minds. It seems to work pretty well. And so far we have been doing just that, but this is, as you just mentioned, our eleventh conversation and it feels pretty special to me."
L.P. "Really? In what way?"
C.G. "It's the number, Leslie. At least that's part of it."
L.P. "Of course..... numerology, or Sacred Sequence, as you call it. One of our first conversations was about esoteric numerology, remember?"
C.G. "Yes, very vividly. You have a natural affinity with mathematics, Leslie, in all its forms, which gives you an edge in this study. 11 is a very special number; the first master number."
L.P. "I remember. The master numbers are 11, 22, 33, 44, etc. Are we going to talk about numerology again?
C.G. "Not really, but numbers are part of a stream of thinking I want to share, and it seems appropriate for this, our eleventh conversation, our first master conversation. I want to discuss the very important "Golden Mean Rectangle' and 'Phi Ratio."
L.P. "I have heard and read those words before, Charles, but I must admit, I don't really know much about them, except they are very mysterious and esoteric."
C.G. "Yes they are, Leslie, but when you finally see the perfect simplicity of this geometry, and its resulting mathematics; when you see just how profound this aspect of 'God Mind' truly is, I believe your human mind will open in an extraordinary way."
L.P. "I can't wait!"
The Phi Ratio and The Golden Mean Rectangle
C.G. "Good! The 'Phi Ratio (1.6180339 and on and on forever) is one of the most esoteric and profound mathematical ratios of the universe, which is naturally found in the sacred geometric archetype known as 'The Golden Mean Rectangle. In other words, the 'Phi Ratio' IS the mathematical equivalent of the geometric Golden Mean Rectangle. I have some graphics to show you (Fig 1)."

Fig. 1
L.P. "That's interesting. I didn't realize the two were connected. So we have this special Phi Ratio, or Golden Mean Rectangle, and if the short side measures one unit tall, that makes the longer side 1.61880339... units wide."
C.G. "Exactly. But the thing that makes these proportions so incredible is this fact: If you create a square within the Golden Mean rectangle, in this case seen in the horizontal position (Fig. 2), the resulting smaller vertical rectangle is also in the same exact Golden Mean proportion. And this transcendental relationship continues to spiral in and out of itself through micro cosmic and macro cosmic infinity (Fig. 3)."
The Golden Mean Rectangle Formula
C.G. "Yes, Leslie, That is the obvious question. Where does it come from? This fantastic cosmic relationship begins with the square (an obvious early archetype of Sacred Geometry) which we bisect vertically. Our square is now divided in half, and that new half way point located on the bottom line becomes the center of a circle we are going to strike (Fig. 4)."
Fig. 4

Creating The Radius
L.P. "All right, I've got it.... very simple. What's next?"
C.G. "With the point of a compass set at the center of the bottom line, adjust it so that the marking part of the compass bisects both opposite corners of the square. Now we strike a circle. Next, extend the bottom line of the square until it crosses one side of our new circle, and we have it. That's it! It's just that simple! We now have two Golden Mean Rectangles (horizontal and vertical) created with the aid of our first square and the new corner created by the intersection of the extended base line and the circle (Fig. 5)."
Fig. 5

The Golden Mean Rectangle Formula
L.P. "I see some simple math formulas at the bottom. Let's see, Unit A divided by Unit B, equals the Phi Ratio (1.6180339). And Unit C divided by Unit A, also equals the Phi Ratio (1.6180339). Wow! You are right, Charles, it is beautifully simple."
C.G. "Don't let the simplicity fool you, Leslie. You could spend lifetimes contemplating this archetype. It is a tremendously potent visual and intellectual mantra which naturally facilitates very deep levels of meditation. It is intrinsically transcendental."
Roots
L.P. "I get the impression this information has been around for a very long time. Who discovered it?"
C.G. "I have no idea, Leslie, and I don't believe anyone knows that for sure. The Golden Mean Rectangle and the related Phi Ratio have been perceived in numerous cultures dating back to the roots of recorded history, especially in the Middle East and the Far East. But the information was a guarded part of esoteric wisdom and was not shared with anyone but the most privileged initiates of very obscure and secret societies."
L.P. "Charles, I've always believed those guys were unbelievably paranoid, but I suppose they had good reason to be afraid. After all, most people believed the earth was a large flat rock resting at the center of the universe, with all the stars and heavenly bodies revolving around us, and if you had the cheek to suggest the earth was a sphere which rotated around the Sun, you were very likely to find yourself being burned at the stake for your demonic blasphemy."
C.G. "Yes, that's true, Leslie. Just imagine their dilemma. The ancient sages far surpassed the intellectual understanding of their day and had to work in the utmost secrecy. But despite the obvious difficulties, they managed to make, without the use of any particularly sophisticated technology, staggering metaphysical and scientifically valid breakthroughs. These masters were intimately connected to the universe through direct and very focused observation. These brilliant mystical minds perceived the Golden Mean relationship and realized it was a repeating architectural formula of God the creator. They also observed this Sacred Geometry manifesting in countless natural forms."
Natural Forms and Sacred Geometry
L.P. "All right, good. What about these natural forms and the Phi Ratio? I love facts. Give me facts."
C.G. "You could write a very thick book on that subject, Leslie. There are countless examples. Let me see if I can come up with something particularly interesting. Of course skeletal structures comes immediately to mind. The skeletal structures of virtually all living creatures follow the golden mean formulas."
L.P. "Really? In what way?"
C.G. "Look at length of the bones in your fingers. They are progressively larger at the Phi Ratio. Your toes are created in the same way. There are numerous examples in the human skeleton, arms and legs, etc."
L.P. "Wow, I never heard that one before. Is that a fact?"
C.G. "Yes. As I am sure you remember, Leslie, and as I have told you many times before: Sacred Geometry is the architecture of the universe. You can observe these proportions everywhere; in fish and shells, birds and beasts of all kinds, many forms of insects, it's in plants and trees; God's Golden Mean is to be found everywhere in nature."
L.P. "Fantastic, Charles. This is so much more important than I suspected. How did this information get so lost? How could we let that happen?"
Suppressed Information
C.G. "This information was never part of ordinary human consciousness, but this kind of esoteric truth was severely suppressed in the Dark Ages, and forced very far underground. The controlling political powers were extremely intimidated by any new and deeper truth, fearing the loss of control. "
L.P. "It's interesting you would mentioned the Dark Ages, because I just read an article which suggested the entire Renaissance began because a Greek Scholar, a monk I believe, translated a large block of lost esoteric information from ancient Middle Eastern manuscripts."
C.G. "Yes, I know that theory. But if the Renaissance actually began because of those translations, or whether that work was only part of a larger flowering of truth, who can say. Nevertheless, the great Italian Renaissance did happen, and mankind began to wake up.
L.P. "Thank God for the Renaissance."
Leonardo Da Vinci
C.G. "Leonardo Da Vinci is the quintessential example of Renaissance Man, and it is obvious he was privy to extremely esoteric information. His exquisite and famous little drawing 'Study Of Human Proportion,' after Vitruvius, came straight from the very arcane wisdom of the Dionysian Architects. And of course, the Fraternity Of Dionysian Architects was one of those mysterious secret societies. Those initiates were most intimately aware of Sacred Geometry and its archetypes, including the Golden Mean reality. Many of the ancient buildings which we still perceive as miraculous human achievements, are based on sacred geometric principles. That includes Egyptian architecture, Classical Greek sculpture and architecture, The fantastic Gothic Cathedrals, and even the wonderfully elegant Japanese Pagoda, to name a few."
L.P. "Charles, Are you suggesting Leonardo was a member of one of those intellectual fraternities?"
C.G. "Most probably, and I believe more than one. A number of the earlier masters were initiated to the inner-circles of several of the mystery schools. Why not Leonardo Da Vinci? And here is an interesting point about Leonardo's little drawing as it relates to our discussion: It has a direct relationship to the Golden Mean Rectangle which can be proven geometrically (Fig. 6)."

Fig. 6: LDV Phi Ratio Geometry... by Charles Gilchrist
L.P. "This is interesting, Charles, one of your pieces on LDV' s beautiful drawing."
C.G. "Yes, I have created over fifty investigative drawings based on this Leonardo Da Vinci master piece. This one illustrates its clear connection to the Golden Mean Rectangle."
L.P. "Wait a minute, Charles. Let me look at this. Ah ha, I see it. This time you have the Golden Mean Rectangle revealed in its vertical position. The square is halved at the man's hips and the radius strikes across his legs. I see it. But how does that prove anything?"
C.G. "It doesn't. But notice Leonardo's small circle and the way it relates to the larger, Golden Mean creating circle. They just barely touch, Leslie. In fact, they are perfectly tangent which is far beyond coincidence, and does prove a direct geometric and mathematical relationship ."
L.P. "How curious! My mind is boggled again. How far and how deep does this Phi Ratio stuff go?"
C.G. "All the way to the source of ALL, Leslie. But let me make just one more point and then we'll let it go for today."
L.P. "Of Course."
Spiraling
C.G. "Sometimes this conceptual two dimensional thinking we are engaged in, does not directly translate into three dimensional space and so Mother Nature must reinterpret, especially as it relates to growing sequences. Mother Nature can not grow 1.6180339 parts of a leaf."
L.P. "I've got you this time, Charles. You're pointing at the Fibonacci Sequence. Does that natural phenomenon also have something to do with the Golden Mean?"
C.G. "Yes, very good, Leslie. The Fibonacci Sequence is Mother Nature's way of interpreting the Golden Mean relationship. When a little bush sprouts, it first grows one leaf. In a little while it grows one more leaf and then two more. And then three, five, eight, thirteen, twenty one, thirty four, and on and on. The Fibonacci Sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, etc. Each new number is a combination of the current and previous number. Numerous flowers use one of these Fibonacci numbers for the number of its pedals, 3, 5, 8, 13, 21, etc. One of my CX-48 Mandalas is on the Fibonacci Sequence (Fig. 7)."

Fig. 7 The Fibonacci Sequence
The Fibonacci Sequence is 1-1-2-3-5-8-13-21-34-55-89-144-233-377- 610-etc.
L.P. "I've already studied this Fibonacci Sequence phenomena, Charles, and I think it's fascinating, but what does that have to do with the Phi Ratio."
The Phi Ratio and The Fibonacci Sequence
C.G. "Get ready for this, Leslie. The mathematical relationship between sequential numbers of the Fibonacci Sequence very quickly comes extremely close to the Golden Mean Phi Ratio (1.6180339) and the larger the numbers. the closer they get. For example:
The Fibonacci Sequence is:
1-1-2-3-5-8-13-21-34-55-89-144-233-377-610-etc.
The Phi Ratio =
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1.6180339......
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Phi Ratio = 1.6180339......
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13 divided by 8 =
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1.6250000......
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Phi Ratio = 1.6180339......
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21 divided by 13 =
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1.615384........
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Phi Ratio = 1.6180339......
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34 divided by 21 =
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1.619048........
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Phi Ratio = 1.6180339......
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55 divided by 34 =
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1.617647........
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Phi Ratio = 1.6180339......
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89 divided by 55 =
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1.618182........
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Phi Ratio = 1.6180339......
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144 divided by 89 =
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1.617978........
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Phi Ratio = 1.6180339......
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233 divided by 144 =
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1.618056........
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Phi Ratio = 1.6180339......
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377 divided by 233 =
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1.618026........
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Phi Ratio = 1.6180339......
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Another of my CX-48 Mandalas makes reference to the intimate relationship between the Golden Mean spiral and the Fibonacci Sequence spiral (Fig. 8)."

Fig. 8
L.P. "I am totally convinced."
C.G. "The Fibonacci Sequence also manifests in beautiful natural spirals of all types, like the pine cone, the nautilus shell, and the face of a sun flower (Fig 9). There are thousands of examples."

Fig. 9
L.P. "Not only am I totally convinced, I am almost overwhelmed."
C.G. "Yes, the incredible significance of the Golden Mean reality is undeniable, which proves my point: Sacred Geometry IS the architecture of the universe. Think about it Leslie. We'll talk again."
A New Painting Plus:
An Interview With The Artist - Number 11 Part 2 - by Leslie Page

1.6180339: an acrylic painting by Gilchrist © 2003
The low wholesale price is:
|
One print - only $39.95 - plus $14.00 I/S/H
|
This print is shipped flat
|
See (Order Form)
|

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