STEP 1. The artist starts with a convenient square ABCD and marks the midpoints of each side. Then he draws the smaller square EFGH: Note that the artist will do this geometry work on a preliminary drawing sheet, often bigger than the final canvas. Vermeer often uses only part of the total geometry – i.e. the complete Grail Geometry often extends beyond the confines of Vermeer’s canvases. This is governed by the artist’s compositional ideas.

This diagram comes from an ancient proof of Plato's Theorem:

STEP 2. A circle is inscribed within the smaller square so as to be tangent to all sides.

This diagram is the Templar Variation of Plato's Theorem:

The TILTED TRIANGLE:

STEP 3. The all-important equilateral triangle (all sides equal, all angles equaling 60 degrees) A—V1—V2 is drawn with all sides tangent to the circle. This is the so-called “Tilted Triangle” that one searches for among the painted features in a composition. The letters J and M identify the fact that this triangle is “tilted” downwards 15 degrees from the horizontal line AB. The point A is often called “The Northwest Point” when the Grail Geometry, hidden a painting, is transferred to the landscape at the proper scale and in registration with certain predetermined permanent landmarks on the map

The TILTED SQUARE:

STEP 4. Using the top side of the triangle A—V1--V2, the "Tilted Square" AMNO is drawn. This square is important, because in some cases, the artist includes a symbolic reference to a crypt, or burial chamber. The location of a symbolic burial site is indicated at the intersection of the diagonals of the tilted square. Note how diagonals AN and MO intersect at the point designated as PX – or “X marks the Spot”. This is the basic Grail Geometry that emerged in the 13th century in The Templar Map of Jerusalem (now in the Royal Library, The Hague)

The HEXAGRAM:

Step 5. To the original tilted triangle A—V1—V2 is added a second equilateral triangle VH1—VH2—VH3. This makes a regular hexagram – an important feature of the Grail Geometry. This geometric pattern remained secret since the 13th century until it was divulged in the 20th.

Note that at “V1”, the lower vertex of the first tilted triangle, the angles 75, 60, and 45 degrees are shown. These angles, made with the horizontal base line of the pattern, are the sine qua non of the Grail Geometry. If these angles are discovered (as shown) in a painting, it is a sure sign that the artist knew and employed this secret geometry.

Vermeer was devoted to using the hexagram as a compositional guide. In some paintings he used two and even three hexagrams to achieve the effect he wished.

The GRID:

Step 6. A grid has been superimposed on the geometric pattern to show that J falls at a point such that the line segment B-J is equal to line segments A-E and E-B. This fact is very useful when trying to discover how the artist laid out his work – if, indeed, he employed the Grail Geometry, as did Vermeer in at least nine of his paintings.

In the painting “Lady Standing at the Virginals”, Vermeer employed the full 16 square grid and two hexagrams to achieve a harmonious composition. His work has been called “sphinx--like”, because its harmony derives from the riddle of a hidden geometric skeleton.

The question as to why Vermeer employed this complicated pattern is often asked. The short answer is that many artists believe that basing their compositions on geometric figures and patterns lends a harmony and structure to their work that is at once pleasing, but at the same time strangely captivating.

This writer is of the opinion that, since Vermeer knew and used a then secret geometric formula -- secret in the opinion of sources believed as reliable by this writer -- he must have been taught it in secret. Therefore, Vermeer must have been at least apprenticed to -- and possibly might even have been a member of -- the secret group or society that preserved this iconic pattern.    Many such groups existed in the 17th century.  Two important secret societies were “The Priory of Sion -- Prieure de Sion” and "The ILLUMINATI -- Los ALUMBRADOS".  20th and 21st century non-fiction and fictional writings ( for example, the best selling novels -- in 2003 -- "The Da Vinci Code"; and -- in 2000 -- "Angels and Demons" ) refer to the above-mentioned secret societies as historical fact. A search of the World Wide Web will yield some fascinating information about these and many other secret societies, in addition to their connection to The Knights Templar and the Freemasons among other well-known ancient and contemporary organizations.

Admittedly, there is a tinge of the sinister in all of this. Consider that Vermeer's wife is said to have lamented after his untimely death at age forty-three -- (1632-1675) -- only 43 years old at death!   (Paraphrasing): "One day he was walking around healthy and happy -- the next day -- dead!"   Make of this what you will . . . For my own part, I suspect foul play.  Vermeer's haunting images presented a clear and present danger to many of his contemporaries in those perilous times.

From the book   "VERMEER'S RIDDLE REVEALED"   by  Robert A diCurcio,  2001      ISBN  0917358139

Note -- Many other artists, some before Vermeer (1632-1675) and some after him, employed exactly the same Grail Geometry (GG) in some of their paintings. At this writing I have personally (and uniquely -- as far as I am aware) discovered the GG in "St. Peter" by El Greco (1541-1614); "Las Meninas" by Velazquez (1599-1660); "Bullfight" by Goya (1746-1828); "Virgin of the Rocks" (Paris 1483-86 and London 1503-06 versions) and "The Mona Lisa" by Leonardo da Vinci (1452-1519); "Sposalizio" by Raphael (1483-1520) . . .

These, and my latest Vermeer analyses, may be found at the "Spider Web" button on the left of these website pages. What other conclusion can possibly be drawn, but that this GG -- then secret and forbidden by "the powers that existed then" -- was passed down from master to apprentice from its inception with the Templar Map of Jerusalem of the 13th Century (The Royal Library, The Hague)?

Several original investigators, other than myself, have identified this type of tilted hexagram, tilted square geometry in the work of many other artists: e.g. Rene I, Duke of Anjou -- King of Naples, Sicily and Jerusalem [!] (1409-1480, "La Fontaine de Fortune"); Sir Anthony Van Dyck (1599-1641, "Lord George Stuart"); Nicolas Poussin (1594-1665, "Et In Arcadia Ego I & II"); David Teniers The Younger (1610-1690, "St. Antony and St. Paul") -- and in other works, some recently discovered -- some yet to be discovered.  RAdiC 12/15/2003.

The Grail Geometry is a hexagonal geometry -- involving the hexagram (6--pointed star -- "Seal of Solomon" -- "Star of David") and the equilateral triangle and the multiples and divisors of the associated hexagram angle of sixty (60) degrees; -- as opposed to the pentagonal geometry -- involving the pentagram (the 5--pointed star of, for example, "Vitruvian Man") and it involves the "divine proportion" (other names include "golden section" and "golden ratio" and the Greek letter PHI for the ratio 1.618 to 1) and the multiples and the divisors of the associated pentagram angle of seventy-two (72) degrees. The paintings I have investigated are composed according to the hexagram (two equilateral triangles superimposed), and I find no evidence of the use of pentagonal geometry or the "golden section" in those paintings.

 But, of course, some artists have used the PHI ratio and its proportions in their compositions.  N. Poussin, (French, 1594-1665) may have used it in his famous painting "Et In Arcadia Ego, II" -- in addition to his obvious emphasis of the Grail Geometry.  I will recommend to the interested viewer Chapter 7 of the recent (2002) book "The Golden Ratio" by Mario Livio.  Livio seems to know nothing of the Grail Geometry -- but his seventh chapter does make the argument that serious artists have often attempted to perfect their compositions by basing them on geometric proportions, and a few have used the "golden ratio".  But he is quite blunt in his debunking the contention that the use of the "golden ratio" is as widespread in art and esthetics as commonly imagined, however.  RAdiC  01/21/2004.

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