The Pentagram

  • I suggest that 1)the pentagram (located just north of the WH) is the prime feature in the DC planning map in that it produces both the 52 degree triangle and the 23+ degeree grid lines, 2)the lines that make up the pentagram comprise the correct section of the map, and 3) 'idealizing' the rest of the map entails bringing it into line with the pentagram.

    Conceptually, the DC map can be seen as a two-layered drawing, with the first layer consisting of an orthogonal grid of square N-S, E-W streets. On top of this grid of regular streets is a layer of diagonal streets, most named for the first states in the Union. The pentagram serves as the core of the diagonal grid and provides the scale for the map plan in general.

  • The distance between the wide points of the pentagram (Wash Cr and Mt Vernon Sq) is the same as the E-W distance between the WH and the CB. Because the map is symmetrical around 16th St we can establish a N-S axis in the west the same distance from the WH as the CB is. [You can use a circle centered on 16th St to do that.]

  • You can use a circle to prove that the point where Penn Ave meets the E Branch is the same distance east of the CB that the WH is west of it. The width of the pentagram sets the east-west scale in the map (as seen by the regular spacing of the N-S lines above).

  • You can also use a circle to prove that Scott Cr is halfway between the WH and the peak of the triangle. The re-located CB is the same distance south of the WH as Scott Cr is north of it, and a re-aligned Jeff Mml (below) falls the same distance south of the WH as the tip of the triangle is north of it. The distance between the WH and SC sets the N-S scale in the map, as witnessed by the E-W lines below.

    The Rhombus Grid

  • As you can see in the image above, the four points that we have just named (Wash Cr, Scott Cr, Mt Vernon Sq and the WH) represent the four corners of a horizontal rhombus. It is actually this rhombus (within the pentagram figure) that defines the scale (both N-S and E-W) in the map. This is the central rhombus (bounded by Penn, NY, Mass and RI Ave's) of the rhombus grid spoken of on the previous page.

    Interesting visual things happen when we project the sides of the rhombus within the context of the (black) N-S E-W grid lines. NY Ave (in red) crosses where N St (through Scott Cr) and N Capitol St meet (both in black). (A straightened) Penn Ave provides the opposite diagonal from corner to corner of a rectangle reaching from Georgetown to the East Branch. Rhode Island parallels NY Ave, and also crosses where black axiis intersect. (A straightened) Mass Ave parallels PA.

    The rhombus within the pentagram determines both the rectangular and diagonal grids in the map.

  • The 'movement' of the red grid lines is two over and one down. With the blue triangle it is three up and two over. All three of these grids (red, blue and black) coincide with one another at the CB and the point opposite that in Arlington, marking this as one potential base of the triangle.

    Fractals

  • Note how the triangle on top of the diagonal grid produces an expanding pentagonal fractal, with each new pentagram featuring a larger rhombus.

    A pentagonal fractal

  • You may recognize this as Metatron's Cube which consists of sets of triangles, hexagons and hexagrams nested inside of one another (a hexagonal fractal). The lower triangle runs from the Jeff Mml to where Penn Ave crosses N St in the west and where NY crosses N St in the east (2 over and 3 up), with these two points being where all three axiis cross like at the CB.

    Metatron's Cube - a hexagonal fractal

    The three axiis in the hexagon match the angles of the two red diagonals and the vertical black lines. [Remember that the triangle in the map has 52 degree base angles, not the equilateral angles that make up the cube, so that the cube image depicted in the map appears short and wide.]

    The Tree of Life

  • You may also recognize this as the Tree of Life. As you can see, the difference between the tree and cube figures is that the tree uses the top two points of the pentagram but not the four corners of the rectangle, and the spheres of the tree fit inside the upright rhombus created by the overlapping isoceles triangles. [The right half of the figure below is the first published image of the Tree of Life (TOL).]

    The Tree of Life

  • Note how the central rhombus (mentioned above) results from the overlap of a pentagon and a hexagon in the TOL, and how projecting the top of the hexagon onto the triangle locates the top two points of the pentagram (which are not used in Metatron's Cube). [Remember that the pentgram and hexagram (symbols of the microcosm and the macrocosm), like the triangle in the map, are short and wide.]

    The Canon

    In 1897 William Stirling published a book entitled "Canon" about the ancient canon of the building arts, the rules and regulations for building temples and cathedrals. In the book there are copies of the floor and elevation plans of the Milan Cathedral that appeared in the 1622 edition of Cesarino's translation of Vitruvius (a 1st century Roman architect).

    According to Sterling, (1) the plan of the Milan Cathedral (seen below) is a most important illustration of the principles of architecture common to all ancient peoples, and (2) the geometrical figure forming the basis of the plan is the rhombus or vesica, invisibly introduced as the mystical complement of the cross, as emblems of the double soul of the Universe. [Other symbols that depict this 'double soul' notion are the yin/yang symbol, the Rebis and the double headed eagle.]

    Stirling is pointing out that the cathedral designers used a rhombus and cross to determine the regulating grids in the plan, with the cross supplying the orthogonal grid of crossing horizontal and vertical lines, and the sides of the rhombus determining the diagonal angles. Symbolically the cross represented the masculine half of the universe and the rhombus represented the feminine half. [The cross in this context is the diagonals of the rhombus.]

    Above we see the rhombus as a product of two equilateral triangles placed base to base (rather than overlapping like we see in the hexagram). Below we see a depiction of 13th Century architect Hugh Libergier holding a model in his hand that features rhombus on the roof.

    Stirling suggests that the 'secret' of the Milan Cathedral plan was that a rhombus and cross were used to determine the dimensions and placement of the elements of it's design, and that the geometrical principles of design found in the Gothic cathedrals formed the secrets of Masonry. While he and others like to depict the use of this as a lost art, I suggest that the geometrical principles of design found in the Gothic cathedrals were used for the design of Washington DC.

  • Below we can see that the plan of the DC map is analogous to the plan for the Milan Cathedral with the only difference being that in the map the template used for the cathedral was reduced in height in order to match the pyramid cross-section image. [Note that the pentagon implicit in the template is not exemplified in the cathedral elevation as it is in the map.]

    By the Triangle

  • The use of the 60 degree triangle and 30 degree diagonals as regulating lines in architectural design is referred to as 'ad triangulum' (by the triangle). Technically you could call it by the rhombus, since the long and short diagonals of a rhombus determine the dimensions and angles of the design, as in the map. Or you could call it by the hexagon, since the angles also derive from a hexagon inscribed in a circle, which derives from the vesica piscis.

    The image above begins with a hexagon in a circle, the diagonals of which give us an equilateral triangle and the upright rhombus grid. Inserting the Pythagorean 'Y' in the hexagon produces 3 rhombus and the 3D image of a cube. The sides of the hexagon and rhombus are either at 90 or 30 degrees, while the base of the triangle is 60 degrees.

    Projecting the top sides of the hexagon to the sides of the big triangle generates a pentagon overlapping the hexagon, although the pentagon is not emphasized in the cathedral plan. The internal angles of the pentagram are the same as those in the hexagon, 30 degrees, in the cathedral plan.

    To produce the DC plan, the equilateral triangle was reduced in height until it's base angles were ~52 degrees, like those of the GP and reducing the angles of the diagonals from 30 to 23+ degrees.

    In the image above, the left-hand side shows red lines bisecting the angles and sides of an equilateral triangle, showing that the short side of a 30-60-90 triangle is half as long as it's hypotenuse, and producing an over-head view of a tetrahedron. These diagonals cross at the center of the triangle.

    In the right-hand side of the image the base angles have been reduced to ~52 degrees, so that the diagonal angles are now 23+ degreees to the horizontal, which is, of course, the image at the heart of the DC map. Designating the right-hand corner the CB puts the WH at the center, making the diagonals Penn and NY Aves (if they are straightened). If we imagine the bottom of the triangle to be K Street between Wash Cr and Mt Vernon Sq, and the centerline to be 16th St, then the center is Scott Circle (seen below) and the diagonals are Mass and RI Aves.

    Scott Cr illustrates what happens to a Star of David when it is reduced in height and 'pinched'.

    Ironically, the Great Pyramid, with 52 degree base angles is located just below 30 degrees north latitude, the point where a hexagon or hexagram (star of david) inscribed within a circle touches that circle; and if an equilateral triangle is reduced to 52 degrees, the internal diagonals form 23 degree angles like the tropics of the earth.

        

    You will note that the Stonehenge lies at 51+ degrees north latitude. In the image on the left above, you can see that projecting a 'regular' hexagram onto the image does not match the Station Stones to the points of the hexagram. In the second image the shortened hexagon from the DC map is overlayed on the (4) Station Stones in the Stonehenge plan.


    Before going on, I would ask that you now take a look at this web page and see how many of my ideas you see being presented there. Note that he does not mention the bend in Pennsylvania Ave and incorrectly places the Capitol Building on the line connecting the right points of the pentagram, claiming that "the Capitol Building is right on the spot where the other 52-degree line ends at the base", of the triangle that is.

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