Washington DC's Isometric Map Grid
Understanding the following will be a lot easier if you are familiar with a few geometric notions like grids, tesselation (or tiling), and stellation. If these are brand new to you, you might want to google around and do some reading about them. Grids A grid is a type of two-dimensional pattern. The term 'grid' usually refers to a series of intersecting axes or to the points of intersection of these axes (called nodes), but actually grids result from tessellating regular polygons. Tessellation refers to filling a plane by repeating a form so that there are no overlaps or gaps. M. C. Escher uses this in his work a lot. The three regular polygons that tessellate are the square, the triangle and the hexagon. Other polygons that tesselate are the rectangle (a modified square) and the rhombus (two equilateral triangles). Stellation refers to extending the sides of a regular polygon until they meet, in this way a pentagon produces a pentagram, and a hexagon produces a hexagram. Triangles and squares do not stellate. Squares A grid of squares, the sides of which are horizontal and vertical, is called an orthogonal grid (left below). The DC street grid is centered on the Capitol Building. North-south streets are numbered and east-west streets are lettered. A square grid rotated 45 degrees produces what is called an oblique grid (right above). [Note that the diagonals of the oblique squares form the orthogonal grid and visa versa.] The District of Columbia is a 10 x 10 mile oblique square within which we find an orthogonal grid of streets. * Good carpenters and builders know that the diagonal of a square produces two isoceles triangles with 45 degree base angles, that the rise and run of a 45 degree angle (roof) are equal (tan 45 equals 1), and that the diagonal of a 12 inch square is 17 inches (16.99). [Note that the 45 degree triangle is the only isoceles triangle with a 'square' (90 degree) corner and the only 'right' triangle with two equal sides.] Triangles A tesselation of triangles forms a whole host of other figures, like the rhombus, the hexagon, and the 3D cube. Here are two versions that are rotated 30 degrees to one another. Notice how in one the long axis of the rhombus is vertical and in the other it is horizontal; in one the bases of the triangles form horizontal axes and in the other they are vertical, and these axes represent the short digonal of the rhombus. In the image on the left below the relative positions of the WH and CB are marked with red circles and New Hampshire Avenue is symbolized by the red line (one side of the big triangle). In the image on the right the relative positions of the WH, CB and Scott Circle are marked as are Penn, NY, Mass and Rh Is Aves. Just like with the square grids, both versions of the triangular grid are represented in the DC map. In the image on the right above, the 'right' axes are vertical and the diagonals are on thirty degrees, like the drafting tool below. Looking at this figure composed of three rhombus, we see a 3D cube. In the image on the left above, the right axes are horizontal and the diagonals are on sixty degrees, forming 'upright' equilateral triangles. In this view we see a hexagon composed of six triangles. 2:1 There are two right triangles with unequal sides that illustrate the 2:1 ratio. One is the 30-60-90 degree triangle (seen above) which is is half of an equilateral one. The sine of 30 degrees is .5 as the side opposite to the angle is half the hypotneuse. [A triangle with a 19.5 degree angle in this corner has a short side of 1 and a hypotenuse of three, the sine of 19.5 is .333; the 14.5 degree triangle is 1:4, for a sine of .25, etc.] The diagonal of half of a square (or a double square) produces 26.5 degree angles. Tan 26.5 = 1/2 or .5. The hypotenuse in this figure is equal to the square root of 5. [You may recognize this as the angle of ascending passage in the Great Pyramid. Also note that DC is at 39 degrees west and 78 degrees north. Two over and one up.] A rectangle results from changing the ratio of the length of the sides of a square, while changing the ratio of the length of the diagonals produces a rhombus. The square grid can be quickly converted to either version of the triangular grid, just as the 26.5 degree triangle can be converted to the 30 degree one by keeping the width the same and adjusting the height. In one case we rotate the grid and insert vertical diagonals producing 45 degree angles. Next we adjust the aspect ratio of the image (increase it's width while keeping the same height) converting the 45 degree triangles to equilateral ones. The short axis of the rhombus produced by these triangles is equal to the sides of that rhombus. A square with 1 inch sides has 2 diagonals equal to the sqrt of 2 (1.41). The image above has been 'adjusted' so that one of the diagonals is now equal to the length of the sides, or 1; making the long diagonal equal to the sqrt of 3 (1.732). The ratio of the length of the short axis to the long one in this rhombus is 15:26 (30:52). The image below uses horizontal diagonals and increases the height producing a rotated version of the above image. The Vesica Piscis Here we see that another way to generate the equilateral triangle, rhombus, hexagon, cube, etc is with the vesica piscis. The side of the blue triangle is equal to the diameter of the circles. The short axis of the rhombus is equal to the radius. A line from a corner to the mid-point of the opposite side bisects that angle. If the radius of the circles (the short axis of the rhombus) is 1 then the long axis is 1.732 (sqrt of 3). We can determine the diameter of the circle that produces any vesica by dividing the long axis by the sqrt of three and doubling it. By doing this we know that a vesica that is 481 feet tall (the height of the GP) is produced by circles 555.5 feet across (the height of the Washington Monument). Below we see the 45, 30 and 26.5 degree angles within the figure of the vesica. The sqrt of 2 is the diagonal of a square with sides of 1 (a unit square), which as we saw above generates 45 degree angles. The sqrt of 5 is the diagonal of a 1x2 rectangle; 26.5 degrees. As was shown above the 30 degree angle results from a diagonal of a rectangle that is 1 by the sqrt of 3 (the dimensions of the vesica/rhombus). While both are 1 wide at the base, the 30 degree triangle is 1.732 tall and the 26.5 triangle is 2 tall. Modified Note that in the DC map, the triangular grid has been modified (the aspect ratio was changed by keeping the width the same and shortening the height) so that the equilateral triangle has become an isoceles triangle with 52 degree base angles like the Great Pyramid cross-section, and what were 30 degree diagonals are now 23.5 degrees like the latitude of the tropics. Interestingly, a hexagram composed of equilateral triangles inscribed within a circle touches that circle at 30 degrees of latitude, very close to the current location of the Great Pyramid. If we shorten that figure until the triangles have 52 degree base angles, we get an image that depicts the tropics (below left). Compare the shortened hexagram to Scott Circle which shows the ellipse, six points of the hexagram and 23 degree angles. On the right we see that the points of the shortened hexagram fit the Station Stones at the Stonehenge. [Note the elliptical figure at the center of that.] Larger Image While you can also use the vesica to generate a 52 degree angle (below), I believe that the ellipses, and the 52 and 23 degree angles in the DC map show that the 'shortened' image above was used as the model for the plan. Continue

Washington DC's Isometric Map Grid

Understanding the following will be a lot easier if you are familiar with a few geometric notions like grids, tesselation (or tiling), and stellation. If these are brand new to you, you might want to google around and do some reading about them.

Grids

A grid is a type of two-dimensional pattern. The term 'grid' usually refers to a series of intersecting axes or to the points of intersection of these axes (called nodes), but actually grids result from tessellating regular polygons.

Tessellation refers to filling a plane by repeating a form so that there are no overlaps or gaps. M. C. Escher uses this in his work a lot. The three regular polygons that tessellate are the square, the triangle and the hexagon. Other polygons that tesselate are the rectangle (a modified square) and the rhombus (two equilateral triangles).

Stellation refers to extending the sides of a regular polygon until they meet, in this way a pentagon produces a pentagram, and a hexagon produces a hexagram. Triangles and squares do not stellate.

Squares

A grid of squares, the sides of which are horizontal and vertical, is called an orthogonal grid (left below). The DC street grid is centered on the Capitol Building. North-south streets are numbered and east-west streets are lettered.



A square grid rotated 45 degrees produces what is called an oblique grid (right above). [Note that the diagonals of the oblique squares form the orthogonal grid and visa versa.] The District of Columbia is a 10 x 10 mile oblique square within which we find an orthogonal grid of streets.

* Good carpenters and builders know that the diagonal of a square produces two isoceles triangles with 45 degree base angles, that the rise and run of a 45 degree angle (roof) are equal (tan 45 equals 1), and that the diagonal of a 12 inch square is 17 inches (16.99). [Note that the 45 degree triangle is the only isoceles triangle with a 'square' (90 degree) corner and the only 'right' triangle with two equal sides.]

Triangles

A tesselation of triangles forms a whole host of other figures, like the rhombus, the hexagon, and the 3D cube. Here are two versions that are rotated 30 degrees to one another. Notice how in one the long axis of the rhombus is vertical and in the other it is horizontal; in one the bases of the triangles form horizontal axes and in the other they are vertical, and these axes represent the short digonal of the rhombus.


In the image on the left below the relative positions of the WH and CB are marked with red circles and New Hampshire Avenue is symbolized by the red line (one side of the big triangle). In the image on the right the relative positions of the WH, CB and Scott Circle are marked as are Penn, NY, Mass and Rh Is Aves. Just like with the square grids, both versions of the triangular grid are represented in the DC map.


In the image on the right above, the 'right' axes are vertical and the diagonals are on thirty degrees, like the drafting tool below. Looking at this figure composed of three rhombus, we see a 3D cube.

In the image on the left above, the right axes are horizontal and the diagonals are on sixty degrees, forming 'upright' equilateral triangles. In this view we see a hexagon composed of six triangles.

2:1

There are two right triangles with unequal sides that illustrate the 2:1 ratio. One is the 30-60-90 degree triangle (seen above) which is is half of an equilateral one. The sine of 30 degrees is .5 as the side opposite to the angle is half the hypotneuse. [A triangle with a 19.5 degree angle in this corner has a short side of 1 and a hypotenuse of three, the sine of 19.5 is .333; the 14.5 degree triangle is 1:4, for a sine of .25, etc.]



The diagonal of half of a square (or a double square) produces 26.5 degree angles. Tan 26.5 = 1/2 or .5. The hypotenuse in this figure is equal to the square root of 5. [You may recognize this as the angle of ascending passage in the Great Pyramid. Also note that DC is at 39 degrees west and 78 degrees north. Two over and one up.]


A rectangle results from changing the ratio of the length of the sides of a square, while changing the ratio of the length of the diagonals produces a rhombus. The square grid can be quickly converted to either version of the triangular grid, just as the 26.5 degree triangle can be converted to the 30 degree one by keeping the width the same and adjusting the height.
In one case we rotate the grid and insert vertical diagonals producing 45 degree angles. Next we adjust the aspect ratio of the image (increase it's width while keeping the same height) converting the 45 degree triangles to equilateral ones. The short axis of the rhombus produced by these triangles is equal to the sides of that rhombus.

A square with 1 inch sides has 2 diagonals equal to the sqrt of 2 (1.41). The image above has been 'adjusted' so that one of the diagonals is now equal to the length of the sides, or 1; making the long diagonal equal to the sqrt of 3 (1.732). The ratio of the length of the short axis to the long one in this rhombus is 15:26 (30:52).
The image below uses horizontal diagonals and increases the height producing a rotated version of the above image.

The Vesica Piscis
Here we see that another way to generate the equilateral triangle, rhombus, hexagon, cube, etc is with the vesica piscis. The side of the blue triangle is equal to the diameter of the circles. The short axis of the rhombus is equal to the radius. A line from a corner to the mid-point of the opposite side bisects that angle. If the radius of the circles (the short axis of the rhombus) is 1 then the long axis is 1.732 (sqrt of 3).

We can determine the diameter of the circle that produces any vesica by dividing the long axis by the sqrt of three and doubling it. By doing this we know that a vesica that is 481 feet tall (the height of the GP) is produced by circles 555.5 feet across (the height of the Washington Monument).
Below we see the 45, 30 and 26.5 degree angles within the figure of the vesica. The sqrt of 2 is the diagonal of a square with sides of 1 (a unit square), which as we saw above generates 45 degree angles. The sqrt of 5 is the diagonal of a 1x2 rectangle; 26.5 degrees.

As was shown above the 30 degree angle results from a diagonal of a rectangle that is 1 by the sqrt of 3 (the dimensions of the vesica/rhombus). While both are 1 wide at the base, the 30 degree triangle is 1.732 tall and the 26.5 triangle is 2 tall.

Modified
Note that in the DC map, the triangular grid has been modified (the aspect ratio was changed by keeping the width the same and shortening the height) so that the equilateral triangle has become an isoceles triangle with 52 degree base angles like the Great Pyramid cross-section, and what were 30 degree diagonals are now 23.5 degrees like the latitude of the tropics.

Interestingly, a hexagram composed of equilateral triangles inscribed within a circle touches that circle at 30 degrees of latitude, very close to the current location of the Great Pyramid. If we shorten that figure until the triangles have 52 degree base angles, we get an image that depicts the tropics (below left). Compare the shortened hexagram to Scott Circle which shows the ellipse, six points of the hexagram and 23 degree angles. On the right we see that the points of the shortened hexagram fit the Station Stones at the Stonehenge. [Note the elliptical figure at the center of that.]

         Larger Image
While you can also use the vesica to generate a 52 degree angle (below), I believe that the ellipses, and the 52 and 23 degree angles in the DC map show that the 'shortened' image above was used as the model for the plan.

Continue