The Pythagorean Theorem tells us, that for a right triangle, the sum of the squares of the two short sides is equal to the sum of the hypotenuse; this is expressed as "a squared + b squared = c squared". The triangle that epitomizes this theorem is the 3-4-5 triangle, where 9 + 16 = 25. Because 4 is associated with the earth, it symbolizes the mother; the three, a symbol of diviinity is deemed to be the father, and the five, symbol of the microcosm becomes the son, in certain depictions of problem.
Since the 3-4-5 triangle is half a 3x4 square, we also know that the area that it encloses in 6 square units, six being the number of the macrocosm.
Consider the following painting and compare to the notions presented above. Here we see 1 figure clothed in white in the bottom right hand corner, and 2 angels in the top corners of the square, the triangle of the Father, the square containing the mother and son, and a pair of pentagonal figures featuring a large circle in the center. You will note that the way that the triangle and square are conjoined in a continuous loop produces seven corners on that. In the dental work at the top, there are 16 spaces and 17 tabs, for a total of 33.
In this M.C.Escher print entitled "Reptiles" we see a presentation of the journey of the divine spark. The notion is analogous to D.T.Suzuki's waterfall metaphor, where he suggests that, in one sense, life is like a river where at first (in the beginning) nothing is differentiated from anything else, until the fall...the waterfall that is. As the river goes over the falls, the droplets differentiate for a while, being pulled down and apart by gravity, the weight of matter, until everything is united again at the bottom.
In this case, the mat represents the river, the realm of ideas, while the reptile on the dodecahedron represents the world of forms, 3D reality. The goemetric impulse here appears to parallel that in the painting to include several plane figures, including the circle (or ellipse), the triangle, the rectangle, the pentagon, and the hexagon. Counting the head of one and the tail of the other reptile stuck in the map, we can see that there are seven of them.
Now take a look at the elements of Metatron's Cube, where we see circles, nested hexagons/hexagrams, pairs of equilateral triangles, and rectangles. That's 3, 4 and 6, but no 5. The location of the top two spheres in the A-shaped Tree, and therefore the position of two corners of the pentagon is determined by extending the top sides of the hexagons (blue, middle), allowing us to insert the pentagon (red, right).
The following image compares the plan for the Milan Cathedral (left) with the Washington DC planning map (right), where we can plainly see the triangle, rectangle, pentagon and hexagon. Read more about the similarites between the cathedral plan and the DC map.
