Ad Quadratum Construction and Study of the Regular Polyhedra

15
A simple way of constructing each individual pyramid is to plot four times C ˆ i along a
great circle of the circumsphere (fig. 11). When cut and folded, we obtain the pyramid
(including the arc of the great circle subtended by the cube edge). Six such constructions
will yield the cube.
The Ad Quadratum Construction
The Ad Quadratum construction begins with the construction of a double square each of
side unity (fig. 12).
We therefore begin with a circle of unit radius to which we add two intersecting and
equal circles, centered at the extremity of a diameter, respectively, so as to form a double
vesica. The intersection of the center line and of the tangents to the circles with the
secants through the 2 vesicas determine the double squares: A B O G.
It is on the basis of these two squares that the construction evolves (fig. 13). Together
they form a rectangle ABOG with sides equal to 1 and 2 respectively.
Diagonal GB 5 .
We can already note in passing that tan A ˆ G B 2
and therefore that A ˆ G B A ˆ O B ˆ I i , the internal angle of the icosahedron.
Now swing GB down in the extension of GA to point D.
In triangle OGD, GD 5, GO 2.
Therefore DO 3.
And therefore,
G ˆ D O D ˆ O B ˆ D i ,
the internal angle of the dodecahedron, since
sinG ˆ D O GO 2
DO 3 .
Now describe the circle centered at O with radius OD and extend AB to S; OB to R and
U; OG to V and OA to W, where these extensions intercept the circle.
Then draw OS.
It will be seen that in triangle BOS,
cosB ˆ
O S 1
3
and therefore B ˆ
O S ˆ
C i , the internal angle of the cube.