Ad Quadratum Construction and Study of the Regular Polyhedra

21
As indicated for the cube (fig. 10 and fig. 11), we can now proceed with construction of
each regular polyhedra.
Thus the tetrahedron will be made up of four pyramids (fig. 19A) built from the
development shown (fig. 19B).
For the cube, as already seen, we have 6 pyramids (fig. 20A) with the following
development shown (fig. 20B). (Repeats of fig. 10 and 11 respectively).
The octahedron will be made up of 8 pyramids (fig. 21A), developed as shown: (fig. 21B)
The dodecahedron will have 12 pyramids (fig. 22A) developed as shown (fig. 22B):
Finally, the icosahedron will be made up of 20 pyramids (fig. 23A), with the
development as shown: (fig. 23B)
The Challenge of Abul Wefa
Abul Wefa was a tenth century Islamic philosopher 9 credited with “the feat of drawing all
five Platonic solids using only a straightedge and a pair of compass at a fixed setting.”
Such fixed compasses (known as “rusty” compasses), adds Hersey 10 , have been the tools
of virtuoso geometrical draftsmanship in many periods.
We would like to show here, without laying claim to virtuosity eleven centuries after
Abul Wefa, that our Adquadratum method can easily be modified to accomplish the deed.
We’ll consider having met the challenge if we can draw the adquadratum diagram with a
“rusty” compass and a straightedge only.
We begin by tracing a line xy with the straightedge (fig. 24A). Setting our compass
opening at R, radius of the circumsphere common to all the regular forms, we then
proceed by drawing four intersecting circles whose centers O 1 are on line xy and
the circumferences of their neighbors as shown on fig. 24A. The three vesicas determine
both a square ABCD of side equal to the circle diameter and the median EF of the square
to which we add diagonals AC and BD and square EHFG.
,O2 ,O3 ,O4 Now, draw IJ (it passes through O 2) and join AO2 and DO2 cutting GE and GF at K and
L respectively. Join KL cutting XY at M. Call center point of square ABCD, .
Then MO 5 2
3 GO 5
9 George L. Hersey: Architecture and Geometry in the Age of the Baroque, Chicago U.P. 2000, p. 88
10 Hersey, op. cit. p. 88
O 5