Mathematics and OrigamiTRAPEZOHEDRON
It is a pseudorregular polyhedron with 24 trapezoid shaped faces, 26 vertices and 48
sides (half of them large and half small). So it fulfils Euler ́s theorem: 24 + 26 = 48 + 2.
It is also a form under which the pyrite (iron disulphide S 2 Fe) crystallises.
Out of the 26 vertices, 18 are equidistantly inscribed on the three orthogonal maximum
circles of a sphere. That sphere circumscribes an octahedron whose 6 vertices are among the
18 mentioned above. These 18 vertices are equidistant from the center of the polyhedron: that
distance R is greater than the other 8 vertices ́; these 8 vertices are associated to the 8 faces of
the octahedron.The dihedrals formed by the polyhedron ́s faces are all equal (regardeless of the length
of their sides) and its measure is 138,118º.
The polyhedron is shown in Fig. 1; we can see in it the section EAFGB that has been
segregated and taken into Fig.2: it is integrated there within one of the 8 co-ordinate trihe-
drals. It follows that the polyhedron results inscribed in a sphere of radius R = OE = OF = OG
= OA = OB. Let ́s determine the face AEBP as a function of R; if R = 1, RA = 1 and RP =
0,9473.
∆AHB and AHO in Fig. 2 are congruent for they are both isosceles right triangles with a
leg in common. Therefore, their hypotenuses will be equal: AB = AO = R = 1.10
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