The Harmonious Pythagorean Tetrahedra
Delineated is base right triangle ABC with proportions given as ratio a : b. A point D1 exists above the plane where for any proportion ¼ : 1 < ◿ABC < 4 : 1, the surface area of ▽BCD1 = a 2, ▽ACD1 = b 2, and ▽ABD1 = c 2. Thus, occurs a four-sided polyhedron, being an oblique triangular pyramid, with right triangle base and corresponding lateral faces consonant with the Pythagorean Theorem (Fig. 1 of original manuscript).
Areas, corresponding to the catheti and hypotenuse, need not necessarily hold similar shape to conform to a 2 + b 2 = c 2. Thought of here as dissimilar triangles, their areas and base catheti lengths are used to compute respective altitudes. While altitude alone is insufficient to describe their final shape, requisite development follows:
Using catheti a, b, and hypotenuse c, each as an independent axis, cylinders with radii equal to their associated triangular altitude can be constructed (Fig. 2 & 3 of original manuscript). Knowing point D1 exists within the locus of points found at the intersection of the perpendicular cylinders, its discrete location determined by a cylinder about the hypotenuse, the intersection of the three define the apex of a tetrahedron. Lateral faces, with slant altitudes of their respective cylinder radii, have areas corresponding to the application of the Pythagorean Theorem to ◿ABC. Point D1 is unique for any given right triangle base, as specified prior, and the evolved tetrahedron.
Noted here, the development can provide two solutions: points D1 above and D2 below the plane of ◿ABC create a bi-pyramid, symmetrical about the medial axis (Fig. 2 of original manuscript). Topologically, this is an irregular, triangular face, concave hexahedron with an Euler Characteristic of 2 having 5 vertices, 9 edges and 6 faces. Further, should ◿ABC = ¼ : 1 or 4 : 1, Point D1 lies in the plane of ◿ABC and no longer yields a three-dimensional solution, the same being true for ¼ : 1 > ◿ABC > 4 : 1. While the ratio of pyramidal volume to height increases with a : b, maximal volume occurs at ⅓ : 1 or 3 : 1 and minimums precipitate as the limits of ¼ : 1 and 4 : 1 are approached. Total pyramidal surface area as well grows with a : b, likewise limited at the two-dimensional threshold.
Analogous to de Gua’s Theorem and unsurprisingly, the formulation to describe the area relationship (Fig. 1 of original manuscript) of the lateral faces to that of ◿ABC follows as:
The same, using catheti a and b, and hypotenuse c:
Creating three points to define a vector, each equidistant to point D1 and perpendicular to each lateral face, a ray through the apex can be struck. Where it pierces ◿ABC, the proportional center, point E is established (Fig. 1 & 4 of original manuscript). Point E, used as a center of dilation and in the trisection of ◿ABC to produce the following:
whereas:
Should line D1E be used to trisect the pyramid in concurrence with the manner above, resulting volumes follow an arrangement of the same; A + B = C.
Of sustained interest:
Sectioning the extended tetrahedron perpendicular to line D1E produces a unique obtuse scalene triangle (Fig. 5 of original manuscript). Point F, being the intersection of line D1E and the scalene triangle, is coincident with its incenter. An exclusive dihedral exists between the plane of ◿ABC and that of the scalene triangle.
Point E, the proportional center of ◿ABC, continues to reveal its significance in regards to the degenerate tetrahedron of zero volume. Should a ◿ABC of ¼ : 1 be used, line BD1 passes through point E. And, if a ◿ABC of 4 : 1 is used, line AD1 will pass through point E. For both, their occurrence is two-dimensional.
The conventional depiction of the Pythagorean Theorem (Fig. 7 of original manuscript) illustrates that the three associated lateral faces which are square cannot be folded or otherwise arranged in a three-dimensional solid object. Only using dissimilar convex polygonal shapes (e.g., triangles) can a polyhedron of Pythagorean conformity be constructed. The net (i.e., flat pattern) of Fig. 6 & 8 (of original manuscript), when folded, forms a Harmonious Pythagorean Tetrahedron.
