Sum of Cubes Gives Rise to Tetrahedral Spires | Steve Wait – March 6, 2024

Select two non-zero random volumes for a and b. I prefer to normalize with b=1 and a as some value ≤ 1. This affords quick and easy comprehension of the a:b ratio. The sum of a and b then equals c. This is the initial step necessary in establishing the proportional center of the base. Take the cube root of the three volumes, each being the corresponding side length of the base triangle. With a along the x-axis, the intersection of a and b defines the origin in the xy plane. Heron’s formula provides the total area of the base and together with the total volume, the height of the composite tetrahedron can be found. Using the height and corresponding volume for each side, the area of the individual tetrahedra bases may be calculated. These individual areas and their associated side lengths yield for each their altitude, the intersection of which defines the proportional center of the composite base triangle. Establishing a line in the z+ direction from the proportional center, with length equal to the height of the composite tetrahedron, delineates each composite tetrahedra and marks the apex for all at its terminus. Thus, a tetrahedral spire who’s composite tetrahedra volumes conforms to a³+b³=c³. While infinite in number, they exist in a finite region as constrained by their composite base triangle; from the isosceles of a:b=1, through the acute scalene as a:b approaches zero. Of note is the non-linear path of the apex as a is manipulated; most specifically as it moves in the z-axis, as function of base area and total volume. The minima extremum of this occurs at an a:b volume ratio of approximately .4933…  Angle C of the composite base triangle can be found and used with the length of b to establish the xy coordinates of point A as needed.

Corresponding Sides: a,b,c

Individual Tetrahedra: BCDE, ACDE, ABDE

Volumes: V_a <input>, V_b <input>, V_c=V_a+V_b

Composite Base Triangle: ABC

Side Lengths: L_n=3√V_n for a,b, and c

Semi-Perimeter: s=(L_a+L_b+L_c)/2

Area: A_ABC=√(s(s-L_a)(s-L_b)(s-L_c))

Composite Tetrahedron: ABCE

                  Volume: V_T=V_a+V_b+V_c

                  Height: E_z=3V_T/A_ABC

Individual Base Triangles: BCD, ACD, ABD

                  Areas: A_n=(3V_n)/E_z

                  Altitudes: T_n=(2A_n)/L_n

Point Coordinates: A

                  x=√(1-y²)

                  y=(2A_ABC)/(L_aL_b)

Tags: sum of cubestetrahedronvolume