Harmonious Pythagorean Tetrahedra Apex Singularity D1 for tetrahedron ABCD1
(The Z negative solution D2 is not explicitly shown here but may be inferred.)
For base right triangle ABC:
b = n
b/4 < a > 4b
c = √(a2+b2)
Point A = (0,b,0)
Point B = (a,0,0)
Point C = (0,0,0)
Line g is y = x
Curve f(x)=√(4a2+(√(4c2+(c-√(4b2+(b±√(x2-4b2))2-4c2))2-4a2)-a)2) **
x (or y) component of f,g intersection is hx (this is length of CD1)
-or-
Solve 0=(√(4a2+(√(4c2+(c-√(4b2+(b±√(x2-4b2))2-4c2))2-4a2)-a)2))-x
Point D1X = if a < 8.0622575 then -√(hx2-(2a)2) else √(hx2-(2a)2)
Point D1Y = if a < 1.98455575342734 then √(-(4b2-4a2)+X2) else -√(-(4b2-4a2)+X2)
Point D1Z = √(4b2-X2)
(Point D1 traverses the x-axis at a = 8.06225750 and the y-axis at a = 1.98455575342734)
Thus, volume of the tetrahedron can be realized by ((.5ab)D1Z)/3
I’ve created a GeoGebra animation (Harmonious Pythagorean Tetrahedra – GeoGebra) to illustrate the path of D1 between the limits of the a:b ratio 1:4 and 4:1 where the 2D degenerates occur. *Note that the animation is computationally demanding. Thus, it is rather slow and coarse in nature.
Physical Model – All Harmonious Pythagorean Tetrahedra exists within this volume, its unique shape of continued interest.
**Many thanks to Stylax for helping me see the forest amongst the trees with this equation, it proved invaluable to my exploration.
Please reference the following for full context regarding this post:
SEQUENCING and MAPPING PYTHAGOREAN TRIPLES by Steve Wait – October 8, 2021 – The Cognitive Condition
ANALOGOUS to PYTHAGORAS (et al.) by Steve Wait – February 23, 2021 – The Cognitive Condition
PYTHAGOREAN PYRAMIDS: A Contribution by Walter Trump – February 2, 2021 – The Cognitive Condition
WHY DOES the PYTHAGOREAN THEOREM WORK? by Steve Wait – January 25, 2021 – The Cognitive Condition
ADDENDUM – REGION of 3D SOLUTIONS by Steve Wait – November 16, 2020 – The Cognitive Condition
