Locating the Apex Singularity for Harmonious Pythagorean Tetrahedra | Steve Wait – May 18, 2021

As with most any problem, this too may be approached from a multitude of directions.

The tack I have chosen distills to the determination of the discrete among a locus of points at the coping intersection of the two cylinders at right angles.

For reference and context please see ALL prior posts associated with A Precursory Study of Pythagorean Theorem Generalization a Consequence of Three-Dimensions

Example:

Catheti

a = 3

b = 4

Hypotenuse

c  = √(a² + b²) = 5

To a requisite number of decimal places, exhaustive iteration of D_1z against the defined a,b will produce values for D_1x,D_1y and allow for the application of Heron’s formula to lateral face ABD₁.

Apex Singularity X,Y Coordinates

D1_z = 5.58670988566614

D_1y = -√ ((2a)² – D_1z²) = -2.18830360174273

D_1x = -√ ((2b)² – D_1z²) = -5.72613941966140

Lateral Face ABD₁ Edge Lengths

BD₁ = √ ((a + |D_1x|)² + D_1y² + D_1z²) = 10.5898776749955

AD₁ = √ (D_1x² + (b + |D_1y|)² + D_1z²) = 10.1141040862422

ABD₁ Semi-Perimeter

s = (AD₁ + BD₁ + c) / 2 = 12.8519908806189

ABD₁ Area

√ (s (s – AD₁) (s – BD₁) (s – c)) = a² + b² = 25.00000000000000

ABCD₁ Volume

V_P = (.5abD_1z) / 3 = 11.1734197713323

Where this area converges with that of c², the coordinates of the apex singularity for a Pythagorean compliant tetrahedron are known. (Additionally, pyramidal volume is derived.)

While this approach is conceptually simple, the large number of terms resulting from the expanded equation have prevented me from solving for D_1z should that be possible.

Alternatively, the apex singularity can be viewed as residing on a hyperbolic paraboloid surface for a given D_1x. D_1zon the cylindrical surface is simply a projection of D_1zhp from the hyperbolic paraboloid surface. An obvious deficiency in this approach remains the lack of relation between a and D_1x.

Apex Singularity Coordinates on Hyperbolic Paraboloid Surface

D_1x = -5.72613941966140

D_1y = ±√ (-(4b² – 4a²) + D_1x²) = -2.18830360174273

D_1zhp = D_1x² – D_1y² = 4b² – 4a² = 28.00000000000000

Apex Singularity Coordinate on Cylindrical surface

D_1z = √ (4b² – D_1x²) = 5.58670988566614

BCD₁ Lateral Face Height/Altitude

h_BCD1 = 2a = 6.00000000000000

Lateral Face ABD₁ Edge Lengths

CD₁ = √ (D₁ D_1x² + h_BCD1²) = 8.29389369677477

BD₁ = ±√ (a² + CD₁² ±2√ (a² CD₁² – 4a²)) = 10.5898776749955

AD₁ = ±√ (b²+CD₁² ±2√ (b² CD₁² – 4b²)) = 10.1141040862422

Whether the position of this vertex or its usefulness in determining volume, the consequence of knowing the apex singularity thus allows for further examination of Harmonious Pythagorean Tetrahedra.

Tags: geometry