Trirectangular tetrahedron

In geometry, a trirectangular tetrahedron is a tetrahedron where all three face angles at one vertex are right angles. That vertex is called the right angle of the trirectangular tetrahedron and the face opposite it is called the base. The three edges that meet at the right angle are called the legs and the perpendicular from the right angle to the base is called the altitude of the tetrahedron.

Only the bifurcating graph of the $B_{3}$ Affine Coxeter group has a Trirectangular tetrahedron fundamental domain.

Metric formulas If the legs have lengths x, y, z then the trirectangular tetrahedron has the

A trirectangular tetrahedron can be constructed by a coordinate octant and a plane crossing all 3 axes away from the origin, like:
x>0 y>0 z>0
and x/a +y/b +z/c <1

volume

V={\frac {xyz}{6}}

The altitude h satisfies^[1]

{\frac {1}{h^{2}}}={\frac {1}{x^{2}}}+{\frac {1}{y^{2}}}+{\frac {1}{z^{2}}}

The area $T_{0}$ of the base is given by^[2]

T_{0}={\frac {xyz}{2h}}

Here are 5 useful relationships in Tri-Rectangular Tetrahedrons

Use the color coded illustration to the lower right.

In the 1st equation presented below you can see Heron's constant [S] has been modified such each of the 3 diagonal elements |a,b,c| have been squared & have become |a²+b²+c²| & [S] is replaced by [K]

And Pythagorean's theorem has been modified such that all three rectangular legs |x²+y²+z²| are used simultaneously instead of just 2 at a time & you don't take the square root. When the relationship of diagonals & axials (as shown below) equal each other then the tetrahedron is a Trirectangular Tetrahedron.

$\ \ \ \ \ K={\Biggl [}{\frac {a^{2}+b^{2}+c^{2}}{2}}{\Biggr ]}={\Bigl [}x^{2}+y^{2}+z^{2}{\Bigr ]}$

Below are the 3 diagonal elements matched with their 3 corresponding axial legs & each matched pair equals the above 2 equations.

$\ \ \ \ \ K={\Bigl [}x^{2}+c^{2}{\Bigr ]}={\Bigl [}y^{2}+b^{2}{\Bigr ]}={\Bigl [}z^{2}+a^{2}{\Bigr ]}$

Then $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Given{\Bigl [}a,b,c{\Bigr ]}\ \ \ \ \ \ \ \ \ \ \ \ \ Given{\Bigl [}x,y,z{\Bigr ]}$

$K={\Bigl [}Constant{\Bigr ]}\ \ \ \ \ \ \ \ \ K={\Biggl [}{\frac {a^{2}+b^{2}+c^{2}}{2}}{\Biggr ]}\ \ \ \ \ \ \ \ \ K={\Bigl [}x^{2}+y^{2}+z^{2}{\Bigr ]}$

$Use\ {\Bigl [}x^{2}+c^{2}{\Bigr ]}\ \ \ \ \ \ \ \ x={\sqrt {_{\ }K-c^{2}\ \ }}\ \ \ \ \ \ \ \ \ \ \ c={\sqrt {_{\ }K-x^{2}\ \ }}$

$Use\ {\Bigl [}y^{2}+b^{2}{\Bigr ]}\ \ \ \ \ \ \ \ \ y={\sqrt {_{\ }K-b^{2}\ \ }}\ \ \ \ \ \ \ \ \ \ \ b={\sqrt {_{\ }K-y^{2}\ \ }}$

$Use\ {\Bigl [}z^{2}+a^{2}{\Bigr ]}\ \ \ \ \ \ \ \ \ z={\sqrt {_{\ }K-a^{2}\ \ }}\ \ \ \ \ \ \ \ \ \ \ a={\sqrt {_{\ }K-z^{2}\ \ }}$

The Volume of a Tri-Rectangular Tetrahedron & the box it'll fit in.

$\ \ \ \ V_{tet}={\frac {\ xyz\ }{6}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ V_{box}=xyz$

Internal height of a Tri-Rectangular Tetrahedron from the point of origin at |x,y,z| to its base bounded by |a,b,c|

$h={\frac {xyz}{\ \ {\sqrt {\ x^{2}y^{2}+z^{2}{\Bigl [}x^{2}+y^{2}{\Bigr ]}\ }}\ \ }}$

Area of the base ( A_abc ) bounded by |a,b,c| - 2 formulas given here use the axial elements |x,y,z| - Herons Theorem does the same thing using the 3 diagonals |a,b,c|.

$\ \ \ \ \ A_{abc}={\frac {\ xyz\ }{\ 2h}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A_{abc}={\frac {\ \ {\sqrt {x^{2}y^{2}+z^{2}{\Bigl [}x^{2}+y^{2}{\Bigr ]}\ }}\ \ \ }{2}}$

Heron's original 2000 year old [2 equation] solution is shown below & a 1 piece transposed form is shown beneath it. Heron's system is better.

$\ \ \ \ \ S={\frac {\ a+b+c\ }{2}}\ \ \ \ \ \ \ \ A_{abc}={\sqrt {\ S{\Bigl (}S-a{\Bigr )}{\Bigl (}S-b{\Bigr )}{\Bigl (}S-c{\Bigr )}\ }}$

$\ \ \ \ \ A_{abc}={\sqrt {\ {\Bigl (}a^{2}b^{2}+c^{2}{\Bigl (}a^{2}+b^{2}{\Bigr )}{\Bigl )}-a^{4}-b^{4}-c^{4}\ }}$

Area of all 4 surfaces of a Tri-Rectangular Tetrahedron

$\ \ \ \ \ A_{abc}={\frac {\ xyz\ }{2h}}\ \ \ \ A_{xy}={\frac {\ xy\ }{2}}\ \ \ \ A_{xz}={\frac {\ xz\ }{2}}\ \ \ \ A_{yz}={\frac {\ yz\ }{2}}$

Total area of a Tri-Rectangular Tetrahedron

$\ \ \ \ \ A_{tet}={\frac {\ xy+z{\Bigl [}x+y{\Bigr ]}+{\sqrt {x^{2}y^{2}+z^{2}{\Bigl [}x^{2}+y^{2}{\Bigr ]}\ }}\ \ }{2}}$

Example Dimensions

$\ \ \ \ \ a=14.4\ \ \ \ \ b=10\ \ \ \ \ c=12$

$\ \ \ \ \ x=9.037698822156002781875821128798$

$\ \ \ \ \ y=11.210709165793214885544254958172$

$\ \ \ \ \ z=4.2801869118065393196504607419769$

These work but are not as slick as the above interlocking equations.

$\ \ \ \ \ \ x^{2}+c^{2}={\frac {\ a^{2}+b^{2}+c^{2}}{2}}\ \ therefore\ \ \ x={\sqrt {{\frac {\ a^{2}+b^{2}-c^{2}}{2}}\ }}$

$\ \ \ \ \ \ y^{2}+b^{2}={\frac {\ a^{2}+b^{2}+c^{2}}{2}}\ \ \ therefore\ \ \ y={\sqrt {{\frac {\ a^{2}+c^{2}-b^{2}}{2}}\ }}$

$\ \ \ \ \ \ z^{2}+a^{2}={\frac {\ a^{2}+b^{2}+c^{2}}{2}}\ \ \ therefore\ \ \ z={\sqrt {{\frac {\ b^{2}+c^{2}-a^{2}}{2}}\ }}$

De Gua's theorem

If the area of the base is $T_{0}$ and the areas of the three other (right-angled) faces are $T_{1}$ , $T_{2}$ and $T_{3}$ , then

T_{0}^{2}=T_{1}^{2}+T_{2}^{2}+T_{3}^{2}.

This is a generalization of the Pythagorean theorem to a tetrahedron.

Integer solution

Perfect body

The area of the base (a,b,c) is always (Gua) an irrational number. Thus a trirectangular tetrahedron with integer edges is never a perfect body. The trirectangular bipyramid (6 faces, 9 edges, 5 vertices) built from these trirectangular tetrahedrons and the related left-handed ones connected on their bases have rational edges, faces and volume, but the inner space-diagonal between the two trirectangular vertices is still irrational. The later one is the double of the altitude of the trirectangular tetrahedron and a rational part of the (proved)^[3] irrational space-diagonal of the related Euler-brick (bc, ca, ab).

Integer edges

Trirectangular tetrahedrons with integer legs $a,b,c$ and sides $d={\sqrt {b^{2}+c^{2}}},e={\sqrt {a^{2}+c^{2}}},f={\sqrt {a^{2}+b^{2}}}$ of the base triangle exist, e.g. $a=240,b=117,c=44,d=125,e=244,f=267$ (discovered 1719 by Halcke). Here are a few more examples with integer legs and sides.

    a        b        c        d        e        f

   240      117       44      125      244      267
   275      252      240      348      365      373
   480      234       88      250      488      534
   550      504      480      696      730      746
   693      480      140      500      707      843
   720      351      132      375      732      801
   720      132       85      157      725      732
   792      231      160      281      808      825
   825      756      720     1044     1095     1119
   960      468      176      500      976     1068
  1100     1008      960     1392     1460     1492
  1155     1100     1008     1492     1533     1595
  1200      585      220      625     1220     1335
  1375     1260     1200     1740     1825     1865
  1386      960      280     1000     1414     1686
  1440      702      264      750     1464     1602
  1440      264      170      314     1450     1464

Notice that some of these are multiples of smaller ones. Note also A031173.

Integer faces

Trirectangular tetrahedrons with integer faces $T_{c},T_{a},T_{b},T_{0}$ and altitude h exist, e.g. $a=42,b=28,c=14,T_{c}=588,T_{a}=196,T_{b}=294,T_{0}=686,h=12$ without or $a=156,b=80,c=65,T_{c}=6240,T_{a}=2600,T_{b}=5070,T_{0}=8450,h=48$ with coprime $a,b,c$ .

References

^ Eves, Howard Whitley, "Great moments in mathematics (before 1650)", Mathematical Association of America, 1983, p. 41.
^ Gutierrez, Antonio, "Right Triangle Formulas"
^ Walter Wyss, "No Perfect Cuboid", arXiv:1506.02215

External links

Weisstein, Eric W. "Trirectangular tetrahedron". MathWorld.

[1] Eves, Howard Whitley, "Great moments in mathematics (before 1650)", Mathematical Association of America, 1983, p. 41.

[2] Gutierrez, Antonio, "Right Triangle Formulas"

[3] Walter Wyss, "No Perfect Cuboid", arXiv:1506.02215

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