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 or apex 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 (analogous to the altitude of a triangle).

An example of a trirectangular tetrahedron is a truncated solid figure near the corner of a cube or an octant at the origin of Euclidean space. Kepler discovered the relationship between the cube, regular tetrahedron and trirectangular tetrahedron.^[1]

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

If the legs have lengths a, b, c, then the trirectangular tetrahedron has the volume^[2][3]

${\displaystyle V={\frac {abc}{6}}.}$

The altitude h satisfies^[4]

${\displaystyle {\frac {1}{h^{2}}}={\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}+{\frac {1}{c^{2}}}.}$

The area ${\displaystyle T_{0}}$ of the base is given by^[5]

${\displaystyle T_{0}={\frac {abc}{2h}}.}$

The solid angle at the right-angled vertex, from which the opposite face (the base) subtends an octant, has measure π/2  steradians, one eighth of the surface area of a unit sphere.

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

${\displaystyle T_{0}^{2}=T_{1}^{2}+T_{2}^{2}+T_{3}^{2}.}$

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

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)^[6] irrational space-diagonal of the related Euler-brick (bc, ca, ab).

Trirectangular tetrahedrons with integer legs ${\displaystyle a,b,c}$ and sides ${\displaystyle d={\sqrt {b^{2}+c^{2}}},e={\sqrt {a^{2}+c^{2}}},f={\sqrt {a^{2}+b^{2}}}}$ of the base triangle exist, e.g. ${\displaystyle a=240,b=117,c=44,d=125,e=244,f=267}$ (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.

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

• Irregular tetrahedra
• Standard simplex
• Euler Brick

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