- Source: 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.
Only the bifurcating graph of the
B
3
{\displaystyle B_{3}}
Affine Coxeter group has a Trirectangular tetrahedron fundamental domain.
Metric formulas
If the legs have lengths a, b, c, then the trirectangular tetrahedron has the volume
V
=
a
b
c
6
.
{\displaystyle V={\frac {abc}{6}}.}
The altitude h satisfies
1
h
2
=
1
a
2
+
1
b
2
+
1
c
2
.
{\displaystyle {\frac {1}{h^{2}}}={\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}+{\frac {1}{c^{2}}}.}
The area
T
0
{\displaystyle T_{0}}
of the base is given by
T
0
=
a
b
c
2
h
.
{\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.
De Gua's theorem
If the area of the base is
T
0
{\displaystyle T_{0}}
and the areas of the three other (right-angled) faces are
T
1
{\displaystyle T_{1}}
,
T
2
{\displaystyle T_{2}}
and
T
3
{\displaystyle T_{3}}
, then
T
0
2
=
T
1
2
+
T
2
2
+
T
3
2
.
{\displaystyle 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) irrational space-diagonal of the related Euler-brick (bc, ca, ab).
= Integer edges
=Trirectangular tetrahedrons with integer legs
a
,
b
,
c
{\displaystyle a,b,c}
and sides
d
=
b
2
+
c
2
,
e
=
a
2
+
c
2
,
f
=
a
2
+
b
2
{\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.
a
=
240
,
b
=
117
,
c
=
44
,
d
=
125
,
e
=
244
,
f
=
267
{\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.
= Integer faces
=Trirectangular tetrahedrons with integer faces
T
c
,
T
a
,
T
b
,
T
0
{\displaystyle 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
{\displaystyle 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
{\displaystyle 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
{\displaystyle a,b,c}
.
See also
Irregular tetrahedra
Standard simplex
Euler Brick
References
External links
Weisstein, Eric W. "Trirectangular tetrahedron". MathWorld.