geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a trirectangular tetrahedron is a

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

where all three face angles at one

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

are

right angles In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...

. 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 In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...

has a Trirectangular tetrahedron fundamental domain.

Metric formulas

If the legs have lengths ''a, b, c'', then the trirectangular tetrahedron has the

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...

V=\frac.

The altitude ''h'' satisfies :

\frac=\frac+\frac+\frac.

The area

T_0

of the base is given by :

T_0=\frac.

De Gua's theorem

If the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open su ...

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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...

to a tetrahedron.

Integer solution

Perfect body

Smallest_perfect_trirectangular_biparamyd,_Jan_2018

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)Walter Wyss, "No Perfect Cuboid", 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, e=\sqrt, f=\sqrt

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 .

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

External links

*{{MathWorld , title = Trirectangular tetrahedron , urlname = TrirectangularTetrahedron Polyhedra