Trirectangular tetrahedron

{{short description|Tetrahedron where all three face angles at one vertex are right angles}}

File:2D-simplex.svg and a plane crossing all 3 axes away from the origin (x>0; y>0; z>0) and x/a+y/b+z/c<1]]

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).

File:Kepler's tetrahedron in cube.png inscribed in a cube (on the left), and one of the four trirectangular tetrahedra that surround it (on the right), filling the cube.]]

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.{{cite book|title=Harmonices Mundi|first=Johannes|last=Kepler|author-link=Johannes Kepler|title-link=Harmonices Mundi|year=1619|language=la|page=[https://archive.org/details/ioanniskepplerih00kepl/page/180 181]}}

Only the bifurcating graph of the $B_3$ 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{{cite book |title=An Excursion through Elementary Mathematics, Volume II: Euclidean Geometry |author1=Antonio Caminha Muniz Neto |edition= |publisher=Springer |year=2018 |isbn=978-3-319-77974-4 |page=437 |url=https://books.google.com/books?id=AtpVDwAAQBAJ}} [https://books.google.com/books?id=AtpVDwAAQBAJ&pg=PA437 Problem 3 on page 437]

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

The altitude h satisfiesEves, Howard Whitley, "Great moments in mathematics (before 1650)", Mathematical Association of America, 1983, p. 41.

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

The area $T_0$ of the base is given by[https://gogeometry.com/pythagoras/right_triangle_formulas_facts.htm Gutierrez, Antonio, "Right Triangle Formulas"]

: $T_0=\frac{abc}{2h}.$

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

De Gua's theorem

If the area of the base is $T_0$ 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

=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 {{OEIS link|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

External links

{{MathWorld | title = Trirectangular tetrahedron | urlname = TrirectangularTetrahedron}}

Category:Tetrahedra