Euler brick

An Euler brick is a cuboid in which the lengths of the edges and surface diagonals have integer values. This special parallelepiped is named after Leonhard Euler . It is spanned by three triangles whose edge lengths are Pythagorean triples and whose right angles meet at a corner.

definition

An Euler brick is primitive if the three edge lengths do not have a common factor .

The geometrical definition of the Euler brick is equivalent to a solution of the following system of Diophantine equations :

${\displaystyle {\begin{cases}a^{2}+b^{2}=d^{2}\\a^{2}+c^{2}=e^{2}\\b^{2}+c^{2}=f^{2}\end{cases}}}$

where a, b, c are the edges and d, e, f are the surface diagonals. Euler found at least two parametric solutions to the problem, but none provides all the solutions.

properties

If ( a , b , c ) is a solution, then ( ka , kb , kc ) is also a solution for some k . Hence the solutions in rational numbers are obtained by multiplying integer solutions by a factor k .

For an Euler brick with the edge lengths ( a , b , c ) the triple ( bc , ac , ab ) also yields an Euler brick.

• At least two edges of an Euler brick can be divided by 3.
• At least two edges of an Euler brick are divisible by 4.
• At least one edge of an Euler brick is divisible by 11.

Generating formulas

An infinite number of Euler tiles can be generated with the following formula: Let ( u , v , w ) be a Pythagorean triple (that is, ). Then has a cuboid with the edges ${\displaystyle u^{2}+v^{2}=w^{2}.}$

${\displaystyle a=u|4v^{2}-w^{2}|,\quad b=v|4u^{2}-w^{2}|,\quad c=4uvw}$

the surface diagonals

${\displaystyle d=w^{3},\quad e=u(4v^{2}+w^{2}),\quad f=v(4u^{2}+w^{2}).}$

These formulas were derived by Nicholas Saunderson in 1740 .

Exercise No. 289 by Paul Halcke with the solution 44 ² = 1936, 240 ² = 57600, 117 ² = 13689

Examples

The squares of the edge lengths of the smallest Euler brick were specified by Paul Halcke as early as 1719 . The first primitive solutions (see the OEIS sequences OEIS A031173 , A031174 , A031175 ) are:

The five primitive Euler bricks with edge lengths less than 1000

(a, b, c)          (d, e, f)
(44, 117, 240)     (125, 244, 267)   (Paul Halcke)
(85, 132, 720)     (157, 725, 732)
(140, 480, 693)    (500, 707, 843)
(160, 231, 792)    (281, 808, 825)
(187, 1020, 1584)  (1037, 1595, 1884)
(195, 748, 6336)   (773, 6339, 6380)
(240, 252, 275)    (348, 365, 373)
(429, 880, 2340)   (979, 2379, 2500)
(495, 4888, 8160)  (4913, 8175, 9512)
(528, 5796, 6325)  (5820, 6347, 8579)

Perfect Euler brick

An Euler brick is called perfect if the space diagonal also has an integral length, i.e. the following Diophantine equation is added to the above system:

${\displaystyle a^{2}+b^{2}+c^{2}=g^{2},\,}$

where g is the space diagonal. No example of a perfect Euler brick has yet been found, nor has it been proven that none exist. Computer-aided searches show that in a perfect Euler brick one of the edges should be larger than 3 · 10 ¹² . In addition, its smallest edge should be greater than 10 ¹⁰ .

A primitive perfect Euler brick, if it existed, would have to have the following properties:

• The lengths of an edge, two surface diagonals and those of the room diagonals must be odd; one edge length and the length of the remaining surface diagonal must be divisible by 4, and the length of the third edge must be divisible by 16.
• Two edge lengths must be divisible by 3, and at least one of these edge lengths must be divisible by 9.
• An edge length must be divisible by 5.
• An edge length must be divisible by 7.
• An edge length must be divisible by 11.
• An edge length must be divisible by 19.
• An edge length or the length of the room diagonal must be divisible by 13.
• An edge, surface diagonal or room diagonal length must be divisible by 17.
• An edge, surface diagonal or room diagonal length must be divisible by 29.
• An edge, surface diagonal or room diagonal length must be divisible by 37.
• The space diagonal length cannot be a power of two or five times a power of two.

Solutions have been found for weakened conditions, for example at

${\displaystyle (a,b,c)=(672,153,104).\,}$

the space diagonal and only two of the three surface diagonals whole-numbered lengths, or at

${\displaystyle (a,b,c)=(18720,{\sqrt {211773121}},7800)}$

and

${\displaystyle (a,b,c)=(520,576,{\sqrt {618849}}).}$

Although all four diagonals have integer lengths, only two of the three edges.

There is no cuboid with an integer space diagonal length and successive edge lengths.

The proof that there is no such thing as a perfect Euler brick may be incomplete.

Perfect parallelepiped

A perfect parallelepiped is a parallelepiped with integer lengths of the edges, surface diagonals and space diagonals, but which does not necessarily have all right angles. A perfect Euler brick is a special case of a perfect parallelepiped. In 2009 it was shown that dozens of perfect parallelepipeds existed, which answered an open-ended question from Richard Guy . Some of these parallelepipeds have two rectangular faces.

literature

• John Leech: The Rational Cuboid Revisited . In: American Mathematical Monthly . 84, No. 7, 1977, pp. 518-533. doi : 10.2307 / 2320014 .
• Richard K. Guy : Unsolved Problems in Number Theory . Springer-Verlag , 2004, ISBN 0-387-20860-7 , pp. 275-283.
• Tim Roberts: Some constraints on the existence of a perfect cuboid . In: Australian Mathematical Society Gazette . 37, 2010, ISSN  1326-2297 , pp. 29-31.

Individual evidence

1. ↑ L. Euler: Fragmenta commentationis cuiusdam maioris, de invenienda relatione enter latera triangulorum, quorum area rationaliter exprimi possit. (PDF) In: Opera posthuma. Retrieved July 12, 2015 . 
2. ↑ Eric W. Weisstein : Euler Brick . In: MathWorld (English).
3. ↑ ^{a b c d e f g} Waclaw Sierpinski : Pythagorean Triangles . Dover Publications, 2003 (orig. Ed. 1962).
4. ↑ Nicholas Saunderson, John Saunderson, Abraham de Moivre: Diophantine Problems . In: The Elements of Algebra, in Ten Books, Volume the Second . University Press, Cambridge 1740, 257. To find three square numbers such, that the sum of every two of them shall be a square, p. 429 ff . (English, google.com - Art. 257, Problem 27, Being a case of the thirtieth of the fifth book of Diophantus).
5. ^ Halcken, Paul: Deliciae mathematicae or Mathematisches Sinnen-Confect . Consisting of five hundred and seventy-four selected, sometimes even artistic algebra, geometry and astronomical tasks, adorned with many artificial solutions and rules ... Nicolaus Sauer, Hamburg, 1719, p. approx. 420 ( mpg.de [accessed on July 12, 2015] p. 256, problem no. 289, the squares 1936, 57600, 13689 are given).
6. ^ Bill Durango: The “Integer Brick” Problem .
7. ↑ Eric W. Weisstein : Perfect Cuboid . In: MathWorld (English).
8. ↑ Randall Rathbun: Perfect Cuboid search to 1e10 completed - none found . NMBRTHRY maillist, November 28, 2010 [accessed July 16, 2017]
9. ^ Walter Wyss: No Perfect Cuboid . arxiv : 1506.02215
10. ↑ Ruslan Sharipov Abdulovich: On Walter Wyss's No Perfect Cuboid Paper . arxiv : 1704.00165v1
11. ^ Jorge F. Sawyer, Clifford A. Reiter: Perfect parallelepipeds exist . In: Mathematics of Computation . 80, 2011, pp. 1037-1040. arxiv : 0907.0220 . doi : 10.1090 / s0025-5718-2010-02400-7 .