Template:Short description In mathematics, an Euler brick, named after Leonhard Euler, is a rectangular cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime. A perfect Euler brick is one whose space diagonal is also an integer, but such a brick has not yet been found.
DefinitionEdit
The definition of an Euler brick in geometric terms is equivalent to a solution to the following system of Diophantine equations:
- <math>\begin{cases} a^2 + b^2 = d^2\\ a^2 + c^2 = e^2\\ b^2 + c^2 = f^2\end{cases}</math>
where Template:Math are the edges and Template:Math are the diagonals.
PropertiesEdit
- If Template:Math is a solution, then Template:Math is also a solution for any Template:Math. Consequently, the solutions in rational numbers are all rescalings of integer solutions. Given an Euler brick with edge-lengths Template:Math, the triple Template:Math constitutes an Euler brick as well.<ref name=Sierpinski>Wacław Sierpiński, Pythagorean Triangles, Dover Publications, 2003 (orig. ed. 1962).</ref>Template:Rp
- Exactly one edge and two face diagonals of a primitive Euler brick are odd.
- At least two edges of an Euler brick are divisible by 3.<ref name=Sierpinski/>Template:Rp
- At least two edges of an Euler brick are divisible by 4.<ref name=Sierpinski/>Template:Rp
- At least one edge of an Euler brick is divisible by 11.<ref name=Sierpinski/>Template:Rp
ExamplesEdit
The smallest Euler brick, discovered by Paul Halcke in 1719, has edges Template:Math and face diagonals Template:Math.<ref>Visions of Infinity: The Great Mathematical Problems By Ian Stewart, Chapter 17</ref> Some other small primitive solutions, given as edges Template:Math — face diagonals Template:Math, are below:
( 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 )
Generating formulaEdit
Euler found at least two parametric solutions to the problem, but neither gives all solutions.<ref>Template:Mathworld</ref>
An infinitude of Euler bricks can be generated with Saunderson's<ref name=Saunderson>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> parametric formula. Let Template:Math be a Pythagorean triple (that is, Template:Math.) Then<ref name=Sierpinski/>Template:Rp the edges
- <math> a=u|4v^2-w^2| ,\quad b=v|4u^2-w^2|, \quad c=4uvw </math>
give face diagonals
- <math>d=w^3, \quad e=u(4v^2+w^2), \quad f=v(4u^2+w^2).</math>
There are many Euler bricks which are not parametrized as above, for instance the Euler brick with edges Template:Math and face diagonals Template:Math.
Perfect cuboidEdit
Template:Unsolved A perfect cuboid (also called a perfect Euler brick or perfect box) is an Euler brick whose space diagonal also has integer length. In other words, the following equation is added to the system of Diophantine equations defining an Euler brick:
- <math>a^2 + b^2 + c^2 = g^2,</math>
where Template:Math is the space diagonal. Template:As of, no example of a perfect cuboid had been found and no one has proven that none exist.<ref>Template:Cite journal</ref>
Exhaustive computer searches show that, if a perfect cuboid exists,
- the odd edge must be greater than 2.5 × 1013,<ref name=Matson>{{#invoke:citation/CS1|citation
|CitationClass=web }}</ref>
- the smallest edge must be greater than Template:Val,<ref name=Matson/> and
- the space diagonal must be greater than 9 × 1015.<ref name=Belogourov>Alexander Belogourov, Distributed search for a perfect cuboid, https://www.academia.edu/39920706/Distributed_search_for_a_perfect_cuboid</ref>
Some facts are known about properties that must be satisfied by a primitive perfect cuboid, if one exists, based on modular arithmetic:<ref>M. Kraitchik, On certain Rational Cuboids, Scripta Mathematica, volume 11 (1945).</ref>
- One edge, two face diagonals and the space diagonal must be odd, one edge and the remaining face diagonal must be divisible by 4, and the remaining edge must be divisible by 16.
- Two edges must have length divisible by 3 and at least one of those edges must have length divisible by 9.
- One edge must have length divisible by 5.
- One edge must have length divisible by 7.
- One edge must have length divisible by 11.
- One edge must have length divisible by 19.
- One edge or space diagonal must be divisible by 13.
- One edge, face diagonal or space diagonal must be divisible by 17.
- One edge, face diagonal or space diagonal must be divisible by 29.
- One edge, face diagonal or space diagonal must be divisible by 37.
In addition:
- The space diagonal is neither a prime power nor a product of two primes.<ref name=Korec/>Template:Rp
- The space diagonal can only contain prime divisors that are congruent to 1 modulo 4.<ref name=Korec>I. Korec, Lower bounds for Perfect Rational Cuboids, Math. Slovaca, 42 (1992), No. 5, p. 565-582.</ref>Template:Rp<ref>Ronald van Luijk, On Perfect Cuboids, June 2000</ref>
Heronian trianglesEdit
If a perfect cuboid exists with edges <math>a, b, c</math>, corresponding face diagonals <math>d, e, f</math>, and space diagonal <math>g</math>, then the following Heronian triangles exist:
- A Heronian triangle with side lengths <math>(d^2, e^2, f^2)</math>, an area of <math>abcg</math>, and rational angle bisectors.<ref name="Luca">Florian Luca (2000). "Perfect Cuboids and Perfect Square Triangles". Mathematics Magazine, 73(5), 400–401.</ref>
- An acute Heronian triangle with side lengths <math>(af, be, cd)</math> and an area of <math>\frac{abcg}{2}</math>.
- Obtuse Heronian triangles with side lengths <math>(bf, ae, gd)</math>, <math>(ad, cf, ge)</math>, and <math>(ce, bd, gf)</math>, each with an area of <math>\frac{abcg}{2}</math>.
- Right Heronian triangles with side lengths <math>(ab, cg, ef)</math>, <math>(ac, bg, df)</math>, and <math>(bc, ag, de)</math>, each with an area of <math>\frac{abcg}{2}</math>.
Cuboid conjecturesEdit
Three cuboid conjectures are three mathematical propositions claiming irreducibility of three univariate polynomials with integer coefficients depending on several integer parameters. The conjectures are related to the perfect cuboid problem.<ref name=shr_01>Template:Cite journal</ref><ref name=shr_02>Template:Cite journal</ref> Though they are not equivalent to the perfect cuboid problem, if all of these three conjectures are valid, then no perfect cuboids exist. They are neither proved nor disproved.
Cuboid conjecture 1. For any two positive coprime integer numbers <math>a \neq u</math> the eighth degree polynomial Template:NumBlk is irreducible over the ring of integers <math>\mathbb Z</math>.
Cuboid conjecture 2. For any two positive coprime integer numbers <math>p \neq q</math> the tenth-degree polynomial
is irreducible over the ring of integers <math>\mathbb Z</math>.
Cuboid conjecture 3. For any three positive coprime integer numbers <math>a</math>, <math>b</math>, <math>u</math> such that none of the conditions
are fulfilled, the twelfth-degree polynomial
is irreducible over the ring of integers <math>\mathbb Z</math>.
Almost-perfect cuboidsEdit
An almost-perfect cuboid has 6 out of the 7 lengths as rational. Such cuboids can be sorted into three types, called body, edge, and face cuboids.<ref>Rathbun R. L., Granlund Т., The integer cuboid table with body, edge, and face type of solutions // Math. Comp., 1994, Vol. 62, P. 441-442.</ref> In the case of the body cuboid, the body (space) diagonal Template:Math is irrational. For the edge cuboid, one of the edges Template:Math is irrational. The face cuboid has one of the face diagonals Template:Math irrational.
The body cuboid is commonly referred to as the Euler cuboid in honor of Leonhard Euler, who discussed this type of cuboid.<ref>Euler, Leonhard, Vollst¨andige Anleitung zur Algebra, Kayserliche Akademie der Wissenschaften, St. Petersburg, 1771</ref> He was also aware of face cuboids, and provided the (104, 153, 672) example.<ref> Euler, Leonhard, Vollst¨andige Anleitung zur Algebra, 2, Part II, 236, English translation: Euler, Elements of Algebra, Springer-Verlag 1984</ref> The three integer cuboid edge lengths and three integer diagonal lengths of a face cuboid can also be interpreted as the edge lengths of a Heronian tetrahedron that is also a Schläfli orthoscheme. There are infinitely many face cuboids, and infinitely many Heronian orthoschemes.<ref>Template:Citation</ref>
The smallest solutions for each type of almost-perfect cuboids, given as edges, face diagonals and the space diagonal Template:Math, are as follows:
- Body cuboid: Template:Math
- Edge cuboid: Template:Math
- Face cuboid: Template:Math
Template:Asof, there are 167,043 found cuboids with the smallest integer edge less than 200,000,000,027: 61,042 are Euler (body) cuboids, 16,612 are edge cuboids with a complex number edge length, 32,286 were edge cuboids, and 57,103 were face cuboids.<ref>Template:Cite arXiv</ref>
Template:Asof, an exhaustive search counted all edge and face cuboids with the smallest integer space diagonal less than 1,125,899,906,842,624: 194,652 were edge cuboids, 350,778 were face cuboids.<ref name=Belogourov/>
Perfect parallelepipedEdit
A perfect parallelepiped is a parallelepiped with integer-length edges, face diagonals, and body diagonals, but not necessarily with all right angles; a perfect cuboid is a special case of a perfect parallelepiped. In 2009, dozens of perfect parallelepipeds were shown to exist,<ref>Template:Cite journal.</ref> answering an open question of Richard Guy. Some of these perfect parallelepipeds have two rectangular faces. The smallest perfect parallelepiped has edges 271, 106, and 103; short face diagonals 101, 266, and 255; long face diagonals 183, 312, and 323; and body diagonals 374, 300, 278, and 272.
Connection to elliptic curvesEdit
In 2022, Aubrey de Grey published<ref>Template:Cite journal</ref> an exploration of perfect isosceles rectangular frusta, which he termed "plinths". These are hexahedra with two rectangular faces of the same aspect ratio and four faces that are isosceles trapezia. Thus, as for almost-perfect cuboids and perfect parallelepipeds, a perfect cuboid would be a special case of a perfect plinth. Perfect plinths exist, but are much rarer for a given size than perfect parallelepipeds or almost-perfect cuboids. In a subsequent paper,<ref>Template:Cite journal</ref> de Grey, Philip Gibbs and Louie Helm built on this finding to explore classes of elliptic curve that correspond to perfect plinths, almost-perfect cuboids, and other generalisations of perfect cuboids. By this means they dramatically increased the range up to which perfect cuboids can be sought computationally. They also showed that a large proportion of Pythagorean triples cannot form a face of a perfect cuboid, by identifying several families of elliptic curves that must have positive rank if a perfect cuboid exists. Independently, Paulsen and West showed<ref>Template:Cite journal</ref> that a perfect cuboid must correspond to a congruent number elliptic curve of rank at least 2.