Finding the perfect Euler brick

Megan Grywalski

An Euler brick is a cuboid whose side lengths and face diagonals are integers.

For example, a cuboid with dimensions (a, b, c) = (44, 117, 240) is an Euler brick. It is actually the smallest Euler brick, which was found by Paul Halcke in 1719. This Euler brick has face diagonals (d, e, f) = (125, 244, 267).

An Euler brick satisfies the following equations:

a2 + b2 = d2

a2 + c2 = e2

b2 + c2 = f2

d = dab e = dac f = dbc

Euler brick

22

22

22

cb

ca

ba

Must be integers

A perfect cuboid, or perfect Euler brick, is an Euler brick whose space diagonal is also an integer.

A perfect Euler brick

A perfect Euler brick must satisfy the equations for sides (a, b, c), face diagonals (d, e, f), and space diagonal g with a, b, c, d, e, f, and g ∈ ℤ:

a2 + b2 = d2

a2 + c2 = e2

b2 + c2 = f2 a2 + b2 + c2 = g2

d = dab e = dac f = dbc g = dabc

It is unknown if a perfect Euler brick exists, nor has anyone proved that one does not exist.

222

22

22

22

cba

cb

ca

ba

In 1740, Nicholas Saunderson, a blind mathematician, came up with a parameterization that produced Euler bricks.

If (u, v, w) is a Pythagorean triple u2 + v2 = w2 then

(a, b, c) = (|u(4v2 - w2)|, |v(4u2 - w2)|, |4uvw|)

if u = 2st, v = s2 - t2, w = s2 +t2

a =6ts5 − 20t3s3 + 6t5s, b = −s6 + 15t2s4 − 15t4s2 + t6, c = 8s5t

− 8st5.

The surface r(s, t) = <6ts5 − 20t3s3 + 6t5s,−s6 + 15t2s4 − 15t4s2 +

t6, 8s5t − 8st5> is an Euler brick.

a2 +b2 +c2 = f(t, s)(s2 +t2)2

f(t, s) = s8 + 68s6t2 - 122s4t4 + 68s2t6 + t8,

if you found s, t for which f(t, s) is a square then this would be a

perfect Euler brick.

Using these parameters with the help of a computer it was found that there exists an (a, b, c) with a having 68162 digits, b with 56802 digits, and c with 56803 digits so that the space diagonal √(a2 + b2 + c2) is 10-60589 close to an integer.

Exhaustive computer searches show that, if a perfect cuboid exists, one of its sides must be greater than 1012.

Solutions have been found where two of the three face diagonals and the space diagonal are integers, such as:

(a, b, c) = (672, 153, 104) Some solutions have been found where all

four diagonals but only two of the three sides are integers, such as: (a, b, c) = (18720, √211773121, 7800) and

(a, b, c) = (520, 576, √618849)