Editing Euler brick (section)

== Generating formula ==

Euler found at least two [[parametric solution]]s to the problem, but neither gives all solutions.<ref>{{mathworld|urlname=EulerBrick|title=Euler Brick}}</ref>

An infinitude of Euler bricks can be generated with [[Nicholas Saunderson|Saunderson]]'s<ref name=Saunderson>{{cite web |series=Math table |title=Treasure Hunting Perfect Euler bricks |date=February 24, 2009 |first=Oliver |last=Knill |url=http://www.math.harvard.edu/~knill/various/eulercuboid/lecture.pdf |publisher=[[Harvard University]]}}</ref> [[parametric formula]]. Let {{math|(''u'', ''v'', ''w'')}} be a [[Pythagorean triple]] (that is, {{math|''u''{{sup|2}} + ''v''{{sup|2}} {{=}} ''w''{{sup|2}}}}.) Then<ref name=Sierpinski/>{{rp|105}} 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 {{math|(''a'', ''b'', ''c'') <nowiki>=</nowiki> (240, 252, 275)}} and face diagonals {{math|(''d'', ''e'', ''f'' ) <nowiki>=</nowiki> (348, 365, 373)}}.