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If we have $a^4=b^4+c^4+d^2$ then $a=(q^2+q+1)^2$ is the greatest diagonal. $b=(q^2+q)^2, (q+1)^2$, and so on. The other two sides derive from the above material.
@AaronMeyerowitz.The first product converges to 2/pi. This product is well known, the Vieta product. The second product converges to the square root of e.
