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  "body": "\n## Statement\n\nLet $\\operatorname{Li}_2$ be the [[dilogarithm|G0X0NWSDB7]], and for $0 \u003c x \u003c 1$ let\n$$L(x) = \\operatorname{Li}_2(x) + \\tfrac12 \\log x \\, \\log(1 - x)$$\nbe the Rogers dilogarithm, where $\\log$ is the [[natural logarithm|04DXAFN2ES]].\nLet $D$ be the [[Bloch–Wigner function|Q8K5MWEXB9]].\n\n1. For all real $x, y$ with $0 \u003c x \u003c 1$ and $0 \u003c y \u003c 1$,\n$$L(x) + L(y) = L(xy) + L\\left(\\frac{x(1-y)}{1-xy}\\right) + L\\left(\\frac{y(1-x)}{1-xy}\\right).$$\n2. For all complex $x, y$ such that $x$, $y$, $1 - xy$, $\\frac{1-x}{1-xy}$ and $\\frac{1-y}{1-xy}$ all lie in $\\C \\setminus \\set{0, 1}$,\n$$D(x) + D(1 - xy) + D(y) + D\\left(\\frac{1-x}{1-xy}\\right) + D\\left(\\frac{1-y}{1-xy}\\right) = 0.$$\n\n## Remarks\n\nFor the real form, fix $y$ and differentiate both sides in $x$.\nUsing $\\frac{d}{dx}\\operatorname{Li}_2(x) = -\\frac{\\log(1-x)}{x}$, the derivatives of the two sides agree, and both sides tend to $L(y)$ as $x \\to 0^+$, since $L(x) \\to 0$.\nThe complex form follows in the same way from the differential of $D$, which is an explicit real-analytic $1$-form on $\\C \\setminus \\set{0, 1}$.\n\nExample: with $y = x$, the third argument is $\\frac{x(1-x)}{1-x^2} = \\frac{x}{1+x}$, so the relation gives\n$$2L(x) = L(x^2) + 2L\\left(\\frac{x}{1+x}\\right), \\qquad 0 \u003c x \u003c 1.$$\n\nThe five arguments in the complex form are cross-ratios of five points on the Riemann sphere taken four at a time.\nGeometrically, $D(z)$ is the hyperbolic volume of the ideal tetrahedron with vertices $0, 1, \\infty, z$.\nThe relation says that two ways of cutting an ideal polyhedron with five vertices into ideal tetrahedra give the same volume.\n\nThe correction term $\\frac12 \\log x \\log(1-x)$ is needed: the plain dilogarithm $\\operatorname{Li}_2$ satisfies a five-term relation only with extra products of logarithms.\n\n## Explanations\n\n### L1 human\n\nThe dilogarithm is a function defined by a power series, much like the logarithm.\nThe logarithm turns products into sums; the dilogarithm has a more complicated rule that ties together its values at five related points.\nThis five-term rule is the basic functional equation of the dilogarithm, and many other identities for it can be derived from it.\n\n### L2 human\n\nThe relation is usually stated for the Rogers dilogarithm on $(0, 1)$ or for the Bloch–Wigner function on $\\C \\setminus \\set{0, 1}$, because these versions have no elementary correction terms.\nIt is the defining relation of the Bloch group in algebraic K-theory, and it explains why sums of values $D(z_i)$ compute volumes of hyperbolic $3$-manifolds.\nIts quantum analogue is the pentagon identity of the [[quantum dilogarithm|6HMT17DVX7]].\n",
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