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Penrose introduced the one complex parameter family of $(2:2)$ holomorphic correspondences $\\mathcal{F}_a$: $$\\left(\\frac{aw-1}{w-1}\\right)^2+\\left(\\frac{aw-1}{w-1}\\right)\\left(\\frac{az+1}{z+1}\\right) +\\left(\\frac{az+1}{z+1}\\right)^2=3$$ and proved that for every value of $a \\in [4,7] \\subset \\mathbb{R}$ the correspondence $\\mathcal{F}_a$ is a mating between a quadratic polynomial $Q_c(z)=z^2+c,\\,\\,c \\in \\mathbb{R}$ and the modular group $\\Gamma=PSL(2,\\mathbb{Z})$. 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