This work studies the problem of GPU thread mapping for a Sierpi\'nski gasketfractal embedded in a discrete Euclidean space of $n \times n$. A block-spacemap $\lambda: \mathbb{Z}_{\mathbb{E}}^{2} \mapsto \mathbb{Z}_{\mathbb{F}}^{2}$is proposed, from Euclidean parallel space $\mathbb{E}$ to embedded fractalspace $\mathbb{F}$, that maps in $\mathcal{O}(\log_2 \log_2(n))$ time and usesno more than $\mathcal{O}(n^\mathbb{H})$ threads with $\mathbb{H} \approx1.58...$ being the Hausdorff dimension, making it parallel space efficient.When compared to a bounding-box map, $\lambda(\omega)$ offers a sub-exponentialimprovement in parallel space and a monotonically increasing speedup once $n >n_0$. Experimental performance tests show that in practice $\lambda(\omega)$can produce performance improvement at any block-size once $n > n_0 = 2^8$,reaching approximately $10\times$ of speedup for $n=2^{16}$ under optimal blockconfigurations.
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