The purpose of this paper is to construct a Quasi-volume-filling surface and study its properties. We start with the construction of a volume-filling surface, the PÃƒÆ’³lya surface, based on PÃƒÆ’³lya's curve, by rotating the PÃƒÆ’³lya's curve in 3-dimensional space. Then we construct a Quasi-space-filling curve in 2-dimensions, the Quasi- PÃƒÆ’³lya curve, which approximates the PÃƒÆ’³lya's curve and fills a triangle up to a residual small surface of arbitrary size. We prove that the Quasi-PÃƒÆ’³lya curve satisfies the open set condition, and there exists a unique invariant (self-similar) measure consistent with the normalized Hausdorff measure on it. Moreover, the energy form constructed on Quasi-PÃƒÆ’³lya curve is proved to be a closed & regular form, and we prove that the Quasi-PÃƒÆ’³lya curve is a variational fractal in the end. Next, we use the same idea, by rotating the Quasi-PÃƒÆ’³lya curve in 3-dimensional space, to construct the Quasi-PÃƒÆ’³lya surface, which is a Quasi-volume-filling surface and approximates to PÃƒÆ’³lya surface in some sense.
Worcester Polytechnic Institute
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Liu, Pan, "On Quasi-Volume-Filling Surfaces" (2012). Masters Theses (All Theses, All Years). 259.
Fractal, filling surface, Quasi-filling surface