TL;DR
This paper explores the mathematical conditions for Gomboc shapes, resulting in the discovery of two specific analytical Gomboc forms with infinitely differentiable surfaces, and provides their explicit formulas.
Contribution
It introduces the first analytical Gomboc shapes with smooth surfaces and derives their explicit mathematical formulas.
Findings
Discovered two specific analytical Gomboc shapes.
Provided explicit formulas for these shapes.
Confirmed their infinite differentiability.
Abstract
Investigation of the mathematical requirements for a three dimensional geometrical object to qualify as a Gomboc (mono-monostatic) has resulted in the discovery of two specific, analytical Gomboc shapes. Analytical in that the function describing the Gomboc surface is infinitely differentiable. In this brief note, the analysis undertaken is summarized and the formulae for the two specific shapes provided.
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Mathematics and Applications