article published yes Ranita Biswas author 3C2B033E-F248-11E8-B48F-1D18A9856A870000-0002-5372-7890 Partha Bhowmick author This paper presents a novel study on the functional gradation of coordinate planes in connection with the thinnest and tunnel-free (i.e., naive) discretization of sphere in the integer space. For each of the 48-symmetric quadraginta octants of naive sphere with integer radius and integer center, we show that the corresponding voxel set forms a bijection with its projected pixel set on a unique coordinate plane, which thereby serves as its functional plane. We use this fundamental property to prove several other theoretical results for naive sphere. First, the quadraginta octants form symmetry groups and subgroups with certain equivalent topological properties. Second, a naive sphere is always unique and consists of fewest voxels. Third, it is efficiently constructible from its functional-plane projection. And finally, a special class of 4-symmetric discrete 3D circles can be constructed on a naive sphere based on back projection from the functional plane. Springer Nature2017 eng 0924990710.1007/s10851-017-0718-4 59169-83 yes Biswas, R., &#38; Bhowmick, P. (2017). On the functionality and usefulness of Quadraginta octants of naive sphere. <i>Journal of Mathematical Imaging and Vision</i>. Springer Nature. <a href="https://doi.org/10.1007/s10851-017-0718-4">https://doi.org/10.1007/s10851-017-0718-4</a> R. Biswas, P. Bhowmick, Journal of Mathematical Imaging and Vision 59 (2017) 69–83. Biswas R, Bhowmick P. On the functionality and usefulness of Quadraginta octants of naive sphere. <i>Journal of Mathematical Imaging and Vision</i>. 2017;59(1):69-83. doi:<a href="https://doi.org/10.1007/s10851-017-0718-4">10.1007/s10851-017-0718-4</a> Biswas, Ranita, and Partha Bhowmick. “On the Functionality and Usefulness of Quadraginta Octants of Naive Sphere.” <i>Journal of Mathematical Imaging and Vision</i>. Springer Nature, 2017. <a href="https://doi.org/10.1007/s10851-017-0718-4">https://doi.org/10.1007/s10851-017-0718-4</a>. Biswas R, Bhowmick P. 2017. On the functionality and usefulness of Quadraginta octants of naive sphere. Journal of Mathematical Imaging and Vision. 59(1), 69–83. Biswas, Ranita, and Partha Bhowmick. “On the Functionality and Usefulness of Quadraginta Octants of Naive Sphere.” <i>Journal of Mathematical Imaging and Vision</i>, vol. 59, no. 1, Springer Nature, 2017, pp. 69–83, doi:<a href="https://doi.org/10.1007/s10851-017-0718-4">10.1007/s10851-017-0718-4</a>. R. Biswas and P. Bhowmick, “On the functionality and usefulness of Quadraginta octants of naive sphere,” <i>Journal of Mathematical Imaging and Vision</i>, vol. 59, no. 1. Springer Nature, pp. 69–83, 2017. 58002019-01-08T20:42:08Z2021-01-12T08:03:34Z