Igor Pak对立方体进行改造,构造出了表面和立方体等距同构的一个非凸多面体且围出更大的体积,本文将此构造方法称之为“Pak折叠”,在其启发下,本文将Pak折叠思想运用于对正六面体以外的正多面体的改造:(1) 正四面体、(2) 正八面体、(3)...Igor Pak对立方体进行改造,构造出了表面和立方体等距同构的一个非凸多面体且围出更大的体积,本文将此构造方法称之为“Pak折叠”,在其启发下,本文将Pak折叠思想运用于对正六面体以外的正多面体的改造:(1) 正四面体、(2) 正八面体、(3) 正十二面体、(4) 正二十面体,我们进行了类似的改造,详细计算了经过改造后所围体积,最终给出相应体积关于参数的渐近展开公式,从而发现决定体积增减的决定性原因。我们推测,对于其他凸多面体,都可能有类似于Pak折叠的通用改造方法,但是由于非正多面体的复杂性,该问题有待进一步解决证明。由于这样的改造在不改变表面积的情况下,会增加体积,而且改造后的形状可能适用于一些特殊需要,由此我们也指出一些潜在的实际应用并计算了改造的有效范围。Igor Pak transformed the cube and constructed a non-convex polyhedron with anisometric surface and a larger volume. This construction method is named “Pak bending” in this paper. Inspired by it, we study the application of the idea of Pak bending to the transformation of regular polyhedron other than regular hexahedron: (1) Regular tetrahedron, (2) Regular octahedron, (3) Regular dodecahedron, (4) Regular icosahedron. We carry out similar transformations, and calculate in detail the volume after transformation. Finally, we give the formula of the corresponding volume with asymptotic expansion of the parameter, and find the decisive factor that determines the change of volume. We speculate that for other convex polyhedron, there may be a general transformation method similar to Pak bending, but due to the complexity of non-regular polyhedron, this problem needs to be further solved and proved. Since such a transformation increases volume without changing surface area, and the modified shape may be suitable for some special needs, we also indicate some potential practical applications and calculate the effective range of the transformation.展开更多
文摘Igor Pak对立方体进行改造,构造出了表面和立方体等距同构的一个非凸多面体且围出更大的体积,本文将此构造方法称之为“Pak折叠”,在其启发下,本文将Pak折叠思想运用于对正六面体以外的正多面体的改造:(1) 正四面体、(2) 正八面体、(3) 正十二面体、(4) 正二十面体,我们进行了类似的改造,详细计算了经过改造后所围体积,最终给出相应体积关于参数的渐近展开公式,从而发现决定体积增减的决定性原因。我们推测,对于其他凸多面体,都可能有类似于Pak折叠的通用改造方法,但是由于非正多面体的复杂性,该问题有待进一步解决证明。由于这样的改造在不改变表面积的情况下,会增加体积,而且改造后的形状可能适用于一些特殊需要,由此我们也指出一些潜在的实际应用并计算了改造的有效范围。Igor Pak transformed the cube and constructed a non-convex polyhedron with anisometric surface and a larger volume. This construction method is named “Pak bending” in this paper. Inspired by it, we study the application of the idea of Pak bending to the transformation of regular polyhedron other than regular hexahedron: (1) Regular tetrahedron, (2) Regular octahedron, (3) Regular dodecahedron, (4) Regular icosahedron. We carry out similar transformations, and calculate in detail the volume after transformation. Finally, we give the formula of the corresponding volume with asymptotic expansion of the parameter, and find the decisive factor that determines the change of volume. We speculate that for other convex polyhedron, there may be a general transformation method similar to Pak bending, but due to the complexity of non-regular polyhedron, this problem needs to be further solved and proved. Since such a transformation increases volume without changing surface area, and the modified shape may be suitable for some special needs, we also indicate some potential practical applications and calculate the effective range of the transformation.
