HCR’s Rank Formula is used to compute the correct hierarchical rank of any article such as correct alphabetic rank of a word or correct (increasing or decreasing) order of a number in a set of all linear permutations obtained by permuting all the articles (without repetitions)
A regular n-gonal right antiprism is a semiregular convex polyhedron that has 2n identical vertices all lying on a sphere, 4n edges, and (2n+2) faces out of which 2 are congruent regular n-sided polygons, and 2n are congruent equilateral triangles such that all the faces have equal side. The equilateral triangular faces meet the regular polygonal faces at the common edges and vertices alternatively such that three equilateral triangular faces meet at each of 2n vertices. This paper presents, in detail, the mathematical derivations of the generalized and analytic formulas which are used to determine the different important parameters in terms of edge length, such as normal distances of faces, normal height, the radius of the circumscribed sphere, surface area, volume, dihedral angles between adjacent faces, solid angle subtended by each face at the center, and the solid angle subtended by polygonal antiprism at each of its 2n vertices using HCR’s Theory of Polygon. All the generalized formulae have been derived using simple trigonometry, and 2D geometry which are difficult to derive using any other methods.
A regular pentagonal right antiprism is a convex polyhedron that has 10 identical vertices all lying on a sphere, 20 edges, and 12 faces out of which 2 are congruent regular pentagons, and 10 are congruent equilateral triangles such that all the faces have equal side. This paper presents, in detail, the mathematical derivations of the analytic formula to determine the different parameters in terms of side, such as normal distances of faces, normal height, the radius of the circumscribed sphere, surface area, volume, dihedral angles between adjacent faces, and the solid angle subtended by each face at the center, using the known results of a regular icosahedron. All the analytic formulae have been derived using simple trigonometry, and 2-D geometry which are difficult to derive using any other methods. A paper model of a regular pentagonal right antiprism with an edge length of 4 cm has been made by folding the net of faces made from an A4 white sheet of paper.
The analytic and generalized formula and recurrence relations were mathematically derived to determine the radii of n umber of circles inscribed or packed in the plane region bounded by circular arcs (including sectors, semi and quarter circles) & the straight lines. The values of radii obtained using analytic formula and recurrence relations have been verified by comparing with those obtained using MATLAB codes. The methods used in this paper for packing circles are deterministic unlike heuristic strategies and optimization techniques. The analytic formulae derived for plane packing of tangent circles can be generalized and used for packing of spheres in 3D space and packing of circles on the spherical surface which is analogous to distribution of non-point charges. The packing density of identical circles, externally tangent to each other, the most densely packed on the regular hexagonal and the infinite planes have been formulated and analysed. This study paves the way for mathematically solving the problems of dense packing of circles in 2D containers, the packing of spheres in the voids (tetrahedral and octahedral) and finding the planar density on crystallographic plane.
Link to full article: https://www.academia.edu/85243125/Mathematical_analysis_of_2D_packing_of_circles_on_bounded_and_unbounded_planes_Analytic_Formulation_and_Simulation
The circumscribed and the inscribed polygons are well known and mathematically well defined in the context of 2D-Geometry. The term ‘Circum-inscribed Polygon’ has been proposed by the author and used as a new definition of the polygon which satisfies the conditions of a circumscribed polygon and an inscribed polygon together. In other words, the circum-inscribed polygon is a polygon which has both the inscribed and circumscribed circles. The newly defined circum-inscribed polygon has each of its sides touching a circle and each of its vertices lying on another circle. The most common examples of circum-inscribed polygon are triangle, regular polygon, trapezium with each of its non-parallel sides equal to the Arithmetic Mean (AM) of its parallel sides (called circum-inscribed trapezium) and right kite. This paper describes the mathematical derivations of the analytic formula to find out the different parameters in terms of AM and GM of known sides such as radii of circumscribed & inscribed circles, unknown sides, interior angles, diagonals, angle between diagonals, ratio of intersecting diagonals, perimeter, area, and distance between circum-centre and in-centre of circum-inscribed trapezium. Like an inscribed polygon, a circum-inscribed polygon always has all of its vertices lying on infinite number of spherical surfaces. All the analytic formulae have been derived using simple trigonometry and 2-dimensional geometry which can be used to analyse the complex 2D and 3D geometric figures such as cyclic quadrilateral and trapezohedron, and other polyhedrons.
The maximum possible packing fraction of an infinite plane using identical circles of finite a radius is π√3/6≈ 90.69%.
The author has discovered a new polyhedron called truncated rhombic dodecahedron (right) by truncating a rhombic dodecahedron (left) from all its 24 edges so that newly generated 24 identical vertices exactly lie on a spherical surface. A truncated rhombic dodecahedron is a non uniform convex polyhedron having 12 congruent rectangular faces, 6 congruent square faces, 8 congruent equilateral triangular faces, 48 edges & 24 identical vertices.
The author has derived the radius of circumscribed sphere passing through all 24 identical vertices of a rhombicuboctahedron with given edge length applying ‘HCR’s Theory of Polygon’ & subsequently derived various formula to analytically compute the normal distances of equilateral triangular & square faces from the centre of rhombicuboctahedron, radius of mid-sphere, surface area, volume, solid angles subtended by each equilateral triangular face & each square face at the centre by using ‘HCR’s Theory of Polygon’, dihedral angle between each two faces meeting at any of 24 identical vertices (i.e. truncated rhombic dodecahedron), solid angle subtended by truncated rhombic dodecahedron at any of its 24 identical vertices.
The author H C Rajpoot has mathematically analysed & derived analytic formula for a rhombic dodecahedron having 12 congruent faces each as a rhombus, 24 edges & 14 vertices lying on a spherical surface with a certain radius. ‘HCR’s Theory of Polygon’ is used to derive formula to analytically compute the angles & diagonals of rhombic face, radii of circumscribed sphere, inscribed sphere & midsphere, surface area & volume of rhombic dodecahedron in terms of edge length, solid angles subtended at the vertices and dihedral angle between adjacent faces. This convex polyhedron can be constructed by joining 12 congruent elementary-right pyramids with rhombic base & certain normal height.
The author H C Rajpoot has discovered a new polyhedron by truncating a rhombic dodecahedron from all its 24 edges so that newly generated 24 identical vertices exactly lie on a spherical surface. A truncated rhombic dodecahedron is a non uniform convex polyhedron having 12 congruent rectangular faces, 6 congruent square faces, 8 congruent equilateral triangular faces, 48 edges & 24 identical vertices. The author, applying his theory of polygon, derives the formula to analytically compute the radius of circumscribed sphere passing through all 24 identical vertices, normal distances of rectangular, square & equilateral triangular faces from the centre of polyhedron, surface area, volume, solid angles subtended by rectangular, square & equilateral triangular faces at the centre of polyhedron by using ‘HCR’s Theory of Polygon’, dihedral angle between each two faces meeting at any of 24 identical vertices (i.e. truncated rhombic dodecahedron), solid angle subtended by truncated rhombic dodecahedron at any of its 24 identical vertices.
![H. C. Rajpoot [PhD, IITB]](https://harishchandrarajpoot.home.blog/wp-content/uploads/2022/08/cropped-cropped-cropped-polish_20220725_123717302-1.jpg)
H. C. Rajpoot [PhD, IITB]