HarishChandraRajpoot

A regular n-gonal right antiprism is a semiregular convex polyhedron which 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 details, the mathematical derivations of the generalized and analytic formula which are used to determine the different important parameters in terms of edge length, such as normal distances of faces, normal height, radius of circumscribed sphere, surface area, volume, dihedral angles between adjacent faces, solid angle subtended by each face at the centre, and 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.…

HarishChandraRajpoot

A regular pentagonal right antiprism is a convex polyhedron which 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 details, the mathematical derivations of the analytic formula to determine the different parameters in term of side, such as normal distances of faces, normal height, radius of circumscribed sphere, surface area, volume, dihedral angles between adjacent faces, and solid angle subtended by each face at the centre, 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 regular pentagonal right antiprism with edge length of 4 cm has been made by folding the net of faces made from a A4 white sheet paper.…

HarishChandraRajpoot

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.…

HarishChandraRajpoot

The author has derived a new theorem in 3D-Geometry for rotation of two co-planes about their intersecting edges. In this theorem, the author derives mathematical formula to analytically compute the V-cut angle (δ) required for rotating through the same angle (θ) the two co-planar planes, initially meeting at a common edge bisecting the angle (α) between their intersecting straight edges, about their intersecting straight edges until their new straight edges (generated after removing V-shaped planar region) coincide. As a result, we get a point (apex) where three planes intersect one another out of which two are original planes (rotated) & third one is their co-plane (fixed).This theorem is very useful for making specific pyramidal flat containers with polygonal (regular or irregular) base using sheet of paper, polymer, metal or alloy which can be easily bent and butt-joined at the mating edges, closed right pyramids/bi-pyramids & polyhedrons having two regular n-gonal & 2n congruent trapezoidal faces. The author has also presented some paper models for making pyramidal flat containers with regular pentagonal, heptagonal and octagonal bases.…

HarishChandraRajpoot

Model of Solar Dome: A regular Penta-decahedral solar dome of 5mm thick acrylic sheet with 15 regular triangular faces each with side 15 cm crafted by H C Rajpoot.The important parameters of this model are as follows1. Area of regular pentagonal base with each side 15 cm = 387.1074 sq.cm,2. Vertical height of dome = 20.646cm,3. Total surface area = 1461.42 sq.cm, 4. Volume of dome =6345.675 cubic cm., 5. Angle between any two adjacent faces = 138.19 degrees, 6. Percentage increase in surface area of the dome compared to a spherical dome for given volume and vertical height = 15.7%…

HarishChandraRajpoot

Mathematical Proof: It is impossible to pack an infinite plane using identical circles of a finite radius by more than π√3/6≈ 90.69%.…

HarishChandraRajpoot

This theorem was derived by Mr H. C. Rajpoot @IIT Delhi…

HarishChandraRajpoot

This book mainly deals with the new articles based on research work of the author in Electro-Magnetism. The research articles in this book are related to the derivation of mathematical formula to analytically compute the magnetic field & magnetic dipole moment generated by electric charge moving on circular paths. The electric charge in circular motion is concentrated as well as distributed over line-segments, planar-laminae, curved surfaces and throughout the volume of solids like cylinder, sphere, cube etc. All the topics in this book are new and innovative to the learners and academicians studying Theoretical Physics at any level. This makes the book unique one. All the articles discussed here are the complementary parts of Electro-Magnetism.There are solved numerical examples related to the topics discussed in this book which signifies the applications of analytic formula mathematically derived in this book.…

HarishChandraRajpoot

In this theorem, the author Mr H. C. Rajpoot derives mathematical formula to analytically compute the V-cut angle (δ) required for rotating through the same angle (θ) the two co-planar planes, initially meeting at a common edge bisecting the angle (α) between their intersecting straight edges, about their intersecting straight edges until their new straight edges (generated after removing V-shaped planar region) coincide. As a result, we get a point (apex) where three planes intersect one another out of which two are original planes (rotated) & third one is their co-plane (fixed).This theorem is very useful for making specific pyramidal flat containers with polygonal (regular or irregular) base using sheet of paper, polymer, metal or alloy which can be easily bent and butt-joined at the mating edges, closed right pyramids/bi-pyramids & polyhedrons having two regular n-gonal & 2n congruent trapezoidal faces. The author has also presented some paper models for making pyramidal flat containers with regular pentagonal, heptagonal and octagonal bases.…

HarishChandraRajpoot

The author Mr H. C. Rajpoot derives a corollary from HCR's Theorem to analytically compute the di-hedral angle between two planes rotated through the same angle (θ), initially meeting at a common edge bisecting the angle (α) between their intersecting straight edges,, about their intersecting straight edges until their new straight edges (generated after removing V-shaped planar region) coincide.…

HarishChandraRajpoot

Author Harish Chandra Rajpoot (HCR) has derived a general formula to compute all the important parameters of any regular spherical polygon having each side as a great circle arc.…

HarishChandraRajpoot

Application of HCR's Formula for constructions of all five platonic solids…

HarishChandraRajpoot

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