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FeaturedHCR’s Rank Formula
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)
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Regular N-gonal Right Antiprism
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… Read more
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Mathematical analysis of regular pentagonal right antiprism
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… Read more
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Mathematical Analysis of 2D Packing of Circles
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… Read more
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Mathematical Analysis of Circum-inscribed (C-I) Polygons
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… Read more
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Packing Fraction of an Infinite Plane using Identical Circles
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Packing Fraction of an Infinite Plane using Identical Circles
The maximum possible packing fraction of an infinite plane using identical circles of finite a radius is π√3/6≈ 90.69%.
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Truncation of a rhombic dodecahedron
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,… Read more
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Mathematical Analysis of Rhombicuboctahedron
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,… Read more
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Rhombic Dodecahedron
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… Read more
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Truncated Rhombic Dodecahedron
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… Read more
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Electro-Magnetism
Magnetic Field Generated by Rotating Electric Charge by H. C. Rajpoot
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Electro-Magnetism by H C Rajpoot
The author Mr H. C. Rajpoot (Master of Technology at I.I.T. Delhi) has derived numerous mathematical formula used to analytically compute the strength of magnetic field produced by (point or uniformly distributed) electric charge rotating at a constant angular speed about a given axis. The author has derived and proposed a generalized mathematical formula (i.e. HCR’s formula for… Read more
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Electro-Magnetism by H. C. Rajpoot
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 path. The electric charge in circular motion… Read more
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HCR’s Theorem
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Applications of HCR’s Theorem and Corollary
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HCR’s Corollary
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HCR’s Theorem
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Mathematical analysis of Disphenoid (Isosceles Tetrahedron) by HCR
The author Mr H. C. Rajpoot has derived various Analytic Formula to analytically compute volume, surface area, vertical height, in-radius, circum-radius, centroid etc.
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Three Proofs of Apollonius Theorem by HCR
The author gives three proofs of Apollonius Theorem by using Trigonometry & Pythagorean Theorem.
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Solid angle subtended by a rectangular right pyramid at its apex
The author Mr H. C. Rajpoot represents the derivation of General Formula to analytically compute the solid angle subtended at the apex by a rectangular right pyramid given the apex angles (i.e. angles between two pairs of consecutive lateral edges meeting at the apex). These formula are very important in 3D Geometry for case studies… Read more
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Solid angle subtended by a regular n-gonal right pyramid at its apex
The author Mr H. C. Rajpoot represents General Formula to analytically compute the solid angle subtended by a regular n-gonal right pyramid (solid or hollow) at its apex when either normal height, number of sides & length of side of regular n-gonal base are known or apex angle & the number of sides of regular… Read more
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HCR’s Generalized Formula for uniform polyhedrons with right kite faces
The author H. C. Rajpoot has derived Generalized Formula for uniform polyhedrons (trapezohedrons) with congruent right kite faces to compute various parameters such as volume, surface area, radii of inscribed & circumscribed spheres, relation between unequal edges etc.
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Mathematical Analysis of three externally touching circles
The author H. C. Rajpoot has derived General Formula for analytically computing the radii of circles inscribing & circumscribing three externally touching circles of known radii
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HCR’s Cosine Formula for computing great-arc distance on sphere/globe
HCR’s Cosine Formula is an analytic formula used to compute correct value of minimum (great arc) distance between any two points on the sphere or globe given the latitudes & longitudes.
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HCR’s Corollary of Solid Angle
Solid angle subtended by any plane at any point in 3D space is given by HCR’s corollary (detailed explanation & derivation are done in Advanced Geometry by H. C. Rajpoot)
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HCR’s Formula for Regular Polyhedrons/platonic solids
The vertex angle (i.e. angle between two lateral edges meeting at the vertex) of elementary right pyramid of any regular polyhedron (platonic solid) having n number of faces each as regular n-gon, is given by HCR’s Formula for Regular Polyhedrons/Platonic Solids
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HCR’s Formula for Regular Spherical Polygons
Mathematical Analysis of Regular Spherical Polygons
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HCR’s Theory of Polygon
The Solid angle subtended at any point in 3D space by any polygonal plane can be computed by using Master formula derived by the author H. C. Rajpoot in his proposed Theory of Polygon The Master/Standard Formula is used to find the solid angle subtended by a right triangle at any point lying on perpendicular… Read more
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H. C. Rajpoot [PhD, IITB] 