A Mediaeval Arabic Addendum to Euclid's Elements
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A New Calculable Orbital-Model of the Atomic Nucleus Based on a Platonic-Solid Framework
A new calculable Orbital-Model of the Atomic Nucleus based on a Platonic-Solid framework and based on constant Phi by Dipl. Ing. (FH) Harry Harry K. K.Hahn Hahn / Germany ------------------------------ 12. December 2020 Abstract : Crystallography indicates that the structure of the atomic nucleus must follow a crystal -like order. Quasicrystals and Atomic Clusters with a precise Icosahedral - and Dodecahedral structure indi cate that the five Platonic Solids are the theoretical framework behind the design of the atomic nucleus. With my study I advance the hypothesis that the reference for the shell -structure of the atomic nucleus are the Platonic Solids. In my new model of the atomic nucleus I consider the central space diagonals of the Platonic Solids as the long axes of Proton - or Neutron Orbitals, which are similar to electron orbitals. Ten such Proton- or Neutron Orbitals form a complete dodecahedral orbital-stru cture (shell), which is the shell -type with the maximum number of protons or neutrons. An atomic nucleus therefore mainly consists of dodecahedral shaped shells. But stable Icosahedral- and Hexagonal-(cubic) shells also appear in certain elements. Consta nt PhI which directly appears in the geometry of the Dodecahedron and Icosahedron seems to be the fundamental constant that defines the structure of the atomic nucleus and the structure of the wave systems (orbitals) which form the atomic nucelus. Albert Einstein wrote in a letter that the true constants of an Universal Theory must be mathematical constants like Pi (π) or e. My mathematical discovery described in chapter 5 shows that all irrational square roots of the natural numbers and even constant Pi (π) can be expressed with algebraic terms that only contain constant Phi (ϕ) and 1 Therefore it is logical to assume that constant Phi, which also defines the structure of the Platonic Solids must be the fundamental constant that defines the structure of the atomic nucleus. -
The Roundest Polyhedra with Symmetry Constraints
Article The Roundest Polyhedra with Symmetry Constraints András Lengyel *,†, Zsolt Gáspár † and Tibor Tarnai † Department of Structural Mechanics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary; [email protected] (Z.G.); [email protected] (T.T.) * Correspondence: [email protected]; Tel.: +36-1-463-4044 † These authors contributed equally to this work. Academic Editor: Egon Schulte Received: 5 December 2016; Accepted: 8 March 2017; Published: 15 March 2017 Abstract: Amongst the convex polyhedra with n faces circumscribed about the unit sphere, which has the minimum surface area? This is the isoperimetric problem in discrete geometry which is addressed in this study. The solution of this problem represents the closest approximation of the sphere, i.e., the roundest polyhedra. A new numerical optimization method developed previously by the authors has been applied to optimize polyhedra to best approximate a sphere if tetrahedral, octahedral, or icosahedral symmetry constraints are applied. In addition to evidence provided for various cases of face numbers, potentially optimal polyhedra are also shown for n up to 132. Keywords: polyhedra; isoperimetric problem; point group symmetry 1. Introduction The so-called isoperimetric problem in mathematics is concerned with the determination of the shape of spatial (or planar) objects which have the largest possible volume (or area) enclosed with given surface area (or circumference). The isoperimetric problem for polyhedra can be reflected in a question as follows: What polyhedron maximizes the volume if the surface area and the number of faces n are given? The problem can be quantified by the so-called Steinitz number [1] S = A3/V2, a dimensionless quantity in terms of the surface area A and volume V of the polyhedron, such that solutions of the isoperimetric problem minimize S. -
1 Point for the Player Who Has the Shape with the Longest Side. Put
Put three pieces together to create a new 1 point for the player who has the shape. 1 point for the player who has shape with the longest side. the shape with the smallest number of sides. 1 point for the player who has the 1 point for the player with the shape that shape with the shortest side. has the longest perimeter. 1 point for the player with the shape that 1 point for each triangle. has the largest area. 1 point for each shape with 1 point for each quadrilateral. a perimeter of 24 cm. Put two pieces together to create a new shape. 1 point for the player with 1 point for each shape with the shape that has the longest a perimeter of 12 cm. perimeter. Put two pieces together to create a new shape. 1 point for the player with 1 point for each shape with the shape that has the shortest a perimeter of 20 cm. perimeter. 1 point for each 1 point for each shape with non-symmetrical shape. a perimeter of 19 cm. 1 point for the shape that has 1 point for each the largest number of sides. symmetrical shape. Count the sides on all 5 pieces. 1 point for each pair of shapes where the 1 point for the player who has the area of one is twice the area of the other. largest total number of sides. Put two pieces together to create a new shape. 1 point for the player who has 1 point for each shape with at least two the shape with the fewest number of opposite, parallel sides. -
Extremal Problems for Convex Polygons ∗
Extremal Problems for Convex Polygons ∗ Charles Audet Ecole´ Polytechnique de Montreal´ Pierre Hansen HEC Montreal´ Fred´ eric´ Messine ENSEEIHT-IRIT Abstract. Consider a convex polygon Vn with n sides, perimeter Pn, diameter Dn, area An, sum of distances between vertices Sn and width Wn. Minimizing or maximizing any of these quantities while fixing another defines ten pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research on these problems, which uses geometrical reasoning increasingly complemented by global optimization meth- ods. Numerous open problems are mentioned, as well as series of test problems for global optimization and nonlinear programming codes. Keywords: polygon, perimeter, diameter, area, sum of distances, width, isoperimeter problem, isodiametric problem. 1. Introduction Plane geometry is replete with extremal problems, many of which are de- scribed in the book of Croft, Falconer and Guy [12] on Unsolved problems in geometry. Traditionally, such problems have been solved, some since the Greeks, by geometrical reasoning. In the last four decades, this approach has been increasingly complemented by global optimization methods. This allowed solution of larger instances than could be solved by any one of these two approaches alone. Probably the best known type of such problems are circle packing ones: given a geometrical form such as a unit square, a unit-side triangle or a unit- diameter circle, find the maximum radius and configuration of n circles which can be packed in its interior (see [46] for a recent survey and the site [44] for a census of exact and approximate results with up to 300 circles). -
Of Great Rhombicuboctahedron/Archimedean Solid
Mathematical Analysis of Great Rhombicuboctahedron/Archimedean Solid Mr Harish Chandra Rajpoot M.M.M. University of Technology, Gorakhpur-273010 (UP), India March, 2015 Introduction: A great rhombicuboctahedron is an Archimedean solid which has 12 congruent square faces, 8 congruent regular hexagonal faces & 6 congruent regular octagonal faces each having equal edge length. It has 72 edges & 48 vertices lying on a spherical surface with a certain radius. It is created/generated by expanding a truncated cube having 8 equilateral triangular faces & 6 regular octagonal faces. Thus by the expansion, each of 12 originally truncated edges changes into a square face, each of 8 triangular faces of the original solid changes into a regular hexagonal face & 6 regular octagonal faces of original solid remain unchanged i.e. octagonal faces are shifted radially. Thus a solid with 12 squares, 8 hexagonal & 6 octagonal faces, is obtained which is called great rhombicuboctahedron which is an Archimedean solid. (See figure 1), thus we have Figure 1: A great rhombicuboctahedron having 12 ( ) ( ) congruent square faces, 8 congruent regular ( ) ( ) hexagonal faces & 6 congruent regular octagonal faces each of equal edge length 풂 ( ) ( ) We would apply HCR’s Theory of Polygon to derive a mathematical relationship between radius of the spherical surface passing through all 48 vertices & the edge length of a great rhombicuboctahedron for calculating its important parameters such as normal distance of each face, surface area, volume etc. Derivation of outer (circumscribed) radius ( ) of great rhombicuboctahedron: Let be the radius of the spherical surface passing through all 48 vertices of a great rhombicuboctahedron with edge length & the centre O. -
A Technology to Synthesize 360-Degree Video Based on Regular Dodecahedron in Virtual Environment Systems*
A Technology to Synthesize 360-Degree Video Based on Regular Dodecahedron in Virtual Environment Systems* Petr Timokhin 1[0000-0002-0718-1436], Mikhail Mikhaylyuk2[0000-0002-7793-080X], and Klim Panteley3[0000-0001-9281-2396] Federal State Institution «Scientific Research Institute for System Analysis of the Russian Academy of Sciences», Moscow, Russia 1 [email protected], 2 [email protected], 3 [email protected] Abstract. The paper proposes a new technology of creating panoramic video with a 360-degree view based on virtual environment projection on regular do- decahedron. The key idea consists in constructing of inner dodecahedron surface (observed by the viewer) composed of virtual environment snapshots obtained by twelve identical virtual cameras. A method to calculate such cameras’ projec- tion and orientation parameters based on “golden rectangles” geometry as well as a method to calculate snapshots position around the observer ensuring synthe- sis of continuous 360-panorama are developed. The technology and the methods were implemented in software complex and tested on the task of virtual observing the Earth from space. The research findings can be applied in virtual environment systems, video simulators, scientific visualization, virtual laboratories, etc. Keywords: Visualization, Virtual Environment, Regular Dodecahedron, Video 360, Panoramic Mapping, GPU. 1 Introduction In modern human activities the importance of the visualization of virtual prototypes of real objects and phenomena is increasingly grown. In particular, it’s especially required for training specialists in space and aviation industries, medicine, nuclear industry and other areas, where the mistake cost is extremely high [1-3]. The effectiveness of work- ing with virtual prototypes is significantly increased if an observer experiences a feeling of being present in virtual environment. -
Polygons and Convexity
Geometry Week 4 Sec 2.5 to ch. 2 test section 2.5 Polygons and Convexity Definitions: convex set – has the property that any two of its points determine a segment contained in the set concave set – a set that is not convex concave concave convex convex concave Definitions: polygon – a simple closed curve that consists only of segments side of a polygon – one of the segments that defines the polygon vertex – the endpoint of the side of a polygon 1 angle of a polygon – an angle with two properties: 1) its vertex is a vertex of the polygon 2) each side of the angle contains a side of the polygon polygon not a not a polygon (called a polygonal curve) polygon Definitions: polygonal region – a polygon together with its interior equilateral polygon – all sides have the same length equiangular polygon – all angels have the same measure regular polygon – both equilateral and equiangular Example: A square is equilateral, equiangular, and regular. 2 diagonal – a segment that connects 2 vertices but is not a side of the polygon C B C B D A D A E AC is a diagonal AC is a diagonal AB is not a diagonal AD is a diagonal AB is not a diagonal Notation: It does not matter which vertex you start with, but the vertices must be listed in order. Above, we have square ABCD and pentagon ABCDE. interior of a convex polygon – the intersection of the interiors of is angles exterior of a convex polygon – union of the exteriors of its angles 3 Polygon Classification Number of sides Name of polygon 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon -
Englewood Public School District Geometry Second Marking Period
Englewood Public School District Geometry Second Marking Period Unit 2: Polygons, Triangles, and Quadrilaterals Overview: During this unit, students will learn how to prove two triangles are congruent, relationships between angle measures and side lengths within a triangle, and about different types of quadrilaterals. Time Frame: 35 to 45 days (One Marking Period) Enduring Understandings: • Congruent triangles can be visualized by placing one on top of the other. • Corresponding sides and angles can be marked using tic marks and angle marks. • Theorems can be used to prove triangles congruent. • The definitions of isosceles and equilateral triangles can be used to classify a triangle. • The Midpoint Formula can be used to find the midsegment of a triangle. • The Distance Formula can be used to examine relationships in triangles. • Side lengths of triangles have a relationship. • The negation statement can be proved and used to show a counterexample. • The diagonals of a polygon can be used to derive the formula for the angle measures of the polygon. • The properties of parallel and perpendicular lines can be used to classify quadrilaterals. • Coordinate geometry can be used to classify special parallelograms. • Slope and the distance formula can be used to prove relationships in the coordinate plane. Essential Questions: • How do you identify corresponding parts of congruent triangles? • How do you show that two triangles are congruent? • How can you tell if a triangle is isosceles or equilateral? • How do you use coordinate geometry -
Midsegment of a Trapezoid Parallelogram
Vocabulary Flash Cards base angles of a trapezoid bases of a trapezoid Chapter 7 (p. 402) Chapter 7 (p. 402) diagonal equiangular polygon Chapter 7 (p. 364) Chapter 7 (p. 365) equilateral polygon isosceles trapezoid Chapter 7 (p. 365) Chapter 7 (p. 402) kite legs of a trapezoid Chapter 7 (p. 405) Chapter 7 (p. 402) Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved. Vocabulary Flash Cards The parallel sides of a trapezoid Either pair of consecutive angles whose common side is a base of a trapezoid A polygon in which all angles are congruent A segment that joins two nonconsecutive vertices of a polygon A trapezoid with congruent legs A polygon in which all sides are congruent The nonparallel sides of a trapezoid A quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved. Vocabulary Flash Cards midsegment of a trapezoid parallelogram Chapter 7 (p. 404) Chapter 7 (p. 372) rectangle regular polygon Chapter 7 (p. 392) Chapter 7 (p. 365) rhombus square Chapter 7 (p. 392) Chapter 7 (p. 392) trapezoid Chapter 7 (p. 402) Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved. Vocabulary Flash Cards A quadrilateral with both pairs of opposite sides The segment that connects the midpoints of the parallel legs of a trapezoid PQRS A convex polygon that is both equilateral and A parallelogram with four right angles equiangular A parallelogram with four congruent sides and four A parallelogram with four congruent sides right angles A quadrilateral with exactly one pair of parallel sides Copyright © Big Ideas Learning, LLC Big Ideas Math Geometry All rights reserved. -
Side of the Polygon Polygon Vertex of the Diagonal Sides Angles Regular
King Geometry – Chapter 6 – Notes and Examples Section 1 – Properties and Attributes of Polygons Each segment that forms a polygon is a _______side__ _ of ____ _the____ ______________. polygon The common endpoint of two sides is a ____________vertex _____of _______ the ________________polygon . A segment that connects any two nonconsecutive vertices is a _________________.diagonal Number of Sides Name of Polygon You can name a polygon by the number of its 3 Triangle sides. The table shows the names of some common polygons. 4 Quadrilateral 5 Pentagon If the number of sides is not listed in the table, then the polygon is called a n-gon where n 6 Hexagon represents the number of sides. Example: 16 sided figure is called a 16-gon 7 Heptagon 8 Octagon Remember! A polygon is a closed plane figure formed by 9 Nonagon three or more segments that intersect only at their endpoints. 10 Decagon 12 Dodecagon n n-gon All of the __________sides are congruent in an equilateral polygon. All of the ____________angles are congruent in an equiangular polygon. A __________________________regular polygon is one that is both equilateral and equiangular. If a polygon is ______not regular, it is called _________________irregular . A polygon is ______________concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in the exterior, then the polygon is ____________convex . A regular polygon is always _____________convex . Examples: Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. not a not a polygon, not a polygon, heptagon polygon hexagon polygon polygon Tell whether the polygon is regular or irregular. -
A Method for Predicting Multivariate Random Loads and a Discrete Approximation of the Multidimensional Design Load Envelope
International Forum on Aeroelasticity and Structural Dynamics IFASD 2017 25-28 June 2017 Como, Italy A METHOD FOR PREDICTING MULTIVARIATE RANDOM LOADS AND A DISCRETE APPROXIMATION OF THE MULTIDIMENSIONAL DESIGN LOAD ENVELOPE Carlo Aquilini1 and Daniele Parisse1 1Airbus Defence and Space GmbH Rechliner Str., 85077 Manching, Germany [email protected] Keywords: aeroelasticity, buffeting, combined random loads, multidimensional design load envelope, IFASD-2017 Abstract: Random loads are of greatest concern during the design phase of new aircraft. They can either result from a stationary Gaussian process, such as continuous turbulence, or from a random process, such as buffet. Buffet phenomena are especially relevant for fighters flying in the transonic regime at high angles of attack or when the turbulent, separated air flow induces fluctuating pressures on structures like wing surfaces, horizontal tail planes, vertical tail planes, airbrakes, flaps or landing gear doors. In this paper a method is presented for predicting n-dimensional combined loads in the presence of massively separated flows. This method can be used both early in the design process, when only numerical data are available, and later, when test data measurements have been performed. Moreover, the two-dimensional load envelope for combined load cases and its discretization, which are well documented in the literature, are generalized to n dimensions, n ≥ 3. 1 INTRODUCTION The main objectives of this paper are: 1. To derive a method for predicting multivariate loads (but also displacements, velocities and accelerations in terms of auto- and cross-power spectra, [1, pp. 319-320]) for aircraft, or parts of them, subjected to a random excitation. -
Bridges Conference Proceedings Guidelines
Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture Exploring the Vertices of a Triacontahedron Robert Weadon Rollings 883 Brimorton Drive, Scarborough, Ontario, M1G 2T8 E-mail: [email protected] Abstract The rhombic triacontahedron can be used as a framework for locating the vertices of the five platonic solids. The five models presented here, which are made of babinga wood and brass rods illustrate this relationship. Introduction My initial interest in the platonic solids began with the work of Cox [1]. My initial explorations of the platonic solids lead to the series of models Polyhedra through the Beauty of Wood [2]. I have continued this exploration by using a rhombic triacontahedron at the core of my models and illustrating its relationship to all five platonic solids. Rhombic Triacontahedron The rhombic triacontahedron has 30 rhombic faces where the ratio of the diagonals is the golden ratio. The models shown in Figures 1, 2, 4, and 5 illustrate that the vertices of an icosahedron, dodecahedron, hexahedron and tetrahedron all lie on the circumsphere of the rhombic triacontahedron. In addition, Figure 3 shows that the vertices of the octahedron lie on the inscribed sphere of the rhombic triacontahedron. Creating the Models Each of the five rhombic triacontahedrons are made of 1/4 inch thick babinga wood. The edges of the rhombi were cut at 18 degrees on a band saw. After the assembly of the triacontahedron 1/8 inch brass rods were inserted a specific vertices and then fitted with 1/2 inch cocabola wood spheres. Different vertices (or faces in the case of the octahedron) were chosen for each platonic solid.