A Treatise on the Circle and the Sphere
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A Genetic Context for Understanding the Trigonometric Functions Danny Otero Xavier University, [email protected]
Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Pre-calculus and Trigonometry Mathematics via Primary Historical Sources (TRIUMPHS) Spring 3-2017 A Genetic Context for Understanding the Trigonometric Functions Danny Otero Xavier University, [email protected] Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_precalc Part of the Curriculum and Instruction Commons, Educational Methods Commons, Higher Education Commons, and the Science and Mathematics Education Commons Click here to let us know how access to this document benefits oy u. Recommended Citation Otero, Danny, "A Genetic Context for Understanding the Trigonometric Functions" (2017). Pre-calculus and Trigonometry. 1. https://digitalcommons.ursinus.edu/triumphs_precalc/1 This Course Materials is brought to you for free and open access by the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) at Digital Commons @ Ursinus College. It has been accepted for inclusion in Pre-calculus and Trigonometry by an authorized administrator of Digital Commons @ Ursinus College. For more information, please contact [email protected]. A Genetic Context for Understanding the Trigonometric Functions Daniel E. Otero∗ July 22, 2019 Trigonometry is concerned with the measurements of angles about a central point (or of arcs of circles centered at that point) and quantities, geometrical and otherwise, that depend on the sizes of such angles (or the lengths of the corresponding arcs). It is one of those subjects that has become a standard part of the toolbox of every scientist and applied mathematician. It is the goal of this project to impart to students some of the story of where and how its central ideas first emerged, in an attempt to provide context for a modern study of this mathematical theory. -
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. -
Examples of Manifolds
Examples of Manifolds Example 1 (Open Subset of IRn) Any open subset, O, of IRn is a manifold of dimension n. One possible atlas is A = (O, ϕid) , where ϕid is the identity map. That is, ϕid(x) = x. n Of course one possible choice of O is IR itself. Example 2 (The Circle) The circle S1 = (x,y) ∈ IR2 x2 + y2 = 1 is a manifold of dimension one. One possible atlas is A = {(U , ϕ ), (U , ϕ )} where 1 1 1 2 1 y U1 = S \{(−1, 0)} ϕ1(x,y) = arctan x with − π < ϕ1(x,y) <π ϕ1 1 y U2 = S \{(1, 0)} ϕ2(x,y) = arctan x with 0 < ϕ2(x,y) < 2π U1 n n n+1 2 2 Example 3 (S ) The n–sphere S = x =(x1, ··· ,xn+1) ∈ IR x1 +···+xn+1 =1 n A U , ϕ , V ,ψ i n is a manifold of dimension . One possible atlas is 1 = ( i i) ( i i) 1 ≤ ≤ +1 where, for each 1 ≤ i ≤ n + 1, n Ui = (x1, ··· ,xn+1) ∈ S xi > 0 ϕi(x1, ··· ,xn+1)=(x1, ··· ,xi−1,xi+1, ··· ,xn+1) n Vi = (x1, ··· ,xn+1) ∈ S xi < 0 ψi(x1, ··· ,xn+1)=(x1, ··· ,xi−1,xi+1, ··· ,xn+1) n So both ϕi and ψi project onto IR , viewed as the hyperplane xi = 0. Another possible atlas is n n A2 = S \{(0, ··· , 0, 1)}, ϕ , S \{(0, ··· , 0, −1)},ψ where 2x1 2xn ϕ(x , ··· ,xn ) = , ··· , 1 +1 1−xn+1 1−xn+1 2x1 2xn ψ(x , ··· ,xn ) = , ··· , 1 +1 1+xn+1 1+xn+1 are the stereographic projections from the north and south poles, respectively. -
An Introduction to Topology the Classification Theorem for Surfaces by E
An Introduction to Topology An Introduction to Topology The Classification theorem for Surfaces By E. C. Zeeman Introduction. The classification theorem is a beautiful example of geometric topology. Although it was discovered in the last century*, yet it manages to convey the spirit of present day research. The proof that we give here is elementary, and its is hoped more intuitive than that found in most textbooks, but in none the less rigorous. It is designed for readers who have never done any topology before. It is the sort of mathematics that could be taught in schools both to foster geometric intuition, and to counteract the present day alarming tendency to drop geometry. It is profound, and yet preserves a sense of fun. In Appendix 1 we explain how a deeper result can be proved if one has available the more sophisticated tools of analytic topology and algebraic topology. Examples. Before starting the theorem let us look at a few examples of surfaces. In any branch of mathematics it is always a good thing to start with examples, because they are the source of our intuition. All the following pictures are of surfaces in 3-dimensions. In example 1 by the word “sphere” we mean just the surface of the sphere, and not the inside. In fact in all the examples we mean just the surface and not the solid inside. 1. Sphere. 2. Torus (or inner tube). 3. Knotted torus. 4. Sphere with knotted torus bored through it. * Zeeman wrote this article in the mid-twentieth century. 1 An Introduction to Topology 5. -
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. -
The Sphere Project
;OL :WOLYL 7YVQLJ[ +XPDQLWDULDQ&KDUWHUDQG0LQLPXP 6WDQGDUGVLQ+XPDQLWDULDQ5HVSRQVH ;OL:WOLYL7YVQLJ[ 7KHULJKWWROLIHZLWKGLJQLW\ ;OL:WOLYL7YVQLJ[PZHUPUP[PH[P]L[VKL[LYTPULHUK WYVTV[LZ[HUKHYKZI`^OPJO[OLNSVIHSJVTT\UP[` /\THUP[HYPHU*OHY[LY YLZWVUKZ[V[OLWSPNO[VMWLVWSLHMMLJ[LKI`KPZHZ[LYZ /\THUP[HYPHU >P[O[OPZ/HUKIVVR:WOLYLPZ^VYRPUNMVYH^VYSKPU^OPJO [OLYPNO[VMHSSWLVWSLHMMLJ[LKI`KPZHZ[LYZ[VYLLZ[HISPZO[OLPY *OHY[LYHUK SP]LZHUKSP]LSPOVVKZPZYLJVNUPZLKHUKHJ[LK\WVUPU^H`Z[OH[ YLZWLJ[[OLPY]VPJLHUKWYVTV[L[OLPYKPNUP[`HUKZLJ\YP[` 4PUPT\T:[HUKHYKZ This Handbook contains: PU/\THUP[HYPHU (/\THUP[HYPHU*OHY[LY!SLNHSHUKTVYHSWYPUJPWSLZ^OPJO HUK YLÅLJ[[OLYPNO[ZVMKPZHZ[LYHMMLJ[LKWVW\SH[PVUZ 4PUPT\T:[HUKHYKZPU/\THUP[HYPHU9LZWVUZL 9LZWVUZL 7YV[LJ[PVU7YPUJPWSLZ *VYL:[HUKHYKZHUKTPUPT\TZ[HUKHYKZPUMV\YRL`SPMLZH]PUN O\THUP[HYPHUZLJ[VYZ!>H[LYZ\WWS`ZHUP[H[PVUHUKO`NPLUL WYVTV[PVU"-VVKZLJ\YP[`HUKU\[YP[PVU":OLS[LYZL[[SLTLU[HUK UVUMVVKP[LTZ"/LHS[OHJ[PVU;OL`KLZJYPILwhat needs to be achieved in a humanitarian response in order for disaster- affected populations to survive and recover in stable conditions and with dignity. ;OL:WOLYL/HUKIVVRLUQV`ZIYVHKV^ULYZOPWI`HNLUJPLZHUK PUKP]PK\HSZVMMLYPUN[OLO\THUP[HYPHUZLJ[VYH common language for working together towards quality and accountability in disaster and conflict situations ;OL:WOLYL/HUKIVVROHZHU\TILYVMºJVTWHUPVU Z[HUKHYKZ»L_[LUKPUNP[ZZJVWLPUYLZWVUZL[VULLKZ[OH[ OH]LLTLYNLK^P[OPU[OLO\THUP[HYPHUZLJ[VY ;OL:WOLYL7YVQLJ[^HZPUP[PH[LKPU I`HU\TILY VMO\THUP[HYPHU5.6ZHUK[OL9LK*YVZZHUK 9LK*YLZJLU[4V]LTLU[ ,+0;065 7KHB6SKHUHB3URMHFWBFRYHUBHQBLQGG The Sphere Project Humanitarian Charter and Minimum Standards in Humanitarian Response Published by: The Sphere Project Copyright@The Sphere Project 2011 Email: [email protected] Website : www.sphereproject.org The Sphere Project was initiated in 1997 by a group of NGOs and the Red Cross and Red Crescent Movement to develop a set of universal minimum standards in core areas of humanitarian response: the Sphere Handbook. -
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. -
Understand the Principles and Properties of Axiomatic (Synthetic
Michael Bonomi Understand the principles and properties of axiomatic (synthetic) geometries (0016) Euclidean Geometry: To understand this part of the CST I decided to start off with the geometry we know the most and that is Euclidean: − Euclidean geometry is a geometry that is based on axioms and postulates − Axioms are accepted assumptions without proofs − In Euclidean geometry there are 5 axioms which the rest of geometry is based on − Everybody had no problems with them except for the 5 axiom the parallel postulate − This axiom was that there is only one unique line through a point that is parallel to another line − Most of the geometry can be proven without the parallel postulate − If you do not assume this postulate, then you can only prove that the angle measurements of right triangle are ≤ 180° Hyperbolic Geometry: − We will look at the Poincare model − This model consists of points on the interior of a circle with a radius of one − The lines consist of arcs and intersect our circle at 90° − Angles are defined by angles between the tangent lines drawn between the curves at the point of intersection − If two lines do not intersect within the circle, then they are parallel − Two points on a line in hyperbolic geometry is a line segment − The angle measure of a triangle in hyperbolic geometry < 180° Projective Geometry: − This is the geometry that deals with projecting images from one plane to another this can be like projecting a shadow − This picture shows the basics of Projective geometry − The geometry does not preserve length -
Circles in Areals
Circles in areals 23 December 2015 This paper has been accepted for publication in the Mathematical Gazette, and copyright is assigned to the Mathematical Association (UK). This preprint may be read or downloaded, and used for any non-profit making educational or research purpose. Introduction August M¨obiusintroduced the system of barycentric or areal co-ordinates in 1827[1, 2]. The idea is that one may attach weights to points, and that a system of weights determines a centre of mass. Given a triangle ABC, one obtains a co-ordinate system for the plane by placing weights x; y and z at the vertices (with x + y + z = 1) to describe the point which is the centre of mass. The vertices have co-ordinates A = (1; 0; 0), B = (0; 1; 0) and C = (0; 0; 1). By scaling so that triangle ABC has area [ABC] = 1, we can take the co-ordinates of P to be ([P BC]; [PCA]; [P AB]), provided that we take area to be signed. Our convention is that anticlockwise triangles have positive area. Points inside the triangle have strictly positive co-ordinates, and points outside the triangle must have at least one negative co-ordinate (we are allowed negative masses). The equation of a line looks like the equation of a plane in Cartesian co-ordinates, but note that the equation lx + ly + lz = 0 (with l 6= 0) is not satisfied by any point (x; y; z) of the Euclidean plane. We use such equations to describe the line at infinity when doing projective geometry. -
A Generalization of the Conway Circle
Forum Geometricorum b Volume 13 (2013) 191–195. b b FORUM GEOM ISSN 1534-1178 A Generalization of the Conway Circle Francisco Javier Garc´ıa Capitan´ Abstract. For any point in the plane of the triangle we show a conic that be- comes the Conway circle in the case of the incenter. We give some properties of the conic and of the configuration of the six points that define it. Let ABC be a triangle and I its incenter. Call Ba the point on line CA in the opposite direction to AC such that ABa = BC = a and Ca the point on line BA in the opposite direction to AB such that ACa = a. Define Cb, Ab and Ac, Bc cyclically. The six points Ab, Ac, Bc, Ba, Ca, Cb lie in a circle called the Conway circle with I as center and squared radius r2 + s2 as indicated in Figure 1. This configuration also appeared in Problem 6 in the 1992 Iberoamerican Mathematical Olympiad. The problem asks to establish that the area of the hexagon CaBaAbCbBcAc is at least 13∆(ABC) (see [4]). Ca Ca Ba Ba a a a a A A C0 B0 I I r Ab Ac Ab Ac b B s − b s − c C c b B s − b s − c C c c c Bc Bc Cb Cb Figure 1. The Conway circle Figure 2 Figure 2 shows a construction of these points which can be readily generalized. The lines BI and CI intersect the parallel of BC through A at B0 and C0. -
Coaxal Pencil of Circles and Spheres in the Pavillet Tetrahedron
17TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS ©2016 ISGG 4–8 AUGUST, 2016, BEIJING, CHINA COAXAL PENCILS OF CIRCLES AND SPHERES IN THE PAVILLET TETRAHEDRON Axel PAVILLET [email protected] ABSTRACT: After a brief review of the properties of the Pavillet tetrahedron, we recall a theorem about the trace of a coaxal pencil of spheres on a plane. Then we use this theorem to show a re- markable correspondence between the circles of the base and those of the upper triangle of a Pavillet tetrahedron. We also give new proofs and new point of view of some properties of the Bevan point of a triangle using solid triangle geometry. Keywords: Tetrahedron, orthocentric, coaxal pencil, circles, spheres polar 2 Known properties We first recall the notations and properties (with 1 Introduction. their reference) we will use in this paper (Fig.1). • The triangle ABC is called the base trian- gle and defines the base plane (horizontal). The orthocentric tetrahedron of a scalene tri- angle [12], named the Pavillet tetrahedron by • The incenter, I, is called the apex of the Richard Guy [8, Ch. 5] and Gunther Weiss [16], tetrahedron. is formed by drawing from the vertices A, B and 0 0 0 C of a triangle ABC, on an horizontal plane, • The other three vertices (A ,B ,C ) form a three vertical segments AA0 = AM = AL = x, triangle called the upper triangle and define BB0 = BK = BM = y, CC0 = CL = CK = z, a plane called the upper plane. where KLM is the contact triangle of ABC. We As a standard notation, all points lying on the denote I the incenter of ABC, r its in-radius, base plane will have (as much as possible) a 0 0 0 and consider the tetrahedron IA B C . -
On a Construction of Hagge
Forum Geometricorum b Volume 7 (2007) 231–247. b b FORUM GEOM ISSN 1534-1178 On a Construction of Hagge Christopher J. Bradley and Geoff C. Smith Abstract. In 1907 Hagge constructed a circle associated with each cevian point P of triangle ABC. If P is on the circumcircle this circle degenerates to a straight line through the orthocenter which is parallel to the Wallace-Simson line of P . We give a new proof of Hagge’s result by a method based on reflections. We introduce an axis associated with the construction, and (via an areal anal- ysis) a conic which generalizes the nine-point circle. The precise locus of the orthocenter in a Brocard porism is identified by using Hagge’s theorem as a tool. Other natural loci associated with Hagge’s construction are discussed. 1. Introduction One hundred years ago, Karl Hagge wrote an article in Zeitschrift fur¨ Mathema- tischen und Naturwissenschaftliche Unterricht entitled (in loose translation) “The Fuhrmann and Brocard circles as special cases of a general circle construction” [5]. In this paper he managed to find an elegant extension of the Wallace-Simson theorem when the generating point is not on the circumcircle. Instead of creating a line, one makes a circle through seven important points. In 2 we give a new proof of the correctness of Hagge’s construction, extend and appl§ y the idea in various ways. As a tribute to Hagge’s beautiful insight, we present this work as a cente- nary celebration. Note that the name Hagge is also associated with other circles [6], but here we refer only to the construction just described.