DOCSLIB.ORG

Algebra and Geometry of Hamilton's Quaternions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algebra and Geometry of Hamilton's Quaternions GENERAL ARTICLE Algebra and Geometry of Hamilton’s Quaternions ‘Well, Papa, Can You Multiply Triplets?’ Govind S Krishnaswami and Sonakshi Sachdev Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three-dimensional geometry. Un- able to multiply and divide triples, he invented a non-commutative division algebra of quadru- ples, in what he considered his most significant (left) Govind Krishnaswami is on the faculty of the work, generalizing the real and complex number Chennai Mathematical systems. We give a motivated introduction to Institute. His research in quaternions and discuss how they are related to theoretical physics spans Pauli matrices, rotations in three dimensions, the various topics three sphere, the group SU(2) and the celebrated from quantum field theory and particle physics to fluid, Hopf fibrations. plasma and non-linear dynamics. 1. Introduction (right) Sonakshi Sachdev is Every morning in the early part of the above-cited month a PhD student at the [October 1843], on my coming down to breakfast, your Chennai Mathematical (then) little brother William Edwin, and yourself, used Institute. She has worked on to ask me, “Well, Papa, can you multiply triplets”? the dynamics of rigid bodies, Whereto I was always obliged to reply, with a sad shake fluids and plasmas. of the head: “No, I can only add and subtract them.” W R Hamilton in a letter dated August 5, 1865 to his son A H Hamilton [1]. Imaginary and complex numbers arose in looking for ‘impossible’ solutions to polynomial equations such as x2 + 1 = 0. Even a sceptic might be convinced of their utility upon seeing the remarkable Ferro–Tartaglia–Car- dano–Bombelli formula for a real root of cubic equa- tions such as x3 =20x + 25, expressed as sums of cube roots of complex numbers! No such general expression in Keywords terms of real radicals is possible. Some of the mystery Quaternions, division algebra, surrounding complex numbers was removed once they rotations, spheres, Hopf map. RESONANCE ⎜ June 2016 529 GENERAL ARTICLE could be visualized as vectors in the plane, following the work of Euler, Wessel, Argand and Gauss. Hamilton entered the scene in 1830 by defining complex numbers z = x + yi as ordered pairs (or ‘couples’) (x, y)ofreal numbers. He abstracted rules for adding, multiplying and dividing ordered pairs obeying the commutative, 1 1 Commutative means z1 z2 = z2 associative and distributive laws of arithmetic .Thus z , associative means z (z z ) = 1 1 2 3 the complex numbers C constitute a ‘field’ just like the (z z ) z while z (z + z ) = z z 1 2 3 1 2 3 1 2 reals R. The manner in which the geometry of the plane + z z is distributivity. 1 3 was encoded in the algebra of complex numbers2 made 2 The parallelogram law for add- a deep impression on Hamilton. The problem of finding ing vectors is just the sum of an algebra of triples (α, β, γ) to describe the geometry complex numbers, a rotation cor- of vectors in three-dimensional (3D) space haunted him responds to multiplication by a for at least fifteen years. While he knew how to add complex number of unit magni- tude, the cosine of the angle and subtract triples, the problem of multiplying them between vectors is given in terms and dividing by non-zero triples seemed insurmountable. of the scalar product of the cor- What is more, by analogy with the complex numbers, responding complex numbers, he wanted the Euclidean length (square-root of the sum etc. of squares of the components) of the product of a pair 3 The cross product of vectors of triples to equal the product of their lengths3.Incur- had not yet been discovered. rent language, Hamilton was looking for a real, three- Besides being non-associative dimensional, normed, associative, division algebra; we a × (b × c) ≠ (a × b) × c, it does not satisfy the condition | a × b| now know that such an algebra does not exist. = |a||b|. Cross products were invented by Grassmann within a In 1843, Hamilton found an ingenious way around this year of Hamilton's discovery of problem, combining two major innovations. The first quaternions. was to drop the commutative law for multiplication. He wrote a triple as t = α+βi+γj and assumed by analogy with the complex numbers, that i2 = j2 = −1. It follows that t2 = α2 − β2 − γ2 +2αβi +2αγj + βγ(ij + ji). (1) Moreover, he wanted the ‘law of moduli’ |t|2 = |t2| to hold. Now |t|2 = α2 +β2 +γ2 would equal |t2| if ij+ji = 4 The quotations in this para- 0, since graph are from a letter dated October 17, 1843 from Hamilton (α2 −β2 −γ2)2 +(2αβ)2 +(2αγ)2 =(α2 +β2 +γ2)2. (2) to J T Graves, which was re- printed in Vol.xxv, pp.489–495 Hamilton was “tempted for a moment to fancy that” of [2]. ij = −ji =04. But this “seemed uncomfortable” and 530 RESONANCE ⎜ June 2016 GENERAL ARTICLE he settled on the “less harsh” condition ij = −ji = k, reserving to himself to “inquire whether k was 0 or not”. The second innovation, was to “admit in some sense a fourth dimension of space for the purpose of calculating 5 k with triples” . In other words, he assumed that was 5 In taking this step, Hamilton linearly independent of 1, i and j. Applying the asso- was partly motivated by specu- ciative law, he inferred that k2 = ijij = −1, ki = j lations on how time and space and jk = i. Hamilton vividly describes his elation upon may be “girdled” together, as 6 well as some vague notions discovering these multiplication rules: “They started (based on Kantian philosophy) into life, or light, full grown, on the 16th of October, of geometry and arithmetic be- 1843, as I was walking with Lady Hamilton to Dublin, ing the sciences of space and and came up to Brougham Bridge, which my boys have time respectively [3]. since called the Quaternion Bridge. That is to say, I 6 The article by N Mukunda in then and there felt the galvanic circuit of thought close; this issue of Resonance places and the sparks which fell from it were the fundamental Hamilton's discovery of quater- equations in i, j, k exactly such as I have used them ever nions in the wider context of his life and work. See also the ar- since.” Extract from a letter dated October 15, 1858 ticle by R Nityananda devoted to from Hamilton to P G Tait [1]. Hamilton's work in optics. In more recent language [4, 5], quaternions are quadru- ples of real numbers q =(α, β, γ, δ) forming a four- dimensional real vector space H (named after Hamilton). In terms of the canonical basis: Figure 1. Plaque on Broome Bridge, Dublin, commemorat- ing Hamilton’s discovery of quaternions. Photograph by Brian Dolan, In- genious Ireland (http://ingeniousireland.ie), re- printed with permission RESONANCE ⎜ June 2016 531 GENERAL ARTICLE e =(1, 0, 0, 0),i=(0, 1, 0, 0),j=(0, 0, 1, 0) and k =(0, 0, 0, 1), (3) q = αe + βi+ γj + δk. It is natural to write q = αe + u, where αe is called the ‘real’ or ‘scalar’ part of q.The other part u = βi+γj+δk may be regarded as a vector in three-dimensional space and is called the ‘imaginary’ or ‘vectorial’ part of q. In fact, the word vector first appears in Hamilton’s writings of 1845. Quaternions derive Quaternions derive their power from the fact that they their power from the can be multiplied with each other, so that they form a fact that real algebra (a vector space where vectors can be mul- they can be multiplied tiplied to get other vectors). Since any quaternion is a linear combination of the basis quaternions e, i, j and k, with each other, so it suffices to specify their products. First, e is taken to that they form a real be the multiplicative identity (sometimes denoted 1), algebra (a vector so eq = qe = q for any quaternion q. In addition, space where vectors i2 = j2 = k2 = −e, while can be multiplied to get other ij = −ji = k, jk = −kj = i and ki = −ik = j. (4) vectors). Notice the analogy with cross products of unit vectors ˆi × jˆ = kˆ in 3D Euclidean space. The formula for the product of two quaternions q = (α, u = βi + γj + δk)andq =(α, u = βi + γj + δk) can be computed using the quaternion multiplication table (Table 1) and written compactly in terms of the dot and cross products of their vectorial parts: e i j k e e i j k i i −e k −j j j −k −e i Table 1. Hamilton’s quater- k k j −i −e nion relations. 532 RESONANCE ⎜ June 2016 GENERAL ARTICLE qq =(αα + ββ i2 + γγ j2 + δδ k2, Two quaternions (βα + αβ )i +(γα + αγ )j +(δα + αδ )k commute if and + βγ (ij)+βδ (ik)+γβ (ji) only if their vector + γδ (jk)+δβ (ki)+δγ (kj)) parts are collinear. =(αα − u · u,αu + αu +(βγ − γβ)k +(δβ − βδ)j +(γδ − δγ)i) =(αα − u · u,αu + αu + u × u) . (5) Two quaternions commute qq = qq iff their vector parts are collinear: u × u = 0. Moreover, the two- sided multiplicative inverse (reciprocal) of q is q−1 = (α2 + u2)−1(α, −u). Quaternion multiplication allows us to define polynomials in a quaternion variable. Inter- estingly, a polynomial of the nth degree may have more than n quaternion roots, see Box 1 for an example.
Load more

Recommended publications
  • Relativistic Dynamics
    Chapter 4 Relativistic dynamics We have seen in the previous lectures that our relativity postulates suggest that the most efficient (lazy but smart) approach to relativistic physics is in terms of 4-vectors, and that velocities never exceed c in magnitude. In this chapter we will see how this 4-vector approach works for dynamics, i.e., for the interplay between motion and forces. A particle subject to forces will undergo non-inertial motion. According to Newton, there is a simple (3-vector) relation between force and acceleration, f~ = m~a; (4.0.1) where acceleration is the second time derivative of position, d~v d2~x ~a = = : (4.0.2) dt dt2 There is just one problem with these relations | they are wrong! Newtonian dynamics is a good approximation when velocities are very small compared to c, but outside of this regime the relation (4.0.1) is simply incorrect. In particular, these relations are inconsistent with our relativity postu- lates. To see this, it is sufficient to note that Newton's equations (4.0.1) and (4.0.2) predict that a particle subject to a constant force (and initially at rest) will acquire a velocity which can become arbitrarily large, Z t ~ d~v 0 f ~v(t) = 0 dt = t ! 1 as t ! 1 . (4.0.3) 0 dt m This flatly contradicts the prediction of special relativity (and causality) that no signal can propagate faster than c. Our task is to understand how to formulate the dynamics of non-inertial particles in a manner which is consistent with our relativity postulates (and then verify that it matches observation, including in the non-relativistic regime).
    [Show full text]
  • How to Show That Various Numbers Either Can Or Cannot Be Constructed Using Only a Straightedge and Compass
    How to show that various numbers either can or cannot be constructed using only a straightedge and compass Nick Janetos June 3, 2010 1 Introduction It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square on one side... -Indiana House Bill No. 246, 1897 Three problems of classical Greek geometry are to do the following using only a compass and a straightedge: 1. To "square the circle": Given a circle, to construct a square of the same area, 2. To "trisect an angle": Given an angle, to construct another angle 1/3 of the original angle, 3. To "double the cube": Given a cube, to construct a cube with twice the area. Unfortunately, it is not possible to complete any of these tasks until additional tools (such as a marked ruler) are provided. In section 2 we will examine the process of constructing numbers using a compass and straightedge. We will then express constructions in algebraic terms. In section 3 we will derive several results about transcendental numbers. There are two goals: One, to show that the numbers e and π are transcendental, and two, to show that the three classical geometry problems are unsolvable. The two goals, of course, will turn out to be related. 2 Constructions in the plane The discussion in this section comes from [8], with some parts expanded and others removed. The classical Greeks were clear on what constitutes a construction. Given some set of points, new points can be defined at the intersection of lines with other lines, or lines with circles, or circles with circles.
    [Show full text]
  • More on Vectors Math 122 Calculus III D Joyce, Fall 2012
    More on Vectors Math 122 Calculus III D Joyce, Fall 2012 Unit vectors. A unit vector is a vector whose length is 1. If a unit vector u in the plane R2 is placed in standard position with its tail at the origin, then it's head will land on the unit circle x2 + y2 = 1. Every point on the unit circle (x; y) is of the form (cos θ; sin θ) where θ is the angle measured from the positive x-axis in the counterclockwise direction. u=(x;y)=(cos θ; sin θ) 7 '$θ q &% Thus, every unit vector in the plane is of the form u = (cos θ; sin θ). We can interpret unit vectors as being directions, and we can use them in place of angles since they carry the same information as an angle. In three dimensions, we also use unit vectors and they will still signify directions. Unit 3 vectors in R correspond to points on the sphere because if u = (u1; u2; u3) is a unit vector, 2 2 2 3 then u1 + u2 + u3 = 1. Each unit vector in R carries more information than just one angle since, if you want to name a point on a sphere, you need to give two angles, longitude and latitude. Now that we have unit vectors, we can treat every vector v as a length and a direction. The length of v is kvk, of course. And its direction is the unit vector u in the same direction which can be found by v u = : kvk The vector v can be reconstituted from its length and direction by multiplying v = kvk u.
    [Show full text]
  • Modular Arithmetic
    CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5 Modular Arithmetic One way to think of modular arithmetic is that it limits numbers to a predefined range f0;1;:::;N ¡ 1g, and wraps around whenever you try to leave this range — like the hand of a clock (where N = 12) or the days of the week (where N = 7). Example: Calculating the day of the week. Suppose that you have mapped the sequence of days of the week (Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday) to the sequence of numbers (0;1;2;3;4;5;6) so that Sunday is 0, Monday is 1, etc. Suppose that today is Thursday (=4), and you want to calculate what day of the week will be 10 days from now. Intuitively, the answer is the remainder of 4 + 10 = 14 when divided by 7, that is, 0 —Sunday. In fact, it makes little sense to add a number like 10 in this context, you should probably find its remainder modulo 7, namely 3, and then add this to 4, to find 7, which is 0. What if we want to continue this in 10 day jumps? After 5 such jumps, we would have day 4 + 3 ¢ 5 = 19; which gives 5 modulo 7 (Friday). This example shows that in certain circumstances it makes sense to do arithmetic within the confines of a particular number (7 in this example), that is, to do arithmetic by always finding the remainder of each number modulo 7, say, and repeating this for the results, and so on.
    [Show full text]
  • Lecture 3.Pdf
    ENGR-1100 Introduction to Engineering Analysis Lecture 3 POSITION VECTORS & FORCE VECTORS Today’s Objectives: Students will be able to : a) Represent a position vector in Cartesian coordinate form, from given geometry. In-Class Activities: • Applications / b) Represent a force vector directed along Relevance a line. • Write Position Vectors • Write a Force Vector along a line 1 DOT PRODUCT Today’s Objective: Students will be able to use the vector dot product to: a) determine an angle between In-Class Activities: two vectors, and, •Applications / Relevance b) determine the projection of a vector • Dot product - Definition along a specified line. • Angle Determination • Determining the Projection APPLICATIONS This ship’s mooring line, connected to the bow, can be represented as a Cartesian vector. What are the forces in the mooring line and how do we find their directions? Why would we want to know these things? 2 APPLICATIONS (continued) This awning is held up by three chains. What are the forces in the chains and how do we find their directions? Why would we want to know these things? POSITION VECTOR A position vector is defined as a fixed vector that locates a point in space relative to another point. Consider two points, A and B, in 3-D space. Let their coordinates be (XA, YA, ZA) and (XB, YB, ZB), respectively. 3 POSITION VECTOR The position vector directed from A to B, rAB , is defined as rAB = {( XB –XA ) i + ( YB –YA ) j + ( ZB –ZA ) k }m Please note that B is the ending point and A is the starting point.
    [Show full text]
  • Discrete Mathematics
    Slides for Part IA CST 2016/17 Discrete Mathematics <www.cl.cam.ac.uk/teaching/1617/DiscMath> Prof Marcelo Fiore [email protected] — 0 — What are we up to ? ◮ Learn to read and write, and also work with, mathematical arguments. ◮ Doing some basic discrete mathematics. ◮ Getting a taste of computer science applications. — 2 — What is Discrete Mathematics ? from Discrete Mathematics (second edition) by N. Biggs Discrete Mathematics is the branch of Mathematics in which we deal with questions involving finite or countably infinite sets. In particular this means that the numbers involved are either integers, or numbers closely related to them, such as fractions or ‘modular’ numbers. — 3 — What is it that we do ? In general: Build mathematical models and apply methods to analyse problems that arise in computer science. In particular: Make and study mathematical constructions by means of definitions and theorems. We aim at understanding their properties and limitations. — 4 — Lecture plan I. Proofs. II. Numbers. III. Sets. IV. Regular languages and finite automata. — 6 — Proofs Objectives ◮ To develop techniques for analysing and understanding mathematical statements. ◮ To be able to present logical arguments that establish mathematical statements in the form of clear proofs. ◮ To prove Fermat’s Little Theorem, a basic result in the theory of numbers that has many applications in computer science. — 16 — Proofs in practice We are interested in examining the following statement: The product of two odd integers is odd. This seems innocuous enough, but it is in fact full of baggage. — 18 — Proofs in practice We are interested in examining the following statement: The product of two odd integers is odd.
    [Show full text]
  • EE 595 (PMP) Advanced Topics in Communication Theory Handout #1
    EE 595 (PMP) Advanced Topics in Communication Theory Handout #1 Introduction to Cryptography. Symmetric Encryption.1 Wednesday, January 13, 2016 Tamara Bonaci Department of Electrical Engineering University of Washington, Seattle Outline: 1. Review - Security goals 2. Terminology 3. Secure communication { Symmetric vs. asymmetric setting 4. Background: Modular arithmetic 5. Classical cryptosystems { The shift cipher { The substitution cipher 6. Background: The Euclidean Algorithm 7. More classical cryptosystems { The affine cipher { The Vigen´erecipher { The Hill cipher { The permutation cipher 8. Cryptanalysis { The Kerchoff Principle { Types of cryptographic attacks { Cryptanalysis of the shift cipher { Cryptanalysis of the affine cipher { Cryptanalysis of the Vigen´erecipher { Cryptanalysis of the Hill cipher Review - Security goals Last lecture, we introduced the following security goals: 1. Confidentiality - ability to keep information secret from all but authorized users. 2. Data integrity - property ensuring that messages to and from a user have not been corrupted by communication errors or unauthorized entities on their way to a destination. 3. Identity authentication - ability to confirm the unique identity of a user. 4. Message authentication - ability to undeniably confirm message origin. 5. Authorization - ability to check whether a user has permission to conduct some action. 6. Non-repudiation - ability to prevent the denial of previous commitments or actions (think of a con- tract). 7. Certification - endorsement of information by a trusted entity. In the rest of today's lecture, we will focus on three of these goals, namely confidentiality, integrity and authentication (CIA). In doing so, we begin by introducing the necessary terminology. 1 We thank Professors Radha Poovendran and Andrew Clark for the help in preparing this material.
    [Show full text]
  • Operations with Complex Numbers Adding & Subtracting: Combine Like Terms (풂 + 풃풊) + (풄 + 풅풊) = (풂 + 풄) + (풃 + 풅)풊 Examples: 1
    Name: __________________________________________________________ Date: _________________________ Period: _________ Chapter 2: Polynomial and Rational Functions Topic 1: Complex Numbers What is an imaginary number? What is a complex number? The imaginary unit is defined as 풊 = √−ퟏ A complex number is defined as the set of all numbers in the form of 푎 + 푏푖, where 푎 is the real component and 푏 is the coefficient of the imaginary component. An imaginary number is when the real component (푎) is zero. Checkpoint: Since 풊 = √−ퟏ Then 풊ퟐ = Operations with Complex Numbers Adding & Subtracting: Combine like terms (풂 + 풃풊) + (풄 + 풅풊) = (풂 + 풄) + (풃 + 풅)풊 Examples: 1. (5 − 11푖) + (7 + 4푖) 2. (−5 + 7푖) − (−11 − 6푖) 3. (5 − 2푖) + (3 + 3푖) 4. (2 + 6푖) − (12 − 4푖) Multiplying: Just like polynomials, use the distributive property. Then, combine like terms and simplify powers of 푖. Remember! Multiplication does not require like terms. Every term gets distributed to every term. Examples: 1. 4푖(3 − 5푖) 2. (7 − 3푖)(−2 − 5푖) 3. 7푖(2 − 9푖) 4. (5 + 4푖)(6 − 7푖) 5. (3 + 5푖)(3 − 5푖) A note about conjugates: Recall that when multiplying conjugates, the middle terms will cancel out. With complex numbers, this becomes even simpler: (풂 + 풃풊)(풂 − 풃풊) = 풂ퟐ + 풃ퟐ Try again with the shortcut: (3 + 5푖)(3 − 5푖) Dividing: Just like polynomials and rational expressions, the denominator must be a rational number. Since complex numbers include imaginary components, these are not rational numbers. To remove a complex number from the denominator, we multiply numerator and denominator by the conjugate of the Remember! You can simplify first IF factors can be canceled.
    [Show full text]
  • Quaternion Algebra and Calculus
    Quaternion Algebra and Calculus David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Created: March 2, 1999 Last Modified: August 18, 2010 Contents 1 Quaternion Algebra 2 2 Relationship of Quaternions to Rotations3 3 Quaternion Calculus 5 4 Spherical Linear Interpolation6 5 Spherical Cubic Interpolation7 6 Spline Interpolation of Quaternions8 1 This document provides a mathematical summary of quaternion algebra and calculus and how they relate to rotations and interpolation of rotations. The ideas are based on the article [1]. 1 Quaternion Algebra A quaternion is given by q = w + xi + yj + zk where w, x, y, and z are real numbers. Define qn = wn + xni + ynj + znk (n = 0; 1). Addition and subtraction of quaternions is defined by q0 ± q1 = (w0 + x0i + y0j + z0k) ± (w1 + x1i + y1j + z1k) (1) = (w0 ± w1) + (x0 ± x1)i + (y0 ± y1)j + (z0 ± z1)k: Multiplication for the primitive elements i, j, and k is defined by i2 = j2 = k2 = −1, ij = −ji = k, jk = −kj = i, and ki = −ik = j. Multiplication of quaternions is defined by q0q1 = (w0 + x0i + y0j + z0k)(w1 + x1i + y1j + z1k) = (w0w1 − x0x1 − y0y1 − z0z1)+ (w0x1 + x0w1 + y0z1 − z0y1)i+ (2) (w0y1 − x0z1 + y0w1 + z0x1)j+ (w0z1 + x0y1 − y0x1 + z0w1)k: Multiplication is not commutative in that the products q0q1 and q1q0 are not necessarily equal.
    [Show full text]
  • Concept of a Dyad and Dyadic: Consider Two Vectors a and B Dyad: It Consists of a Pair of Vectors a B for Two Vectors a a N D B
    1/11/2010 CHAPTER 1 Introductory Concepts • Elements of Vector Analysis • Newton’s Laws • Units • The basis of Newtonian Mechanics • D’Alembert’s Principle 1 Science of Mechanics: It is concerned with the motion of material bodies. • Bodies have different scales: Microscropic, macroscopic and astronomic scales. In mechanics - mostly macroscopic bodies are considered. • Speed of motion - serves as another important variable - small and high (approaching speed of light). 2 1 1/11/2010 • In Newtonian mechanics - study motion of bodies much bigger than particles at atomic scale, and moving at relative motions (speeds) much smaller than the speed of light. • Two general approaches: – Vectorial dynamics: uses Newton’s laws to write the equations of motion of a system, motion is described in physical coordinates and their derivatives; – Analytical dynamics: uses energy like quantities to define the equations of motion, uses the generalized coordinates to describe motion. 3 1.1 Vector Analysis: • Scalars, vectors, tensors: – Scalar: It is a quantity expressible by a single real number. Examples include: mass, time, temperature, energy, etc. – Vector: It is a quantity which needs both direction and magnitude for complete specification. – Actually (mathematically), it must also have certain transformation properties. 4 2 1/11/2010 These properties are: vector magnitude remains unchanged under rotation of axes. ex: force, moment of a force, velocity, acceleration, etc. – geometrically, vectors are shown or depicted as directed line segments of proper magnitude and direction. 5 e (unit vector) A A = A e – if we use a coordinate system, we define a basis set (iˆ , ˆj , k ˆ ): we can write A = Axi + Ay j + Azk Z or, we can also use the A three components and Y define X T {A}={Ax,Ay,Az} 6 3 1/11/2010 – The three components Ax , Ay , Az can be used as 3-dimensional vector elements to specify the vector.
    [Show full text]
  • Theory of Trigintaduonion Emanation and Origins of Α and Π
    Theory of Trigintaduonion Emanation and Origins of and Stephen Winters-Hilt Meta Logos Systems Aurora, Colorado Abstract In this paper a specific form of maximal information propagation is explored, from which the origins of , , and fractal reality are revealed. Introduction The new unification approach described here gives a precise derivation for the mysterious physics constant (the fine-structure constant) from the mathematical physics formalism providing maximal information propagation, with being the maximal perturbation amount. Furthermore, the new unification provides that the structure of the space of initial ‘propagation’ (with initial propagation being referred to as ‘emanation’) has a precise derivation, with a unit-norm perturbative limit that leads to an iterative-map-like computed (a limit that is precisely related to the Feigenbaum bifurcation constant and thus fractal). The computed can also, by a maximal information propagation argument, provide a derivation for the mathematical constant . The ideal constructs of planar geometry, and related such via complex analysis, give methods for calculation of to incredibly high precision (trillions of digits), thereby providing an indirect derivation of to similar precision. Propagation in 10 dimensions (chiral, fermionic) and 26 dimensions (bosonic) is indicated [1-3], in agreement with string theory. Furthermore a preliminary result showing a relation between the Feigenbaum bifurcation constant and , consistent with the hypercomplex algebras indicated in the Emanator Theory, suggest an individual object trajectory with 36=10+26 degrees of freedom (indicative of heterotic strings). The overall (any/all chirality) propagation degrees of freedom, 78, are also in agreement with the number of generators in string gauge symmetries [4].
    [Show full text]
  • WORKSHEET # 7 QUATERNIONS the Set of Quaternions Are the Set Of
    WORKSHEET # 7 QUATERNIONS The set of quaternions are the set of all formal R-linear combinations of \symbols" i; j; k a + bi + cj + dk for a; b; c; d 2 R. We give them the following addition rule: (a + bi + cj + dk) + (a0 + b0i + c0j + d0k) = (a + a0) + (b + b0)i + (c + c0)j + (d + d0)k Multiplication is induced by the following rules (λ 2 R) i2 = j2 = k2 = −1 ij = k; jk = i; ki = j ji = −k kj = −i ik = −j λi = iλ λj = jλ λk = kλ combined with the distributive rule. 1. Use the above rules to carefully write down the product (a + bi + cj + dk)(a0 + b0i + c0j + d0k) as an element of the quaternions. Solution: (a + bi + cj + dk)(a0 + b0i + c0j + d0k) = aa0 + ab0i + ac0j + ad0k + ba0i − bb0 + bc0k − bd0j ca0j − cb0k − cc0 + cd0i + da0k + db0j − dc0i − dd0 = (aa0 − bb0 − cc0 − dd0) + (ab0 + a0b + cd0 − dc0)i (ac0 − bd0 + ca0 + db0)j + (ad0 + bc0 − cb0 + da0)k 2. Prove that the quaternions form a ring. Solution: It is obvious that the quaternions form a group under addition. We just showed that they were closed under multiplication. We have defined them to satisfy the distributive property. The only thing left is to show the associative property. I will only prove this for the elements i; j; k (this is sufficient by the distributive property). i(ij) = i(k) = ik = −j = (ii)j i(ik) = i(−j) = −ij = −k = (ii)k i(ji) = i(−k) = −ik = j = ki = (ij)i i(jj) = −i = kj = (ij)j i(jk) = i(i) = (−1) = (k)k = (ij)k i(ki) = ij = k = −ji = (ik)i i(kk) = −i = −jk = (ik)k j(ii) = −j = −ki = (ji)i j(ij) = jk = i = −kj = (ji)j j(ik) = j(−j) = 1 = −kk = (ji)k j(ji) = j(−k) = −jk = −i = (jj)i j(jk) = ji = −k = (jj)k j(ki) = jj = −1 = ii = (jk)i j(kj) = j(−i) = −ji = k = ij = (jk)j j(kk) = −j = ik = (jk)k k(ii) = −k = ji = (ki)i k(ij) = kk = −1 = jj = (ki)j k(ik) = k(−j) = −kj = i = jk = (ki)k k(ji) = k(−k) = 1 = (−i)i = (kj)i k(jj) = k(−1) = −k = −ij = (kj)j k(jk) = ki = j = −ik = (kj)k k(ki) = kj = −i = (kk)i k(kj) = k(−i) = −ki = −j = (kk)j 1 WORKSHEET #7 2 3.
    [Show full text]