Octonion multiplication
In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold . There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H. The octonions are the largest such algebra, with eight dimensions, double the number of the quaternions from which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity, namely they are alternative.
Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Despite this they have some interesting properties and are related to a number of exceptional structures in mathematics, among them the exceptional Lie groups. Additionally, octonions have applications in fields such as string theory, special relativity, and quantum logic.
The octonions were discovered in 1843 by John T. Graves, inspired by his friend William Hamilton's discovery of quaternions. Graves called his discovery octaves. They were discovered independently by Arthur Cayley and are sometimes referred to as Cayley numbers or the Cayley algebra.

This is an excerpt from the article Octonion multiplication from the Wikipedia free encyclopedia. A list of authors is available at Wikipedia.
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Octonion - Wikipedia, the free encyclopedia
Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions. Multiplication is ...
en.wikipedia.org/wiki/Octonion
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Constructing the Octonions
The most elementary way to construct the octonions is to give their multiplication table. The octonions are an 8-dimensional algebra with basis \$1, e_1,e_2,e_3 ...
math.ucr.edu/home/baez/octonions/node3.html
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Octonions
Octonionic Multiplication Table. The octonions O are the nonassociative, noncommutative, normed division algebra over the real numbers. They can be ...
www.physics.orst.edu/~tevian/octonions/
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Octonions
Let us now illustrate how to obtain the Octonion multiplication table using some of the elementary properties of the octonions. Squaring octonions is easy.
homepages.wmich.edu/~drichter/octonions.htm
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Quaternions, octonions, and now, 16-ons and 2n ... - Range Voting
Feb 14, 2004 ... section 8.4 of [68] states that a general octonion left-multiplication may be expressed as 7 consecutive octonion right-multiplications, but no ...
www.scorevoting.net/WarrenSmithPages/homepage/nce2.pdf
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Octonion Introduction - 7Stones
C+ Octonion multiplication: Index cycling and. index doubling are automorphisms. e. 1. 2. 3. 4. 5. 6. 7. What does it mean for (x, y, z) to be a. quaternionic triple (in ...
www.7stones.com/Homepage/Publisher/oct1.html
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Octonions and the Exceptional Lie Algebra g2 - Virginia Tech
Apr 23, 2004 ... We first introduce the octonions as an eight dimensional vector space over a field of characteristic zero with a multiplication defined using a ...
scholar.lib.vt.edu/theses/available/etd-04232004-131642/unrestricted/thesis.pdf
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Octonion Products
Octonion Products and Lattices. WHERE DO THE 480 Octonion multiplication PRODUCTS COME FROM? One point of view is based on Clifford ...
www.tony5m17h.net/480op.html
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Lipschitz minimality of the multiplication maps of unit complex ...
We prove that the multiplication maps S n ×Sn → Sn (n = 1, 3, 7) for unit complex, quaternion and octonion numbers are, up to isometries of domain and range ...
www.math.upenn.edu/~weh/papers/lipS3.pdf
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On Quaternions and Octonions, by John…
On Quaternions and Octonions, by John Conway and Derek Smith
math.ucr.edu/home/baez/octonions/conway_smith/
Search results for "Octonion multiplication"
Octonion multiplication in science
[PDF]Octonions and the Exceptional Lie Algebra g2 - Virginia Tech
Apr 23, 2004 ... State University in partial fulfillment of the requirements for the degree of ... We also show that the multiplication rules for octonions can be ...
Octonion multiplication and Heawood's map
Jun 29, 2011 ... Abstract: In this note, the Octonion multiplication table is recovered from a regular tesselation of the "equilateral" two dimensional torus by seven ...
[PDF]Quaternions, octonions, and now, 16-ons and 2n ... - Range Voting
Feb 14, 2004 ... ∗Some of this research was done at DIMACS, Rutgers University, ..... Fortunately Octonion multiplication obeys11 a weakened as- sociativity ...
[PDF]Algorithm for Multiplying Two Octonions - Springer
Algorithm for Multiplying Two Octonions. A. Cariow and G. Cariowa. West Pomeranian University of Technology, Szczecin, Poland. Received in final form ...
[PDF]The Octonions - University of California, Riverside
May 16, 2001 ... universities they were the only advanced mathematics taught. ..... We should become as comfortable with multiplying octonions as we are with ...
[PDF]The Octonions - University of California, Riverside
Apr 7, 2010 ... followed by addition and subtraction, multiplication and—the hard ... We want to explain the octonions and their role in string theory. But we ...
Octonion Algebra Multiplication Tables - Learning Ace
Multiplication tables for Cayley's octonion numbers incorporate three scalar ... Ball State University MATH 411/511 · Our Lady of the Lake University - San ...
Octonions - Electrical and Computer Engineering - University of ...
[PDF]Lipschitz minimality of the multiplication maps of unit complex ...
We prove that the multiplication maps S n ×Sn → Sn (n = 1, 3, 7) for unit complex, quaternion and octonion numbers are, up to isometries of domain and range ...
Octonion RPN Calculator
The Octonion multiplication table used for this calculator is explained here. John Wayland Bales, Department of Mathematics, Tuskegee University, Tuskegee, ...
Books on the term Octonion multiplication
The Finite Simple Groups
Robert Wilson, 2009
One solution is to take the ordinary integral octonions and adjoin instead ω = 1 2 (−1+i 0 + i1 + i3), ψ = 1 2 (i0 + i1 + i2 + i4), and close under multiplication and addition. Now it is easy to see that this algebra contains elements corresponding to ...
The Princeton Companion to Mathematics
Timothy Gowers, June Barrow-Green, Imre Leader, 2010
Unfortunately, multiplication of octonions is not even associative, but it does have two very good properties: every nonzero octonion has a multiplicative inverse, and two nonzero octonions never multiply together to give zero. (Because ...
Introduction to Octonion and Other Non-Associative Algebras in Physics
Susumo Okubo, 1995
Let e0,e1,...,eN_1 be a basis of the vector space A with the multiplication table N —1 ejek = 2 Cfke/. (3.20) /=0 Setting Lj E L(ej), Rj E R(ej), (3.21) we then calculate N—l Ljek = ejek = Z Cfke; 24 Hurwitz theorems and octonions 3.2 Left and right ...
Topological Geometrodynamics
Matti Pitkänen, 2006
Quaternions and octonions as well as their hyper counterparts obtained by multiplying imaginary units by commuting √ −1 and forming a sub-space of complexified division algebra, are in in a central role in the number theoretical vision about ...
Octonions: a development of Clifford's bi-quaterions
Alexander McAulay, 1898
In other words the definitions of Lator, Rotor, &c. are not complete till we have implied by the definition of the multiplication and addition of octonions what is meant by the multiplication and addition of lators, rotors, &c. A lator is a quantity which ...

Blog posts on the term
Octonion multiplication
Multiplication by Infinity: War of the Octonions
Forgot the String Wars, sometimes called the Loop vs String Wars, or better known in the modern era as the Woit-Motl War? Well, fret not, it's back! At the center of the action is a piece written by John Baez and John Huerta in the current issue of Scientific American. Of course, Peter Woit and Lubos Motl weigh in.
tetrahedral.blogspot.com/2011/04/war-of-octonions.html
This Week’s Hype | Not Even Wrong
Posted on April 28, 2011 by woit.
www.math.columbia.edu/~woit/wordpress/?p=3665
TGD diary: Could one generalize the notion of twistor to 8-D case using the notion of triality?
The basic problem of the twistorial approach is that one cannot represent massive momenta in terms of twistors in elegant manner. I have proposed a possible representation of massive states based on the existence of preferred plane of M2 in the basic definition of theory allowing to express four-momentum as some of two light-like momenta allowing twistor description.
matpitka.blogspot.com/2009/07/could-one-generalize-notion-of-twistor.html
MyToE » The 480 octonions, their Fano planes and multiplication tables
I am pleased to announce the availability of Fano. pdf, a 241 page pdf file with the 480 octonion permutations (with Fano planes and multiplication tables).
theoryofeverything.org/MyToE/?p=1026
Seven-dimensional cross product | Complex Projective 4-Space
In three dimensions, the familiar cross product is a bilinear function expressible in terms of the Levi-Civita alternating tensor. Specifically, a = b × c can be written as ai = εijk bj ck, which has the following beautiful properties: Bilinearity: a is a linear function of b (when c is fixed), and a linear function…
cp4space.wordpress.com/2013/08/04/seven-dimensional-cross-product/
Replacing the Quaternion adjacency with X(Z/nZ, S) | Peter's ruminations
In Turing’s On Permutations manuscript, we saw what I think were random walks on the finite circle: http://turingarchive.org/viewer/?id=133&title=31 http://www.scribd.com/doc/119042559/Fano-plane That is, rotor U1 is in I setting, and rotor U2 is in J setting. Whether U1 is I or J (and vice versa for U2) is 1/2 – to start. Then, one looks…