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Topic: Octonions

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	3x3 Octonion Matrix Models
		But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.
		In the heptagon of imaginary octonions {e1,e2,e3,e4,e5,e6,e7}, there are 7 triangles (6 colors and 1 fl).
		Since the X-product and XY-product were constructed to study octonions of unit norm, that is, the unit 7-spheres in octonionic X-space and in octonionic Y-space, it is natural that some of the most interesting fractal images occur for values of X and Y on or near the two unit 7-spheres.
	valdostamuseum.org /hamsmith/3x3OctCnf.html (3539 words)

	Octonion Encyclopedia (Site not responding. Last check: )
		In mathematics, the octonions are a nonassociative extension of the quaternions.
		Addition of octonions is accomplished by adding corresponding coefficients, as with the complex numbers and quaternions.
		The basis for the octonions given here is not nearly as universal as the standard basis for the quaternions; however, nearly all other choices differ from this one only in order and sign.
	www.hallencyclopedia.com /topic/Octonion.html (883 words)

	Octonion - Encyclopedia, History, Geography and Biography
		In mathematics, the octonions are a nonassociative extension of the quaternions.
		The octonion algebra is often denoted O, or in flboard bold by $\mathbb{O}$ .
		The basis for the octonions given here is not nearly as universal as the standard basis for the quaternions; however, nearly all other choices differ from this one only in order and sign.
	www.arikah.com /encyclopedia/Octonions (996 words)

	Octonion: Definition and Links by Encyclopedian.com
		...Octonion Octonion The octonions are a non- associative extension of the...algebra.
		The octonions are a non-associative extension of the quaternions.
		The octonions form an 8-dimensional algebra over the real numbers, and can therefore be thought of as octets of real numbers.
	www.encyclopedian.com /oc/Octonions.html (255 words)

	NationMaster - Encyclopedia: Hypercomplex number
		Increasing dimensionality introduces algebraic complications: Quaternion multiplication is not commutative anymore, octonion multiplication additionally is non-associative, and sedenions do not form a normed space with multiplicative norm.
		Because quaternions and octonions offer a (multiplicative) norm similar to lengths in four and eight dimensional Euclidean vector space respectively, these numbers can be referred to as points in some higher-dimensional Euclidean space.
		Hypercomplex numbers are extensions of the complex numbers, such as quaternions, octonions and sedenions.
	www.nationmaster.com /encyclopedia/Hypercomplex-number (2461 words)

	NationMaster - Encyclopedia: Bicomplex number
		In mathematics, hypercomplex numbers are extensions of the complex numbers constructed by means of abstract algebra, such as quaternions, tessarines, coquaternions, octonions, biquaternions and sedenions.
		\mathbb{H}, invented by Sir William Rowan Hamilton, in which multiplication is not commutative, and the octonions, in which multiplication is not associative.
		The existence of complex numbers was not completely accepted until the geometrical interpretation had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion.
	www.nationmaster.com /encyclopedia/Bicomplex-number (1182 words)

	Octonion
		Despite this, the octonions retain importance for being related to a number of exceptional structures in mathematics, among them the exceptional Lie groups.
		Every octonion is a real linear combination of the unit octonions {1, i, j, k, l, li, lj, lk}.
		(Note that the basis for the octonions given here is not nearly as universal as the standard basis for the quaternions, however, nearly all other choices differ from this one only in order and sign.)
	www.xasa.com /wiki/en/wikipedia/o/oc/octonion.html (786 words)

	es octoniones In mathematics mathematics the octonions are a non...
		es:octoniones In mathematics mathematics, the "octonions" are a non-associative associative extension of the quaternion quaternions.
		The octonions form an 8-dimensional (non-associative) division algebra division algebra over the real number real numbers, and can therefore be thought of as octets (or 8-tuples) of real numbers.
		Every octonion is a real linear combination linear combination of the "unit octonions" 1, "e"1, "e"2, "e"3, "e"4, "e"5, "e"6 and "e"7.
	www.biodatabase.de /Octonion (274 words)

	[No title]
		Octonions were invented by Cayley later on in the 1800s.
		In case you lose your octonion table, don't worry: you don't really need to remember the names of those 7 square roots of -1 and their positions on the chart.
		However, the octonions remain mysterious, at least to me. They are related to rotations in 7 and 8 dimensions, but not as simply as one might hope.
	www.math.niu.edu /~rusin/known-math/97/divialg2 (2493 words)

	Octonion - InformationBlast
		In mathematics, the octonions are a non-associative extension of the quaternions.
		The octonions form an 8-dimensional (non-associative) division algebra over the real numbers, and can therefore be thought of as octets (or 8-tuples) of real numbers.
		By linearity, multiplication of octonions is completely determined by the multiplication table for the unit octonions; this table is given below.
	www.informationblast.com /Octonion.html (192 words)

	Hypercomplex number - RecipeFacts (Site not responding. Last check: )
		In mathematics, hypercomplex numbers are extensions of the complex numbers constructed by means of abstract algebra, such as quaternions, split-quaternions, tessarines, coquaternions, octonions, split-octonions, biquaternions and sedenions.
		The quaternions, octonions and sedenion can be generated by the Cayley-Dickson construction.
		The hypernumbers program offers hyperbolic, circular, and conic quaternions, octonions, and sedenions, which all have a multiplicative modulus (except for hyperbolic sedenions).
	www.recipeland.com /facts/Hypercomplex_number (255 words)

	Constructing the Octonion
		The most elementary way to construct the octonion is to give their multiplication table.
		The octonions are a non-associative extension of the quaternions.
		The octonions form an 8-dimensional algebra over the real numbers, and can therefore be thought of as octets of real numbers.
	www1.gantep.edu.tr /~olgar/octonion.htm (184 words)

	Cayley-Dickson construction Information
		Inasmuch as octonions consist of two quaternions, the octonions form an 8-dimensional vector space.
		Besides being non-commutative, it is not associative, that is, if p, q, and r are octonions, it is not generally true that (pq)r = p(qr).
		After the octonions, though, the algebras even contain zero divisors, that is, if p and q are elements of one of these algebras, then pq = 0 no longer implies p = 0 or q = 0.
	www.bookrags.com /Cayley-Dickson_construction (799 words)

	[No title]
		Well, there are not too many places in physics yet where the octonions reach out and grab one with the force the reals, complexes, and quaternions do.
		Now we can define an octonion to be a pair of quaternions; as before, we add these component-wise and multiply them as follows: (a,b)(c,d) = (ac - db, da + bc).
		I am learning about octonions just for fun, since I just finished writing some rather technical papers, and my idea of fun does not presently include multiplying two hexadecanions together to see why the norm-squared law (a,b) (a,b)* = (a,b)* (a,b) = K (1,0) breaks down.
	www.math.niu.edu /~rusin/known-math/95/octonions.phys (2154 words)

	[No title] (Site not responding. Last check: )
		It is argued that the abstract structure of states of consciousness is given by the octonionic division algebra.
		The octonions are generated by an eight-element basis; the seven non-unity elements of this basis are equated with the "magic number" 7 +/- 2 of short-term memory capacity.
		On the other hand, if the probabilities are biased in a way similar to one of the many possible octonion multiplication tables, then one has found a valuable piece of evidence in favor of the present model of short-term memory and, indirectly, in favor of the finite division algebra theory of consciousness.
	www.goertzel.org /dynapsyc/1996/consalg.html (12356 words)

	[No title]
		Despite this, the octonions retain importance for being related to a number of exceptional structures in mathematics, among them the
		linear combination of the unit octonions {1, i, j, k, l, li, lj, lk}.
		This group is the smallest of the five exceptional Lie groups.
	www.en-cyclopedia.com /wiki/Octonion (622 words)

	Title page for ETD etd-04232004-131642
		We first introduce the octonions as an eight dimensional vector space over a field of characteristic zero with a multiplication defined using a table.
		We prove that the algebra of derivations on the octonions is a Lie algebra of type G_2.
		The proof proceeds by showing the set of derivations on the octonions is a Lie algebra, has dimension fourteen, and is semisimple.
	scholar.lib.vt.edu /theses/available/etd-04232004-131642 (227 words)

	What IS a Lie Group?
		F4 is the automorphism group of 3x3 matrices of octonions o11 o12 o13 o21 o22 o23 o31 o32 o33 such that o11, o22, and o33 are real (have no imaginary part), and o12, o13, o23 are the octonion conjugates of o21, o31, o32 respectively.
		We have seen that the 7-dim imaginary octonion commutator algebra lives inside the 28-dim Lie algebra of Spin(0,8) and that it is not a Lie algebra (It belongs to the class of algebras called Malcev algebras).
		It seems to me that: octonion X- and XY-product structure is a logarithm of z to zz + c structure; and z to zz + c structure is an exponential of octonion X- and XY-product structure.
	akbar.marlboro.edu /~mahoney/groups/Lie.html (2525 words)

	Octonions
		Octonions, like quaternions, are a relative of complex numbers.
		Addition and a multiplication is defined on the set of octonions, which generalize their quaternionic counterparts.
		Octonions (and their kin) are described in far more details in this other document (with errata and addenda).
	www.boost.org /doc/html/boost_math/octonions.html (144 words)

	Octonions
		The octonions O are the nonassociative, noncommutative, normed division algebra over the real numbers.
		The octonions are thus a generalization of the complex numbers, with 7 imaginary units rather than just one.
		Octonionic conjugation is given by reversing the sign of the imaginary basis units.
	www.physics.orst.edu /~tevian/octonions (280 words)

	on quaternions and octonions
		He defines an octonion to be a pair of "motors", where a = "motor" consists of a "scalar" and a "rotor"...
		He defines an octonion to be a pair of "motors", where a "motor" consists of a "scalar" and a "rotor"...
		It's possible that he was describing the octonions or biquaternions in terms of the exterior algebra of R^3.
	www.forum-one.org /new-6596145-4346.html (3187 words)

	Maths - Octonion - Martin Baker
		Octonions are a superset of quaternions in the same way that quaternions are a superset of complex numbers.
		Octonions have been used in fields such as abstract algebra and topology but have not really made a big impact.
		More recently octonions have been used as the basis for String Theory so octonions may yet prove to be important in describing the structure of the universe.
	www.euclideanspace.com /maths/algebra/realNormedAlgebra/octonion/index.htm (1026 words)

	Board on Physics and Astronomy
		Perhaps the most important success of the octonionic viewpoint, however, was the explanation [2] of the known result that supersymmetric theories are only consistent in certain dimensions.
		This sort of simplification is typical of the use, and the beauty, of formulating quantum physics using octonions.
		A hint that this may be correct is that this naturally leads to the symmetry group E6, which has often been proposed as the fundamental symmetry group of nature.
	www7.nationalacademies.org /bpa/1projects_cpu_proposals_oct_fermions.html (1248 words)

	Cayley-Dickson Construction of Quaternions, Octonions, etc.
		Note, however, that because the quaternions are not commutative, the order of the factors in the multiplication formula becomes important--if the last factor in the multiplication formula were b c rather than c b, the formula for the conjugate wouldn’t yield a real number.
		Inasmuch as octonions consist of a pair of quaternions, they form an 8-dimensional vector space with with real scalars.
		The multiplication of octonions is even stranger than that of quaternions.
	www.zipcon.net /~swhite/docs/math/quaternions/Cayley-Dickson.html (719 words)

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