The Cayley-Dickson Calculator
Software © (2009) John Wayland Bales under the GNU General Public License
The universal real Cayley-Dickson algebra
x0, x1, . . .xn, 0, 0, 0, . . .
The canonical basis vectors for this space are denoted i0, i1, i2, i3, . . . where
i0 = 1, 0, 0, 0, . . .
i1 = 0, 1, 0, 0, . . .
i2 = 0, 0, 1, 0, . . .
i3 = 0, 0, 0, 1, . . .
Real numbers are identified with sequences of the form
x0, 0, 0, 0, . . .
Complex numbers are identified with sequences of the form
x0, x1, 0, 0, 0, . . .
Quaternions are identified with sequences of the form
x0, x1, x2, x3, 0, 0, 0, . . .
Octonions are identified with sequences of the form
x0, x1, x2, x3, x4, x5, x6, x7, 0, 0, 0, . . .
The Sedenions are identified with sequences of the form
x0, x1, . . .x15, 0, 0, 0, . . .
There is an infinite sequence of Cayley-Dickson algebras, each with twice the dimension of the previous algebra and each containing all previous algebras as proper sub-algebras.
If the ordered pair (x, y) of two sequences
x = x0, x1, x2, . . .
y = y0, y1, y2, . . .
is identified with the "shuffled" sequence
(x, y) = x0, y0, x1, y1, x2, y2, . . .
then each algebra in the sequence beginning with the complex numbers consists of all ordered pairs of elements of the previous algebra.
And for each element a,b ε
The basis vectors obey the identities
i2p = (ip, 0)
i2p+1 = (0, ip)
The conjugate of an element
x = x0, x1, . . .xn, 0, 0, 0, . . .
is the sequence
x* = x0, –x1, . . .–xn, 0, 0, 0, . . .
Thus for sequences x and y it follows that (x,y)* = (x*,–y).
For sequences a, b, c and d for which multiplication is defined, the Cayley-Dickson product of the shuffled sequences (a,b) and (c,d) is defined by
(a,b)·(c,d) = (ac-db*,a*d+cb).
Using this product definition (there are alternate definitions), the product of the basis vectors is determined recursively as follows:
If p and q are non-negative integers then
i2p·i2q = (ip·iq, 0)
i2p·i2q+1 = (0, ip*·iq)
i2p+1·i2q = (0, iq·ip)
i2p+1·i2q+1 = –(iq·ip*, 0)
The ultimate result of this recursive definition is a product of Cayley-Dickson basis vectors ip and iq given by the formula
ip· iq = γ(p,q) ip^q
where γ(p,q) is either +1 or -1 and where p^q is the bit-wise 'exclusive or' of the binary representations of p and q.
[For example if p = 5=101B and q = 6=110B then p^q=101B^110B=011B=3. So i5·i6 = γ(5,6)i3.]
The sign function γ is called a "twist" and can be recursively defined as follows:
The proof of the Cayley-Dickson twist is given in "Cayley-Dickson and Clifford Algebras as Twisted Group Algebras (2003)" by John W. Bales.
For the universal Cayley-Dickson algebra, if 0 ≠ p≠ q≠ 0 and if ip· iq = ir then
These may be called the "quaternion" properties.
If 0 ≠ p≠ q≠ 0 and if ip· iq = ir then we denote this by the ordered number triple (p,q,r).
For the universal Cayley-Dickson algebra (p,q,r) implies (q,r,p) and (r,p,q) by the quaternion properties, and by the properties of γ implies (2p,2q,2r), (2q,2p+1,2r+1), (2q+1,2p,2r+1) and (2q+1,2p+1,2r). Beginning with (1,2n,2n+1) for all n, these recursively generate the multiplication table for the universal Cayley-Dickson algebra when 0 ≠ p≠ q≠ 0. To complete the table one needs only to note that i0 = 1 and ip· ip = –1 for p ≠ 0.
and for 0 ≠ p≠ q≠ 0:
The last four identities are the inductive identities and are implied by the nature of γ. The inductive identities are summarized in the following table:
[ Note to Octonion specialists: This indexing of the octonion basis vectors, automatically satisfies index doubling, but not index cycling. For more on this topic go here.]
Given the Euclidean inner product <x,y>, the universal Cayley-Dickson algebra
<x·y, z> = <x, z·y*>
<x·y, z> = <y, x*·z>
Thus the kth component of the product x·y, is <x·y, ik> = <x, ik·y*>.
Also check out the Sedenion Reverse Polish Notation (RPN) Calculator,
the Octonion RPN calculator,
and the Sedenion Norm Comparison Calculator.
And the Color-coded Clifford Algebra Multiplication Table
Reference "Cayley-Dickson and Clifford Algebras as Twisted Group Algebras" (by J. W. Bales (2003); Dept. of Mathematics, Tuskegee University
Reference "Properly twisted groups and their algebras" by J. W. Bales (2006); Dept. of Mathematics, Tuskegee University
Reference "A catalog of Cayley-Dickson-like products" by J. W. Bales (2011); Dept. of Mathematics, Tuskegee University
Reference "A tree for computing the Cayley-Dickson twist" by J. W. Bales (2007) published in the Missouri Journal of Mathematical Sciences Vol. 21 No. 2 (2009)
© John Wayland Bales (2009)
Reference "The eight Cayley-Dickson doubling products" by J. W. Bales (2015); Advanced in Applied Clifford Algebras (Springer)