mbish/cayley-dickson

A script for investigating the Cayley-Dickson construction

Cayley-Dickson Shell

This is pretty much just a perl eval loop so: if you can perl-it you can do it.

Usage

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Controls

If you have readline installed you should get most of the tools for looking through your command history other wise hit return to end a command.

Variable declerations are done with:

our $x = 2

and are always refered to with a sigil ($) in front of their name:

$a = $b

and things like this are perfectly valid.

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Element

• Elements can be specified as real numbers or pairs of the form [a,b] where a and b are elements of the same dimension
• Note: In a pair, you must pair an element with another element of the same dimension
• Examples: 1 = 1, [1,2] = 1 + 2i, [[1,2],[1,2]] = 1 + 2i + 1j + 2k

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Functions

   * mult(a,b) - multiplies two elements
   * conj(a)   - conjugates an element
   * neg(a)    - returns negation of an element
   * add(a,b)  - adds two elements

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Notation

ei is used as the basis of the i'th dimension in a vector space with e0 being the real part

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Examples

Basics:
    >> our $a = mult(2,2)
    4e0
    >> mult([1,2],[3,$a])
    -5e0 + 10e1

Observing loss of commutitivty in quaternions:
    >> our $a = [[1,2],[3,4]]
    1e0 + 2e1 + 3e2 + 4e3

    >> our $b = [[5,6],[7,8]]
    5e0 + 6e1 + 7e2 + 8e3

    >> mult($a, $b)
    9e0 + 12e1 + 13e2 + 24e3 

    >> mult($b, $a)
    9e0 + 20e1 + 29e2 + 32e3

Dependencies

A readline package for perl is very helpful for editing and keeping track of history.