# clifford.Layout¶

class clifford.Layout(sig, bladeTupList, firstIdx=0, names=None)[source]

Layout stores information regarding the geometric algebra itself and the internal representation of multivectors.

It is constructed like this:

The arguments to be passed are interpreted as follows:

signature – the signature of the vector space. This should be

a list of positive and negative numbers where the sign determines the sign of the inner product of the corresponding vector with itself. The values are irrelevant except for sign. This list also determines the dimensionality of the vectors. Signatures with zeroes are not permitted at this time.

Examples:
signature = [+1, -1, -1, -1] # Hestenes’, et al. Space-Time Algebra signature = [+1, +1, +1] # 3-D Euclidean signature
bladeTupList – list of tuples corresponding to the blades in the whole

algebra. This list determines the order of coefficients in the internal representation of multivectors. The entry for the scalar must be an empty tuple, and the entries for grade-1 vectors must be singleton tuples. Remember, the length of the list will be 2**dims.

Example:
bladeTupList = [(), (0,), (1,), (0,1)] # 2-D
firstIdx – the index of the first vector. That is, some systems number
the base vectors starting with 0, some with 1. Choose by passing the correct number as firstIdx. 0 is the default.
names – list of names of each blade. When pretty-printing multivectors,

use these symbols for the blades. names should be in the same order as bladeTupList. You may use an empty string for scalars. By default, the name for each non-scalar blade is ‘e’ plus the indices of the blade as given in bladeTupList.

Example:
names = [‘’, ‘s0’, ‘s1’, ‘i’] # 2-D

Layout’s Members:

dims – dimensionality of vectors (== len(signature))

sig – normalized signature (i.e. all values are +1 or -1)

firstIdx – starting point for vector indices

names – pretty-printing symbols for the blades

even – dictionary of even permutations of blades to the canonical blades

odd – dictionary of odd permutations of blades to the canonical blades

gmt – multiplication table for geometric product [1]

imt – multiplication table for inner product [1]

omt – multiplication table for outer product [1]

lcmt – multiplication table for the left-contraction [1]

[1] The multiplication tables are NumPy arrays of rank 3 with indices like
the tensor g_ijk discussed above.

Attributes

 I the psuedoScalar basis_names basis_vectors basis_vectors_lst blades blades_list Ordered list of blades in this layout (with scalar as [0]) metric pseudoScalar the psuedoScalar scalar the scalar of value 1, for this GA (a MultiVector object)

Methods

 MultiVector create a multivector in this layout __init__ Initialize self. bases Returns a dictionary mapping basis element names to their MultiVector instances, optionally for specific grades blades_of_grade return all blades of a given grade, dict_to_multivector Takes a dictionary of coefficient values and converts it into a MultiVector object get_grade_projection_matrix Returns the matrix M_g that performs grade projection via left multiplication eg. get_left_gmt_matrix This produces the matrix X that performs left multiplication with x eg. gmt_func_generator imt_func_generator lcmt_func_generator load_ga_file Takes a ga file Checks it is the same signature as this layout Loads the data into an MVArray omt_func_generator parse_multivector Parses a multivector string into a MultiVector object randomMV Convenience method to create a random multivector. randomRotor generate a random Rotor. randomV generate n random 1-vector s