clifford.Layout

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

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

Parameters

• signature (List[int]) –

  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[Tuple[int, ..]]) –

  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 (int) – 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[str]) –

  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

dims

dimensionality of vectors (== len(signature))

sig

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

firstIdx

starting point for vector indices

bladeTupList

list of blades

gradeList

corresponding list of the grades of each blade

gaDims

2**dims

einf

if conformal returns einf

eo

if conformal returns eo

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	List of blades in this layout matching the order of self.bladeTupList
metric
pseudoScalar	the psuedoScalar
rotor_mask
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
gen_dual_func	Generates the dual function for the pseudoscalar
gen_right_complement_func	Generates the right complement of a multivector
gen_vee_func	Generates the vee product function
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.
get_right_gmt_matrix	This produces the matrix X that performs right multiplication with x eg.
gmt_func_generator
grade_mask
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