Category:Single 3-bit Walsh permutations; transform; JF

This category is an image set. It should contain only images that have the same style, and should have a parent category that is purely topical.

	137 in conjugacy class 2+4	531 in conjugacy class 3+3
binary
as inverse

There are 330 images of transformed cubes in this category: 168 for the invertible binary matrices,
and 162 for the inverses that contain negative 1s. (The 6 matrices in the symmetric subgroup do not.)

The tranformation matrix with vector $v$ as inverse is not to be confused with the inverse
of the binary transformation matrix with vector $v$ .

The Walsh permutations 137 and 136 are inverse to each other.
The inverse matrix of 137 as binary is 136 as inverse.
The inverse matrix of 136 as binary is 137 as inverse.

matrix multiplication example

Python console with NumPy (as np)

The matrices are multiplied from the left, so the coordinates are column vectors.
The neutral coordinates are of the cube (±1, ±1, ±1).

coordinates_neutral
array([[-1,  1, -1,  1, -1,  1, -1,  1],
       [-1, -1,  1,  1, -1, -1,  1,  1],
       [-1, -1, -1, -1,  1,  1,  1,  1]])

The two matrices used are the inverses 137 as binary and 136 as inverse.

matrix_137_bin
array([[1, 1, 1],
       [0, 1, 1],
       [0, 0, 1]])
       
matrix_136_inv
array([[ 1, -1,  0],
       [ 0,  1, -1],
       [ 0,  0,  1]])
       
np.dot(matrix_137_bin, matrix_136_inv)
array([[1, 0, 0],
       [0, 1, 0],
       [0, 0, 1]])

The matrices are applied (like functions) to the neutral coordinates.
The resulting coordinates are those shown in 137 as binary and 136 as inverse.
(Here for mathematical reasons as column vectors, there for practical reasons as row vectors.)

coordinates_137_bin = np.dot(matrix_137_bin, coordinates_neutral)
coordinates_137_bin
array([[-3, -1, -1,  1, -1,  1,  1,  3],
       [-2, -2,  0,  0,  0,  0,  2,  2],
       [-1, -1, -1, -1,  1,  1,  1,  1]])
       
       
coordinates_136_inv = np.dot(matrix_136_inv, coordinates_neutral)
coordinates_136_inv
array([[ 0,  2, -2,  0,  0,  2, -2,  0],
       [ 0,  0,  2,  2, -2, -2,  0,  0],
       [-1, -1, -1, -1,  1,  1,  1,  1]])

When $a$ and $b$ are inverses, then $a\cdot b\cdot x~=~b\cdot a\cdot x~=~x$ .
So applying one matrix to the coordinates created with the other one gives the neutral coordinates.

np.array_equal(
    np.dot(matrix_137_bin, coordinates_136_inv),
    coordinates_neutral
)
True

np.array_equal(
    np.dot(matrix_136_inv, coordinates_137_bin),
    coordinates_neutral
)
True

Subcategories

This category has the following 2 subcategories, out of 2 total.

S

Single 3-bit Walsh permutations; transform binary; JF‎ (2 C, 2 P)
Single 3-bit Walsh permutations; transform inverse; JF‎ (2 C, 1 P)

Pages in category "Single 3-bit Walsh permutations; transform; JF"

The following 2 pages are in this category, out of 2 total.

W

Media in category "Single 3-bit Walsh permutations; transform; JF"

The following 3 files are in this category, out of 3 total.

JF in skeletonic cube.png 2,704 × 2,637; 123 KB
JF in skeletonic parallelepiped.png 6,376 × 2,208; 172 KB
Walsh permutation 534 JF as inverse; screenshot of details.png 1,626 × 807; 248 KB

Category:Single 3-bit Walsh permutations; transform; JF

Subcategories

S

Pages in category "Single 3-bit Walsh permutations; transform; JF"

W

Media in category "Single 3-bit Walsh permutations; transform; JF"

Navigation menu

Search