# 3-bit Walsh permutation/seeds

Cluster of the neighbor graph, on the right the matrix sums and vertex types

There are 25 transforms that look similar to the neutral position.
This is the case when the view from one or two axes remains the same or almost the same.
Almost the same means, that the original square is sheared into a (simple) parallelogram.

124
125 (as inverse)
136 (binary)
125 and 136 look similar to the neutral position.
125 looks the same from x and almost the same from y (here shown from −y).
136 looks almost the same from x.

These are the 25 permutations in the middle cluster of the positive component of the neighbor graph.
Each of their matrices has a different set of columns. (I.e., each of their vectors has entries from a different set of integers.)

There are 3*6=18 transform that do not look like a square or simple parallelogram from any side, namely those who's matrices have seven 1s.
They are also shown in three rows of the table below. The choice which to put in the left column is somewhat random.
The ones chosen are those not connected to the central cluster in the positive component. (I.e. the three remaining ones with black circles.)

The table shows some properties of the permutations in the left column:

 conjugacy class cycle shape sum quantity neut. 2+2 2+4 London Rome Florence Lima 5 Lima 6 Rio + 3 4 5a 5b 6 7 1 6 3 3 6 6 3 12 12

Permutations in the same row have the same complement pattern:

2+2
Rome
2+4
Lima 5
7a
Rio +
1
2
4
2+2
London
2+4
Lima 6
3
5
6

neut.
2+2
Florence
7

Each row contains the transform with the binary matrix above and that with an inverse matrix below.
(The latter is the same pattern of non-zero entries, but some 1s are negative.)
(The matrices for the first row are self-inverse.)

cc cs cp t stretch stretched seeds
neut. neut. 3 37 124

124

214

142

412

241

421
2+2 London 4 26 134

134

314

143

413

341

431
2+2 London 4 26 125

125

215

152

512

251

521
2+2 London 4 25 234

324

234

342

432

243

423
2+2 London 4 25 126

126

216

162

612

261

621
2+2 London 4 23 245

524

254

542

452

245

425
2+2 London 4 23 146

164

614

146

416

641

461
2+2 Rome 5a 11 247

724

274

742

472

247

427
2+2 Rome 5a 12 147

174

714

147

417

741

471
2+2 Rome 5a 14 127

127

217

172

712

271

721
2+2 Florence 5a 37 135

135

315

153

513

351

531
2+2 Florence 5a 37 236

326

236

362

632

263

623
2+2 Florence 5a 37 456

564

654

546

456

645

465
2+4 Lima 5 5b 12 156

165

615

156

516

651

561
2+4 Lima 5 5b 14 136

136

316

163

613

361

631
2+4 Lima 5 5b 14 235

325

235

352

532

253

523
2+4 Lima 5 5b 11 256

526

256

562

652

265

625
2+4 Lima 5 5b 12 345

534

354

543

453

345

435
2+4 Lima 5 5b 11 346

364

634

346

436

643

463
2+4 Lima 6 6 23 157

175

715

157

517

751

571
2+4 Lima 6 6 25 137

137

317

173

713

371

731
2+4 Lima 6 6 23 267

726

276

762

672

267

627
2+4 Lima 6 6 26 237

327

237

372

732

273

723
2+4 Lima 6 6 25 467

764

674

746

476

647

467
2+4 Lima 6 6 26 457

574

754

547

457

745

475
7a Rio +7 11 357

753

735

573

537

375

357
7a Rio +7 12 367

673

637

763

736

367

376
7a Rio +7 14 567

657

675

567

576

765

756