Nothing much to do with your great post, but I almost REALLY liked that first pyramid, but the last line being off threw me visually, so I had to straighten it out:

                                    1
                                  1   1
                                1   0   1
                              1   1   1   1
                            1   0   0   0   1
                          1   1   0   0   1   1
                        1   0   1   0   1   0   1
                      1   1   1   1   1   1   1   1
                    1   0   0   0   0   0   0   0   1
                  1   1   0   0   0   0   0   0   1   1
                1   0   1   0   0   0   0   0   1   0   1
              1   1   1   1   0   0   0   0   1   1   1   1
            1   0   0   0   1   0   0   0   1   0   0   0   1
          1   1   0   0   1   1   0   0   1   1   0   0   1   1
        1   0   1   0   1   0   1   0   1   0   1   0   1   0   1
      1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
    1   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   1
  1   1   0   0   0   0   0   0   0   0   0   0   0   0   0   0   1   1