I tried, although I don't speak J so it's a rough transliteration. I couldn't do the last GUI animated part without more knowledge of J's tools, but it can compute multiple generations.

Code summary:

   r =. (3 3 $ i. 9) e. 1 2 3 4 7
   R =. _1 |. _2 |."1 (5 7 {. r)
   RR =. 15 35 {.  _10 _20 {. R
   life =. verb : '+./ > (1;y) *.each 3 4 =each +each/ +each/ 1 0 _1 |.each/  1 0 _1 |."1 each <y'
   gen =. dyad : 'life ^: y x'

The 20th generation:

    (RR gen 20) { '.X'

    ...................................
    ...................................
    ...................................
    ...................................
    .........X....XXX..................
    ........XX.....XX..................
    .......XX.....XX...................
    ........X.X...X.X..................
    .........XX....XXXX................
    ..........X...X..XX................
    ..............X.XX.................
    ..............XXX..................
    ...................................
    ...................................
    ...................................

Transcript of the whole talk with line by line translation here: https://dpaste.de/OoEM/raw (expires in 7 days).