Fripp described the Fractalisation of King Crimson thus:

"A particular vocabulary, repertoire or leitmotif has currency for a particular period. Then the wind changes direction and everything is different. The traditional response of King Crimson, when faced with the end of a cycle, is to break up. And in this KC's timing has been impeccable. 

Our current approach is different. Rather than disband and cease to exist for a period, we are fractalising into smaller units within the Double Trio and working together, privately and publicly; this, rather than for all six of us to clatter and bang away simultaneously - which is wonderful and frequently invigorating, for some of the time. ProjeKcting is loosening up the band's view of itself and our sense of possible futures.”

On 20th August, he had written to the other band members, proposing this approach

“Almost since we were last together I have been concerned, twitching and even desperate trying to figure out the next step for Crim and Crim material.
I know that rock music has changed, and I know that guitar playing has changed. But I don't know to what in either case. So it's a question of adopting whatever strategies get us there.
Adrian and I are both changing our equipment and hitting program default buttons; essentially we are beginning again. We intend to begin our new writing program supported by a new battery of sounds and textures, which is how Ade and I began on "Discipline".
I can't recommend that we meet in November and spend $30,000 on recording / writing / rehearsals - we're not ready for that.
An alternative approach is for units within the Sextet to work together, privately and publicly, rather than all six of us  to clatter and bang away simultaneously - which is wonderful for some of the time. This is down to the small units and members to use their own initiative rather than waiting to be called.
Pat, Trey and myself are already committed to playing 3-7 nights in Austin or Nashville. We don't know exactly when, but soon-ish. Meanwhile, to appease the savage Bruford breast, I suggested that Bill and I do dates in England as a duo. He suggested a trio with Tony, and I suggested a quartet with Trey.
It is important to me that the declared aim of these sub-groups of Crimson is to advance the interest of the overall and Greater Crim. Then, the whole band benefits. Otherwise, the projects might distract.
My suggestion for the name/s of these smaller projects would be PROJEkCT One, Two, etc., and the names of the individual members. I would anticipate recording the performances for possible individual release, and / or incorporation of less / more on a full Crim album. These projeKcts can become as much and / or as little as they might become. My main own main interest is that they function as Research & Development for the Greater Crim.”

PROJEkCT TWO was chronologically the first of the smaller units into action, featuring Adrian Belew, Robert Fripp & Trey Gunn, which recorded the double album "Space Groove" at Studio Belewbeloible in the Nashville Sector during November 19, 20 & 21st. 1997. 

PROJEkCT ONE, the first King Crimson sub-group planned, was the second into action. Bill Bruford, Robert Fripp, Trey Gunn and Tony Levin improvised four nights of music at the Jazz Cafe in London, December 1- 4th. 1997. 

PROJEkCT FOUR featured Tony Levin, Pat Mastelotto, Trey Gunn and Robert Fripp. It began rehearsals in Nashville, and performed seven shows in Colorado and the West Coast of the United States, from 23 October – 2 November 1998

PROJEkCT THREE, Pat Mastelotto, Trey Gunn and Robert Fripp, performed five shows from 21 March through 25 March 1999 in Texas.

All sixty-one shows played by the Projekcts are available for download. 

The fractalisation continued thereafter with Projekct X, Projeckt Six, and the Jakszyk, Fripp, Collins King Crimson Projeckt

ProjeKcts Discography :

ProjeKct One 
Live at the Jazz Café 
Jazz Café Suite (2003, King Crimson Collector's Club)

ProjeKct Two
Space Groove (1998)
Live Groove
Live in Northampton, MA (2001, King Crimson Collector's Club)
Live in Pittsburgh, Pennsylvania (2005, King Crimson Collector's Club)
Live in Chicago, IL (2006, King Crimson Collector's Club)

ProjeKct Three
Masque 
Live in Austin, TX (2004, King Crimson Collectors' Club)
Live in Alexandria, VA (2007, King Crimson Collectors' Club)

ProjeKct Four
West Coast Live (1999, Japan-only)
Live in San Francisco/The Roar of P4 (1999, King Crimson Collectors' Club)

Compilations:
The ProjeKcts (1999). 4-CD live boxed set
The Deception of the Thrush: A Beginners' Guide to ProjeKcts (1999)

ProjeKct X
Heaven and Earth (2000)

ProjeKct One, Two, Three, Four, X, Six
Heaven and Earth (Box Set) (2019)